Split (1)

train · 275k rows

id stringlengths 6 15	solution stringlengths 8 36.1k	answer stringlengths 0 543 ⌀	metadata dict	problem stringlengths 15 7.84k	tags listlengths 1 56	idea stringlengths 29 420	problem_embedding listlengths 2.56k 2.56k
aops_99039	The answer is $\frac{1}{4}$. The worst curve is the broken line consisting of 2 perpendicular intervals of length $\frac{1}{2}$. Here is a [hide="hint."] To show that every curve of length $1$ is contained in a rectangle of area $\frac{1}{4}$, consider the least rectangle containing our curve with one side parallel to the line containing the endpoints of the curve. Then the projection of the curve to the perpendicular side of the rectangle covers it at least twice. This is all you need to make the estimate. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $L$ be a continuous (make it $C^1879$ if you want) curve in $R^2$. Let $A(L)$ the set of rectangles $R$ of $R^2$ such that $L$ is contained in the interiour of $R$. What is the $sup$ value over all curves $L$ of length 1 of the $inf$ value over $R$ in $A(L)$ of the area of $R$ (sorry for the very bad explanation).", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png\" class=\"latex\" alt=\"$L$\" width=\"12\" height=\"12\" > be a continuous (make it <img src=\"//latex.artofproblemsolving.com/d/4/5/d45523075d5a9b5bdc166ddfa0323d0adbfb1585.png\" class=\"latex\" alt=\"$C^1879$\" width=\"48\" height=\"15\" > if you want) curve in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/2/6/62623d70eaa11458b42450ab31277cc88f4dbd6b.png\" class=\"latex\" alt=\"$R^2$\" width=\"20\" height=\"15\" >.</span> Let <img src=\"//latex.artofproblemsolving.com/e/a/0/ea0e68bb93c5edfee2a89e950da2f434a2f41ade.png\" class=\"latex\" alt=\"$A(L)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > the set of rectangles <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > of <img src=\"//latex.artofproblemsolving.com/6/2/6/62623d70eaa11458b42450ab31277cc88f4dbd6b.png\" class=\"latex\" alt=\"$R^2$\" width=\"20\" height=\"15\" > such that <img src=\"//latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png\" class=\"latex\" alt=\"$L$\" width=\"12\" height=\"12\" > is contained in the interiour of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >.</span> What is the <img src=\"//latex.artofproblemsolving.com/1/1/4/114c5c9a86126ac58d9fc7273bf2d443cafa1fbb.png\" class=\"latex\" alt=\"$sup$\" style=\"vertical-align: -3px\" width=\"28\" height=\"11\" > value over all curves <img src=\"//latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png\" class=\"latex\" alt=\"$L$\" width=\"12\" height=\"12\" > of length 1 of the <img src=\"//latex.artofproblemsolving.com/7/e/5/7e569ae3308714e3a274718c6d248a0a92859fc7.png\" class=\"latex\" alt=\"$inf$\" style=\"vertical-align: -3px\" width=\"28\" height=\"16\" > value over <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > in <img src=\"//latex.artofproblemsolving.com/e/a/0/ea0e68bb93c5edfee2a89e950da2f434a2f41ade.png\" class=\"latex\" alt=\"$A(L)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > of the area of <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > (sorry for the very bad explanation).", "post_id": 558934, "post_number": 1, "post_time_unix": 1151424871, "post_time_utc": "2006-06-27 16:14:31 UTC", "thanks_received": 1, "user_id": 285, "username": "harazi" }, { "attachments": [], "content_bbcode": "The answer is $\\frac{1}{4}$. The worst curve is the broken line consisting of 2 perpendicular intervals of length $\\frac{1}{2}$. \r\nHere is a [hide=\"hint.\"] To show that every curve of length $1$ is contained in a rectangle of area $\\frac{1}{4}$, consider the least rectangle containing our curve with one side parallel to the line containing the endpoints of the curve. Then the projection of the curve to the perpendicular side of the rectangle covers it at least twice. This is all you need to make the estimate.\n[/hide]", "content_html": "The answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c339afa03e060d959e4942be7522c0c992f4c41c.png\" class=\"latex\" alt=\"$\\frac{1}{4}$\" style=\"vertical-align: -13px\" width=\"11\" height=\"37\" >.</span> The worst curve is the broken line consisting of 2 perpendicular intervals of length <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/f/9/2f960094315d60883495f9b74148e17487ee9584.png\" class=\"latex\" alt=\"$\\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" >.</span><br>\nHere is a <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">hint.</a><div class=\"cmty-hide-content\" style=\"display:none\">To show that every curve of length <img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" > is contained in a rectangle of area <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c339afa03e060d959e4942be7522c0c992f4c41c.png\" class=\"latex\" alt=\"$\\frac{1}{4}$\" style=\"vertical-align: -13px\" width=\"11\" height=\"37\" >,</span> consider the least rectangle containing our curve with one side parallel to the line containing the endpoints of the curve. Then the projection of the curve to the perpendicular side of the rectangle covers it at least twice. This is all you need to make the estimate.</div>", "post_id": 572799, "post_number": 2, "post_time_unix": 1152788862, "post_time_utc": "2006-07-13 11:07:42 UTC", "thanks_received": 2, "user_id": 6542, "username": "fedja" } ], "source": null }	Let $L$ be a continuous curve in $\mathbb{R}^2$. Let $A(L)$ denote the set of rectangles $R\subset\mathbb{R}^2$ such that $L$ is contained in the interior of $R$. For a given curve $L$ of length 1, define \[ m(L)=\inf_{R\in A(L)}\operatorname{area}(R). \] Determine \[ \sup\{\,m(L):L\subset\mathbb{R}^2\text{ is a curve of length }1\,\}. \]	[ "/Mathematics/CalculusandAnalysis/Inequalities/UpperBound", "/Mathematics/Geometry/Curves/PlaneCurves/GeneralPlaneCurves", "/Mathematics/Geometry/GeometricInequalities", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle" ]	Project the curve onto the sides of its minimal enclosing rectangle; one side is bounded by half the total length and the opposite side is covered at least twice, limiting the area to 1/4.	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aops_99042	Let $\{e_{\alpha}\}_{\alpha\in A}$ be a Hamel basis for the given vector space $V.$ We are assuming that $A$ is an infinite set. Then an arbitrary element of $V$ can be written uniquely as $\sum_{\alpha\in A}f(\alpha)e_{\alpha}$ where $f: A\mapsto F$ is any function of finite support. An arbritrary element $\Lambda$ of the dual of $V$ can be associated uniquely to an arbitrary function $g: A\mapsto F.$ The functional is then given by $\Lambda\left(\sum_{\alpha\in A}f(\alpha)e_{\alpha}\right) = \sum_{\alpha\in A}g(\alpha)f(\alpha)$ What is left is a cardinality argument: arguing that there are more functions than there are finitely supported functions. I think this is going to split into some casework depending on the cardinality of $A$ and the cardinality of $F.$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove that the dual of the infinite-dimensional vector space(over any field) isn't isomorphic to this space.\r\n\r\n(The dual here is supposed in it's purely algebraic sense, i.e. the set of all linear functionals on the vector space).", "content_html": "Prove that the dual of the infinite-dimensional vector space(over any field) isn't isomorphic to this space.<br>\n<br>\n(The dual here is supposed in it's purely algebraic sense, i.e. the set of all linear functionals on the vector space).", "post_id": 558939, "post_number": 1, "post_time_unix": 1151425235, "post_time_utc": "2006-06-27 16:20:35 UTC", "thanks_received": 1, "user_id": 902, "username": "eugene" }, { "attachments": [], "content_bbcode": "Let $\\{e_{\\alpha}\\}_{\\alpha\\in A}$ be a Hamel basis for the given vector space $V.$ We are assuming that $A$ is an infinite set. Then an arbitrary element of $V$ can be written uniquely as\r\n\r\n$\\sum_{\\alpha\\in A}f(\\alpha)e_{\\alpha}$\r\n\r\nwhere $f: A\\mapsto F$ is any function of finite support.\r\n\r\nAn arbritrary element $\\Lambda$ of the dual of $V$ can be associated uniquely to an arbitrary function $g: A\\mapsto F.$ The functional is then given by \r\n\r\n$\\Lambda\\left(\\sum_{\\alpha\\in A}f(\\alpha)e_{\\alpha}\\right) = \\sum_{\\alpha\\in A}g(\\alpha)f(\\alpha)$\r\n\r\nWhat is left is a cardinality argument: arguing that there are more functions than there are finitely supported functions. I think this is going to split into some casework depending on the cardinality of $A$ and the cardinality of $F.$", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/9/4/6/9466ddb239ab4c4fee51c1d91a7af5aafd9f4d88.png\" class=\"latex\" alt=\"$\\{e_{\\alpha}\\}_{\\alpha\\in A}$\" style=\"vertical-align: -4px\" width=\"63\" height=\"18\" > be a Hamel basis for the given vector space <img src=\"//latex.artofproblemsolving.com/4/a/0/4a0370c751a4b0f88b2c9b6a364f304ad9258d53.png\" class=\"latex\" alt=\"$V.$\" width=\"15\" height=\"12\" > We are assuming that <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > is an infinite set. Then an arbitrary element of <img src=\"//latex.artofproblemsolving.com/1/2/d/12d58aa29201da09d8e620f8698e3a37547f6b4a.png\" class=\"latex\" alt=\"$V$\" width=\"14\" height=\"12\" > can be written uniquely as<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/4/e/b4e1a8afd751be94f46d82dea245ecdc9f587c7a.png\" class=\"latex\" alt=\"$\\sum_{\\alpha\\in A}f(\\alpha)e_{\\alpha}$\" style=\"vertical-align: -21px\" width=\"83\" height=\"39\" ><br>\n<br>\nwhere <img src=\"//latex.artofproblemsolving.com/b/4/9/b4901ea0d811ecd20ad755dbb44dcb7de8bc36ce.png\" class=\"latex\" alt=\"$f: A\\mapsto F$\" style=\"vertical-align: -3px\" width=\"82\" height=\"16\" > is any function of finite support.<br>\n<br>\nAn arbritrary element <img src=\"//latex.artofproblemsolving.com/2/d/2/2d295c3b71504a216422e5fffdfbad742c7add64.png\" class=\"latex\" alt=\"$\\Lambda$\" style=\"vertical-align: 0px\" width=\"12\" height=\"13\" > of the dual of <img src=\"//latex.artofproblemsolving.com/1/2/d/12d58aa29201da09d8e620f8698e3a37547f6b4a.png\" class=\"latex\" alt=\"$V$\" width=\"14\" height=\"12\" > can be associated uniquely to an arbitrary function <img src=\"//latex.artofproblemsolving.com/2/9/a/29a1d3df4146889b58be569a94f9ad059a78f3a7.png\" class=\"latex\" alt=\"$g: A\\mapsto F.$\" style=\"vertical-align: -3px\" width=\"82\" height=\"16\" > The functional is then given by<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/c/5/9c5e6f1a8f7183344f356abbfddcfb86b80f9a3d.png\" class=\"latex\" alt=\"$\\Lambda\\left(\\sum_{\\alpha\\in A}f(\\alpha)e_{\\alpha}\\right) = \\sum_{\\alpha\\in A}g(\\alpha)f(\\alpha)$\" style=\"vertical-align: -22px\" width=\"254\" height=\"53\" ><br>\n<br>\nWhat is left is a cardinality argument: arguing that there are more functions than there are finitely supported functions. I think this is going to split into some casework depending on the cardinality of <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > and the cardinality of <img src=\"//latex.artofproblemsolving.com/f/2/9/f2920fa2bc40ed68149a854f8b32c09f619b0d70.png\" class=\"latex\" alt=\"$F.$\" width=\"15\" height=\"12\" >", "post_id": 559135, "post_number": 2, "post_time_unix": 1151433520, "post_time_utc": "2006-06-27 18:38:40 UTC", "thanks_received": 2, "user_id": 2948, "username": "Kent Merryfield" }, { "attachments": [], "content_bbcode": "This problem is related to [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=97458]that one[/url].\r\n\r\nIf $V$ is a $F$-vectorspace with an infinite basis $A$ then, as [b]Kent Merryfield[/b] noted, the dual space $V^$ is isomorphic to $F^A$. So the dimension of $V^$ is $\|F\|^{\|A\|}$ which is greater than $\|A\|=dim V$.\r\n\r\nNote, that it is possible to have $\|V^\|=\|V\|$ (e.g. when $\|F\|=2^{\|A\|}$).", "content_html": "This problem is related to <a href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=97458\" class=\"bbcode_url\" target=\"_blank\">that one</a>.<br>\n<br>\nIf <img src=\"//latex.artofproblemsolving.com/1/2/d/12d58aa29201da09d8e620f8698e3a37547f6b4a.png\" class=\"latex\" alt=\"$V$\" width=\"14\" height=\"12\" > is a <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/0/5/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" >-</span>vectorspace with an infinite basis <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > then, as <b>Kent Merryfield</b> noted, the dual space <img src=\"//latex.artofproblemsolving.com/2/d/9/2d98c2bcba96160b6d03f6ebfb7f6586d6e4cab3.png\" class=\"latex\" alt=\"$V^$\" width=\"20\" height=\"13\" > is isomorphic to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/7/9/d79fd3d0bb24ef35e6243c24a2f2d82e55779444.png\" class=\"latex\" alt=\"$F^A$\" width=\"23\" height=\"15\" >.</span> So the dimension of <img src=\"//latex.artofproblemsolving.com/2/d/9/2d98c2bcba96160b6d03f6ebfb7f6586d6e4cab3.png\" class=\"latex\" alt=\"$V^$\" width=\"20\" height=\"13\" > is <img src=\"//latex.artofproblemsolving.com/7/e/b/7ebde6f9d9c3699f52b8286bcd763c503f7b7713.png\" class=\"latex\" alt=\"$\|F\|^{\|A\|}$\" style=\"vertical-align: -4px\" width=\"41\" height=\"21\" > which is greater than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/0/d204de9890ffa5f8b6759b4f955077cfad51d732.png\" class=\"latex\" alt=\"$\|A\|=dim V$\" style=\"vertical-align: -4px\" width=\"94\" height=\"18\" >.</span><br>\n<br>\nNote, that it is possible to have <img src=\"//latex.artofproblemsolving.com/a/e/8/ae89ba1479a2b94a548fb902ab4f5186348a14da.png\" class=\"latex\" alt=\"$\|V^\|=\|V\|$\" style=\"vertical-align: -4px\" width=\"79\" height=\"18\" > (e.g. when <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/d/5/4d54d65b468c63010c9bb18e20a3f9dd55dcc4c7.png\" class=\"latex\" alt=\"$\|F\|=2^{\|A\|}$\" style=\"vertical-align: -4px\" width=\"74\" height=\"21\" >)</span>.", "post_id": 559567, "post_number": 3, "post_time_unix": 1151450228, "post_time_utc": "2006-06-27 23:17:08 UTC", "thanks_received": 2, "user_id": 19435, "username": "lofar" } ], "source": null }	Prove that the dual of an infinite-dimensional vector space (over any field) is not isomorphic to the space itself. (Here the dual is meant in the purely algebraic sense, i.e. the set of all linear functionals on the vector space.)	[ "/Mathematics/Algebra/LinearAlgebra", "/Mathematics/Algebra/VectorAlgebra/AbstractVectorSpace", "/Mathematics/Algebra/VectorAlgebra/HamelBasis", "/Mathematics/Algebra/VectorAlgebra/Vector", "/Mathematics/Algebra/VectorAlgebra/VectorSpace" ]	Identify V with finite‑support functions on a basis and V* with all functions, then use a cardinality comparison to show V* is strictly larger.	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aops_99074	I'll try: Let's introduce a cartesian system with the origin at the bottom of the wall, and the current position of the top of the wall at $(0,\ y)$. Suppose that we are in an innertial system in which the bottom of the wall doesn't move. Let $\tau_0 = 0$ be the moment when a ray of light starts from the top of the wall. It will reach the floor at the moment $\tau_1 = y/c\sin\theta$. Now consider another ray which started from the top of the wall at time $\tau'_0 = \Delta t$, when the height of the wall was $y - \beta c\Delta t$. This ray will reach the floor at the moment $\tau'_1 = (y-\beta c\Delta t)/c\sin\theta$, so the shadow traversed a distance $v\Delta t'$, where $v$ is the speed of the shadow and $\Delta t' = \tau'_0 - \tau_0 + \tau'_1 - \tau_1 = \Delta t \frac{\sin\theta - \beta}{\sin\theta}$. We also have the equality \[ \frac{y-\beta c \Delta t}{x - v\Delta t'} = \frac{y}{x}(= \tan\theta) \Rightarrow \beta c\Delta t = v \tan\theta \Delta t' \Rightarrow v = \frac{\cos\theta}{\sin\theta - \beta}\beta c. \] This reduces to the expected result $v = \cot\theta \beta c$ when $\beta \rightarrow 0$.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [ { "name": "image.gif", "url": "https://cdn.artofproblemsolving.com/attachments/b/4/2ca6f37269f62d1deff47614f22b623bd29f37.gif" } ], "content_bbcode": "I have a problem now.\r\nImagine a wall which is decreasing with a relativistic velocity $\\upsilon_w=\\beta c$. Light ray falls on this wall at an angle $\\theta$ to the horizontal. Find the shadow speed.", "content_html": "I have a problem now.<br>\nImagine a wall which is decreasing with a relativistic velocity <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/4/4a4716b57fb5e31db666409bbfd57c114baa4170.png\" class=\"latex\" alt=\"$\\upsilon_w=\\beta c$\" style=\"vertical-align: -3px\" width=\"64\" height=\"16\" >.</span> Light ray falls on this wall at an angle <img src=\"//latex.artofproblemsolving.com/5/2/e/52e8ed7a3ba22130ad3984eb2cd413406475a689.png\" class=\"latex\" alt=\"$\\theta$\" width=\"8\" height=\"13\" > to the horizontal. Find the shadow speed.", "post_id": 559147, "post_number": 1, "post_time_unix": 1151433984, "post_time_utc": "2006-06-27 18:46:24 UTC", "thanks_received": 2, "user_id": 20298, "username": "azereyvazov" }, { "attachments": [], "content_bbcode": "I'll try:\r\n\r\nLet's introduce a cartesian system with the origin at the bottom of the wall, and the current position of the top of the wall at $(0,\\ y)$. Suppose that we are in an innertial system in which the bottom of the wall doesn't move. \r\n\r\nLet $\\tau_0 = 0$ be the moment when a ray of light starts from the top of the wall. It will reach the floor at the moment $\\tau_1 = y/c\\sin\\theta$.\r\n\r\nNow consider another ray which started from the top of the wall at time $\\tau'_0 = \\Delta t$, when the height of the wall was $y - \\beta c\\Delta t$. This ray will reach the floor at the moment $\\tau'_1 = (y-\\beta c\\Delta t)/c\\sin\\theta$, so the shadow traversed a distance $v\\Delta t'$, where $v$ is the speed of the shadow and $\\Delta t' = \\tau'_0 - \\tau_0 + \\tau'_1 - \\tau_1 = \\Delta t \\frac{\\sin\\theta - \\beta}{\\sin\\theta}$. We also have the equality \\[ \\frac{y-\\beta c \\Delta t}{x - v\\Delta t'} = \\frac{y}{x}(= \\tan\\theta) \\Rightarrow \\beta c\\Delta t = v \\tan\\theta \\Delta t' \\Rightarrow v = \\frac{\\cos\\theta}{\\sin\\theta - \\beta}\\beta c. \\]\r\n\r\nThis reduces to the expected result $v = \\cot\\theta \\beta c$ when $\\beta \\rightarrow 0$.", "content_html": "I'll try:<br>\n<br>\nLet's introduce a cartesian system with the origin at the bottom of the wall, and the current position of the top of the wall at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/d/84d44f8bbb6f5fa8711da2b8f38b6d5a9608f702.png\" class=\"latex\" alt=\"$(0,\\ y)$\" style=\"vertical-align: -4px\" width=\"45\" height=\"18\" >.</span> Suppose that we are in an innertial system in which the bottom of the wall doesn't move.<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/d/6/8/d688bee5821dda9a3ecbeeb0e29a3d39996e2808.png\" class=\"latex\" alt=\"$\\tau_0 = 0$\" style=\"vertical-align: -2px\" width=\"48\" height=\"15\" > be the moment when a ray of light starts from the top of the wall. It will reach the floor at the moment <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/3/0/a305f0dfca825284554e130d73ecc0080eb22b76.png\" class=\"latex\" alt=\"$\\tau_1 = y/c\\sin\\theta$\" style=\"vertical-align: -4px\" width=\"103\" height=\"18\" >.</span><br>\n<br>\nNow consider another ray which started from the top of the wall at time <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/c/0ecdac317cea8f21ca3f80989658677da7950996.png\" class=\"latex\" alt=\"$\\tau'_0 = \\Delta t$\" style=\"vertical-align: -4px\" width=\"60\" height=\"18\" >,</span> when the height of the wall was <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/4/4/5441ebe76cacfe37f2810733ca1552b29928f79e.png\" class=\"latex\" alt=\"$y - \\beta c\\Delta t$\" style=\"vertical-align: -3px\" width=\"72\" height=\"16\" >.</span> This ray will reach the floor at the moment <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/9/5/d95aa668892c3cadd366e310fcbb62da6cef53af.png\" class=\"latex\" alt=\"$\\tau'_1 = (y-\\beta c\\Delta t)/c\\sin\\theta$\" style=\"vertical-align: -4px\" width=\"180\" height=\"18\" >,</span> so the shadow traversed a distance <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/3/8e3095a525fb13844dd5d690a5b7489427936640.png\" class=\"latex\" alt=\"$v\\Delta t'$\" width=\"34\" height=\"14\" >,</span> where <img src=\"//latex.artofproblemsolving.com/a/9/f/a9f23bf124b6b2b2a993eb313c72e678664ac74a.png\" class=\"latex\" alt=\"$v$\" width=\"9\" height=\"8\" > is the speed of the shadow and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/8/9/889e0018fab41b04770560fdc431e2d07e7f0284.png\" class=\"latex\" alt=\"$\\Delta t' = \\tau'_0 - \\tau_0 + \\tau'_1 - \\tau_1 = \\Delta t \\frac{\\sin\\theta - \\beta}{\\sin\\theta}$\" style=\"vertical-align: -12px\" width=\"295\" height=\"38\" >.</span> We also have the equality <img src=\"//latex.artofproblemsolving.com/f/4/4/f44adb57f3948a351f2d2ac08cc9c84ac27d7a40.png\" class=\"latexcenter\" alt=\"\\[ \\frac{y-\\beta c \\Delta t}{x - v\\Delta t'} = \\frac{y}{x}(= \\tan\\theta) \\Rightarrow \\beta c\\Delta t = v \\tan\\theta \\Delta t' \\Rightarrow v = \\frac{\\cos\\theta}{\\sin\\theta - \\beta}\\beta c. \\]\" width=\"512\" height=\"41\" ><br>\n<br>\nThis reduces to the expected result <img src=\"//latex.artofproblemsolving.com/e/f/b/efbcbbe4184970831de3594f2d773d5a8ec3dd2b.png\" class=\"latex\" alt=\"$v = \\cot\\theta \\beta c$\" style=\"vertical-align: -3px\" width=\"89\" height=\"16\" > when <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/2/ee2972c53a2123f6cb19038c68288de229702249.png\" class=\"latex\" alt=\"$\\beta \\rightarrow 0$\" style=\"vertical-align: -3px\" width=\"48\" height=\"16\" >.</span>", "post_id": 559309, "post_number": 2, "post_time_unix": 1151440025, "post_time_utc": "2006-06-27 20:27:05 UTC", "thanks_received": 2, "user_id": 16186, "username": "Djole" } ], "source": null }	Imagine a wall moving with relativistic velocity $v_w=\beta c$. A light ray falls on this wall at an angle $\theta$ to the horizontal. Find the shadow speed.	[ "/Mathematics/AppliedMathematics", "/Mathematics/CalculusandAnalysis/Functions/Variable", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Cosine", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Cotangent", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Ctg", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine" ]	Analyze two successive light rays, accounting for the wall's motion, to relate the shadow’s displacement to the time difference using similar‑triangle geometry.	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aops_99084	Nifty [hide]Note that $\displaystyle\sum_{1}^{p-1}{i^2}= \frac{(p-1)(p)(2p-1)}{6}$. Let $q = \frac{(p-1)(2p-1)}{6}$. We show that q is an integer. Case 1: $p \equiv 1 (mod 6)$. Then $6\mid(p-1)$, therefore q is an integer. Case 2: $p \equiv 5 (mod 6)$. Then $2 \mid (p-1)$ and $3 \mid (2p-1)$ (since $2p-1 \equiv 3 (mod 6)$), therefore q is an integer. Hence, the sum on the right is equal to pq, which is clearly divisible by p. $QED$ [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $p>3$ be a prime. Prove that\r\n\\[ p\\mid 1^2 + 2^2+\\ldots+(p-1)^2 \\]", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/4/8/4/4843d240c7b2c54b45cfc9ed28b13887dfd36adb.png\" class=\"latex\" alt=\"$p>3$\" style=\"vertical-align: -3px\" width=\"43\" height=\"16\" > be a prime. Prove that<br>\n<img src=\"//latex.artofproblemsolving.com/b/5/2/b5238deef89a99c3ad8cba828f126e26592f5e53.png\" class=\"latexcenter\" alt=\"\\[ p\\mid 1^2 + 2^2+\\ldots+(p-1)^2 \\]\" width=\"206\" height=\"19\" >", "post_id": 559183, "post_number": 1, "post_time_unix": 1151435433, "post_time_utc": "2006-06-27 19:10:33 UTC", "thanks_received": 2, "user_id": 10530, "username": "Yimin Ge" }, { "attachments": [], "content_bbcode": "Either use the formula for the sum of the first $p-1$ squares or see http://www.mathlinks.ro/Forum/viewtopic.php?t=40171 with $k=2$.", "content_html": "Either use the formula for the sum of the first <img src=\"//latex.artofproblemsolving.com/a/b/0/ab0727c08c4aaa6fa119bd3fafe17851e431fd70.png\" class=\"latex\" alt=\"$p-1$\" style=\"vertical-align: -3px\" width=\"40\" height=\"15\" > squares or see <a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=40171\">http://www.mathlinks.ro/Forum/viewtopic.php?t=40171</a> with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/f/6/4f681bd84648c174d45eed0037702cae77ed0686.png\" class=\"latex\" alt=\"$k=2$\" width=\"43\" height=\"12\" >.</span>", "post_id": 559198, "post_number": 2, "post_time_unix": 1151436069, "post_time_utc": "2006-06-27 19:21:09 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "Nifty\r\n\r\n[hide]Note that $\\displaystyle\\sum_{1}^{p-1}{i^2}= \\frac{(p-1)(p)(2p-1)}{6}$. Let $q = \\frac{(p-1)(2p-1)}{6}$. We show that q is an integer.\n\nCase 1: $p \\equiv 1 (mod 6)$. Then $6\\mid(p-1)$, therefore q is an integer.\n\nCase 2: $p \\equiv 5 (mod 6)$. Then $2 \\mid (p-1)$ and $3 \\mid (2p-1)$ (since $2p-1 \\equiv 3 (mod 6)$), therefore q is an integer.\n\nHence, the sum on the right is equal to pq, which is clearly divisible by p. $QED$\n[/hide]", "content_html": "Nifty<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Note that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/7/b/d7bba9fa8b3b1ba91bb033622e8930db7373cb2d.png\" class=\"latex\" alt=\"$\\displaystyle\\sum_{1}^{p-1}{i^2}= \\frac{(p-1)(p)(2p-1)}{6}$\" style=\"vertical-align: -20px\" width=\"210\" height=\"51\" >.</span> Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/7/c7716ba918c978a8d63479c597a00d851986f355.png\" class=\"latex\" alt=\"$q = \\frac{(p-1)(2p-1)}{6}$\" style=\"vertical-align: -12px\" width=\"153\" height=\"38\" >.</span> We show that q is an integer.<br>\n<br>\nCase 1: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/1/7/1176fba7b54f057c6b2c3c5d75095461b7f369e5.png\" class=\"latex\" alt=\"$p \\equiv 1 (mod 6)$\" style=\"vertical-align: -4px\" width=\"99\" height=\"18\" >.</span> Then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/1/2/f1296187fe888bc47d99dda517712c4d4592d4fb.png\" class=\"latex\" alt=\"$6\\mid(p-1)$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" >,</span> therefore q is an integer.<br>\n<br>\nCase 2: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/6/b/96b2c7f4a28f19bf619f2cf7ece782f4465f82a8.png\" class=\"latex\" alt=\"$p \\equiv 5 (mod 6)$\" style=\"vertical-align: -4px\" width=\"99\" height=\"18\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/c/f/a/cfa3f80413cc496fa7cf2746fe355a614403128a.png\" class=\"latex\" alt=\"$2 \\mid (p-1)$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/3/6/3/3631fdf3e86fbad98eab7af2f5dae1cb2bd87fc8.png\" class=\"latex\" alt=\"$3 \\mid (2p-1)$\" style=\"vertical-align: -4px\" width=\"86\" height=\"18\" > (since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/d/dad3642167fa2a743cfa125c7a52b9867ba7750a.png\" class=\"latex\" alt=\"$2p-1 \\equiv 3 (mod 6)$\" style=\"vertical-align: -4px\" width=\"139\" height=\"18\" >)</span>, therefore q is an integer.<br>\n<br>\nHence, the sum on the right is equal to pq, which is clearly divisible by p. <img src=\"//latex.artofproblemsolving.com/4/9/6/4965e3d73b33f66d3c40556be29a9ccdae238e9d.png\" class=\"latex\" alt=\"$QED$\" style=\"vertical-align: -3px\" width=\"44\" height=\"16\" ></div>", "post_id": 559229, "post_number": 3, "post_time_unix": 1151437777, "post_time_utc": "2006-06-27 19:49:37 UTC", "thanks_received": 2, "user_id": 2336, "username": "justdudxit" }, { "attachments": [], "content_bbcode": "[quote=\"justdudxit\"]Nifty\n\n[hide]Note that $\\displaystyle\\sum_{1}^{p-1}{i^2}= \\frac{(p-1)(p)(2p-1)}{6}$. Let $q = \\frac{(p-1)(2p-1)}{6}$. We show that q is an integer.\n\nCase 1: $p \\equiv 1 (mod 6)$. Then $6\\mid(p-1)$, therefore q is an integer.\n\nCase 2: $p \\equiv 5 (mod 6)$. Then $2 \\mid (p-1)$ and $3 \\mid (2p-1)$ (since $2p-1 \\equiv 3 (mod 6)$), therefore q is an integer.\n\nHence, the sum on the right is equal to pq, which is clearly divisible by p. $QED$\n[/hide][/quote]\r\nI think we can make it more simple:\r\n$6\|(p-1)(2p-1)p$\r\n$p\|(p-1)(2p-1)p$\r\n$(6,p)=1$\r\nthen$6p\|(p-1)(2p-1)p$", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">justdudxit wrote:</div>\n<div class=\"bbcode_quote_body\">Nifty<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Note that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/7/b/d7bba9fa8b3b1ba91bb033622e8930db7373cb2d.png\" class=\"latex\" alt=\"$\\displaystyle\\sum_{1}^{p-1}{i^2}= \\frac{(p-1)(p)(2p-1)}{6}$\" style=\"vertical-align: -20px\" width=\"210\" height=\"51\" >.</span> Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/7/c7716ba918c978a8d63479c597a00d851986f355.png\" class=\"latex\" alt=\"$q = \\frac{(p-1)(2p-1)}{6}$\" style=\"vertical-align: -12px\" width=\"153\" height=\"38\" >.</span> We show that q is an integer.<br>\n<br>\nCase 1: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/1/7/1176fba7b54f057c6b2c3c5d75095461b7f369e5.png\" class=\"latex\" alt=\"$p \\equiv 1 (mod 6)$\" style=\"vertical-align: -4px\" width=\"99\" height=\"18\" >.</span> Then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/1/2/f1296187fe888bc47d99dda517712c4d4592d4fb.png\" class=\"latex\" alt=\"$6\\mid(p-1)$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" >,</span> therefore q is an integer.<br>\n<br>\nCase 2: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/6/b/96b2c7f4a28f19bf619f2cf7ece782f4465f82a8.png\" class=\"latex\" alt=\"$p \\equiv 5 (mod 6)$\" style=\"vertical-align: -4px\" width=\"99\" height=\"18\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/c/f/a/cfa3f80413cc496fa7cf2746fe355a614403128a.png\" class=\"latex\" alt=\"$2 \\mid (p-1)$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/3/6/3/3631fdf3e86fbad98eab7af2f5dae1cb2bd87fc8.png\" class=\"latex\" alt=\"$3 \\mid (2p-1)$\" style=\"vertical-align: -4px\" width=\"86\" height=\"18\" > (since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/d/dad3642167fa2a743cfa125c7a52b9867ba7750a.png\" class=\"latex\" alt=\"$2p-1 \\equiv 3 (mod 6)$\" style=\"vertical-align: -4px\" width=\"139\" height=\"18\" >)</span>, therefore q is an integer.<br>\n<br>\nHence, the sum on the right is equal to pq, which is clearly divisible by p. <img src=\"//latex.artofproblemsolving.com/4/9/6/4965e3d73b33f66d3c40556be29a9ccdae238e9d.png\" class=\"latex\" alt=\"$QED$\" style=\"vertical-align: -3px\" width=\"44\" height=\"16\" ></div></div>\n</div>\nI think we can make it more simple:<br>\n<img src=\"//latex.artofproblemsolving.com/4/2/5/425a581f3844a1906d54f6f9fa0271216dab3547.png\" class=\"latex\" alt=\"$6\|(p-1)(2p-1)p$\" style=\"vertical-align: -4px\" width=\"141\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/0/9/2/092fde30b580bb8677274ed4b5d8b04e67b4eadf.png\" class=\"latex\" alt=\"$p\|(p-1)(2p-1)p$\" style=\"vertical-align: -4px\" width=\"142\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/8/e/f8eb461e63af39711f7706e32338d7bba0bb4a9c.png\" class=\"latex\" alt=\"$(6,p)=1$\" style=\"vertical-align: -4px\" width=\"72\" height=\"18\" ><br>\nthe<span style=\"white-space:nowrap;\">n<img src=\"//latex.artofproblemsolving.com/4/a/7/4a7c87849d588b341c3d9955bc125433211f23bc.png\" class=\"latex\" alt=\"$6p\|(p-1)(2p-1)p$\" style=\"vertical-align: -4px\" width=\"150\" height=\"18\" ></span>", "post_id": 559873, "post_number": 4, "post_time_unix": 1151475218, "post_time_utc": "2006-06-28 06:13:38 UTC", "thanks_received": 2, "user_id": 14130, "username": "Hawk Tiger" } ], "source": null }	Let $p>3$ be a prime. Prove that \[ p \mid 1^2 + 2^2 + \cdots + (p-1)^2. \]	[ "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/PrimeNumbers/PrimeSumsandProducts" ]	Write the sum using the closed form and prove the non‑p factor is an integer by checking p modulo 6.	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aops_99096	First, he can make 25/5 = 5 cigars, and then he will smoke them. After that, he can take the 5 remaining cigar butts to make a new cigar and smoke that one. Then he only has 1 cigar butt left and can't make any new cigars, so the answer is 5 + 1 = [b]6[/b]. [quote][hide]25+5+1=31[/hide][/quote] The first 25 he finds are cigar butts, not cigars. [quote][hide]$\frac{25}{5}=5$ [/hide][/quote] After he smokes those 5 cigars, he has 5 cigar butts left over.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?", "content_html": "Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?", "post_id": 559341, "post_number": 1, "post_time_unix": 1151441282, "post_time_utc": "2006-06-27 20:48:02 UTC", "thanks_received": 1, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "[hide=\"easy division\"]$\\frac{25}{5}=5$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">easy division</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/c/3/b/c3b8e246f097d04649637ffd244fb2f9f10b57fb.png\" class=\"latex\" alt=\"$\\frac{25}{5}=5$\" style=\"vertical-align: -12px\" width=\"55\" height=\"37\" ></div>", "post_id": 559356, "post_number": 2, "post_time_unix": 1151441608, "post_time_utc": "2006-06-27 20:53:28 UTC", "thanks_received": 2, "user_id": 17780, "username": "redcomet46" }, { "attachments": [], "content_bbcode": "[hide]5+1=6[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">5+1=6</div>", "post_id": 559358, "post_number": 3, "post_time_unix": 1151441620, "post_time_utc": "2006-06-27 20:53:40 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "First, he can make 25/5 = 5 cigars, and then he will smoke them. After that, he can take the 5 remaining cigar butts to make a new cigar and smoke that one. Then he only has 1 cigar butt left and can't make any new cigars, so the answer is 5 + 1 = [b]6[/b].\r\n\r\n[quote][hide]25+5+1=31[/hide][/quote]\nThe first 25 he finds are cigar butts, not cigars.\n\n[quote][hide]$\\frac{25}{5}=5$\n[/hide][/quote]\r\nAfter he smokes those 5 cigars, he has 5 cigar butts left over.", "content_html": "First, he can make 25/5 = 5 cigars, and then he will smoke them. After that, he can take the 5 remaining cigar butts to make a new cigar and smoke that one. Then he only has 1 cigar butt left and can't make any new cigars, so the answer is 5 + 1 = <b>6</b>.<br>\n\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">25+5+1=31</div></div>\n</div>\nThe first 25 he finds are cigar butts, not cigars.<br>\n\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/c/3/b/c3b8e246f097d04649637ffd244fb2f9f10b57fb.png\" class=\"latex\" alt=\"$\\frac{25}{5}=5$\" style=\"vertical-align: -12px\" width=\"55\" height=\"37\" ></div></div>\n</div>\nAfter he smokes those 5 cigars, he has 5 cigar butts left over.", "post_id": 559360, "post_number": 4, "post_time_unix": 1151441646, "post_time_utc": "2006-06-27 20:54:06 UTC", "thanks_received": 2, "user_id": 20563, "username": "nebula42" }, { "attachments": [], "content_bbcode": "[hide]We could also just subtract 4 as many times as possible from 25, without reaching 0. This can be done 6 times.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">We could also just subtract 4 as many times as possible from 25, without reaching 0. This can be done 6 times.</div>", "post_id": 559385, "post_number": 5, "post_time_unix": 1151442385, "post_time_utc": "2006-06-27 21:06:25 UTC", "thanks_received": 1, "user_id": 1171, "username": "MCrawford" }, { "attachments": [], "content_bbcode": "Interestingly, this problem is equivalent to finding the number of zeros in $25!$ (but that's only because every [b]5[/b] cigar butts becomes one cigar).\r\n\r\nBy the way, how do you use LaTeX to write the symbols for the floor and ceiling functions?", "content_html": "Interestingly, this problem is equivalent to finding the number of zeros in <img src=\"//latex.artofproblemsolving.com/4/0/7/4078fa6376a28e8b3ceb9795766a8d5f4a4f31cb.png\" class=\"latex\" alt=\"$25!$\" width=\"21\" height=\"13\" > (but that's only because every <b>5</b> cigar butts becomes one cigar).<br>\n<br>\nBy the way, how do you use LaTeX to write the symbols for the floor and ceiling functions?", "post_id": 559410, "post_number": 6, "post_time_unix": 1151443288, "post_time_utc": "2006-06-27 21:21:28 UTC", "thanks_received": 2, "user_id": 20563, "username": "nebula42" }, { "attachments": [], "content_bbcode": "[quote=\"nebula42\"]Interestingly, this problem is equivalent to finding the number of zeros in $25!$ (but that's only because every [b]5[/b] cigar butts becomes one cigar).\n\nBy the way, how do you use LaTeX to write the symbols for the floor and ceiling functions?[/quote]\r\n\r\n\\lfloor=$\\lfloor$ , \\rfloor=$\\rfloor$, \\lceil=$\\lceil$ , \\rceil=$\\rceil$\r\n\r\n$\\lfloor \\frac{25}{4} \\rfloor=6$", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nebula42 wrote:</div>\n<div class=\"bbcode_quote_body\">Interestingly, this problem is equivalent to finding the number of zeros in <img src=\"//latex.artofproblemsolving.com/4/0/7/4078fa6376a28e8b3ceb9795766a8d5f4a4f31cb.png\" class=\"latex\" alt=\"$25!$\" width=\"21\" height=\"13\" > (but that's only because every <b>5</b> cigar butts becomes one cigar).<br>\n<br>\nBy the way, how do you use LaTeX to write the symbols for the floor and ceiling functions?</div>\n</div>\n<br>\n\\lfloor<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/a/5/0/a501a210c4f22118f43d09317c17f75c006c1e55.png\" class=\"latex\" alt=\"$\\lfloor$\" style=\"vertical-align: -5px\" width=\"8\" height=\"19\" ></span> , \\rfloor<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/6/2/5/625f626a06c6ff567dfc57da93337aeccacae0a7.png\" class=\"latex\" alt=\"$\\rfloor$\" style=\"vertical-align: -5px\" width=\"5\" height=\"19\" >,</span> \\lceil<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/a/5/e/a5e03343fb25f6532be263808489d53b2d914600.png\" class=\"latex\" alt=\"$\\lceil$\" style=\"vertical-align: -5px\" width=\"8\" height=\"19\" ></span> , \\rceil<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/2/6/7/26757e7d8125a2df54a341bbe9d10fa5348ce72c.png\" class=\"latex\" alt=\"$\\rceil$\" style=\"vertical-align: -5px\" width=\"5\" height=\"19\" ></span><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/c/a/7ca0ecac73b114fd7ea819c722aad564b275528c.png\" class=\"latex\" alt=\"$\\lfloor \\frac{25}{4} \\rfloor=6$\" style=\"vertical-align: -13px\" width=\"71\" height=\"38\" >", "post_id": 559665, "post_number": 7, "post_time_unix": 1151456889, "post_time_utc": "2006-06-28 01:08:09 UTC", "thanks_received": 2, "user_id": 11029, "username": "ch1n353ch3s54a1l" }, { "attachments": [], "content_bbcode": "25/5= 5 \r\n5/5=1 \r\n5+1=6", "content_html": "25/5= 5<br>\n5/5=1<br>\n5+1=6", "post_id": 599090, "post_number": 8, "post_time_unix": 1155256147, "post_time_utc": "2006-08-11 00:29:07 UTC", "thanks_received": 2, "user_id": 21713, "username": "Challenge_24_Champ" }, { "attachments": [], "content_bbcode": "Was this an old POW problem from MC.", "content_html": "Was this an old POW problem from MC.", "post_id": 600514, "post_number": 9, "post_time_unix": 1155416338, "post_time_utc": "2006-08-12 20:58:58 UTC", "thanks_received": 2, "user_id": 8960, "username": "bpms" }, { "attachments": [], "content_bbcode": "because he has 25 butts,so he can makes 5 cigars,but then he'll makes another 5 butts,so he'll makes 5 plus 1=6", "content_html": "because he has 25 butts,so he can makes 5 cigars,but then he'll makes another 5 butts,so he'll makes 5 plus 1=6", "post_id": 608712, "post_number": 10, "post_time_unix": 1156249416, "post_time_utc": "2006-08-22 12:23:36 UTC", "thanks_received": 2, "user_id": 21819, "username": "mathsfun_Tony" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?[/quote]\r\n\r\n$\\frac{25}{5}= 5+1 = \\fbox{6}$\r\nI'm probably wrong...there must be a catch :rotfl: :rotfl:", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?</div>\n</div>\n<br>\n<img src=\"//latex.artofproblemsolving.com/1/7/0/1708e1659c735571c07aabe814b190c420c74401.png\" class=\"latex\" alt=\"$\\frac{25}{5}= 5+1 = \\fbox{6}$\" style=\"vertical-align: -12px\" width=\"131\" height=\"37\" ><br>\nI'm probably wrong...there must be a catch <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" />", "post_id": 608919, "post_number": 11, "post_time_unix": 1156267084, "post_time_utc": "2006-08-22 17:18:04 UTC", "thanks_received": 2, "user_id": 9401, "username": "#H34N1" }, { "attachments": [], "content_bbcode": "[hide=\"tricky part\"]After he smokes 5 cigars, there are 5 cigar butts, so he makes one more[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">tricky part</a><div class=\"cmty-hide-content\" style=\"display:none\">After he smokes 5 cigars, there are 5 cigar butts, so he makes one more</div>", "post_id": 609866, "post_number": 12, "post_time_unix": 1156348259, "post_time_utc": "2006-08-23 15:50:59 UTC", "thanks_received": 2, "user_id": 15223, "username": "1=2" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?[/quote]\r\n\r\n[hide]25/5=5, so 1 left over 5+1=6[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">25/5=5, so 1 left over 5+1=6</div>", "post_id": 615228, "post_number": 13, "post_time_unix": 1156821554, "post_time_utc": "2006-08-29 03:19:14 UTC", "thanks_received": 1, "user_id": 11899, "username": "moogra" } ], "source": null }	Hobo Billy Bob likes to smoke cigars. From 5 cigar butts, Billy Bob can make one cigar. He finds 25 cigar butts in an ashtray. How many cigars will he smoke, assuming that he smokes every one he makes?	[ "/Mathematics/Algebra/RateProblems", "/Mathematics/RecreationalMathematics/MathematicalHumor/Self-Recursion", "/Mathematics/RecreationalMathematics/Puzzles/Cigarettes" ]	Repeatedly exchange every 5 butts for a new cigar, including butts obtained from smoking, until fewer than 5 remain.	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aops_99109	[hide]Well, 10 quarters means 2.50, 10 nickels means 0.50, and 10 pennies, is, of course, 0.10. Adding these up, they total to 3.10. Well, since the total is 5.00, the dimes add up to 1.90, so there are [b]19 dimes[/b].[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Aldebaran has $\\$$5.00 in quarters, nickels, dimes, and pennies. If Aldebaran has 10 quarters, 10 nickels, and 10 pennies, how many dimes does Aldebaran have?", "content_html": "Aldebaran has <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >5</span>.00 in quarters, nickels, dimes, and pennies. If Aldebaran has 10 quarters, 10 nickels, and 10 pennies, how many dimes does Aldebaran have?", "post_id": 559537, "post_number": 1, "post_time_unix": 1151449131, "post_time_utc": "2006-06-27 22:58:51 UTC", "thanks_received": 2, "user_id": 17514, "username": "Ignite168" }, { "attachments": [], "content_bbcode": "[hide]$5-2.5-.5-.1=1.9$\n$\\boxed{19}$ dimes[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/1/0/6/10602124f3c5645c8d5b12d64210e4c83ab35853.png\" class=\"latex\" alt=\"$5-2.5-.5-.1=1.9$\" style=\"vertical-align: 0px\" width=\"174\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/e/6/3e6b4391f25789ac9419bc73abb91174108ef344.png\" class=\"latex\" alt=\"$\\boxed{19}$\" style=\"vertical-align: -5px\" width=\"30\" height=\"23\" > dimes</div>", "post_id": 559542, "post_number": 2, "post_time_unix": 1151449238, "post_time_utc": "2006-06-27 23:00:38 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[hide]Well, 10 quarters means 2.50, 10 nickels means 0.50, and 10 pennies, is, of course, 0.10. Adding these up, they total to 3.10. Well, since the total is 5.00, the dimes add up to 1.90, so there are [b]19 dimes[/b].[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Well, 10 quarters means 2.50, 10 nickels means 0.50, and 10 pennies, is, of course, 0.10. Adding these up, they total to 3.10. Well, since the total is 5.00, the dimes add up to 1.90, so there are <b>19 dimes</b>.</div>", "post_id": 559752, "post_number": 3, "post_time_unix": 1151460305, "post_time_utc": "2006-06-28 02:05:05 UTC", "thanks_received": 2, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "[hide=\"The no decimal way!\"]You add up the number of cents he has not including the dimes and get 250+10+50=310. Subtract that from 500 to get 190. [b][i][u]19[/u][/i][/b][/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">The no decimal way!</a><div class=\"cmty-hide-content\" style=\"display:none\">You add up the number of cents he has not including the dimes and get 250+10+50=310. Subtract that from 500 to get 190. <b><i><u>19</u></i></b></div>", "post_id": 560026, "post_number": 4, "post_time_unix": 1151495958, "post_time_utc": "2006-06-28 11:59:18 UTC", "thanks_received": 2, "user_id": 15223, "username": "1=2" } ], "source": null }	Aldebaran has \$5.00 in quarters, nickels, dimes, and pennies. If Aldebaran has 10 quarters, 10 nickels, and 10 pennies, how many dimes does Aldebaran have?	[ "/Mathematics/FoundationsofMathematics/MathematicalProblems/SolvedProblems" ]	Subtract the known coin values from the total and convert the remaining amount into dimes.	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aops_99112	[quote="xxxyyyy"] $Question 2$Which value is greater of $1999^{1999}$ or $2000^{1998}$ ? You can not use a calculator or a computer. Logic is expected!!!![/quote] [hide] $\frac{2000^{1998}}{1999^{1999}}=\left(\frac{2000}{1999}\right)^{1999}\times \frac{1}{2000}=(1+\frac{1}{1999})^{1999}\times \frac{1}{1999}<3\times\frac{1}{1999}<1$. Thus $1999^{1999}>2000^{1998}$. Here I used the fact that $(1+\frac{1}{x})^x<3$.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "$Question 1$ Express $997$ as the sum of natural numbers i.e. $997 = n_1 + n_2 + n_3 + ... + n_x$ so that $P = n_1 * n_2 * n_3 ... n_x$ is the maximum.\r\n\r\nPlease, take a note that there is no constraint that you have to divide $997$ in $2$ parts only. If that were the case then the answer would have been $997 = 498 + 499$ and the Productmax would have been $498499 = 248502$.\r\n\r\n$Question 2$Which value is greater of $1999^{1999}$ or $2000^{1998}$ ? You can not use a calculator or a computer. Logic is expected!!!!", "content_html": "<img src=\"//latex.artofproblemsolving.com/e/e/0/ee087ccec87c62a48368c90fa53d97e82b1dadf9.png\" class=\"latex\" alt=\"$Question 1$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > Express <img src=\"//latex.artofproblemsolving.com/9/0/7/90782e3f18f6a430876413c43e3d163e0e8ae3f8.png\" class=\"latex\" alt=\"$997$\" width=\"27\" height=\"12\" > as the sum of natural numbers i.e. <img src=\"//latex.artofproblemsolving.com/e/a/7/ea78a5fe7dd6157d7a2772e5a34b8c2ce16375a7.png\" class=\"latex\" alt=\"$997 = n_1 + n_2 + n_3 + ... + n_x$\" style=\"vertical-align: -2px\" width=\"228\" height=\"15\" > so that <img src=\"//latex.artofproblemsolving.com/c/1/a/c1a35542d07c8f8b0bbc06e60db3c795544ba29a.png\" class=\"latex\" alt=\"$P = n_1 n_2 * n_3 ... n_x$\" style=\"vertical-align: -2px\" width=\"195\" height=\"15\" > is the maximum.<br>\n<br>\nPlease, take a note that there is no constraint that you have to divide <img src=\"//latex.artofproblemsolving.com/9/0/7/90782e3f18f6a430876413c43e3d163e0e8ae3f8.png\" class=\"latex\" alt=\"$997$\" width=\"27\" height=\"12\" > in <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > parts only. If that were the case then the answer would have been <img src=\"//latex.artofproblemsolving.com/5/3/e/53e497e18e425bf11a9103e1f6fb81edd47dbb4a.png\" class=\"latex\" alt=\"$997 = 498 + 499$\" style=\"vertical-align: -1px\" width=\"127\" height=\"14\" > and the Productmax would have been <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/5/9/0591a524926fd7eeef5dabae468f372fdec9ebc1.png\" class=\"latex\" alt=\"$498*499 = 248502$\" style=\"vertical-align: 0px\" width=\"149\" height=\"13\" >.</span><br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/1/ee1a290540e1f27cdde089a4aa1e5c938a21906f.png\" class=\"latex\" alt=\"$Question 2$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" >W</span>hich value is greater of <img src=\"//latex.artofproblemsolving.com/f/0/f/f0f4a95149501f0be0f3b347743993bb482560a9.png\" class=\"latex\" alt=\"$1999^{1999}$\" style=\"vertical-align: 0px\" width=\"61\" height=\"15\" > or <img src=\"//latex.artofproblemsolving.com/5/c/8/5c81dca2dc5bc605461db52be66fb56f6508f6ca.png\" class=\"latex\" alt=\"$2000^{1998}$\" width=\"61\" height=\"15\" > ? You can not use a calculator or a computer. Logic is expected!!!!", "post_id": 559573, "post_number": 1, "post_time_unix": 1151450429, "post_time_utc": "2006-06-27 23:20:29 UTC", "thanks_received": 2, "user_id": 16952, "username": "xxxyyyy" }, { "attachments": [], "content_bbcode": "[quote=\"xxxyyyy\"]\n$Question 2$Which value is greater of $1999^{1999}$ or $2000^{1998}$ ? You can not use a calculator or a computer. Logic is expected!!!![/quote]\r\n[hide]\n$\\frac{2000^{1998}}{1999^{1999}}=\\left(\\frac{2000}{1999}\\right)^{1999}\\times \\frac{1}{2000}=(1+\\frac{1}{1999})^{1999}\\times \\frac{1}{1999}<3\\times\\frac{1}{1999}<1$. Thus $1999^{1999}>2000^{1998}$.\nHere I used the fact that $(1+\\frac{1}{x})^x<3$.[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">xxxyyyy wrote:</div>\n<div class=\"bbcode_quote_body\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/1/ee1a290540e1f27cdde089a4aa1e5c938a21906f.png\" class=\"latex\" alt=\"$Question 2$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" >W</span>hich value is greater of <img src=\"//latex.artofproblemsolving.com/f/0/f/f0f4a95149501f0be0f3b347743993bb482560a9.png\" class=\"latex\" alt=\"$1999^{1999}$\" style=\"vertical-align: 0px\" width=\"61\" height=\"15\" > or <img src=\"//latex.artofproblemsolving.com/5/c/8/5c81dca2dc5bc605461db52be66fb56f6508f6ca.png\" class=\"latex\" alt=\"$2000^{1998}$\" width=\"61\" height=\"15\" > ? You can not use a calculator or a computer. Logic is expected!!!!</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/8/3/c8323e6f7aee5a6c773644435df2a99fbdfb256e.png\" class=\"latex\" alt=\"$\\frac{2000^{1998}}{1999^{1999}}=\\left(\\frac{2000}{1999}\\right)^{1999}\\times \\frac{1}{2000}=(1+\\frac{1}{1999})^{1999}\\times \\frac{1}{1999}<3\\times\\frac{1}{1999}<1$\" style=\"vertical-align: -17px\" width=\"576\" height=\"46\" >.</span> Thus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/9/c399eba4b019138ad3649ea8c69e9a6dee52242d.png\" class=\"latex\" alt=\"$1999^{1999}>2000^{1998}$\" style=\"vertical-align: 0px\" width=\"149\" height=\"15\" >.</span><br>\nHere I used the fact that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/2/5/225d747f0d9edfd7e39443b203a5ebe92921e7d2.png\" class=\"latex\" alt=\"$(1+\\frac{1}{x})^x<3$\" style=\"vertical-align: -12px\" width=\"101\" height=\"37\" >.</span></div>", "post_id": 559603, "post_number": 2, "post_time_unix": 1151453232, "post_time_utc": "2006-06-28 00:07:12 UTC", "thanks_received": 2, "user_id": 9197, "username": "kimby_102" }, { "attachments": [], "content_bbcode": "[hide]We assign values to the variables in any way. Then, for each $n_{k}$ such that $n_{k}\\ge 5$, replace $n_{k}$ with $3$ and $n_{k}-3$. Since $2n_{k}>9$, we have $3(n_{k}-3)>n_{k}$, so the product increases. Therefore, our product consists of no integers greater than $4$.\n\nFor convenience, we replace each $4$ with two $2$'s, so that the product remains the same. Our product now consists only of $2$'s and $3$'s.\n\nSince $2^{3}<3^{2}$, we replace groups of three $2$'s with two $3$'s until we have less than three $2$'s. Therefore, $n_{1}=n_{2}=\\cdots =n_{331}=3$ and $n_{332}=n_{333}=2$.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">We assign values to the variables in any way. Then, for each <img src=\"//latex.artofproblemsolving.com/b/2/d/b2d3e21607da8e3e0a0a4d9700f3d03b2ec4f8f8.png\" class=\"latex\" alt=\"$n_{k}$\" style=\"vertical-align: -2px\" width=\"18\" height=\"10\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/6/f/e6f0aee75f3e2d2387a91b9ab0b7f4d97c5f4db5.png\" class=\"latex\" alt=\"$n_{k}\\ge 5$\" style=\"vertical-align: -2px\" width=\"52\" height=\"15\" >,</span> replace <img src=\"//latex.artofproblemsolving.com/b/2/d/b2d3e21607da8e3e0a0a4d9700f3d03b2ec4f8f8.png\" class=\"latex\" alt=\"$n_{k}$\" style=\"vertical-align: -2px\" width=\"18\" height=\"10\" > with <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/f/c/8fcde86fe174ce72a532c40c55a32e2a2a7f42e9.png\" class=\"latex\" alt=\"$n_{k}-3$\" style=\"vertical-align: -2px\" width=\"50\" height=\"15\" >.</span> Since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/d/8eda920ee1673f57a864728dddaa7dcb04dba29b.png\" class=\"latex\" alt=\"$2n_{k}>9$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" >,</span> we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/e/30ee2a59fc99a6bd49165e05dc24895314d1d60a.png\" class=\"latex\" alt=\"$3(n_{k}-3)>n_{k}$\" style=\"vertical-align: -4px\" width=\"116\" height=\"18\" >,</span> so the product increases. Therefore, our product consists of no integers greater than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png\" class=\"latex\" alt=\"$4$\" style=\"vertical-align: 0px\" width=\"9\" height=\"12\" >.</span><br>\n<br>\nFor convenience, we replace each <img src=\"//latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png\" class=\"latex\" alt=\"$4$\" style=\"vertical-align: 0px\" width=\"9\" height=\"12\" > with two <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >'</span>s, so that the product remains the same. Our product now consists only of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >'</span>s and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" >'</span>s.<br>\n<br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/8/7/e87889cbc81ce0e4107112562aafd14b16be2eb8.png\" class=\"latex\" alt=\"$2^{3}<3^{2}$\" style=\"vertical-align: 0px\" width=\"56\" height=\"15\" >,</span> we replace groups of three <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >'</span>s with two <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" >'</span>s until we have less than three <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >'</span>s. Therefore, <img src=\"//latex.artofproblemsolving.com/6/2/9/6294d921991216b998bf685628b5464571127579.png\" class=\"latex\" alt=\"$n_{1}=n_{2}=\\cdots =n_{331}=3$\" style=\"vertical-align: -2px\" width=\"195\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/8/7/3873e180be7cae4d03e8223b1960d7ed6516000c.png\" class=\"latex\" alt=\"$n_{332}=n_{333}=2$\" style=\"vertical-align: -2px\" width=\"120\" height=\"14\" >.</span></div>", "post_id": 559610, "post_number": 3, "post_time_unix": 1151453809, "post_time_utc": "2006-06-28 00:16:49 UTC", "thanks_received": 2, "user_id": 9526, "username": "matt276eagles" }, { "attachments": [], "content_bbcode": "[quote=\"kimby_102\"][quote=\"xxxyyyy\"]\n$Question 2$Which value is greater of $1999^{1999}$ or $2000^{1998}$ ? You can not use a calculator or a computer. Logic is expected!!!![/quote]\n[hide]\n$\\frac{2000^{1998}}{1999^{1999}}=\\left(\\frac{2000}{1999}\\right)^{1999}\\times \\frac{1}{2000}=(1+\\frac{1}{1999})^{1999}\\times \\frac{1}{1999}<3\\times\\frac{1}{1999}<1$. Thus $1999^{1999}>2000^{1998}$.\nHere I used the fact that $(1+\\frac{1}{x})^x<3$.[/hide][/quote]\r\n\r\nI got it....", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">kimby_102 wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">xxxyyyy wrote:</div>\n<div class=\"bbcode_quote_body\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/1/ee1a290540e1f27cdde089a4aa1e5c938a21906f.png\" class=\"latex\" alt=\"$Question 2$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" >W</span>hich value is greater of <img src=\"//latex.artofproblemsolving.com/f/0/f/f0f4a95149501f0be0f3b347743993bb482560a9.png\" class=\"latex\" alt=\"$1999^{1999}$\" style=\"vertical-align: 0px\" width=\"61\" height=\"15\" > or <img src=\"//latex.artofproblemsolving.com/5/c/8/5c81dca2dc5bc605461db52be66fb56f6508f6ca.png\" class=\"latex\" alt=\"$2000^{1998}$\" width=\"61\" height=\"15\" > ? You can not use a calculator or a computer. Logic is expected!!!!</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/8/3/c8323e6f7aee5a6c773644435df2a99fbdfb256e.png\" class=\"latex\" alt=\"$\\frac{2000^{1998}}{1999^{1999}}=\\left(\\frac{2000}{1999}\\right)^{1999}\\times \\frac{1}{2000}=(1+\\frac{1}{1999})^{1999}\\times \\frac{1}{1999}<3\\times\\frac{1}{1999}<1$\" style=\"vertical-align: -17px\" width=\"576\" height=\"46\" >.</span> Thus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/9/c399eba4b019138ad3649ea8c69e9a6dee52242d.png\" class=\"latex\" alt=\"$1999^{1999}>2000^{1998}$\" style=\"vertical-align: 0px\" width=\"149\" height=\"15\" >.</span><br>\nHere I used the fact that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/2/5/225d747f0d9edfd7e39443b203a5ebe92921e7d2.png\" class=\"latex\" alt=\"$(1+\\frac{1}{x})^x<3$\" style=\"vertical-align: -12px\" width=\"101\" height=\"37\" >.</span></div></div>\n</div>\n<br>\nI got it....", "post_id": 559711, "post_number": 4, "post_time_unix": 1151458033, "post_time_utc": "2006-06-28 01:27:13 UTC", "thanks_received": 2, "user_id": 16952, "username": "xxxyyyy" } ], "source": null }	Question 1. Express 997 as a sum of natural numbers \[ 997 = n_1 + n_2 + n_3 + \dots + n_x \] so that \[ P = n_1 n_2 n_3 \cdots n_x \] is maximal. Question 2. Which is greater, $1999^{1999}$ or $2000^{1998}$?	[ "/Mathematics/Algebra/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/Algebra/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/N", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/RationalInteger", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/Products/Product", "/Mathematics/Algebra/Sums/Sum" ]	Compare the powers by forming their ratio and bounding (1+1/n)^n with a constant (e.g., <3).	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aops_99137	[hide]Each group has 4 people, so there would be 6 matches in each group making $6*8=\boxed{48}$ so far. Now there's 16 teams and single elimination. Therefore, there are 15 games and the game between the losing teams in the semi-finals, which makes $\boxed{16}$. Final answer: $\boxed{64}$ games.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "There are 32 teams in the tournament, and they are divided into eight groups with the same number of teams in each group. Every team in a group plays the other teams in their group exactly once. This is the first round. The top two teams from each group then advance to the second round, where all of these first-place and second-place teams are put into a bracket and a single-elimination tournament takes place until the winner is determined. (There is also a play-off game between the losing teams in the semi-finals to determine third and fourth place.) According to this set-up, how many total games are played during the World Cup?\r\n\r\nAlong with the answer, you must include an explanation. And, also, as a random add-on, Brazil has no chance of winning this year because England is da bomb. :D", "content_html": "There are 32 teams in the tournament, and they are divided into eight groups with the same number of teams in each group. Every team in a group plays the other teams in their group exactly once. This is the first round. The top two teams from each group then advance to the second round, where all of these first-place and second-place teams are put into a bracket and a single-elimination tournament takes place until the winner is determined. (There is also a play-off game between the losing teams in the semi-finals to determine third and fourth place.) According to this set-up, how many total games are played during the World Cup?<br>\n<br>\nAlong with the answer, you must include an explanation. And, also, as a random add-on, Brazil has no chance of winning this year because England is da bomb. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 559743, "post_number": 1, "post_time_unix": 1151459930, "post_time_utc": "2006-06-28 01:58:50 UTC", "thanks_received": 1, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "[hide]Each group has 4 people, so there would be 6 matches in each group making $68=\\boxed{48}$ so far.\n\nNow there's 16 teams and single elimination. Therefore, there are 15 games and the game between the losing teams in the semi-finals, which makes $\\boxed{16}$.\n\nFinal answer: $\\boxed{64}$ games.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Each group has 4 people, so there would be 6 matches in each group making <img src=\"//latex.artofproblemsolving.com/7/c/9/7c9ccd6e4cafae16fc30a5457ec43ac3fdc54076.png\" class=\"latex\" alt=\"$68=\\boxed{48}$\" style=\"vertical-align: -5px\" width=\"89\" height=\"23\" > so far.<br>\n<br>\nNow there's 16 teams and single elimination. Therefore, there are 15 games and the game between the losing teams in the semi-finals, which makes <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/2/2/122be32b3f050309c88e59fbbfdda0fd5e32a498.png\" class=\"latex\" alt=\"$\\boxed{16}$\" style=\"vertical-align: -5px\" width=\"30\" height=\"23\" >.</span><br>\n<br>\nFinal answer: <img src=\"//latex.artofproblemsolving.com/3/3/0/330f919429e279b05cb9cd753567d7b6e0bb5218.png\" class=\"latex\" alt=\"$\\boxed{64}$\" style=\"vertical-align: -5px\" width=\"30\" height=\"23\" > games.</div>", "post_id": 560048, "post_number": 2, "post_time_unix": 1151498962, "post_time_utc": "2006-06-28 12:49:22 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[hide]\n32/8 = 4 in each group.\n\n(323)/2 = 48. This because each team plays every other team in its group, but I counted each game twice, once for each team, so I divided by 2.\n\nThen you have the knockout round! 8 games in first round of knockout (16/2), 4 games in quarterfinals (8/2), and 2 games in semifinals (4/2), then the final which is 1 game, and the 3rd place game, which is 1 game.\n\nSo you have 48 + 8 + 4 + 2 + 1 + 1 = 64 games in the World Cup!\n\n[/hide]\r\n\r\nI'm currently not happy with Brazil because they beat Ghana 3-0 (I like underdogs), and Germany and Italy have each won 3 cups and Brazil has won 5. So I'll root for France.\r\n\r\nnumberdance", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">32/8 = 4 in each group.<br>\n<br>\n(323)/2 = 48. This because each team plays every other team in its group, but I counted each game twice, once for each team, so I divided by 2.<br>\n<br>\nThen you have the knockout round! 8 games in first round of knockout (16/2), 4 games in quarterfinals (8/2), and 2 games in semifinals (4/2), then the final which is 1 game, and the 3rd place game, which is 1 game.<br>\n<br>\nSo you have 48 + 8 + 4 + 2 + 1 + 1 = 64 games in the World Cup!</div><br>\n<br>\nI'm currently not happy with Brazil because they beat Ghana 3-0 (I like underdogs), and Germany and Italy have each won 3 cups and Brazil has won 5. So I'll root for France.<br>\n<br>\nnumberdance", "post_id": 560510, "post_number": 3, "post_time_unix": 1151521659, "post_time_utc": "2006-06-28 19:07:39 UTC", "thanks_received": 2, "user_id": 14877, "username": "numberdance" }, { "attachments": [], "content_bbcode": "[hide]Yes, 64 is the correct answer, as other posters have demonstrated[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Yes, 64 is the correct answer, as other posters have demonstrated</div>", "post_id": 560783, "post_number": 4, "post_time_unix": 1151539013, "post_time_utc": "2006-06-28 23:56:53 UTC", "thanks_received": 1, "user_id": 15534, "username": "José" } ], "source": null }	There are 32 teams divided into 8 groups with the same number of teams in each group. Every team in a group plays every other team in that group exactly once. The top two teams from each group advance to the second round; these 16 teams are placed into a single-elimination bracket and play until a winner is determined. There is also a third-place play-off between the losing semi-finalists. How many total games are played? Provide an explanation.	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]	Count games by adding intra‑group pairings (nC2 per group) to knockout games (teams − 1 plus the third‑place match).	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aops_99138	[hide]Put three on each side, then there's two case: Case 1: They weigh the same, that means the lighter one is in the other group of three. Weigh two of the other group and you can easily find out. Case 2: One group weighs less. That group has the lighter statue. Weigh two statues from that group and you can find the lighter statue.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Billy Bob is the local king. He got 9 golden statues from the neighboring king, the Burger King. The Burger King also tells Billy Bob that one statue weighs a little less than all the others. Billy Bob has only a weighing scale (that kind where there are two sides, and you put stuff on either side, and the scale tilts and stuff) to tell the difference. An old man tells Billy Bob that he can find that statue in two weighings of the scale. How did he do it? Explain thouroughly.", "content_html": "Billy Bob is the local king. He got 9 golden statues from the neighboring king, the Burger King. The Burger King also tells Billy Bob that one statue weighs a little less than all the others. Billy Bob has only a weighing scale (that kind where there are two sides, and you put stuff on either side, and the scale tilts and stuff) to tell the difference. An old man tells Billy Bob that he can find that statue in two weighings of the scale. How did he do it? Explain thouroughly.", "post_id": 559758, "post_number": 1, "post_time_unix": 1151460856, "post_time_utc": "2006-06-28 02:14:16 UTC", "thanks_received": 2, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "[hide=\"just one weighing\"]Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">just one weighing</a><div class=\"cmty-hide-content\" style=\"display:none\">Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.</div>", "post_id": 559768, "post_number": 2, "post_time_unix": 1151462002, "post_time_utc": "2006-06-28 02:33:22 UTC", "thanks_received": 2, "user_id": 15599, "username": "Totally Zealous" }, { "attachments": [], "content_bbcode": "[hide]Put three on each side, then there's two case:\n\nCase 1:\nThey weigh the same, that means the lighter one is in the other group of three. Weigh two of the other group and you can easily find out.\n\nCase 2:\nOne group weighs less. That group has the lighter statue. Weigh two statues from that group and you can find the lighter statue.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Put three on each side, then there's two case:<br>\n<br>\nCase 1:<br>\nThey weigh the same, that means the lighter one is in the other group of three. Weigh two of the other group and you can easily find out.<br>\n<br>\nCase 2:<br>\nOne group weighs less. That group has the lighter statue. Weigh two statues from that group and you can find the lighter statue.</div>", "post_id": 560049, "post_number": 3, "post_time_unix": 1151499120, "post_time_utc": "2006-06-28 12:52:00 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[quote=\"Totally Zealous\"][hide=\"just one weighing\"]Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.[/hide][/quote]\nActually, the question is asking how to [i]guarantee[/i] that you can find the one with the lowest weight for [i]any[/i] configuration, not just a specific one.\n\n[quote=\"SplashD\"][hide]Put three on each side, then there's two case:\n\nCase 1:\nThey weigh the same, that means the lighter one is in the other group of three. Weigh two of the other group and you can easily find out.\n\nCase 2:\nOne group weighs less. That group has the lighter statue. Weigh two statues from that group and you can find the lighter statue.[/hide][/quote]\r\nYes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Totally Zealous wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">just one weighing</a><div class=\"cmty-hide-content\" style=\"display:none\">Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.</div></div>\n</div>\nActually, the question is asking how to <i>guarantee</i> that you can find the one with the lowest weight for <i>any</i> configuration, not just a specific one.<br>\n\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">SplashD wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Put three on each side, then there's two case:<br>\n<br>\nCase 1:<br>\nThey weigh the same, that means the lighter one is in the other group of three. Weigh two of the other group and you can easily find out.<br>\n<br>\nCase 2:<br>\nOne group weighs less. That group has the lighter statue. Weigh two statues from that group and you can find the lighter statue.</div></div>\n</div>\nYes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.", "post_id": 560422, "post_number": 4, "post_time_unix": 1151518639, "post_time_utc": "2006-06-28 18:17:19 UTC", "thanks_received": 2, "user_id": 20563, "username": "nebula42" }, { "attachments": [], "content_bbcode": "[quote=\"Totally Zealous\"][hide=\"just one weighing\"]Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.[/hide][/quote]\r\n\r\nThis only works if he is lucky. If the scale does not balance, then there are 4 which could be the lighter gold statue, thus it does not gaurantee a result.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Totally Zealous wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">just one weighing</a><div class=\"cmty-hide-content\" style=\"display:none\">Actually he can do it in one weighing. Just test them and u'll end up with 4 on either side and they're balanced. The left over is the one that's a little less.</div></div>\n</div>\n<br>\nThis only works if he is lucky. If the scale does not balance, then there are 4 which could be the lighter gold statue, thus it does not gaurantee a result.", "post_id": 560669, "post_number": 5, "post_time_unix": 1151531888, "post_time_utc": "2006-06-28 21:58:08 UTC", "thanks_received": 2, "user_id": 8949, "username": "b-flat" }, { "attachments": [], "content_bbcode": "[quote=\"nebula42\"]\nYes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.[/quote]\r\nI assume we would take the floor of $\\frac{2}{3}x$ and eliminate it.\r\n\r\nIs that true for any one of these problems? For example, pose the same question with 90 statues. Would this mean:\r\n\r\nWeighing 1: 30 statues\r\nWeighing 2: 10 statues\r\nWeighing 3: 4 statues\r\nWeighing 4: 2 statues\r\nWeighing 5: 1 statues\r\n\r\nSo it would take 5 trials for 90 items? Does this work out?", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nebula42 wrote:</div>\n<div class=\"bbcode_quote_body\">Yes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.</div>\n</div>\nI assume we would take the floor of <img src=\"//latex.artofproblemsolving.com/a/3/c/a3c4c37c5debae414b63672cc9607bbc8273a001.png\" class=\"latex\" alt=\"$\\frac{2}{3}x$\" style=\"vertical-align: -12px\" width=\"23\" height=\"37\" > and eliminate it.<br>\n<br>\nIs that true for any one of these problems? For example, pose the same question with 90 statues. Would this mean:<br>\n<br>\nWeighing 1: 30 statues<br>\nWeighing 2: 10 statues<br>\nWeighing 3: 4 statues<br>\nWeighing 4: 2 statues<br>\nWeighing 5: 1 statues<br>\n<br>\nSo it would take 5 trials for 90 items? Does this work out?", "post_id": 562255, "post_number": 6, "post_time_unix": 1151631764, "post_time_utc": "2006-06-30 01:42:44 UTC", "thanks_received": 2, "user_id": 6133, "username": "mysmartmouth" }, { "attachments": [], "content_bbcode": "[quote=\"mysmartmouth\"][quote=\"nebula42\"]\nYes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.[/quote]\nI assume we would take the floor of $\\frac{2}{3}x$ and eliminate it.\n\nIs that true for any one of these problems? For example, pose the same question with 90 statues. Would this mean:\n\nWeighing 1: 30 statues\nWeighing 2: 10 statues\nWeighing 3: 4 statues\nWeighing 4: 2 statues\nWeighing 5: 1 statues\n\nSo it would take 5 trials for 90 items? Does this work out?[/quote]\r\n\r\nYes, I think. 5 weighings is the maximum required to find the lightest one. With 10 left, you could weigh 3 and 3, with 4 left. This could potentially narrow it down to 3, but also possibly 4. You could also weigh 4 and 4, with 2 left.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">mysmartmouth wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nebula42 wrote:</div>\n<div class=\"bbcode_quote_body\">Yes, that's correct. With each weighing you can be guaranteed to eliminate 2/3 of the total number of objects.</div>\n</div>\nI assume we would take the floor of <img src=\"//latex.artofproblemsolving.com/a/3/c/a3c4c37c5debae414b63672cc9607bbc8273a001.png\" class=\"latex\" alt=\"$\\frac{2}{3}x$\" style=\"vertical-align: -12px\" width=\"23\" height=\"37\" > and eliminate it.<br>\n<br>\nIs that true for any one of these problems? For example, pose the same question with 90 statues. Would this mean:<br>\n<br>\nWeighing 1: 30 statues<br>\nWeighing 2: 10 statues<br>\nWeighing 3: 4 statues<br>\nWeighing 4: 2 statues<br>\nWeighing 5: 1 statues<br>\n<br>\nSo it would take 5 trials for 90 items? Does this work out?</div>\n</div>\n<br>\nYes, I think. 5 weighings is the maximum required to find the lightest one. With 10 left, you could weigh 3 and 3, with 4 left. This could potentially narrow it down to 3, but also possibly 4. You could also weigh 4 and 4, with 2 left.", "post_id": 562265, "post_number": 7, "post_time_unix": 1151632109, "post_time_utc": "2006-06-30 01:48:29 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "[hide]Weight 3 against 3. If one group is lighter, it contains the statue. If they are the same, the third group of 3 contains the statue. Repeat with 1 against 1 from that group, and use the same argument.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Weight 3 against 3. If one group is lighter, it contains the statue. If they are the same, the third group of 3 contains the statue. Repeat with 1 against 1 from that group, and use the same argument.</div>", "post_id": 562350, "post_number": 8, "post_time_unix": 1151638434, "post_time_utc": "2006-06-30 03:33:54 UTC", "thanks_received": 1, "user_id": 11295, "username": "pianoforte" } ], "source": null }	Billy Bob is the local king. He got 9 golden statues from the neighboring king, the Burger King. The Burger King also tells Billy Bob that one statue weighs a little less than all the others. Billy Bob has only a two-pan balance scale to tell the difference. An old man tells Billy Bob that he can find that statue in two weighings of the scale. How did he do it? Explain thoroughly.	[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Compare two equal groups of three to isolate the lighter statue to a set of three, then weigh two of those to identify it.	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aops_99141	First you send billy bob and killy bob. It will take 2 min to cross. Then the flashlight on the right side of the bridge. Now billy bod return to the left side taking flash light in 1 min. So the total time is 2+1= 3min completed. Noe the Hilly Bob and Jilly Bob Goes. They take 10 min to cross the bridge. So the total is now 10 + 3 = 13min Now the flashlight is again on the right side. Now killy bob take it and return to the left side of the bridge where his brother billy bob is waiting. His journey take 2 min. So the Total time taken is now 13 + 2 = 15 min. Now Both will Cross the bridge safely in 2 min. The Total time elpased is 15+2 = 17 min. And now they are Safe.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Billy Bob, Killy Kob, Hilly Hob, and Jilly Job were returing from Dilly Dob's house. It was midnight and pure pitch black. They reach a bridge. They have only 1 flashlight. The bridge was made in the USA so it was very weak. Only up to 2 people can cross it at a time. They must use the flashlight for EVERY crossing. At their fastest, Billy Bob can cross it in 1 minute, Killy Kob in 2, Hilly Hob in 5, and Jilly Job in 10. A big bad wolf is waiting in the dark and if they don't cross it in 17 minutes the big bad wolf will eat them. Can you save Billy Bob, Killy Kob, Hilly Hob, and Jilly Job!?!?!?!?\r\n\r\nPlease provide a detailed solution and save Billy Bob, Killy Kob, Hilly Hob, and Jilly Job.\r\n[hide=\"OR ELSE\"]The Big Bad Wolf will come after you next!!![/hide]", "content_html": "Billy Bob, Killy Kob, Hilly Hob, and Jilly Job were returing from Dilly Dob's house. It was midnight and pure pitch black. They reach a bridge. They have only 1 flashlight. The bridge was made in the USA so it was very weak. Only up to 2 people can cross it at a time. They must use the flashlight for EVERY crossing. At their fastest, Billy Bob can cross it in 1 minute, Killy Kob in 2, Hilly Hob in 5, and Jilly Job in 10. A big bad wolf is waiting in the dark and if they don't cross it in 17 minutes the big bad wolf will eat them. Can you save Billy Bob, Killy Kob, Hilly Hob, and Jilly Job!?!?!?!?<br>\n<br>\nPlease provide a detailed solution and save Billy Bob, Killy Kob, Hilly Hob, and Jilly Job.<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">OR ELSE</a><div class=\"cmty-hide-content\" style=\"display:none\">The Big Bad Wolf will come after you next!!!</div>", "post_id": 559774, "post_number": 1, "post_time_unix": 1151462488, "post_time_utc": "2006-06-28 02:41:28 UTC", "thanks_received": 2, "user_id": 15599, "username": "Totally Zealous" }, { "attachments": [], "content_bbcode": "First you send billy bob and killy bob. It will take 2 min to cross. Then the flashlight on the right side of the bridge. Now billy bod return to the left side taking flash light in 1 min. So the total time is 2+1= 3min completed. Noe the Hilly Bob and Jilly Bob Goes. They take 10 min to cross the bridge. So the total is now 10 + 3 = 13min\r\nNow the flashlight is again on the right side. Now killy bob take it and return to the left side of the bridge where his brother billy bob is waiting. His journey take 2 min. So the Total time taken is now \r\n13 + 2 = 15 min. Now Both will Cross the bridge safely in 2 min. The Total time elpased is 15+2 = 17 min. And now they are Safe.", "content_html": "First you send billy bob and killy bob. It will take 2 min to cross. Then the flashlight on the right side of the bridge. Now billy bod return to the left side taking flash light in 1 min. So the total time is 2+1= 3min completed. Noe the Hilly Bob and Jilly Bob Goes. They take 10 min to cross the bridge. So the total is now 10 + 3 = 13min<br>\nNow the flashlight is again on the right side. Now killy bob take it and return to the left side of the bridge where his brother billy bob is waiting. His journey take 2 min. So the Total time taken is now<br>\n13 + 2 = 15 min. Now Both will Cross the bridge safely in 2 min. The Total time elpased is 15+2 = 17 min. And now they are Safe.", "post_id": 559804, "post_number": 2, "post_time_unix": 1151466202, "post_time_utc": "2006-06-28 03:43:22 UTC", "thanks_received": 2, "user_id": 17435, "username": "ashwinrk_jain" } ], "source": null }	Four people—Billy Bob, Killy Kob, Hilly Hob, and Jilly Job—must cross a bridge at night. They have one flashlight. The bridge is weak: at most two people can cross at a time. The flashlight must be used for every crossing. Their individual crossing times are: Billy Bob 1 minute, Killy Kob 2 minutes, Hilly Hob 5 minutes, and Jilly Job 10 minutes. If they do not all cross within 17 minutes, a wolf will eat them. Find a solution that gets all four across within 17 minutes.	[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Use the two fastest people to shuttle the flashlight, sending the two slowest together across.	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aops_99182	I'm lazy, and I won't calculate the constants, but here is my proof. [hide]Firstly, some definitions: $A_n$: the set of $n$-digit integers formed by $1$, $2$ and $3$ which do not contain any consecutive $1$'s. $a_n=\|A_n\|$. $B_n$, $C_n$ and $D_n$: the partitions of $A_n$ with last digit $1$, $2$ and $3$ respectively. $b_n=\|B_n\|$ and so on. (1) Obviously, $a_n=b_n+c_n+d_n$. (2) By construction, it's easy to see that $b_n=a_{n-1}-b_{n-1}$, $c_n=d_n=a_{n-1}$. (3) Using (1) and (2), $a_n=3a_{n-1}-b_{n-1}$. (4) So, we've got the recursive system: $a_n=3a_{n-1}-b_{n-1}$, $b_n=a_{n-1}-b_{n-1}$, with the initial conditions $a_1=3$ and $b_1=1$. (5) Putting (4) in a matricial form and calculating the $n$-th power of the matrix (by diagonalization or induction), we get a solution of the form $a_n=k_1{\lambda_1}^n+k_2{\lambda_2}^n$. The constants can be calculated with some initial values. If $\|\lambda_2\|<1$, soon (for every $n>n_0$) we have $a_n=round(k_1{\lambda_1}^n)$.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $a_{n}$ be the number of $n$-digit integers formed by $1,2$ and $3$ which do not contain any consecutive 1's. Prove that $a_{n}$ is equal to $(\\frac{1}{2}+\\frac{1}{\\sqrt{3}})(\\sqrt{3}+1)^{n}$ rounded off to the nearest integer.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/b/d/2/bd28b278a07bda0ea4d08737743613980c6aa6a3.png\" class=\"latex\" alt=\"$a_{n}$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > be the number of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-</span>digit integers formed by <img src=\"//latex.artofproblemsolving.com/2/d/3/2d338f393d100febd19e4284537ac40517b48340.png\" class=\"latex\" alt=\"$1,2$\" style=\"vertical-align: -3px\" width=\"25\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > which do not contain any consecutive 1's. Prove that <img src=\"//latex.artofproblemsolving.com/b/d/2/bd28b278a07bda0ea4d08737743613980c6aa6a3.png\" class=\"latex\" alt=\"$a_{n}$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > is equal to <img src=\"//latex.artofproblemsolving.com/0/3/d/03dad8628a0fbfd54eb647f38e9fc3145a1d8528.png\" class=\"latex\" alt=\"$(\\frac{1}{2}+\\frac{1}{\\sqrt{3}})(\\sqrt{3}+1)^{n}$\" style=\"vertical-align: -17px\" width=\"155\" height=\"41\" > rounded off to the nearest integer.", "post_id": 559925, "post_number": 1, "post_time_unix": 1151483263, "post_time_utc": "2006-06-28 08:27:43 UTC", "thanks_received": 2, "user_id": 6601, "username": "shobber" }, { "attachments": [], "content_bbcode": "I'm lazy, and I won't calculate the constants, but here is my proof. [hide]Firstly, some definitions:\n$A_n$: the set of $n$-digit integers formed by $1$, $2$ and $3$ which do not contain any consecutive $1$'s. $a_n=\|A_n\|$.\n$B_n$, $C_n$ and $D_n$: the partitions of $A_n$ with last digit $1$, $2$ and $3$ respectively. $b_n=\|B_n\|$ and so on.\n\n(1) Obviously, $a_n=b_n+c_n+d_n$.\n(2) By construction, it's easy to see that\n$b_n=a_{n-1}-b_{n-1}$, $c_n=d_n=a_{n-1}$.\n(3) Using (1) and (2), $a_n=3a_{n-1}-b_{n-1}$.\n(4) So, we've got the recursive system:\n$a_n=3a_{n-1}-b_{n-1}$,\n$b_n=a_{n-1}-b_{n-1}$,\nwith the initial conditions $a_1=3$ and $b_1=1$.\n(5) Putting (4) in a matricial form and calculating the $n$-th power of the matrix (by diagonalization or induction), we get a solution \nof the form $a_n=k_1{\\lambda_1}^n+k_2{\\lambda_2}^n$. The constants can be calculated with some initial values.\nIf $\|\\lambda_2\|<1$, soon (for every $n>n_0$) we have $a_n=round(k_1{\\lambda_1}^n)$.[/hide]", "content_html": "I'm lazy, and I won't calculate the constants, but here is my proof. <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Firstly, some definitions:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/e/c/1eccfbd65334353f671fb71c27b9b9f2c689b0bf.png\" class=\"latex\" alt=\"$A_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" >:</span> the set of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-</span>digit integers formed by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > which do not contain any consecutive <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >'</span>s. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/2/5/325bd12eb1702f21de83c3f5884527d3c63c223a.png\" class=\"latex\" alt=\"$a_n=\|A_n\|$\" style=\"vertical-align: -4px\" width=\"73\" height=\"18\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/b/e/9be2b1de79fff66db67de3b18303bdd2912b7ae6.png\" class=\"latex\" alt=\"$B_n$\" style=\"vertical-align: -2px\" width=\"22\" height=\"15\" >,</span> <img src=\"//latex.artofproblemsolving.com/4/e/1/4e173907ca992482385a50c6f62ede920362b1bf.png\" class=\"latex\" alt=\"$C_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/e/a/2eacbcb991af87d495f8689ac0e952b51317dbaa.png\" class=\"latex\" alt=\"$D_n$\" style=\"vertical-align: -2px\" width=\"23\" height=\"15\" >:</span> the partitions of <img src=\"//latex.artofproblemsolving.com/1/e/c/1eccfbd65334353f671fb71c27b9b9f2c689b0bf.png\" class=\"latex\" alt=\"$A_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > with last digit <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > respectively. <img src=\"//latex.artofproblemsolving.com/0/9/6/096b1995b41c69bdde10482f9e6f3f10c7c7b0ea.png\" class=\"latex\" alt=\"$b_n=\|B_n\|$\" style=\"vertical-align: -4px\" width=\"72\" height=\"18\" > and so on.<br>\n<br>\n(1) Obviously, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/7/c/b7cdcdcdaab364952f216230c44bf7ad2887804b.png\" class=\"latex\" alt=\"$a_n=b_n+c_n+d_n$\" style=\"vertical-align: -2px\" width=\"138\" height=\"15\" >.</span><br>\n(2) By construction, it's easy to see that<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/0/4/70403a29291ef412d73f2c410c93ce6b904c812c.png\" class=\"latex\" alt=\"$b_n=a_{n-1}-b_{n-1}$\" style=\"vertical-align: -2px\" width=\"131\" height=\"15\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/3/bc3b908ec74ffb5217abe1f41f782fe3638ecb88.png\" class=\"latex\" alt=\"$c_n=d_n=a_{n-1}$\" style=\"vertical-align: -2px\" width=\"117\" height=\"15\" >.</span><br>\n(3) Using (1) and (2), <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/0/f/b0f3b3652f54a8e52fec4ba1353d41500a758f26.png\" class=\"latex\" alt=\"$a_n=3a_{n-1}-b_{n-1}$\" style=\"vertical-align: -2px\" width=\"142\" height=\"15\" >.</span><br>\n(4) So, we've got the recursive system:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/0/f/b0f3b3652f54a8e52fec4ba1353d41500a758f26.png\" class=\"latex\" alt=\"$a_n=3a_{n-1}-b_{n-1}$\" style=\"vertical-align: -2px\" width=\"142\" height=\"15\" >,</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/0/4/70403a29291ef412d73f2c410c93ce6b904c812c.png\" class=\"latex\" alt=\"$b_n=a_{n-1}-b_{n-1}$\" style=\"vertical-align: -2px\" width=\"131\" height=\"15\" >,</span><br>\nwith the initial conditions <img src=\"//latex.artofproblemsolving.com/3/8/3/383679febc17d3b8a4563f9010e3eea9caca402a.png\" class=\"latex\" alt=\"$a_1=3$\" style=\"vertical-align: -2px\" width=\"50\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/2/e/22e43540b285099b8d262767192656d487fd6d91.png\" class=\"latex\" alt=\"$b_1=1$\" style=\"vertical-align: -2px\" width=\"47\" height=\"15\" >.</span><br>\n(5) Putting (4) in a matricial form and calculating the <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-</span>th power of the matrix (by diagonalization or induction), we get a solution<br>\nof the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/e/92ec0d61e17583e40a655b592cb6a0e7a0343e5a.png\" class=\"latex\" alt=\"$a_n=k_1{\\lambda_1}^n+k_2{\\lambda_2}^n$\" style=\"vertical-align: -2px\" width=\"152\" height=\"16\" >.</span> The constants can be calculated with some initial values.<br>\nIf <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/d/2/8d2d3a3dd935058da483c944ec4501b6e1209726.png\" class=\"latex\" alt=\"$\|\\lambda_2\|<1$\" style=\"vertical-align: -4px\" width=\"60\" height=\"18\" >,</span> soon (for every <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/b/6cb22bc1bc8a6a00837327cf441745f02835b834.png\" class=\"latex\" alt=\"$n>n_0$\" style=\"vertical-align: -2px\" width=\"52\" height=\"12\" >)</span> we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/d/0/5d0722ff4da8d5bef9c9098e52c733baac6e5ced.png\" class=\"latex\" alt=\"$a_n=round(k_1{\\lambda_1}^n)$\" style=\"vertical-align: -4px\" width=\"148\" height=\"18\" >.</span></div>", "post_id": 560355, "post_number": 2, "post_time_unix": 1151515561, "post_time_utc": "2006-06-28 17:26:01 UTC", "thanks_received": 2, "user_id": 19718, "username": "lordWings" } ], "source": null }	Let $a_n$ be the number of $n$-digit integers formed by $1,2$ and $3$ which do not contain any consecutive 1's. Prove that $a_n$ is equal to \[ \left(\tfrac{1}{2}+\tfrac{1}{\sqrt{3}}\right)(\sqrt{3}+1)^{n} \] rounded to the nearest integer.	[ "/Mathematics/Algebra/LinearAlgebra/Matrices/MatrixDecomposition", "/Mathematics/Algebra/LinearAlgebra/Matrices/MatrixEigenvalues", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation" ]	Encode the no‑consecutive‑1 condition as a linear recurrence by tracking the last digit, then solve it via eigenvalues and keep the dominant term.	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aops_991885	Must be one big fish. [hide=" :ninja: "]1 fish = 2 rocks = 5 pineapples 1 rock = 50 grains Multiply eq (2) by 2: 2 rocks = 100 grains. Hence, 100 grains = 5 pineapples 1 pineapple = 20 grains[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "I am in SF now, heading to Palo Alto to Stanford. Uhhh...\r\n\r\nOn the island of Pohbah, a fish is considered 2 rocks or 5 pineapples. Rocks are considered 50 grains of sand. How many grains of sand is one pineapple worth?", "content_html": "I am in SF now, heading to Palo Alto to Stanford. Uhhh...<br>\n<br>\nOn the island of Pohbah, a fish is considered 2 rocks or 5 pineapples. Rocks are considered 50 grains of sand. How many grains of sand is one pineapple worth?", "post_id": 4402934, "post_number": 1, "post_time_unix": 1183927237, "post_time_utc": "2007-07-08 20:40:37 UTC", "thanks_received": 1, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "Must be one big fish.\r\n\r\n[hide=\" :ninja: \"]1 fish = 2 rocks = 5 pineapples\n1 rock = 50 grains\n\nMultiply eq (2) by 2:\n2 rocks = 100 grains. Hence,\n100 grains = 5 pineapples\n1 pineapple = 20 grains[/hide]", "content_html": "Must be one big fish.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">:ninja:</a><div class=\"cmty-hide-content\" style=\"display:none\">1 fish = 2 rocks = 5 pineapples<br>\n1 rock = 50 grains<br>\n<br>\nMultiply eq (2) by 2:<br>\n2 rocks = 100 grains. Hence,<br>\n100 grains = 5 pineapples<br>\n1 pineapple = 20 grains</div>", "post_id": 4402935, "post_number": 2, "post_time_unix": 1183950996, "post_time_utc": "2007-07-09 03:16:36 UTC", "thanks_received": 1, "user_id": 14262, "username": "vishalarul" }, { "attachments": [], "content_bbcode": "Since when are rocks worth more than pineapples? :roll:", "content_html": "Since when are rocks worth more than pineapples? <img src=\"/assets/images/smilies/rolleyes.gif\" width=\"20\" height=\"20\" alt=\":roll:\" title=\":roll:\" class=\"bbcode_smiley\" />", "post_id": 4402936, "post_number": 3, "post_time_unix": 1184106466, "post_time_utc": "2007-07-10 22:27:46 UTC", "thanks_received": 1, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "On the island of Pohbah.", "content_html": "On the island of Pohbah.", "post_id": 4402937, "post_number": 4, "post_time_unix": 1184551222, "post_time_utc": "2007-07-16 02:00:22 UTC", "thanks_received": 1, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "20 grains of sand\r\ni could actually solve this! :)", "content_html": "20 grains of sand<br>\ni could actually solve this! <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 4402938, "post_number": 5, "post_time_unix": 1184621620, "post_time_utc": "2007-07-16 21:33:40 UTC", "thanks_received": 1, "user_id": 22339, "username": "egghead91" } ], "source": null }	On the island of Pohbah, a fish is considered 2 rocks or 5 pineapples. Rocks are considered 50 grains of sand. How many grains of sand is one pineapple worth?	[ "/Mathematics/RecreationalMathematics/MathematicalHumor", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Translate the chain of equivalences and scale them to connect pineapples directly to grains of sand.	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aops_991910	1st: follows quickly from definition of rational, let $ r\equal{}p/q$ and $ s\equal{}u/v$ 2nd: $ (\sqrt{2}\plus{}(2\minus{}\sqrt{2}))/2 \equal{} 1$ $ (\sqrt{2}\plus{}\sqrt{3})/2\notin\mathbb{Q}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "I have not done this yet, but I will to add Nerd_of_the_Ages to the Contrib status (Oh don't you feel honored).\r\n\r\nSo Stupid question:\r\n\r\nProve: The average of of two rational numbers is a rational number. but the average of two irrational numbers is either rational or irrational.", "content_html": "I have not done this yet, but I will to add Nerd_of_the_Ages to the Contrib status (Oh don't you feel honored).<br>\n<br>\nSo Stupid question:<br>\n<br>\nProve: The average of of two rational numbers is a rational number. but the average of two irrational numbers is either rational or irrational.", "post_id": 4403030, "post_number": 1, "post_time_unix": 1189659390, "post_time_utc": "2007-09-13 04:56:30 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "1st: follows quickly from definition of rational, let $ r\\equal{}p/q$ and $ s\\equal{}u/v$\r\n\r\n2nd:\r\n\r\n$ (\\sqrt{2}\\plus{}(2\\minus{}\\sqrt{2}))/2 \\equal{} 1$\r\n\r\n$ (\\sqrt{2}\\plus{}\\sqrt{3})/2\\notin\\mathbb{Q}$", "content_html": "1st: follows quickly from definition of rational, let <img src=\"//latex.artofproblemsolving.com/3/f/a/3fa1343caa6cc6b1d90999ea15374ddc12383c84.png\" class=\"latex\" alt=\"$ r=p/q$\" style=\"vertical-align: -4px\" width=\"59\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/b/d/0/bd07607607f58a1ff648e3166327e01a34c989ef.png\" class=\"latex\" alt=\"$ s=u/v$\" style=\"vertical-align: -4px\" width=\"61\" height=\"18\" ><br>\n<br>\n2nd:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/1/8/a18de787b9d61af7b08ca12f0dee1fb255fd94a9.png\" class=\"latex\" alt=\"$ (\\sqrt{2}+(2-\\sqrt{2}))/2 = 1$\" style=\"vertical-align: -4px\" width=\"180\" height=\"21\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/6/d/f/6dffd34a50f371d4ca9720aa5ed7de384ad68a37.png\" class=\"latex\" alt=\"$ (\\sqrt{2}+\\sqrt{3})/2\\notin\\mathbb{Q}$\" style=\"vertical-align: -4px\" width=\"138\" height=\"21\" >", "post_id": 4403031, "post_number": 2, "post_time_unix": 1189725437, "post_time_utc": "2007-09-13 23:17:17 UTC", "thanks_received": 2, "user_id": 14262, "username": "vishalarul" }, { "attachments": [], "content_bbcode": "w00t thingy-ma-jigs.", "content_html": "w00t thingy-ma-jigs.", "post_id": 4403032, "post_number": 3, "post_time_unix": 1189732780, "post_time_utc": "2007-09-14 01:19:40 UTC", "thanks_received": 2, "user_id": 28419, "username": "Temperal" } ], "source": null }	Prove: 1. The average of two rational numbers is a rational number. 2. The average of two irrational numbers may be rational or irrational.	[ "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/IrrationalNumber", "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/SquareRootof2", "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/SquareRootof3", "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/Surd", "/Mathematics/Algebra/NumberTheory/RationalNumbers/FieldofRationals", "/Mathematics/Algebra/NumberTheory/RationalNumbers/Q", "/Mathematics/Algebra/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/Algebra/Sums/Sum", "/Mathematics/FoundationsofMathematics/MathematicalProblems/SolvedProblems/GelfondsTheorem", "/Mathematics/FoundationsofMathematics/TheoremProving/Proofs/ElementaryProof", "/Mathematics/FoundationsofMathematics/TheoremProving/Proofs/Proof", "/Mathematics/NumberTheory/IrrationalNumbers/IrrationalNumber", "/Mathematics/NumberTheory/IrrationalNumbers/SquareRootof2", "/Mathematics/NumberTheory/IrrationalNumbers/SquareRootof3", "/Mathematics/NumberTheory/IrrationalNumbers/Surd", "/Mathematics/NumberTheory/RationalNumbers/FieldofRationals", "/Mathematics/NumberTheory/RationalNumbers/Q", "/Mathematics/NumberTheory/RationalNumbers/RationalNumber" ]	Use closure of rationals under addition and division for part 1, and construct explicit irrational pairs whose sum is rational or irrational for part 2.	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aops_991919	Lhopital SPAM to the rescue! The limit is of the form $ \frac{0}{0}$. We spam Lhopital on it. Derivative of top is 2x - 4. Derivative of bottom is 2x. Plug in three and BAM the limit is equal to $ \frac{1}{3}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "So today, I swam a mile, did one-arm pushups, and played soccer. I think I beat Ubemaya. But I have simple limit problem:\r\n$ \\lim_{x\\rightarrow3}\\frac{x^{2}\\minus{}4x\\plus{}3}{x^{2}\\minus{}9}$\r\n\r\nIn Blog news, the shoutbox appears to be shot and crippled for life since it is moving so slow. I am adding pianoforte to the contribs (Even if pianoforte does nothing).", "content_html": "So today, I swam a mile, did one-arm pushups, and played soccer. I think I beat Ubemaya. But I have simple limit problem:<br>\n<img src=\"//latex.artofproblemsolving.com/8/e/b/8eb20b70cedd1f213a113e3647bc9f56c596ed6f.png\" class=\"latex\" alt=\"$ \\lim_{x\\rightarrow3}\\frac{x^{2}-4x+3}{x^{2}-9}$\" style=\"vertical-align: -12px\" width=\"123\" height=\"39\" ><br>\n<br>\nIn Blog news, the shoutbox appears to be shot and crippled for life since it is moving so slow. I am adding pianoforte to the contribs (Even if pianoforte does nothing).", "post_id": 4403051, "post_number": 1, "post_time_unix": 1190863574, "post_time_utc": "2007-09-27 03:26:14 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "Lhopital SPAM to the rescue!\r\n\r\nThe limit is of the form $ \\frac{0}{0}$. We spam Lhopital on it. Derivative of top is 2x - 4.\r\nDerivative of bottom is 2x. Plug in three and BAM the limit is equal to $ \\frac{1}{3}$", "content_html": "Lhopital SPAM to the rescue!<br>\n<br>\nThe limit is of the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/5/f/75f0996a4e11a422a1c073be0a313d56f154874d.png\" class=\"latex\" alt=\"$ \\frac{0}{0}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" >.</span> We spam Lhopital on it. Derivative of top is 2x - 4.<br>\nDerivative of bottom is 2x. Plug in three and BAM the limit is equal to <img src=\"//latex.artofproblemsolving.com/2/6/f/26f15748bda7a66945be27f0effdf2c4fbb35749.png\" class=\"latex\" alt=\"$ \\frac{1}{3}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" >", "post_id": 4403052, "post_number": 2, "post_time_unix": 1190864728, "post_time_utc": "2007-09-27 03:45:28 UTC", "thanks_received": 2, "user_id": 23431, "username": "DiscreetFourierTransform" }, { "attachments": [], "content_bbcode": "Good job.\r\n\r\nI just did\r\n[hide]\n$ \\frac{x^{2}\\minus{}4x\\plus{}3}{x^{2}\\minus{}9}\\equal{}\\frac{(x\\minus{}3)(x\\minus{}1)}{(x\\minus{}3)(x\\plus{}3)}\\equal{}\\frac{x\\minus{}1}{x\\plus{}3}$\n$ \\lim_{x\\rightarrow3}\\frac{x\\minus{}1}{x\\plus{}3}\\equal{}\\frac{2}{6}\\equal{}\\boxed{\\frac{1}{3}}$\n[/hide]", "content_html": "Good job.<br>\n<br>\nI just did<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/b/8/f/b8f51a42e9cb09bc559fe4c57e9579f88d7e6f05.png\" class=\"latex\" alt=\"$ \\frac{x^{2}-4x+3}{x^{2}-9}=\\frac{(x-3)(x-1)}{(x-3)(x+3)}=\\frac{x-1}{x+3}$\" style=\"vertical-align: -17px\" width=\"303\" height=\"43\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/e/3/7e3a94862c0e38a3ae4db3e7ebcf3c0da34b5ee8.png\" class=\"latex\" alt=\"$ \\lim_{x\\rightarrow3}\\frac{x-1}{x+3}=\\frac{2}{6}=\\boxed{\\frac{1}{3}}$\" style=\"vertical-align: -18px\" width=\"162\" height=\"48\" ></div>", "post_id": 4403053, "post_number": 3, "post_time_unix": 1190866049, "post_time_utc": "2007-09-27 04:07:29 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "That's what I did. Much faster.", "content_html": "That's what I did. Much faster.", "post_id": 4403054, "post_number": 4, "post_time_unix": 1190922220, "post_time_utc": "2007-09-27 19:43:40 UTC", "thanks_received": 2, "user_id": 28420, "username": "xpmath" }, { "attachments": [], "content_bbcode": "How is factoring faster than L'hopital's?!", "content_html": "How is factoring faster than L'hopital's?!", "post_id": 4403055, "post_number": 5, "post_time_unix": 1190928042, "post_time_utc": "2007-09-27 21:20:42 UTC", "thanks_received": 2, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "Yea, but what if you get an unfactorable numerator (say, a trig function?) and a factorable denominator... You can factor all you want but eventually it comes down to either the squeeze theorem or Lhopital.", "content_html": "Yea, but what if you get an unfactorable numerator (say, a trig function?) and a factorable denominator... You can factor all you want but eventually it comes down to either the squeeze theorem or Lhopital.", "post_id": 4403056, "post_number": 6, "post_time_unix": 1190944331, "post_time_utc": "2007-09-28 01:52:11 UTC", "thanks_received": 2, "user_id": 23431, "username": "DiscreetFourierTransform" } ], "source": null }	Compute the following limit: \[ \lim_{x\to 3}\frac{x^{2}-4x+3}{x^{2}-9}. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/Limits/LHospitalsRule", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit" ]	Recognize the 0/0 form and apply L'Hôpital's Rule to differentiate numerator and denominator.	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aops_99197	[quote="Kalle"]Let $X$ be a compact Hausdorff space with a sequence of closed, connected subspaces $C_1 \supseteq C_2 \supseteq C_3 ...$. Is it true that $C = \cap^\infty_{n=1} C_n$ is connected? Is this the right forum? I'm sorry if it's not.[/quote] C is connected. Proof: If it is not, we can find open sets $U_1$ and $U_2$ such that ${U_1}\cup{U_2}\supseteq{C}$ and ${U_1}\cap{U_2}=\emptyset$. Now the union of countably infinite open sets $U_1$, $U_2$, and $(C_n)^c (n=1,2,...)$ equals X. Since X is compact, we can choose from them finite sets that still cover X. Obviously, because C must be covered, $U_1$ and $U_2$ must be included in those finite sets. And notice that $(C_1)^c\subseteq(C_2)^c\subseteq...$, so $(C_n)^c\cup{U_1}\cup{U_2}=X$ holds for some n. Thus ${U_1}\cup{U_2}\supseteq{C_n}$, which contradicts with the fact that $C_n$ is connected.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $X$ be a compact Hausdorff space with a sequence of closed, connected subspaces $C_1 \\supseteq C_2 \\supseteq C_3 ...$. Is it true that $C = \\cap^\\infty_{n=1} C_n$ is connected?\r\n\r\nIs this the right forum? I'm sorry if it's not.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > be a compact Hausdorff space with a sequence of closed, connected subspaces <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/0/6/1060b65330f0b92a4379ed01e4b719f850aef60c.png\" class=\"latex\" alt=\"$C_1 \\supseteq C_2 \\supseteq C_3 ...$\" style=\"vertical-align: -2px\" width=\"123\" height=\"15\" >.</span> Is it true that <img src=\"//latex.artofproblemsolving.com/4/d/f/4df40ad09c1accf951f316e1c724e1160bcb5e4b.png\" class=\"latex\" alt=\"$C = \\cap^\\infty_{n=1} C_n$\" style=\"vertical-align: -4px\" width=\"97\" height=\"17\" > is connected?<br>\n<br>\nIs this the right forum? I'm sorry if it's not.", "post_id": 559981, "post_number": 1, "post_time_unix": 1151489810, "post_time_utc": "2006-06-28 10:16:50 UTC", "thanks_received": 1, "user_id": 11334, "username": "Kalle" }, { "attachments": [], "content_bbcode": "[quote=\"Kalle\"]Let $X$ be a compact Hausdorff space with a sequence of closed, connected subspaces $C_1 \\supseteq C_2 \\supseteq C_3 ...$. Is it true that $C = \\cap^\\infty_{n=1} C_n$ is connected?\n\nIs this the right forum? I'm sorry if it's not.[/quote]\r\n\r\nC is connected.\r\nProof:\r\nIf it is not, we can find open sets $U_1$ and $U_2$ such that ${U_1}\\cup{U_2}\\supseteq{C}$ and ${U_1}\\cap{U_2}=\\emptyset$.\r\nNow the union of countably infinite open sets $U_1$, $U_2$, and $(C_n)^c (n=1,2,...)$ equals X.\r\nSince X is compact, we can choose from them finite sets that still cover X.\r\nObviously, because C must be covered, $U_1$ and $U_2$ must be included in those finite sets.\r\nAnd notice that $(C_1)^c\\subseteq(C_2)^c\\subseteq...$, so $(C_n)^c\\cup{U_1}\\cup{U_2}=X$ holds for some n.\r\nThus ${U_1}\\cup{U_2}\\supseteq{C_n}$, which contradicts with the fact that $C_n$ is connected.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Kalle wrote:</div>\n<div class=\"bbcode_quote_body\">Let <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > be a compact Hausdorff space with a sequence of closed, connected subspaces <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/0/6/1060b65330f0b92a4379ed01e4b719f850aef60c.png\" class=\"latex\" alt=\"$C_1 \\supseteq C_2 \\supseteq C_3 ...$\" style=\"vertical-align: -2px\" width=\"123\" height=\"15\" >.</span> Is it true that <img src=\"//latex.artofproblemsolving.com/4/d/f/4df40ad09c1accf951f316e1c724e1160bcb5e4b.png\" class=\"latex\" alt=\"$C = \\cap^\\infty_{n=1} C_n$\" style=\"vertical-align: -4px\" width=\"97\" height=\"17\" > is connected?<br>\n<br>\nIs this the right forum? I'm sorry if it's not.</div>\n</div>\n<br>\nC is connected.<br>\nProof:<br>\nIf it is not, we can find open sets <img src=\"//latex.artofproblemsolving.com/3/c/b/3cb141913b3ab1abe79fb14c5b39a9568faa5543.png\" class=\"latex\" alt=\"$U_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/8/7/b/87b484c85151278e0b91832124fc1e67f2dd2e92.png\" class=\"latex\" alt=\"$U_2$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > such that <img src=\"//latex.artofproblemsolving.com/7/a/b/7abdf2c5a829a1a3954967c13c9e247eceda3dd2.png\" class=\"latex\" alt=\"${U_1}\\cup{U_2}\\supseteq{C}$\" style=\"vertical-align: -2px\" width=\"98\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/1/cc12c1f95499ceba9314adeb6f087324a3975376.png\" class=\"latex\" alt=\"${U_1}\\cap{U_2}=\\emptyset$\" style=\"vertical-align: -2px\" width=\"92\" height=\"17\" >.</span><br>\nNow the union of countably infinite open sets <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/c/b/3cb141913b3ab1abe79fb14c5b39a9568faa5543.png\" class=\"latex\" alt=\"$U_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/7/b/87b484c85151278e0b91832124fc1e67f2dd2e92.png\" class=\"latex\" alt=\"$U_2$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" >,</span> and <img src=\"//latex.artofproblemsolving.com/d/d/d/ddd124abf2c41f4879e1e0c06f84ea63e12d62f6.png\" class=\"latex\" alt=\"$(C_n)^c (n=1,2,...)$\" style=\"vertical-align: -4px\" width=\"140\" height=\"18\" > equals X.<br>\nSince X is compact, we can choose from them finite sets that still cover X.<br>\nObviously, because C must be covered, <img src=\"//latex.artofproblemsolving.com/3/c/b/3cb141913b3ab1abe79fb14c5b39a9568faa5543.png\" class=\"latex\" alt=\"$U_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/8/7/b/87b484c85151278e0b91832124fc1e67f2dd2e92.png\" class=\"latex\" alt=\"$U_2$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > must be included in those finite sets.<br>\nAnd notice that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/8/d/28d541850a26f9fc972b7f2b610fac3159a49e1e.png\" class=\"latex\" alt=\"$(C_1)^c\\subseteq(C_2)^c\\subseteq...$\" style=\"vertical-align: -4px\" width=\"144\" height=\"18\" >,</span> so <img src=\"//latex.artofproblemsolving.com/1/e/8/1e8ada72a9664aa0adb7ffbbe155dd6b7d52d92d.png\" class=\"latex\" alt=\"$(C_n)^c\\cup{U_1}\\cup{U_2}=X$\" style=\"vertical-align: -4px\" width=\"163\" height=\"18\" > holds for some n.<br>\nThus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/8/478b672559b71967c552983a3982616013d23785.png\" class=\"latex\" alt=\"${U_1}\\cup{U_2}\\supseteq{C_n}$\" style=\"vertical-align: -2px\" width=\"105\" height=\"15\" >,</span> which contradicts with the fact that <img src=\"//latex.artofproblemsolving.com/4/e/1/4e173907ca992482385a50c6f62ede920362b1bf.png\" class=\"latex\" alt=\"$C_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > is connected.", "post_id": 560166, "post_number": 2, "post_time_unix": 1151506016, "post_time_utc": "2006-06-28 14:46:56 UTC", "thanks_received": 2, "user_id": 20595, "username": "AlvaroRecoba" }, { "attachments": [], "content_bbcode": "Nice proof. But do you really need the Hausdorff property here?", "content_html": "Nice proof. But do you really need the Hausdorff property here?", "post_id": 560222, "post_number": 3, "post_time_unix": 1151508855, "post_time_utc": "2006-06-28 15:34:15 UTC", "thanks_received": 2, "user_id": 13157, "username": "Ottem" }, { "attachments": [], "content_bbcode": "may I know how can the existence of $U_1, U_2$ be guaranteed?", "content_html": "may I know how can the existence of <img src=\"//latex.artofproblemsolving.com/a/6/4/a64dfba24ee878e2c024e916677d3589620be602.png\" class=\"latex\" alt=\"$U_1, U_2$\" style=\"vertical-align: -3px\" width=\"46\" height=\"16\" > be guaranteed?", "post_id": 560267, "post_number": 4, "post_time_unix": 1151510541, "post_time_utc": "2006-06-28 16:02:21 UTC", "thanks_received": 2, "user_id": 141, "username": "Soarer" }, { "attachments": [], "content_bbcode": "...", "content_html": "...", "post_id": 560319, "post_number": 5, "post_time_unix": 1151513921, "post_time_utc": "2006-06-28 16:58:41 UTC", "thanks_received": 2, "user_id": 13157, "username": "Ottem" }, { "attachments": [], "content_bbcode": "[quote=\"AlvaroRecoba\"]\nIf it is not, we can find open sets $U_1$ and $U_2$ such that ${U_1}\\cup{U_2}\\supseteq{C}$ and ${U_1}\\cap{U_2}=\\emptyset$.\n[/quote]\nDo you skip a whole bunch of steps here? I could work your reasoning out like this:\n\nLet $C = A \\cup B$ be a disjunction of $C$, where $A$ and $B$ are open subsets of $C$. $A$ and $B$ must also be closed in $C$, and since $C$ is closed in $X$, $A$ and $B$ must also be closed as subsets of $X$. Since $X$ is compact and Hausdorff it is normal and we can find open, disjoint sets $U_1$ and $U_2$ in $X$ such that $A \\subseteq U_1$, $B \\subseteq U_2$\n\nIs this what you mean or is there a much easier argument since you omitted it?\n\n[quote=\"AlvaroRecoba\"]\nThus ${U_1}\\cup{U_2}\\supseteq{C_n}$, which contradicts with the fact that $C_n$ is connected.[/quote]\r\nI also do not follow this contradiction... Please clarify", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">AlvaroRecoba wrote:</div>\n<div class=\"bbcode_quote_body\">If it is not, we can find open sets <img src=\"//latex.artofproblemsolving.com/3/c/b/3cb141913b3ab1abe79fb14c5b39a9568faa5543.png\" class=\"latex\" alt=\"$U_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/8/7/b/87b484c85151278e0b91832124fc1e67f2dd2e92.png\" class=\"latex\" alt=\"$U_2$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > such that <img src=\"//latex.artofproblemsolving.com/7/a/b/7abdf2c5a829a1a3954967c13c9e247eceda3dd2.png\" class=\"latex\" alt=\"${U_1}\\cup{U_2}\\supseteq{C}$\" style=\"vertical-align: -2px\" width=\"98\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/1/cc12c1f95499ceba9314adeb6f087324a3975376.png\" class=\"latex\" alt=\"${U_1}\\cap{U_2}=\\emptyset$\" style=\"vertical-align: -2px\" width=\"92\" height=\"17\" >.</span></div>\n</div>\nDo you skip a whole bunch of steps here? I could work your reasoning out like this:<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/e/0/4/e046fc4c160e4efd50535bd16e2fc75a3278ea50.png\" class=\"latex\" alt=\"$C = A \\cup B$\" style=\"vertical-align: 0px\" width=\"86\" height=\"13\" > be a disjunction of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >,</span> where <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > are open subsets of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >.</span> <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > must also be closed in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >,</span> and since <img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > is closed in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > must also be closed as subsets of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" >.</span> Since <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > is compact and Hausdorff it is normal and we can find open, disjoint sets <img src=\"//latex.artofproblemsolving.com/3/c/b/3cb141913b3ab1abe79fb14c5b39a9568faa5543.png\" class=\"latex\" alt=\"$U_1$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/8/7/b/87b484c85151278e0b91832124fc1e67f2dd2e92.png\" class=\"latex\" alt=\"$U_2$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > in <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/b/a/1ba2edc323d7d2665a62635d45a27551e597841a.png\" class=\"latex\" alt=\"$A \\subseteq U_1$\" style=\"vertical-align: -2px\" width=\"56\" height=\"15\" >,</span> <img src=\"//latex.artofproblemsolving.com/3/e/1/3e1fb1c46bebde8b4555a20f651ec5a3b665f61e.png\" class=\"latex\" alt=\"$B \\subseteq U_2$\" style=\"vertical-align: -2px\" width=\"57\" height=\"15\" ><br>\n<br>\nIs this what you mean or is there a much easier argument since you omitted it?<br>\n\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">AlvaroRecoba wrote:</div>\n<div class=\"bbcode_quote_body\">Thus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/8/478b672559b71967c552983a3982616013d23785.png\" class=\"latex\" alt=\"${U_1}\\cup{U_2}\\supseteq{C_n}$\" style=\"vertical-align: -2px\" width=\"105\" height=\"15\" >,</span> which contradicts with the fact that <img src=\"//latex.artofproblemsolving.com/4/e/1/4e173907ca992482385a50c6f62ede920362b1bf.png\" class=\"latex\" alt=\"$C_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > is connected.</div>\n</div>\nI also do not follow this contradiction... Please clarify", "post_id": 560349, "post_number": 6, "post_time_unix": 1151515325, "post_time_utc": "2006-06-28 17:22:05 UTC", "thanks_received": 2, "user_id": 11334, "username": "Kalle" }, { "attachments": [], "content_bbcode": "Sorry I made a terrible mistake here...", "content_html": "Sorry I made a terrible mistake here...", "post_id": 560999, "post_number": 7, "post_time_unix": 1151551900, "post_time_utc": "2006-06-29 03:31:40 UTC", "thanks_received": 1, "user_id": 20595, "username": "AlvaroRecoba" }, { "attachments": [], "content_bbcode": "Kalle is right.\r\n\r\nIf $C$ is not connected, we can find closed sets ${V_1}\\neq{\\emptyset}$ and ${V_2}\\neq{\\emptyset}$ in $C$ such that ${V_1}\\cup{V_2}=C$ and ${V_1}\\cap{V_2}=\\emptyset$. And since $C$ is closed, $V_1$ and $V_2$ are also closed sets in $X$.\r\nBecause compact Hausdorff space has the $T_4$ property, there are open sets ${U_1}\\supseteq{V_1}$, ${U_2}\\supseteq{V_2}$ such that ${U_1}\\cap{U_2}=\\emptyset$.\r\n\r\nAt the end of my proof, we conclude that ${U_1}\\cup{U_2}\\supseteq{C_n}$, then ${U_1}\\cap{C_n}\\neq{\\emptyset}$ and ${U_2}\\cap{C_n}\\neq{\\emptyset}$ are two open sets in ${C_n}$, their union equals ${C_n}$, and their intersection is $\\emptyset$.\r\nThis is impossible since ${C_n}$ is connected.", "content_html": "Kalle is right.<br>\n<br>\nIf <img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > is not connected, we can find closed sets <img src=\"//latex.artofproblemsolving.com/d/b/1/db1f200af9f9e7248d672326a4342a2fb854cfe0.png\" class=\"latex\" alt=\"${V_1}\\neq{\\emptyset}$\" style=\"vertical-align: -4px\" width=\"51\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/7/3/e/73eda9298168c1e2bac657dd2a56ad3971c21516.png\" class=\"latex\" alt=\"${V_2}\\neq{\\emptyset}$\" style=\"vertical-align: -4px\" width=\"51\" height=\"18\" > in <img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > such that <img src=\"//latex.artofproblemsolving.com/2/2/3/223b60152f814e888ab124387d1e8ebd139bb512.png\" class=\"latex\" alt=\"${V_1}\\cup{V_2}=C$\" style=\"vertical-align: -2px\" width=\"94\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/6/cc6af1f7f18a4a336f9483890fec1afce28b9776.png\" class=\"latex\" alt=\"${V_1}\\cap{V_2}=\\emptyset$\" style=\"vertical-align: -2px\" width=\"89\" height=\"17\" >.</span> And since <img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > is closed, <img src=\"//latex.artofproblemsolving.com/7/2/f/72f539a3dc0cea9899543319d030a34ca6c33b0e.png\" class=\"latex\" alt=\"$V_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/e/c/2/ec2fe4ffdca06f7e4f7e2df05465a894eda12eb7.png\" class=\"latex\" alt=\"$V_2$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > are also closed sets in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" >.</span><br>\nBecause compact Hausdorff space has the <img src=\"//latex.artofproblemsolving.com/8/5/7/85746d1f1493a5145f5e46d2a8a07677a21580d8.png\" class=\"latex\" alt=\"$T_4$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > property, there are open sets <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/5/6/8564c3886ae18ee1d92858249d382cb6df43bdaa.png\" class=\"latex\" alt=\"${U_1}\\supseteq{V_1}$\" style=\"vertical-align: -2px\" width=\"60\" height=\"15\" >,</span> <img src=\"//latex.artofproblemsolving.com/c/7/0/c708982aa4c2dc6510a4dac4c4b00d72b0768ee2.png\" class=\"latex\" alt=\"${U_2}\\supseteq{V_2}$\" style=\"vertical-align: -2px\" width=\"60\" height=\"15\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/1/cc12c1f95499ceba9314adeb6f087324a3975376.png\" class=\"latex\" alt=\"${U_1}\\cap{U_2}=\\emptyset$\" style=\"vertical-align: -2px\" width=\"92\" height=\"17\" >.</span><br>\n<br>\nAt the end of my proof, we conclude that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/8/478b672559b71967c552983a3982616013d23785.png\" class=\"latex\" alt=\"${U_1}\\cup{U_2}\\supseteq{C_n}$\" style=\"vertical-align: -2px\" width=\"105\" height=\"15\" >,</span> then <img src=\"//latex.artofproblemsolving.com/6/5/a/65a04acb55e2b45250cc017d4ba28d511e123c5d.png\" class=\"latex\" alt=\"${U_1}\\cap{C_n}\\neq{\\emptyset}$\" style=\"vertical-align: -4px\" width=\"95\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/a/d/e/adece16ff059383dd09935614d6f767933bba752.png\" class=\"latex\" alt=\"${U_2}\\cap{C_n}\\neq{\\emptyset}$\" style=\"vertical-align: -4px\" width=\"95\" height=\"18\" > are two open sets in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/4/a14128185a1adc2785d9140e0a90979ec883042d.png\" class=\"latex\" alt=\"${C_n}$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" >,</span> their union equals <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/4/a14128185a1adc2785d9140e0a90979ec883042d.png\" class=\"latex\" alt=\"${C_n}$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" >,</span> and their intersection is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/2/84241c448cbf497968ba86136beecf766431a3ff.png\" class=\"latex\" alt=\"$\\emptyset$\" style=\"vertical-align: -1px\" width=\"8\" height=\"16\" >.</span><br>\nThis is impossible since <img src=\"//latex.artofproblemsolving.com/a/1/4/a14128185a1adc2785d9140e0a90979ec883042d.png\" class=\"latex\" alt=\"${C_n}$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > is connected.", "post_id": 562405, "post_number": 8, "post_time_unix": 1151644703, "post_time_utc": "2006-06-30 05:18:23 UTC", "thanks_received": 2, "user_id": 20595, "username": "AlvaroRecoba" }, { "attachments": [], "content_bbcode": "Now it works. Thanks!", "content_html": "Now it works. Thanks!", "post_id": 562541, "post_number": 9, "post_time_unix": 1151666999, "post_time_utc": "2006-06-30 11:29:59 UTC", "thanks_received": 2, "user_id": 11334, "username": "Kalle" } ], "source": null }	Let $X$ be a compact Hausdorff space and let $C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots$ be a sequence of closed, connected subspaces of $X$. Is the intersection \[ C=\bigcap_{n=1}^\infty C_n \] necessarily connected?	[ "/Mathematics/Topology/GeneralTopology/Topology", "/Mathematics/Topology/Point-SetTopology/0-Connected", "/Mathematics/Topology/Point-SetTopology/1-Connected", "/Mathematics/Topology/Point-SetTopology/Closed", "/Mathematics/Topology/Point-SetTopology/ClosedSet", "/Mathematics/Topology/Point-SetTopology/CompactSet", "/Mathematics/Topology/Point-SetTopology/ConnectedSet", "/Mathematics/Topology/Point-SetTopology/ConnectedSpace", "/Mathematics/Topology/Spaces/CompactSpace", "/Mathematics/Topology/Spaces/Space", "/Mathematics/Topology/Spaces/T2-Space", "/Mathematics/Topology/Spaces/TopologicalSpace", "/Mathematics/Topology/Spaces/TychonoffSpace" ]	Use compactness to extract a finite subcover from the open cover formed by the two separating opens and the complements of the nested closed sets, forcing some C_n to be split.	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aops_991983	I'm scared hunter? [hide="Solution"] Let the quadrilateral be $ ABCD$ and let $ AC$ and $ BD$ meet at $ X$. Furthermore, let the perpindiculars from $ X$ to $ AB$, $ BC$, $ CD$, and $ AD$ be $ P$, $ Q$, $ R$, and $ S$ respectively and the reflections of $ X$ over $ AB$, $ BC$, $ CD$, and $ AD$ be $ P'$, $ Q'$, $ R'$, and $ S'$ respectively. Let $ \angle ADB \equal{} \alpha$ and $ \angle ACB \equal{} \beta$. Hence, $ XSDR$ is cyclic, so $ \angle XRS \equal{} \angle XDS \equal{} \alpha$. Also, $ XRCQ$ is cyclic, so $ \angle XRQ \equal{} \angle XCQ \equal{} \beta$. Notice that since $ AC\perp BD$, we have that $ \angle DBC \equal{} 90 \minus{} \beta$ and $ \angle CAD \equal{} 90 \minus{} \alpha$. Since $ RPXQ$ and $ APXS$ are cyclic, we have that $ \angle QPX \equal{} \angle QBX \equal{} 90 \minus{} \beta$ and $ \angle SPX \equal{} \angle SAX \equal{} 90 \minus{} \alpha$. Therefore, $ \angle QRS \equal{} \alpha \plus{} \beta$ and $ \angle QPS \equal{} 180 \minus{} \alpha \minus{} \beta$. So, $ PSRQ$ is cyclic. Now, notice that $ XP \equal{} PP'$, $ XS \equal{} SS'$, $ XR \equal{} RR'$, and $ XQ \equal{} QQ'$. Therefore, we have that $ PS\parallel P'S'$, $ RS\parallel R'S'$, $ RQ\parallel R'Q'$, and $ PQ\parallel P'Q'$, so $ PQRS\sim P'Q'R'S'$, which implies that $ P'Q'R'S'$ is cyclic as well. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Today I took an AMC 10. I got a 150.It was the first AMC 10. But nobody really cares. I shall now try AMC 12. And failed miserably.\r\n\r\nA evil geo problem:\r\n\r\nThe diagonals of a convex quadrilateral meet at right angles at X. Show that the four points obtained by reflecting X in each of the sides are cyclic.\r\n\r\nUSAMO 1993.\r\n\r\nQuatto. I know your scared now. Because this was the problem that determines the fact if you are in Geometry Failure Club, Since you answered it correctly. We kick you out.", "content_html": "Today I took an AMC 10. I got a 150.It was the first AMC 10. But nobody really cares. I shall now try AMC 12. And failed miserably.<br>\n<br>\nA evil geo problem:<br>\n<br>\nThe diagonals of a convex quadrilateral meet at right angles at X. Show that the four points obtained by reflecting X in each of the sides are cyclic.<br>\n<br>\nUSAMO 1993.<br>\n<br>\nQuatto. I know your scared now. Because this was the problem that determines the fact if you are in Geometry Failure Club, Since you answered it correctly. We kick you out.", "post_id": 4403282, "post_number": 1, "post_time_unix": 1202062783, "post_time_utc": "2008-02-03 18:19:43 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "[hide=\"First thoughts\"]\nInversion about the circumcircle. There might be a more elementary way, though...[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">First thoughts</a><div class=\"cmty-hide-content\" style=\"display:none\">Inversion about the circumcircle. There might be a more elementary way, though...</div>", "post_id": 4403283, "post_number": 2, "post_time_unix": 1202073034, "post_time_utc": "2008-02-03 21:10:34 UTC", "thanks_received": 2, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "How'd you take the AMC 10? The earliest test day would be Feb 12.", "content_html": "How'd you take the AMC 10? The earliest test day would be Feb 12.", "post_id": 4403284, "post_number": 3, "post_time_unix": 1202074754, "post_time_utc": "2008-02-03 21:39:14 UTC", "thanks_received": 2, "user_id": 23588, "username": "n0vad3m0n" }, { "attachments": [], "content_bbcode": "Practice test.", "content_html": "Practice test.", "post_id": 4403285, "post_number": 4, "post_time_unix": 1202075584, "post_time_utc": "2008-02-03 21:53:04 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "I'm scared hunter?\r\n[hide=\"Solution\"]\nLet the quadrilateral be $ ABCD$ and let $ AC$ and $ BD$ meet at $ X$. Furthermore, let the perpindiculars from $ X$ to $ AB$, $ BC$, $ CD$, and $ AD$ be $ P$, $ Q$, $ R$, and $ S$ respectively and the reflections of $ X$ over $ AB$, $ BC$, $ CD$, and $ AD$ be $ P'$, $ Q'$, $ R'$, and $ S'$ respectively. Let $ \\angle ADB \\equal{} \\alpha$ and $ \\angle ACB \\equal{} \\beta$. Hence, $ XSDR$ is cyclic, so $ \\angle XRS \\equal{} \\angle XDS \\equal{} \\alpha$. Also, $ XRCQ$ is cyclic, so $ \\angle XRQ \\equal{} \\angle XCQ \\equal{} \\beta$. Notice that since $ AC\\perp BD$, we have that $ \\angle DBC \\equal{} 90 \\minus{} \\beta$ and $ \\angle CAD \\equal{} 90 \\minus{} \\alpha$. Since $ RPXQ$ and $ APXS$ are cyclic, we have that $ \\angle QPX \\equal{} \\angle QBX \\equal{} 90 \\minus{} \\beta$ and $ \\angle SPX \\equal{} \\angle SAX \\equal{} 90 \\minus{} \\alpha$. Therefore, $ \\angle QRS \\equal{} \\alpha \\plus{} \\beta$ and $ \\angle QPS \\equal{} 180 \\minus{} \\alpha \\minus{} \\beta$. So, $ PSRQ$ is cyclic. Now, notice that $ XP \\equal{} PP'$, $ XS \\equal{} SS'$, $ XR \\equal{} RR'$, and $ XQ \\equal{} QQ'$. Therefore, we have that $ PS\\parallel P'S'$, $ RS\\parallel R'S'$, $ RQ\\parallel R'Q'$, and $ PQ\\parallel P'Q'$, so $ PQRS\\sim P'Q'R'S'$, which implies that $ P'Q'R'S'$ is cyclic as well. \n[/hide]", "content_html": "I'm scared hunter?<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Solution</a><div class=\"cmty-hide-content\" style=\"display:none\">Let the quadrilateral be <img src=\"//latex.artofproblemsolving.com/3/9/3/39373ad0f592e0aad414c9aebaf981bb327fffc5.png\" class=\"latex\" alt=\"$ ABCD$\" width=\"57\" height=\"13\" > and let <img src=\"//latex.artofproblemsolving.com/0/7/5/0756a0f5f7effcf224e6e829c46c2db34927e86f.png\" class=\"latex\" alt=\"$ AC$\" width=\"27\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/8/f/0/8f060a3b3720e24643ce978a77df88b8d425f0e4.png\" class=\"latex\" alt=\"$ BD$\" width=\"29\" height=\"12\" > meet at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/9/2/19223fc42c69e613a4cdc9c7fddd725b28b68dc6.png\" class=\"latex\" alt=\"$ X$\" width=\"15\" height=\"12\" >.</span> Furthermore, let the perpindiculars from <img src=\"//latex.artofproblemsolving.com/1/9/2/19223fc42c69e613a4cdc9c7fddd725b28b68dc6.png\" class=\"latex\" alt=\"$ X$\" width=\"15\" height=\"12\" > to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/4/e3427a9945d3b1de6a7c0b6862f3228415ea8bca.png\" class=\"latex\" alt=\"$ BC$\" width=\"28\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/4/ba442f9231826f849adfd6b270c42a50da61f82d.png\" class=\"latex\" alt=\"$ CD$\" width=\"29\" height=\"12\" >,</span> and <img src=\"//latex.artofproblemsolving.com/3/f/3/3f3a5cc49a11465eb9aaef9f980ed62b4e696f8c.png\" class=\"latex\" alt=\"$ AD$\" width=\"28\" height=\"13\" > be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/b/1/fb1b7d554ea7d9094511c084a2682639363022f5.png\" class=\"latex\" alt=\"$ P$\" width=\"14\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/b/0/4b073c82664ac30d6b83fc7ef3db6507054c0d95.png\" class=\"latex\" alt=\"$ Q$\" style=\"vertical-align: -3px\" width=\"13\" height=\"16\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/4/e/b4edc0ab313e68210092e53caaf1d36cc7e5b388.png\" class=\"latex\" alt=\"$ R$\" width=\"14\" height=\"12\" >,</span> and <img src=\"//latex.artofproblemsolving.com/c/6/6/c663ecbe181b4e84e91ccbac30187e24917af1ed.png\" class=\"latex\" alt=\"$ S$\" width=\"12\" height=\"12\" > respectively and the reflections of <img src=\"//latex.artofproblemsolving.com/1/9/2/19223fc42c69e613a4cdc9c7fddd725b28b68dc6.png\" class=\"latex\" alt=\"$ X$\" width=\"15\" height=\"12\" > over <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/4/e3427a9945d3b1de6a7c0b6862f3228415ea8bca.png\" class=\"latex\" alt=\"$ BC$\" width=\"28\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/4/ba442f9231826f849adfd6b270c42a50da61f82d.png\" class=\"latex\" alt=\"$ CD$\" width=\"29\" height=\"12\" >,</span> and <img src=\"//latex.artofproblemsolving.com/3/f/3/3f3a5cc49a11465eb9aaef9f980ed62b4e696f8c.png\" class=\"latex\" alt=\"$ AD$\" width=\"28\" height=\"13\" > be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/a/1/2a1415901419afa06f588a2399cb73833aeee277.png\" class=\"latex\" alt=\"$ P'$\" width=\"17\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/1/e/31e1b93d4d6a58781c3603271d32998ec6b3c6e0.png\" class=\"latex\" alt=\"$ Q'$\" style=\"vertical-align: -3px\" width=\"18\" height=\"17\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/1/4/f141c61e33de2aeb2007c09b140ccbedcb15aa24.png\" class=\"latex\" alt=\"$ R'$\" width=\"17\" height=\"14\" >,</span> and <img src=\"//latex.artofproblemsolving.com/4/f/0/4f036e561b0cbe43baccf3bb3bb99ea4e0c602e1.png\" class=\"latex\" alt=\"$ S'$\" width=\"15\" height=\"14\" > respectively. Let <img src=\"//latex.artofproblemsolving.com/7/8/5/78549690bbdcea60ffb8f3ce04f5fbe3b476d48e.png\" class=\"latex\" alt=\"$ \\angle ADB = \\alpha$\" width=\"92\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/c/4/fc4ae815a2aeed7fa86db8a83178948b586de86c.png\" class=\"latex\" alt=\"$ \\angle ACB = \\beta$\" style=\"vertical-align: -3px\" width=\"90\" height=\"16\" >.</span> Hence, <img src=\"//latex.artofproblemsolving.com/1/5/9/1595253e12946eb5440c4df4dbcb3ddca9554f73.png\" class=\"latex\" alt=\"$ XSDR$\" width=\"58\" height=\"12\" > is cyclic, so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/f/eafefed7591a69abc88dcba92df29edc0c6cee18.png\" class=\"latex\" alt=\"$ \\angle XRS = \\angle XDS = \\alpha$\" width=\"172\" height=\"12\" >.</span> Also, <img src=\"//latex.artofproblemsolving.com/6/f/5/6f53b559f63da5f4af88910fb09bb24a8c4f302b.png\" class=\"latex\" alt=\"$ XRCQ$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > is cyclic, so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/4/41416f9058fa2dc7433d076c1150f2faa81a7908.png\" class=\"latex\" alt=\"$ \\angle XRQ = \\angle XCQ = \\beta$\" style=\"vertical-align: -3px\" width=\"174\" height=\"16\" >.</span> Notice that since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/3/d23839c883c229c401f9a61939b2b13185bc62b4.png\" class=\"latex\" alt=\"$ AC\\perp BD$\" width=\"82\" height=\"13\" >,</span> we have that <img src=\"//latex.artofproblemsolving.com/7/3/4/734c32008bc201bc15177c87268e7fb9ad128ffb.png\" class=\"latex\" alt=\"$ \\angle DBC = 90 - \\beta$\" style=\"vertical-align: -3px\" width=\"132\" height=\"16\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/1/b/61b3699152094c3039cf7ac4d5680fa755ea87b2.png\" class=\"latex\" alt=\"$ \\angle CAD = 90 - \\alpha$\" width=\"132\" height=\"13\" >.</span> Since <img src=\"//latex.artofproblemsolving.com/6/6/9/669cad247a80af2fc039bf83aa45ecff7c1de5d6.png\" class=\"latex\" alt=\"$ RPXQ$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > and <img src=\"//latex.artofproblemsolving.com/a/8/e/a8e60e4b383294a57814c1fec05a9ab20b376791.png\" class=\"latex\" alt=\"$ APXS$\" width=\"56\" height=\"13\" > are cyclic, we have that <img src=\"//latex.artofproblemsolving.com/6/4/a/64acb757d622d97141ae58589d05b41bbc693178.png\" class=\"latex\" alt=\"$ \\angle QPX = \\angle QBX = 90 - \\beta$\" style=\"vertical-align: -3px\" width=\"215\" height=\"16\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/a/0/aa0937aa332dcb2c98a30e5d80d4d533b1617ebb.png\" class=\"latex\" alt=\"$ \\angle SPX = \\angle SAX = 90 - \\alpha$\" width=\"211\" height=\"13\" >.</span> Therefore, <img src=\"//latex.artofproblemsolving.com/2/7/c/27c45e9807ba2f271f918ba74fc2845097137f0d.png\" class=\"latex\" alt=\"$ \\angle QRS = \\alpha + \\beta$\" style=\"vertical-align: -3px\" width=\"122\" height=\"16\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/6/6/166940b73282902603fd339fc36ff240523963bb.png\" class=\"latex\" alt=\"$ \\angle QPS = 180 - \\alpha - \\beta$\" style=\"vertical-align: -3px\" width=\"172\" height=\"16\" >.</span> So, <img src=\"//latex.artofproblemsolving.com/1/c/c/1cc132dd6a573b4fdc6311768466fddda88c212c.png\" class=\"latex\" alt=\"$ PSRQ$\" style=\"vertical-align: -3px\" width=\"54\" height=\"16\" > is cyclic. Now, notice that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/0/f/20f5b05bbf0dc2d1d8c14bc91467d7a3d0345d71.png\" class=\"latex\" alt=\"$ XP = PP'$\" width=\"87\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/f/7/7f7c89336d4e66af956c300fd9e8f4d3a2639904.png\" class=\"latex\" alt=\"$ XS = SS'$\" width=\"81\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/e/d8ea43098ca79c6878cb0028ac2d258e15dbc158.png\" class=\"latex\" alt=\"$ XR = RR'$\" width=\"86\" height=\"14\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/3/bc3c62e8fdf62f051f0e36279adf9dbf5d736789.png\" class=\"latex\" alt=\"$ XQ = QQ'$\" style=\"vertical-align: -3px\" width=\"87\" height=\"17\" >.</span> Therefore, we have that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/3/cc33813f255503a1df9d1256e2515be22335bac9.png\" class=\"latex\" alt=\"$ PS\\parallel P'S'$\" style=\"vertical-align: -5px\" width=\"80\" height=\"19\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/4/2/7421aee6c58fad4a2cb073e5b899631458a88fb0.png\" class=\"latex\" alt=\"$ RS\\parallel R'S'$\" style=\"vertical-align: -5px\" width=\"79\" height=\"19\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/f/e/6fe585b756c6be60ef79a7c80f8e9604758a73bb.png\" class=\"latex\" alt=\"$ RQ\\parallel R'Q'$\" style=\"vertical-align: -5px\" width=\"84\" height=\"19\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/a/7/0a721f1f92d5a8bc2910ae6472eaba42845f3ee4.png\" class=\"latex\" alt=\"$ PQ\\parallel P'Q'$\" style=\"vertical-align: -5px\" width=\"84\" height=\"19\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/b/8/eb8f1b268a6f43ba47c5bac46d4056561c605060.png\" class=\"latex\" alt=\"$ PQRS\\sim P'Q'R'S'$\" style=\"vertical-align: -3px\" width=\"151\" height=\"17\" >,</span> which implies that <img src=\"//latex.artofproblemsolving.com/a/e/d/aedea76e45841cc3a6d4a3a896eab18560d80900.png\" class=\"latex\" alt=\"$ P'Q'R'S'$\" style=\"vertical-align: -3px\" width=\"72\" height=\"17\" > is cyclic as well.</div>", "post_id": 4403286, "post_number": 5, "post_time_unix": 1202079495, "post_time_utc": "2008-02-03 22:58:15 UTC", "thanks_received": 2, "user_id": 26129, "username": "The QuattoMaster 6000" }, { "attachments": [], "content_bbcode": "Whoa, non-inversion solution. Good job quatto.", "content_html": "Whoa, non-inversion solution. Good job quatto.", "post_id": 4403287, "post_number": 6, "post_time_unix": 1202093397, "post_time_utc": "2008-02-04 02:49:57 UTC", "thanks_received": 2, "user_id": 28419, "username": "Temperal" } ], "source": null }	The diagonals of a convex quadrilateral meet at right angles at $X$. Show that the four points obtained by reflecting $X$ in each of the sides are concyclic.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Quadrangles/OrthodiagonalQuadrangle", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Quadrilateral", "/Mathematics/Geometry/Points/Point", "/Mathematics/Geometry/Symmetry/MirrorSymmetry", "/Mathematics/Geometry/Transformations/GeometricTransformations", "/Mathematics/Geometry/Transformations/Reflections/MirrorSymmetry", "/Mathematics/Geometry/Transformations/Reflections/Reflection" ]	Prove the feet of the perpendiculars from X to the sides are cyclic, then use the similarity (homothety) between this pedal quadrilateral and the reflected points to deduce their concyclicity.	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aops_99199	I hope I'm not doing anything wrong... The functions $g_1(x)=x^{2005}$ and $h_1(x)=2006x$ are bijective and range over the reals. Thus, the functions $g_2(x)=x^{2005}+1$ and $h_2(x)=2006x+2006$ are bijective, mapping from the reals to the reals. Hence, if $y\in\Re$ such that $y=x^{2005}+1$, then there exists some $t\in\Re$ such that $2006t+2006=y$. Similarly, if $t\in\Re$ such that $t=2006x+2006$, then there exists some $y\in\Re$ such that $y^{2005}+1=t$. Since $f(2006t+2006)\le2006$, then $f(y)=f(x^{2005}+1)\le2006$. Since $f(2006t+2006)=2006$ for some $t$, then $f(y)=f(x^{2005}+1)=2006$ for some $y$. Therefore, the maximum value is $2006$.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "The domain of $f(x)$ is $R$. If the maximum value of $f(2006x+2006)$ is 2006, then find the maximum value of $f(x^{2005}+1)$.", "content_html": "The domain of <img src=\"//latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png\" class=\"latex\" alt=\"$f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >.</span> If the maximum value of <img src=\"//latex.artofproblemsolving.com/9/d/d/9ddb217218bba057d6e3a7f8c466691626f8ceff.png\" class=\"latex\" alt=\"$f(2006x+2006)$\" style=\"vertical-align: -4px\" width=\"128\" height=\"18\" > is 2006, then find the maximum value of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/b/d/7bd184f796dc3741ff86a4a5fb8b90a3d9a4724d.png\" class=\"latex\" alt=\"$f(x^{2005}+1)$\" style=\"vertical-align: -4px\" width=\"92\" height=\"19\" >.</span>", "post_id": 559986, "post_number": 1, "post_time_unix": 1151489935, "post_time_utc": "2006-06-28 10:18:55 UTC", "thanks_received": 1, "user_id": 6601, "username": "shobber" }, { "attachments": [], "content_bbcode": "I hope I'm not doing anything wrong...\r\n\r\nThe functions $g_1(x)=x^{2005}$ and $h_1(x)=2006x$ are bijective and range over the reals.\r\n\r\nThus, the functions $g_2(x)=x^{2005}+1$ and $h_2(x)=2006x+2006$ are bijective, mapping from the reals to the reals.\r\n\r\nHence, if $y\\in\\Re$ such that $y=x^{2005}+1$, then there exists some $t\\in\\Re$ such that $2006t+2006=y$.\r\n\r\nSimilarly, if $t\\in\\Re$ such that $t=2006x+2006$, then there exists some $y\\in\\Re$ such that $y^{2005}+1=t$.\r\n\r\nSince $f(2006t+2006)\\le2006$, then $f(y)=f(x^{2005}+1)\\le2006$.\r\n\r\nSince $f(2006t+2006)=2006$ for some $t$, then $f(y)=f(x^{2005}+1)=2006$ for some $y$.\r\n\r\nTherefore, the maximum value is $2006$.", "content_html": "I hope I'm not doing anything wrong...<br>\n<br>\nThe functions <img src=\"//latex.artofproblemsolving.com/b/8/e/b8efc5a521ab251bbf3c9ee6d5e469f20e3ac45e.png\" class=\"latex\" alt=\"$g_1(x)=x^{2005}$\" style=\"vertical-align: -4px\" width=\"101\" height=\"19\" > and <img src=\"//latex.artofproblemsolving.com/c/a/a/caac7b626503f7f92b34b42ee0510e02d5b9e7a3.png\" class=\"latex\" alt=\"$h_1(x)=2006x$\" style=\"vertical-align: -4px\" width=\"112\" height=\"18\" > are bijective and range over the reals.<br>\n<br>\nThus, the functions <img src=\"//latex.artofproblemsolving.com/e/8/3/e836997af6bc8be212ea1ac1a7633a45ed24fa2a.png\" class=\"latex\" alt=\"$g_2(x)=x^{2005}+1$\" style=\"vertical-align: -4px\" width=\"133\" height=\"19\" > and <img src=\"//latex.artofproblemsolving.com/3/4/d/34d51be4cf1ea593d2a20692e35de5695abbfeeb.png\" class=\"latex\" alt=\"$h_2(x)=2006x+2006$\" style=\"vertical-align: -4px\" width=\"171\" height=\"18\" > are bijective, mapping from the reals to the reals.<br>\n<br>\nHence, if <img src=\"//latex.artofproblemsolving.com/8/2/1/82186cc89d8b229ebf8abd6a379296eb616295f1.png\" class=\"latex\" alt=\"$y\\in\\Re$\" style=\"vertical-align: -3px\" width=\"45\" height=\"17\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/f/7/3f711ebcb3acb42dfadd7eeee6d059954d825ecc.png\" class=\"latex\" alt=\"$y=x^{2005}+1$\" style=\"vertical-align: -3px\" width=\"101\" height=\"18\" >,</span> then there exists some <img src=\"//latex.artofproblemsolving.com/2/9/8/2989a102d03ae3fe14971d4cae33fc786fe2896a.png\" class=\"latex\" alt=\"$t\\in\\Re$\" style=\"vertical-align: -1px\" width=\"42\" height=\"14\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/1/6a1a21f1bc2c9e3d23d3242b8d6dcc4ca0c7f0ca.png\" class=\"latex\" alt=\"$2006t+2006=y$\" style=\"vertical-align: -3px\" width=\"135\" height=\"16\" >.</span><br>\n<br>\nSimilarly, if <img src=\"//latex.artofproblemsolving.com/2/9/8/2989a102d03ae3fe14971d4cae33fc786fe2896a.png\" class=\"latex\" alt=\"$t\\in\\Re$\" style=\"vertical-align: -1px\" width=\"42\" height=\"14\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/1/8/218b5c1e699109daf2841fa4ee0855563c83a53f.png\" class=\"latex\" alt=\"$t=2006x+2006$\" style=\"vertical-align: -1px\" width=\"135\" height=\"14\" >,</span> then there exists some <img src=\"//latex.artofproblemsolving.com/8/2/1/82186cc89d8b229ebf8abd6a379296eb616295f1.png\" class=\"latex\" alt=\"$y\\in\\Re$\" style=\"vertical-align: -3px\" width=\"45\" height=\"17\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/8/a/58a4af55cc6a6476c9bec43367f7ed1a4f08d4d3.png\" class=\"latex\" alt=\"$y^{2005}+1=t$\" style=\"vertical-align: -3px\" width=\"98\" height=\"18\" >.</span><br>\n<br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/8/f/b8f45974eec5d322084e964514814aff9e788a6a.png\" class=\"latex\" alt=\"$f(2006t+2006)\\le2006$\" style=\"vertical-align: -4px\" width=\"186\" height=\"18\" >,</span> then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/a/1/7a18f6921834cb65ec7bb6885ab0a77c57b1e709.png\" class=\"latex\" alt=\"$f(y)=f(x^{2005}+1)\\le2006$\" style=\"vertical-align: -4px\" width=\"212\" height=\"19\" >.</span><br>\n<br>\nSince <img src=\"//latex.artofproblemsolving.com/f/f/8/ff8bf137b1f1c7a10ea99cb2976bfb2a76b6681a.png\" class=\"latex\" alt=\"$f(2006t+2006)=2006$\" style=\"vertical-align: -4px\" width=\"186\" height=\"18\" > for some <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/d/e0d2bf360290fd61d1c1557e763f2622363b3d35.png\" class=\"latex\" alt=\"$t$\" width=\"6\" height=\"11\" >,</span> then <img src=\"//latex.artofproblemsolving.com/e/5/4/e54ce016e58d988b887fc4a629231fcbead250f7.png\" class=\"latex\" alt=\"$f(y)=f(x^{2005}+1)=2006$\" style=\"vertical-align: -4px\" width=\"212\" height=\"19\" > for some <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" >.</span><br>\n<br>\nTherefore, the maximum value is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/2/d1216dd3855e1bb88b6136b4ce49c58b57747417.png\" class=\"latex\" alt=\"$2006$\" width=\"35\" height=\"12\" >.</span>", "post_id": 560313, "post_number": 2, "post_time_unix": 1151513650, "post_time_utc": "2006-06-28 16:54:10 UTC", "thanks_received": 2, "user_id": 2520, "username": "towersfreak2006" } ], "source": null }	The domain of $f(x)$ is $\mathbb{R}$. If the maximum value of $f(2006x+2006)$ is $2006$, then find the maximum value of $f(x^{2005}+1)$.	[ "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialFunction" ]	Use the bijectivity of the transformations (x↦2006x+2006 and x↦x^{2005}+1) to transfer the given maximum value to the new argument form.	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aops_99203	[quote="Heldensheld"] Q2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again. [/quote] [hide] If you think about it, the next occurence must be after 7:00. Let $x$ be the number of minutes after 7 pm. The minute hand is at $6x^{\circ}$. The hour hand is at $(\frac{7}{12}*360+\frac{x}{2})^{\circ}=(210+\frac{x}{2})^{\circ}$, since there are 720 minutes in 12 hours, so each minute is half a degree. $210+\frac{x}{2}-180=6x$. $30+\frac{x}{2}=6x$. $60+x=12x$. $11x=60$. $x=\frac{60}{11}=5+\frac{5}{11}$. So the answer is $\boxed{65+\frac{5}{11}}$. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Here's some questions where I hope you people can help me with :) !\r\n\r\nQ1. If 2^1000 is divided by 5, what will the remainder be?\r\n<Hint: Consider dividing 2^1, 2^2, 2^3.....etc by 5>\r\n\r\nI have absolutely no clue on this one :( .\r\n\r\n\r\nQ2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again.\r\n\r\nThats all for now!", "content_html": "Here's some questions where I hope you people can help me with <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /> !<br>\n<br>\nQ1. If 2^1000 is divided by 5, what will the remainder be?<br>\n<Hint: Consider dividing 2^1, 2^2, 2^3.....etc by 5><br>\n<br>\nI have absolutely no clue on this one <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /> .<br>\n<br>\n<br>\nQ2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again.<br>\n<br>\nThats all for now!", "post_id": 560005, "post_number": 1, "post_time_unix": 1151492965, "post_time_utc": "2006-06-28 11:09:25 UTC", "thanks_received": 2, "user_id": 20648, "username": "Heldensheld" }, { "attachments": [], "content_bbcode": "[hide=\"1\"]You should look for a pattern here, or just do some simple modular arithmetic,\n\n$2^{1000}=16^{250}\\equiv 1^{250}\\equiv1 \\mod 5$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">1</a><div class=\"cmty-hide-content\" style=\"display:none\">You should look for a pattern here, or just do some simple modular arithmetic,<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/4/3/6/4363522d2fdd04bc90645a252d76b53f49203eb5.png\" class=\"latex\" alt=\"$2^{1000}=16^{250}\\equiv 1^{250}\\equiv1 \\mod 5$\" style=\"vertical-align: 0px\" width=\"248\" height=\"15\" ></div>", "post_id": 560018, "post_number": 2, "post_time_unix": 1151494093, "post_time_utc": "2006-06-28 11:28:13 UTC", "thanks_received": 2, "user_id": 17793, "username": "ArcticMonkey" }, { "attachments": [], "content_bbcode": "[quote=\"Heldensheld\"]\nQ2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again.\n[/quote]\r\n[hide] If you think about it, the next occurence must be after 7:00. Let $x$ be the number of minutes after 7 pm. The minute hand is at $6x^{\\circ}$. The hour hand is at $(\\frac{7}{12}360+\\frac{x}{2})^{\\circ}=(210+\\frac{x}{2})^{\\circ}$, since there are 720 minutes in 12 hours, so each minute is half a degree.\n$210+\\frac{x}{2}-180=6x$.\n$30+\\frac{x}{2}=6x$.\n$60+x=12x$.\n$11x=60$.\n$x=\\frac{60}{11}=5+\\frac{5}{11}$. So the answer is $\\boxed{65+\\frac{5}{11}}$. [/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Heldensheld wrote:</div>\n<div class=\"bbcode_quote_body\">Q2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again.</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">If you think about it, the next occurence must be after 7:00. Let <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > be the number of minutes after 7 pm. The minute hand is at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/4/a/94aef9d098dc8811c41678e4886170337afb709e.png\" class=\"latex\" alt=\"$6x^{\\circ}$\" width=\"25\" height=\"13\" >.</span> The hour hand is at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/c/9/4c9b754cfd823433ba29bf0f7f353a97936948cf.png\" class=\"latex\" alt=\"$(\\frac{7}{12}360+\\frac{x}{2})^{\\circ}=(210+\\frac{x}{2})^{\\circ}$\" style=\"vertical-align: -13px\" width=\"233\" height=\"37\" >,</span> since there are 720 minutes in 12 hours, so each minute is half a degree.<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/0/c/d0c403ea9d95806b3177014ee0d71abdf3eec8d6.png\" class=\"latex\" alt=\"$210+\\frac{x}{2}-180=6x$\" style=\"vertical-align: -12px\" width=\"156\" height=\"33\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/a/ffa11bef0a2720e92a766553d4c18f599c465ca9.png\" class=\"latex\" alt=\"$30+\\frac{x}{2}=6x$\" style=\"vertical-align: -12px\" width=\"98\" height=\"33\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/f/d/5fd4d44e385e322c64f59dd65fec6a755f55c69b.png\" class=\"latex\" alt=\"$60+x=12x$\" style=\"vertical-align: -1px\" width=\"103\" height=\"14\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/b/9/2b9d7a0c553e96833dabd0a24c5ad0cbbd572661.png\" class=\"latex\" alt=\"$11x=60$\" style=\"vertical-align: 0px\" width=\"70\" height=\"13\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/5/ba5cd3b8254ac89b2d33b4571ac41ad3f79fa38f.png\" class=\"latex\" alt=\"$x=\\frac{60}{11}=5+\\frac{5}{11}$\" style=\"vertical-align: -13px\" width=\"133\" height=\"38\" >.</span> So the answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/8/0983d6df5f763307c864d3b92ef3e02dc795c37e.png\" class=\"latex\" alt=\"$\\boxed{65+\\frac{5}{11}}$\" style=\"vertical-align: -18px\" width=\"74\" height=\"48\" >.</span></div>", "post_id": 560641, "post_number": 3, "post_time_unix": 1151529624, "post_time_utc": "2006-06-28 21:20:24 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" } ], "source": null }	Here's some questions where I hope you people can help me with :) ! Q1. If 2^1000 is divided by 5, what will the remainder be? <Hint: Consider dividing 2^1, 2^2, 2^3.....etc by 5> I have absolutely no clue on this one :( . Q2. At 6pm, the hands of a clock are 180o apart. How long will it be before the hands of clock are 180o apart again. Thats all for now!	[ "/Mathematics/NumberTheory/Arithmetic/Fractions", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/ClockArithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Numbers/LargeNumbers", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Set up an equation equating the angular difference between hour and minute hands to 180° using their constant angular speeds.	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aops_99211	We will prove that at least one of the following equations has a solution $\mod{p}$: 1)$a^2 \equiv 2$ 2)$a^2 \equiv -2$ 3)$a^2-2a+2 \equiv 0$ Ok, so suppose that first equation doesn't have a solution. Then: $(\frac{-2}{p}) = (\frac{-1}{p})(\frac{2}{p})=(-1)^{\frac{p-1}{2}}(\frac{2}{p})$. It means that $(-1)^{\frac{p-1}{2}}$ has to equal 1. On the other side, $(-1)^{\frac{p-1}{2}}=1$ implies that the equation $x^2 \equiv -1$ has a solution. Plugging $x=a-1$ gives us $a^2-2a+2 \equiv 0$, so it is clear that 1) or 2) or 3) holds. So, it's obvious that $p\|(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2)$. But: $(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2) = a^8 - 16$ Conclusion follows.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove that 16 is always a 8th power modulo $p$ for $p$ prime", "content_html": "Prove that 16 is always a 8th power modulo <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > for <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > prime", "post_id": 560017, "post_number": 1, "post_time_unix": 1151493939, "post_time_utc": "2006-06-28 11:25:39 UTC", "thanks_received": 2, "user_id": 10530, "username": "Yimin Ge" }, { "attachments": [], "content_bbcode": "We will prove that at least one of the following equations has a solution $\\mod{p}$:\r\n\r\n1)$a^2 \\equiv 2$\r\n2)$a^2 \\equiv -2$\r\n3)$a^2-2a+2 \\equiv 0$\r\n\r\nOk, so suppose that first equation doesn't have a solution. Then: $(\\frac{-2}{p}) = (\\frac{-1}{p})(\\frac{2}{p})=(-1)^{\\frac{p-1}{2}}(\\frac{2}{p})$. It means that $(-1)^{\\frac{p-1}{2}}$ has to equal 1. On the other side, $(-1)^{\\frac{p-1}{2}}=1$ implies that the equation $x^2 \\equiv -1$ has a solution. Plugging $x=a-1$ gives us $a^2-2a+2 \\equiv 0$, so it is clear that 1) or 2) or 3) holds.\r\nSo, it's obvious that $p\|(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2)$. But: $(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2) = a^8 - 16$\r\nConclusion follows.", "content_html": "We will prove that at least one of the following equations has a solution <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/8/b/b8b71b5ba176db91f6a4683474cd53aa9912f64f.png\" class=\"latex\" alt=\"$\\mod{p}$\" style=\"vertical-align: -3px\" width=\"62\" height=\"16\" >:</span><br>\n<br>\n1<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/d/6/7/d679176b9793d5d0cc5a26d287d4822f175be54f.png\" class=\"latex\" alt=\"$a^2 \\equiv 2$\" width=\"50\" height=\"15\" ></span><br>\n2<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/6/0/f/60ff1b0306add1bd0f735977bdf722cd4cc5c009.png\" class=\"latex\" alt=\"$a^2 \\equiv -2$\" width=\"64\" height=\"15\" ></span><br>\n3<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/1/5/f/15fdef9e1f165cafdf732aeb91f72bc167a2398e.png\" class=\"latex\" alt=\"$a^2-2a+2 \\equiv 0$\" style=\"vertical-align: -1px\" width=\"122\" height=\"16\" ></span><br>\n<br>\nOk, so suppose that first equation doesn't have a solution. Then: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/e/c/dec0c26431d46c74597552e8e37d3f3cebc6746e.png\" class=\"latex\" alt=\"$(\\frac{-2}{p}) = (\\frac{-1}{p})(\\frac{2}{p})=(-1)^{\\frac{p-1}{2}}(\\frac{2}{p})$\" style=\"vertical-align: -16px\" width=\"281\" height=\"40\" >.</span> It means that <img src=\"//latex.artofproblemsolving.com/6/6/a/66a3ddd25624a6ca97fe8c4f7844a56b13a17252.png\" class=\"latex\" alt=\"$(-1)^{\\frac{p-1}{2}}$\" style=\"vertical-align: -4px\" width=\"60\" height=\"23\" > has to equal 1. On the other side, <img src=\"//latex.artofproblemsolving.com/4/2/d/42d61ee895745193ebd581a795f93b95ba9d0124.png\" class=\"latex\" alt=\"$(-1)^{\\frac{p-1}{2}}=1$\" style=\"vertical-align: -4px\" width=\"95\" height=\"23\" > implies that the equation <img src=\"//latex.artofproblemsolving.com/8/7/c/87cb7e4cfece283c5e43495f32b81503fa3f418c.png\" class=\"latex\" alt=\"$x^2 \\equiv -1$\" style=\"vertical-align: 0px\" width=\"64\" height=\"15\" > has a solution. Plugging <img src=\"//latex.artofproblemsolving.com/2/2/9/2299f794eaa96be01e88ca7142c734cf1e1de64b.png\" class=\"latex\" alt=\"$x=a-1$\" style=\"vertical-align: 0px\" width=\"74\" height=\"12\" > gives us <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/5/f/15fdef9e1f165cafdf732aeb91f72bc167a2398e.png\" class=\"latex\" alt=\"$a^2-2a+2 \\equiv 0$\" style=\"vertical-align: -1px\" width=\"122\" height=\"16\" >,</span> so it is clear that 1) or 2) or 3) holds.<br>\nSo, it's obvious that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/a/4/7a40c2d95f4803e736751510070509b538aea182.png\" class=\"latex\" alt=\"$p\|(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2)$\" style=\"vertical-align: -4px\" width=\"345\" height=\"19\" >.</span> But: <img src=\"//latex.artofproblemsolving.com/1/9/b/19b2153f72d5522912ec577bd6aa7c4b697a81a7.png\" class=\"latex\" alt=\"$(a^2-2)(a^2+2)(a^2-2a+2)(a^2+2a+2) = a^8 - 16$\" style=\"vertical-align: -4px\" width=\"413\" height=\"19\" ><br>\nConclusion follows.", "post_id": 560033, "post_number": 2, "post_time_unix": 1151497048, "post_time_utc": "2006-06-28 12:17:28 UTC", "thanks_received": 2, "user_id": 9092, "username": "TomciO" }, { "attachments": [], "content_bbcode": "Also note that this is more or less the only exception: everything that is an eigtht power $\\mod$ all big primes is of type $a^8$ or $16a^8$.", "content_html": "Also note that this is more or less the only exception: everything that is an eigtht power <img src=\"//latex.artofproblemsolving.com/2/e/c/2ece84ccd8b6babbc8fe4cc51c9f4815ccc5f108.png\" class=\"latex\" alt=\"$\\mod$\" width=\"47\" height=\"12\" > all big primes is of type <img src=\"//latex.artofproblemsolving.com/f/c/7/fc7e7b2127a759252f3fb45c1635fc8bc84a7b6c.png\" class=\"latex\" alt=\"$a^8$\" width=\"15\" height=\"15\" > or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/6/f/d6f1e4addb90c5024ded3c877806c5a0491800be.png\" class=\"latex\" alt=\"$16a^8$\" style=\"vertical-align: 0px\" width=\"33\" height=\"15\" >.</span>", "post_id": 560069, "post_number": 3, "post_time_unix": 1151501099, "post_time_utc": "2006-06-28 13:24:59 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" } ], "source": null }	Prove that $16$ is always an eighth power modulo $p$ for $p$ prime.	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicIdentities/CompletingtheSquare", "/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/Algebra/NumberTheory/Congruences/Congruent", "/Mathematics/Algebra/NumberTheory/Congruences/Mod", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/Congruences/QuadraticCongruenceEquation", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/NumberTheory/PrimeNumbers/MiscellaneousPrimes", "/Mathematics/Algebra/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/Algebra/NumberTheory/ReciprocityTheorems/QuadraticNonresidue", "/Mathematics/Algebra/NumberTheory/ReciprocityTheorems/QuadraticReciprocityTheorem", "/Mathematics/Algebra/NumberTheory/ReciprocityTheorems/QuadraticResidue", "/Mathematics/Algebra/NumberTheory/ReciprocityTheorems/ReciprocityTheorem", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/NumberTheory/Arithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/QuadraticCongruenceEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticReciprocityTheorem", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticResidue" ]	Factor a^8‑16 into four quadratics and use quadratic‑residue arguments to force one factor to vanish mod p, giving a^8≡16.	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aops_99212	an idea: if one knows the values of a polynomial of degree $n$ at $n+1$ points, then one knows exactly this polynomial, and thus all its derivatives. For example, if one knows $P(0)=a_0$, $P(\frac{1}{n})=a_1$,..,$P(\frac{n}{n})a_n$. Then $P(\frac{k}{n})=a_k=P(0)+(\frac{k}{n})P'(0)+..+\frac{(\frac{k}{n})^n}{n!}P^{(n)}(0)$. Hence, for $1 \leq k \leq n$, this gives $n$ equations with $n$ unkown values : $P'(0), P^{(2)}(0),.., P^{(n)}(0)$. This is a Cramer system (Vandermonde system), wich shows that one can find $\alpha_0,..,\alpha_n$, independant of the polynomial $P$, such that $P'(0)=\alpha_0 a_0 + \alpha_1 a_1 + .. + \alpha_n a_n$. One can do the same thing for $P^{(k)}(\frac{k}{n})$. Hence, this shows that the measure $m$ is a linear combination of the dirac measure $\delta_{\frac{k}{n}}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Fix $n\\geq 1$ . Show that there is a measure $m$ on $[0, 1]$ such that for every polynomial $P$ of degree at most $n$ we have that \r\n$\\int P dm = \\sum_{k=1} ^ n P^{(k)}(\\frac{k}{n})$\r\n\r\nwhere $P^{(k)} (\\frac{k}{n})$ is the $k$'th derivative of $P$ evaluated at $\\frac{k}{n}$ for different $k$ values.", "content_html": "Fix <img src=\"//latex.artofproblemsolving.com/9/c/c/9ccd696a5d0cbab8337ceb8a4cdfa202a54a6eae.png\" class=\"latex\" alt=\"$n\\geq 1$\" style=\"vertical-align: -2px\" width=\"43\" height=\"14\" > . Show that there is a measure <img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" > on <img src=\"//latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png\" class=\"latex\" alt=\"$[0, 1]$\" style=\"vertical-align: -5px\" width=\"34\" height=\"18\" > such that for every polynomial <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > of degree at most <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > we have that<br>\n<img src=\"//latex.artofproblemsolving.com/1/d/7/1d7f5dec1ea7e4ad882984f072dbb2a53ec12352.png\" class=\"latex\" alt=\"$\\int P dm = \\sum_{k=1} ^ n P^{(k)}(\\frac{k}{n})$\" style=\"vertical-align: -20px\" width=\"173\" height=\"48\" ><br>\n<br>\nwhere <img src=\"//latex.artofproblemsolving.com/8/5/7/85727c56598e17a5dd9352beb142ee64ee7e554b.png\" class=\"latex\" alt=\"$P^{(k)} (\\frac{k}{n})$\" style=\"vertical-align: -12px\" width=\"60\" height=\"37\" > is the <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" >'</span>th derivative of <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > evaluated at <img src=\"//latex.artofproblemsolving.com/f/a/b/fab87c80bc03be9d21f8b21e2eba0c2b3327e920.png\" class=\"latex\" alt=\"$\\frac{k}{n}$\" style=\"vertical-align: -12px\" width=\"13\" height=\"37\" > for different <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > values.", "post_id": 560019, "post_number": 1, "post_time_unix": 1151494403, "post_time_utc": "2006-06-28 11:33:23 UTC", "thanks_received": 2, "user_id": 902, "username": "eugene" }, { "attachments": [], "content_bbcode": "an idea:\r\nif one knows the values of a polynomial of degree $n$ at $n+1$ points, then one knows exactly this polynomial, and thus all its derivatives.\r\n\r\nFor example, if one knows $P(0)=a_0$, $P(\\frac{1}{n})=a_1$,..,$P(\\frac{n}{n})a_n$. Then\r\n$P(\\frac{k}{n})=a_k=P(0)+(\\frac{k}{n})P'(0)+..+\\frac{(\\frac{k}{n})^n}{n!}P^{(n)}(0)$.\r\nHence, for $1 \\leq k \\leq n$, this gives $n$ equations with $n$ unkown values : $P'(0), P^{(2)}(0),.., P^{(n)}(0)$. This is a Cramer system (Vandermonde system), wich shows that one can find $\\alpha_0,..,\\alpha_n$, independant of the polynomial $P$, such that $P'(0)=\\alpha_0 a_0 + \\alpha_1 a_1 + .. + \\alpha_n a_n$. \r\nOne can do the same thing for $P^{(k)}(\\frac{k}{n})$. Hence, this shows that the measure $m$ is a linear combination of the dirac measure $\\delta_{\\frac{k}{n}}$", "content_html": "an idea:<br>\nif one knows the values of a polynomial of degree <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > at <img src=\"//latex.artofproblemsolving.com/0/e/3/0e3efd9b14723a92c2ae891fe27780d5f8e2b215.png\" class=\"latex\" alt=\"$n+1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" > points, then one knows exactly this polynomial, and thus all its derivatives.<br>\n<br>\nFor example, if one knows <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/c/9/9c91f0932b4b5fc4bf9a4c55a6a5c0019a9a3d97.png\" class=\"latex\" alt=\"$P(0)=a_0$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/1/2/7126e525e597db8e2fb7336aec854bc33e6c2bd4.png\" class=\"latex\" alt=\"$P(\\frac{1}{n})=a_1$\" style=\"vertical-align: -12px\" width=\"82\" height=\"37\" >,</span>..<span style=\"white-space:nowrap;\">,<img src=\"//latex.artofproblemsolving.com/9/5/6/9567fc109122e94cb9d916abb91df38fe35287e3.png\" class=\"latex\" alt=\"$P(\\frac{n}{n})a_n$\" style=\"vertical-align: -12px\" width=\"60\" height=\"33\" >.</span> Then<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/5/b/d5b4eea71ed3c917b90caa59bcec8bba9742cd72.png\" class=\"latex\" alt=\"$P(\\frac{k}{n})=a_k=P(0)+(\\frac{k}{n})P'(0)+..+\\frac{(\\frac{k}{n})^n}{n!}P^{(n)}(0)$\" style=\"vertical-align: -12px\" width=\"389\" height=\"41\" >.</span><br>\nHence, for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/c/f5c86874f9ae7f3e00104cc067b560ccbe8ddc0b.png\" class=\"latex\" alt=\"$1 \\leq k \\leq n$\" style=\"vertical-align: -2px\" width=\"78\" height=\"15\" >,</span> this gives <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > equations with <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > unkown values : <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/9/1/b910b18351e519b881453f3d8761c62394b446b8.png\" class=\"latex\" alt=\"$P'(0), P^{(2)}(0),.., P^{(n)}(0)$\" style=\"vertical-align: -4px\" width=\"187\" height=\"20\" >.</span> This is a Cramer system (Vandermonde system), wich shows that one can find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/5/4/c54214e032cc30e082678e88c08f2e413017d897.png\" class=\"latex\" alt=\"$\\alpha_0,..,\\alpha_n$\" style=\"vertical-align: -3px\" width=\"65\" height=\"11\" >,</span> independant of the polynomial <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >,</span> such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/3/f/23f24d40aecbcd0a92bcb27179ae48e39fc6dc31.png\" class=\"latex\" alt=\"$P'(0)=\\alpha_0 a_0 + \\alpha_1 a_1 + .. + \\alpha_n a_n$\" style=\"vertical-align: -4px\" width=\"253\" height=\"18\" >.</span><br>\nOne can do the same thing for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/7/e/97ea8ac36dc306dc007a272a9229e5f7941bfb92.png\" class=\"latex\" alt=\"$P^{(k)}(\\frac{k}{n})$\" style=\"vertical-align: -12px\" width=\"60\" height=\"37\" >.</span> Hence, this shows that the measure <img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" > is a linear combination of the dirac measure <img src=\"//latex.artofproblemsolving.com/0/f/0/0f0839263ceb0973315e81f261704fb96edea0e8.png\" class=\"latex\" alt=\"$\\delta_{\\frac{k}{n}}$\" style=\"vertical-align: -8px\" width=\"17\" height=\"21\" >", "post_id": 560037, "post_number": 2, "post_time_unix": 1151497627, "post_time_utc": "2006-06-28 12:27:07 UTC", "thanks_received": 2, "user_id": 96, "username": "alekk" } ], "source": null }	Fix n \ge 1. Show that there is a measure m on [0,1] such that for every polynomial P of degree at most n, \[ \int_{0}^{1} P\,dm \;=\; \sum_{k=1}^{n} P^{(k)}\!\biggl(\frac{k}{n}\biggr), \] where $P^{(k)}\!\bigl(\tfrac{k}{n}\bigr)$ denotes the k-th derivative of P evaluated at $\tfrac{k}{n}$.	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/DefiniteIntegrals", "/Mathematics/CalculusandAnalysis/MeasureTheory/Measure", "/Mathematics/CalculusandAnalysis/MeasureTheory/PositiveMeasure", "/Mathematics/CalculusandAnalysis/MeasureTheory/RadonMeasure", "/Mathematics/CalculusandAnalysis/Polynomials/Polynomial", "/Mathematics/CalculusandAnalysis/Polynomials/PolynomialDegree", "/Mathematics/CalculusandAnalysis/Polynomials/UnivariatePolynomial" ]	Apply polynomial interpolation (Vandermonde system) to write each required derivative as a fixed linear combination of values at n+1 points, yielding a Dirac‐measure representation.	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aops_99227	It's not hard: the point is $z=x+iy$ and you have that there is an equation $z^2+az+b=0$. It's clear how to add complex numbers in a constructive way (just draw it when not clear ;) ), so it's enough to be able to multiply (and divide, but not needed here) and to take square roots (since with all that, we could solve quadratics as usual). Both can be done by Thales theorem, try it :)	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Suppose that x and y are real numbers such that [Q(x,y):Q] = 2.\r\nShow that (x,y) is constructible.", "content_html": "Suppose that x and y are real numbers such that [Q(x,y):Q] = 2.<br>\nShow that (x,y) is constructible.", "post_id": 560118, "post_number": 1, "post_time_unix": 1151503451, "post_time_utc": "2006-06-28 14:04:11 UTC", "thanks_received": 2, "user_id": 15730, "username": "Salvador" }, { "attachments": [], "content_bbcode": "It's not hard: the point is $z=x+iy$ and you have that there is an equation $z^2+az+b=0$.\r\nIt's clear how to add complex numbers in a constructive way (just draw it when not clear ;) ), so it's enough to be able to multiply (and divide, but not needed here) and to take square roots (since with all that, we could solve quadratics as usual).\r\nBoth can be done by Thales theorem, try it :)", "content_html": "It's not hard: the point is <img src=\"//latex.artofproblemsolving.com/2/a/5/2a514d19359869e53b6e4be1413344d26ff908ff.png\" class=\"latex\" alt=\"$z=x+iy$\" style=\"vertical-align: -3px\" width=\"81\" height=\"16\" > and you have that there is an equation <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/a/1/1a1dea4f06201f8bc079e2f1864d9a717e34f7c7.png\" class=\"latex\" alt=\"$z^2+az+b=0$\" style=\"vertical-align: -1px\" width=\"120\" height=\"16\" >.</span><br>\nIt's clear how to add complex numbers in a constructive way (just draw it when not clear <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" /> ), so it's enough to be able to multiply (and divide, but not needed here) and to take square roots (since with all that, we could solve quadratics as usual).<br>\nBoth can be done by Thales theorem, try it <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 560139, "post_number": 2, "post_time_unix": 1151504353, "post_time_utc": "2006-06-28 14:19:13 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "[quote=\"ZetaX\"]It's not hard: the point is $z=x+iy$ and you have that there is an equation $z^2+az+b=0$.\nIt's clear how to add complex numbers in a constructive way (just draw it when not clear ;) ), so it's enough to be able to multiply (and divide, but not needed here) and to take square roots (since with all that, we could solve quadratics as usual).\nBoth can be done by Thales theorem, try it :)[/quote]\r\n\r\nok, that's what I thought Zetax. The problem is that I thought that this argument could also be used with [Q(x,y):Q] = 3. Am I wrong?? \r\n\r\nThanks", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ZetaX wrote:</div>\n<div class=\"bbcode_quote_body\">It's not hard: the point is <img src=\"//latex.artofproblemsolving.com/2/a/5/2a514d19359869e53b6e4be1413344d26ff908ff.png\" class=\"latex\" alt=\"$z=x+iy$\" style=\"vertical-align: -3px\" width=\"81\" height=\"16\" > and you have that there is an equation <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/a/1/1a1dea4f06201f8bc079e2f1864d9a717e34f7c7.png\" class=\"latex\" alt=\"$z^2+az+b=0$\" style=\"vertical-align: -1px\" width=\"120\" height=\"16\" >.</span><br>\nIt's clear how to add complex numbers in a constructive way (just draw it when not clear <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" /> ), so it's enough to be able to multiply (and divide, but not needed here) and to take square roots (since with all that, we could solve quadratics as usual).<br>\nBoth can be done by Thales theorem, try it <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\nok, that's what I thought Zetax. The problem is that I thought that this argument could also be used with [Q(x,y):Q] = 3. Am I wrong??<br>\n<br>\nThanks", "post_id": 560155, "post_number": 3, "post_time_unix": 1151505519, "post_time_utc": "2006-06-28 14:38:39 UTC", "thanks_received": 2, "user_id": 15730, "username": "Salvador" }, { "attachments": [], "content_bbcode": "No, exactly those points with algebraic degree a power of $2$ can be constructed (when initially only given some rationals).\r\nThe problem for degree $3$ is that there is no way to construct cube roots (which you would need), whereas this is possible for square roots (needed above).\r\nOne way to see the impossibility:\r\nwhen you intersect line/circle with line/circle, then the degree of the equation of the intersection points will never be higher than $2$, with coefficients the already constructed points; thus adding the new point to our field will either remain it's dimension/degree (over $\\mathbb{Q}$) unchanged or will double it; but by that, we can only reach dimensions/degrees that are powers of $2$.", "content_html": "No, exactly those points with algebraic degree a power of <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > can be constructed (when initially only given some rationals).<br>\nThe problem for degree <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > is that there is no way to construct cube roots (which you would need), whereas this is possible for square roots (needed above).<br>\nOne way to see the impossibility:<br>\nwhen you intersect line/circle with line/circle, then the degree of the equation of the intersection points will never be higher than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >,</span> with coefficients the already constructed points; thus adding the new point to our field will either remain it's dimension/degree (over <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/8/f/18faac56812975e05744319e3c6ac1016643c11b.png\" class=\"latex\" alt=\"$\\mathbb{Q}$\" style=\"vertical-align: -3px\" width=\"13\" height=\"16\" >)</span> unchanged or will double it; but by that, we can only reach dimensions/degrees that are powers of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >.</span>", "post_id": 560173, "post_number": 4, "post_time_unix": 1151506272, "post_time_utc": "2006-06-28 14:51:12 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "[quote=\"ZetaX\"]No, exactly those points with algebraic degree a power of $2$ can be constructed (when initially only given some rationals).\nThe problem for degree $3$ is that there is no way to construct cube roots (which you would need), whereas this is possible for square roots (needed above).\nOne way to see the impossibility:\nwhen you intersect line/circle with line/circle, then the degree of the equation of the intersection points will never be higher than $2$, with coefficients the already constructed points; thus adding the new point to our field will either remain it's dimension/degree (over $\\mathbb{Q}$) unchanged or will double it; but by that, we can only reach dimensions/degrees that are powers of $2$.[/quote]\r\n\r\nAre you saying that a point (x,y) is constructible if and only if [Q(x,y):Q] = power of 2 ??", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ZetaX wrote:</div>\n<div class=\"bbcode_quote_body\">No, exactly those points with algebraic degree a power of <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > can be constructed (when initially only given some rationals).<br>\nThe problem for degree <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > is that there is no way to construct cube roots (which you would need), whereas this is possible for square roots (needed above).<br>\nOne way to see the impossibility:<br>\nwhen you intersect line/circle with line/circle, then the degree of the equation of the intersection points will never be higher than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >,</span> with coefficients the already constructed points; thus adding the new point to our field will either remain it's dimension/degree (over <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/8/f/18faac56812975e05744319e3c6ac1016643c11b.png\" class=\"latex\" alt=\"$\\mathbb{Q}$\" style=\"vertical-align: -3px\" width=\"13\" height=\"16\" >)</span> unchanged or will double it; but by that, we can only reach dimensions/degrees that are powers of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" >.</span></div>\n</div>\n<br>\nAre you saying that a point (x,y) is constructible if and only if [Q(x,y):Q] = power of 2 ??", "post_id": 560186, "post_number": 5, "post_time_unix": 1151506928, "post_time_utc": "2006-06-28 15:02:08 UTC", "thanks_received": 2, "user_id": 15730, "username": "Salvador" }, { "attachments": [], "content_bbcode": "Yes, the only if part is sketched above, the other direction uses a bit more theory.\r\n\r\nEdit: wrong direction ^^ (if <-> only if)", "content_html": "Yes, the only if part is sketched above, the other direction uses a bit more theory.<br>\n<br>\nEdit: wrong direction ^^ (if <-> only if)", "post_id": 560206, "post_number": 6, "post_time_unix": 1151507859, "post_time_utc": "2006-06-28 15:17:39 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "[quote=\"ZetaX\"]Yes, the if part is sketched above, the other direction uses a bit more theory.[/quote]\r\n\r\nThanks ZetaX :)", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ZetaX wrote:</div>\n<div class=\"bbcode_quote_body\">Yes, the if part is sketched above, the other direction uses a bit more theory.</div>\n</div>\n<br>\nThanks ZetaX <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 560210, "post_number": 7, "post_time_unix": 1151507974, "post_time_utc": "2006-06-28 15:19:34 UTC", "thanks_received": 2, "user_id": 15730, "username": "Salvador" } ], "source": null }	Suppose that $x$ and $y$ are real numbers such that $[\mathbb{Q}(x,y):\mathbb{Q}] = 2$. Show that $(x,y)$ is constructible.	[ "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/FieldTheory/AlgebraicExtension", "/Mathematics/Algebra/FieldTheory/ExtensionField", "/Mathematics/Algebra/FieldTheory/ExtensionFieldDegree", "/Mathematics/Algebra/FieldTheory/FiniteExtension", "/Mathematics/Algebra/FieldTheory/QuadraticField", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/Geometry/GeometricConstruction/EuclideanTools", "/Mathematics/Geometry/GeometricConstruction/SteinerConstruction" ]	Treat (x,y) as the complex number z = x+iy which satisfies a quadratic over Q; constructing square roots yields the point.	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aops_992299	Never mind. I choose the one that has the insciption "Exactly one of these two statements is true". Read below (click the spoiler) to find out why. Here are some more useless thoughts: Who cares about the money? It's not going to make you much happier. As for the [i]real[/i] answer (or whatever my stupid brain tells me what the answer is) [hide]The box that claims to have the poison has the treasure. But since I don't care about treasure, I'm choosing the other box, anyway. Here's the explanation. Let Box 1: the one that claims it has the poison Box 2: the one that has "Exactly one of these statements is true" CASE 1: Box 2 is lying So, either both boxes lie or both boxes say the truth. Since, by assumption, we must have Box 2 to lie, Box 1 must lie, too. So Box 1 has the treasure. CASE 2: Box 2 is telling the truth. So, Box 1 must lie. So it (Box 1) has the treasure.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "You see two boxes. One contains fabulous treasure, and the other contains instantly fatal and uncurable poison gas. On each box is an inscription:\r\n\r\nBox 1: This box contains poison\r\nBox 2: One of these two statements is true.\r\n\r\nWhich box do you open?\r\n\r\nEDIT: Hide your answers.", "content_html": "You see two boxes. One contains fabulous treasure, and the other contains instantly fatal and uncurable poison gas. On each box is an inscription:<br>\n<br>\nBox 1: This box contains poison<br>\nBox 2: One of these two statements is true.<br>\n<br>\nWhich box do you open?<br>\n<br>\nEDIT: Hide your answers.", "post_id": 4404348, "post_number": 1, "post_time_unix": 1186079846, "post_time_utc": "2007-08-02 18:37:26 UTC", "thanks_received": 1, "user_id": 26531, "username": "xscapezaer" }, { "attachments": [], "content_bbcode": "[hide]I'm going to take the poison box! I don't care if I die, since it will be instantly fatal.\nEither that, or don't pick at all.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">I'm going to take the poison box! I don't care if I die, since it will be instantly fatal.<br>\nEither that, or don't pick at all.</div>", "post_id": 4404349, "post_number": 2, "post_time_unix": 1186082648, "post_time_utc": "2007-08-02 19:24:08 UTC", "thanks_received": 1, "user_id": 14262, "username": "vishalarul" }, { "attachments": [], "content_bbcode": "OMFG that's not the #$ %@! point of this riddle, #$%@! it!!!\r\n\r\nThe problem is now changed to reflect vishalarul's comments. \r\n\r\n(1) \"Instantly fatal\" is now replaced with \"Excruciatingly painful, but irreversibly fatal.\" \r\n\r\n(2) Assume now that you are in a airtight chamber with only one entrance, where said entrance seals as soon as you enter.\r\n\r\n(3) You cannot escape from the chamber by any other way than opening the correct box. Opening the right box will open the chamber door.\r\n\r\n(4) If you don't choose a box withing 5 minutes, whatever box containing the poison opens automatically, and the entrance to this chamber stays tightly shut.\r\n\r\nEDIT:(5) You don't have anything with you. You cannot blow the door open, you don't have antidotes, absolutely nothing!!\r\n\r\nThe whole point of this is that if you open the right chest, you are very happy. If you don't, you will end up very [i]sad[/i] cough, cough", "content_html": "OMFG that's not the <span style=\"white-space:nowrap;\">#<span class=\"aopscode-error aopscode-latex-error\">$ %@! point of this riddle, #$</span>%</span>@! it!!!<br>\n<br>\nThe problem is now changed to reflect vishalarul's comments.<br>\n<br>\n(1) "Instantly fatal" is now replaced with "Excruciatingly painful, but irreversibly fatal."<br>\n<br>\n(2) Assume now that you are in a airtight chamber with only one entrance, where said entrance seals as soon as you enter.<br>\n<br>\n(3) You cannot escape from the chamber by any other way than opening the correct box. Opening the right box will open the chamber door.<br>\n<br>\n(4) If you don't choose a box withing 5 minutes, whatever box containing the poison opens automatically, and the entrance to this chamber stays tightly shut.<br>\n<br>\nEDIT:(5) You don't have anything with you. You cannot blow the door open, you don't have antidotes, absolutely nothing!!<br>\n<br>\nThe whole point of this is that if you open the right chest, you are very happy. If you don't, you will end up very <i>sad</i> cough, cough", "post_id": 4404350, "post_number": 3, "post_time_unix": 1186245666, "post_time_utc": "2007-08-04 16:41:06 UTC", "thanks_received": 1, "user_id": 26531, "username": "xscapezaer" }, { "attachments": [], "content_bbcode": "Never mind. I choose the one that has the insciption \"Exactly one of these two statements is true\". Read below (click the spoiler) to find out why.\r\n\r\nHere are some more useless thoughts:\r\nWho cares about the money? It's not going to make you much happier. \r\n\r\nAs for the [i]real[/i] answer (or whatever my stupid brain tells me what the answer is)\r\n[hide]The box that claims to have the poison has the treasure. But since I don't care about treasure, I'm choosing the other box, anyway.\n\nHere's the explanation.\nLet\nBox 1: the one that claims it has the poison\nBox 2: the one that has \"Exactly one of these statements is true\"\n\nCASE 1: Box 2 is lying\nSo, either both boxes lie or both boxes say the truth. Since, by assumption, we must have Box 2 to lie, Box 1 must lie, too. So Box 1 has the treasure.\n\nCASE 2: Box 2 is telling the truth.\nSo, Box 1 must lie. So it (Box 1) has the treasure.[/hide]", "content_html": "Never mind. I choose the one that has the insciption "Exactly one of these two statements is true". Read below (click the spoiler) to find out why.<br>\n<br>\nHere are some more useless thoughts:<br>\nWho cares about the money? It's not going to make you much happier.<br>\n<br>\nAs for the <i>real</i> answer (or whatever my stupid brain tells me what the answer is)<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">The box that claims to have the poison has the treasure. But since I don't care about treasure, I'm choosing the other box, anyway.<br>\n<br>\nHere's the explanation.<br>\nLet<br>\nBox 1: the one that claims it has the poison<br>\nBox 2: the one that has "Exactly one of these statements is true"<br>\n<br>\nCASE 1: Box 2 is lying<br>\nSo, either both boxes lie or both boxes say the truth. Since, by assumption, we must have Box 2 to lie, Box 1 must lie, too. So Box 1 has the treasure.<br>\n<br>\nCASE 2: Box 2 is telling the truth.<br>\nSo, Box 1 must lie. So it (Box 1) has the treasure.</div>", "post_id": 4404351, "post_number": 4, "post_time_unix": 1186289549, "post_time_utc": "2007-08-05 04:52:29 UTC", "thanks_received": 2, "user_id": 14262, "username": "vishalarul" }, { "attachments": [], "content_bbcode": "Box 1 must contain the treasure because if box 2 did, then the statement on box 1 would be true, so the statement on box 2 would have to be true iff it was false, or at least that's what logic tells us. However, we can't trust whoever inscribed these not to write a paradox, right? It must be a trick to convince me to open box 1. opens box 2 cough cough gasp dies. Oh, the irony of it all!", "content_html": "Box 1 must contain the treasure because if box 2 did, then the statement on box 1 would be true, so the statement on box 2 would have to be true iff it was false, or at least that's what logic tells us. However, we can't trust whoever inscribed these not to write a paradox, right? It must be a trick to convince me to open box 1. opens box 2 cough cough gasp dies. Oh, the irony of it all!", "post_id": 4404352, "post_number": 5, "post_time_unix": 1194060720, "post_time_utc": "2007-11-03 03:32:00 UTC", "thanks_received": 2, "user_id": 34665, "username": "nenneM xelA" } ], "source": null }	You see two boxes. One contains fabulous treasure, and the other contains instantly fatal and incurable poison gas. On each box is an inscription: Box 1: This box contains poison. Box 2: One of these two statements is true. Which box do you open?	[ "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Assume the ‘exactly one statement is true’ claim and examine both truth possibilities, which forces the other box’s statement to be false, so it contains the treasure.	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aops_992312	Answer: The curvature is $ \frac {(x'y'' \minus{} x''y')}{(x'^2 \plus{} y'^2)}^{\displaystyle\frac {3}{2}}$. So for convexity we require $ x'y'' \ge x''y'$. Putting $ k \equal{} \frac {b}{a}, c \equal{} \cos{\theta}, s \equal{} \sin{\theta}$, we have $ \frac {x}{a} \equal{} c \minus{} k c^{2}, \frac {x'}{a} \equal{} \minus{} s \plus{} 2kcs, \frac {x''}{a} \equal{} \minus{} c \minus{} 2ks^{2} \plus{} 2kc^{2}$ and $ \frac {y}{a} \equal{} s \minus{} ksc, \frac {y'}{a} \equal{} c \minus{} kc^2 \plus{} k s^2, \frac {y''}{a} \equal{} \minus{} s \plus{} 4ksc$. Thus the condition becomes after a little cancellation, $ 1 \plus{} 2k^{2} \minus{} 3kc \ge 0$. $ c$ takes values in the range $ \minus{}1$ to $ 1$, so for the condition to be true for all points of the curve we require $ 1 \plus{} 2k^{2} \minus{} 3k \ge 0$. But $ 1 \plus{} 2k2 \minus{} 3k \equal{} (2k \minus{} 1)(k \minus{} 1)$. We are given that $ k < 1$, so we must have$ k < \frac {1}{2}$. For small $ k$, the curve is approximately a circle centred on the origin. As k increases it develops a flattening near $ x \equal{} a, y \equal{} 0$. For $ k > \frac {1}{2}$, this becomes a dimple in the surface, so that convexity is broken. For $ k \equal{} 1$, the depression in the surface extends as far as the centre (the origin). For$ k > 0$, the curve intersects itself at the origin so that it comprises two ovals one inside the other. 10th Putnam 1950 Problem 1	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "$ a$ and $ b$ are positive reals and $ a > b$. Let $ C$ be the plane curve $ r \\equal{} a \\minus{} b \\cos{\\theta}$. For what values of $ \\frac {b}{a}$ is $ C$ convex?\r\n\r\nLolz. Have fun.", "content_html": "<img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > are positive reals and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/2/d12ade60bc6333446c4df04dde6867d8333f8e80.png\" class=\"latex\" alt=\"$ a > b$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >.</span> Let <img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" > be the plane curve <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/a/9/2a9ce09618d121dfa842b5039b09f608b5fa48dc.png\" class=\"latex\" alt=\"$ r = a - b \\cos{\\theta}$\" width=\"111\" height=\"13\" >.</span> For what values of <img src=\"//latex.artofproblemsolving.com/5/a/b/5abb59dee4abb8ac9596e8e5ddaa654d6a384066.png\" class=\"latex\" alt=\"$ \\frac {b}{a}$\" style=\"vertical-align: -12px\" width=\"12\" height=\"37\" > is <img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" > convex?<br>\n<br>\nLolz. Have fun.", "post_id": 4404404, "post_number": 1, "post_time_unix": 1194144053, "post_time_utc": "2007-11-04 02:40:53 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" }, { "attachments": [], "content_bbcode": "Answer:\r\n\r\nThe curvature is $ \\frac {(x'y'' \\minus{} x''y')}{(x'^2 \\plus{} y'^2)}^{\\displaystyle\\frac {3}{2}}$. So for convexity we require $ x'y'' \\ge x''y'$. Putting $ k \\equal{} \\frac {b}{a}, c \\equal{} \\cos{\\theta}, s \\equal{} \\sin{\\theta}$, we have $ \\frac {x}{a} \\equal{} c \\minus{} k c^{2}, \\frac {x'}{a} \\equal{} \\minus{} s \\plus{} 2kcs, \\frac {x''}{a} \\equal{} \\minus{} c \\minus{} 2ks^{2} \\plus{} 2kc^{2}$ and $ \\frac {y}{a} \\equal{} s \\minus{} ksc, \\frac {y'}{a} \\equal{} c \\minus{} kc^2 \\plus{} k s^2, \\frac {y''}{a} \\equal{} \\minus{} s \\plus{} 4ksc$. Thus the condition becomes after a little cancellation, $ 1 \\plus{} 2k^{2} \\minus{} 3kc \\ge 0$. $ c$ takes values in the range $ \\minus{}1$ to $ 1$, so for the condition to be true for all points of the curve we require $ 1 \\plus{} 2k^{2} \\minus{} 3k \\ge 0$. But $ 1 \\plus{} 2k2 \\minus{} 3k \\equal{} (2k \\minus{} 1)(k \\minus{} 1)$. We are given that $ k < 1$, so we must have$ k < \\frac {1}{2}$.\r\n\r\nFor small $ k$, the curve is approximately a circle centred on the origin. As k increases it develops a flattening near $ x \\equal{} a, y \\equal{} 0$. For $ k > \\frac {1}{2}$, this becomes a dimple in the surface, so that convexity is broken. For $ k \\equal{} 1$, the depression in the surface extends as far as the centre (the origin). For$ k > 0$, the curve intersects itself at the origin so that it comprises two ovals one inside the other.\r\n\r\n10th Putnam 1950 Problem 1", "content_html": "Answer:<br>\n<br>\nThe curvature is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/c/6cc05971b95ef6def44c99ba163459ca6e665426.png\" class=\"latex\" alt=\"$ \\frac {(x'y'' - x''y')}{(x'^2 + y'^2)}^{\\displaystyle\\frac {3}{2}}$\" style=\"vertical-align: -17px\" width=\"118\" height=\"63\" >.</span> So for convexity we require <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/6/f/66ff5730e2989f00e002ce578a7dfa129b46484f.png\" class=\"latex\" alt=\"$ x'y'' \\ge x''y'$\" style=\"vertical-align: -3px\" width=\"89\" height=\"17\" >.</span> Putting <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/5/8/05828db18c6aac65329ef2ebdc7c6c5d85c9dba4.png\" class=\"latex\" alt=\"$ k = \\frac {b}{a}, c = \\cos{\\theta}, s = \\sin{\\theta}$\" style=\"vertical-align: -12px\" width=\"199\" height=\"37\" >,</span> we have <img src=\"//latex.artofproblemsolving.com/c/a/f/caf66468cbe7a786c960307736c1d2c310a7b134.png\" class=\"latex\" alt=\"$ \\frac {x}{a} = c - k c^{2}, \\frac {x'}{a} = - s + 2kcs, \\frac {x''}{a} = - c - 2ks^{2} + 2kc^{2}$\" style=\"vertical-align: -12px\" width=\"415\" height=\"38\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/1/3016dde0dbd62d2102a1f9635d919a14bd01870a.png\" class=\"latex\" alt=\"$ \\frac {y}{a} = s - ksc, \\frac {y'}{a} = c - kc^2 + k s^2, \\frac {y''}{a} = - s + 4ksc$\" style=\"vertical-align: -12px\" width=\"384\" height=\"38\" >.</span> Thus the condition becomes after a little cancellation, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/4/e/c4e5d8f9cb09d98efad652ca8c2b531da8c14d76.png\" class=\"latex\" alt=\"$ 1 + 2k^{2} - 3kc \\ge 0$\" style=\"vertical-align: -2px\" width=\"140\" height=\"17\" >.</span> <img src=\"//latex.artofproblemsolving.com/b/1/4/b144c3decf04b3f7a907c07e4f369f1e02bb9adc.png\" class=\"latex\" alt=\"$ c$\" width=\"8\" height=\"8\" > takes values in the range <img src=\"//latex.artofproblemsolving.com/8/3/7/837d12377ff70746d353a495e0ac3cefb465208e.png\" class=\"latex\" alt=\"$ -1$\" style=\"vertical-align: 0px\" width=\"22\" height=\"12\" > to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/9/0/39064bdd89b3dfa0626ca59d843d926ea072830b.png\" class=\"latex\" alt=\"$ 1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> so for the condition to be true for all points of the curve we require <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/c/30c24a6bd61e3615a2ae0dc30ad78f31d2081b4e.png\" class=\"latex\" alt=\"$ 1 + 2k^{2} - 3k \\ge 0$\" style=\"vertical-align: -2px\" width=\"132\" height=\"17\" >.</span> But <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/7/4/274b3d06a21304abd59744f9ebb71ed42d722dea.png\" class=\"latex\" alt=\"$ 1 + 2k2 - 3k = (2k - 1)(k - 1)$\" style=\"vertical-align: -4px\" width=\"243\" height=\"18\" >.</span> We are given that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/d/2/0d25da72f44cd27ce927888a0c41c3617ae074e2.png\" class=\"latex\" alt=\"$ k < 1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >,</span> so we must hav<span style=\"white-space:nowrap;\">e<img src=\"//latex.artofproblemsolving.com/3/e/b/3eb38cbb5f0349a18289ad603cf9e7e6d84b88b5.png\" class=\"latex\" alt=\"$ k < \\frac {1}{2}$\" style=\"vertical-align: -12px\" width=\"46\" height=\"37\" >.</span><br>\n<br>\nFor small <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/0/d10af8b3fc779f307fe5c87020e433b00a41801d.png\" class=\"latex\" alt=\"$ k$\" width=\"9\" height=\"12\" >,</span> the curve is approximately a circle centred on the origin. As k increases it develops a flattening near <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/5/3/253c89a8d089bd61606926af11150bb1f9db1013.png\" class=\"latex\" alt=\"$ x = a, y = 0$\" style=\"vertical-align: -3px\" width=\"94\" height=\"16\" >.</span> For <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/e/c/4ec7d23ba4a8122876400d29013fcf1b95d7bfad.png\" class=\"latex\" alt=\"$ k > \\frac {1}{2}$\" style=\"vertical-align: -12px\" width=\"46\" height=\"37\" >,</span> this becomes a dimple in the surface, so that convexity is broken. For <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/6/4/364b81edee0376ea9b03316bca07e558d5711557.png\" class=\"latex\" alt=\"$ k = 1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >,</span> the depression in the surface extends as far as the centre (the origin). Fo<span style=\"white-space:nowrap;\">r<img src=\"//latex.artofproblemsolving.com/9/5/1/951169f9cd526641afa641e50051c0d3242ed009.png\" class=\"latex\" alt=\"$ k > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >,</span> the curve intersects itself at the origin so that it comprises two ovals one inside the other.<br>\n<br>\n10th Putnam 1950 Problem 1", "post_id": 4404405, "post_number": 2, "post_time_unix": 1194311721, "post_time_utc": "2007-11-06 01:15:21 UTC", "thanks_received": 2, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "Oh don't you like the copy and paste feature?", "content_html": "Oh don't you like the copy and paste feature?", "post_id": 4404406, "post_number": 3, "post_time_unix": 1194472290, "post_time_utc": "2007-11-07 21:51:30 UTC", "thanks_received": 2, "user_id": 27145, "username": "hunter34" } ], "source": null }	Let a and b be positive real numbers with a > b. Let C be the plane curve given in polar coordinates by r = a − b cos θ. For which values of b/a is C convex?	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/DifferentialCalculus_duplicate", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus", "/Mathematics/CalculusandAnalysis/DifferentialGeometry/DifferentialGeometryofCurves/Curvature", "/Mathematics/Geometry/Curves/PlaneCurves/PolarCurves", "/Mathematics/Geometry/DifferentialGeometry/DifferentialGeometryofCurves/Curvature", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves/GeneralPlaneCurves", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves/PolarCurves" ]	Use the sign of curvature (x' y'' – x'' y') ≥ 0 to obtain an inequality in k = b/a that must hold for all θ.	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aops_99232	[quote="grobber"]Let $(A_i)_{i\in I}$ ($I$ is an index set) be a collection of real sets, all of measure zero, totally ordered with respect to inclusion, such that their union is measurable. Is it true that $\bigcup_{i\in I}A_i$ has measure zero?[/quote] I have an idea, but I'm not sure whether it can be called a proof... If the answer to your question is yes, that is, $\bigcup_{i\in I}A_i$ must have measure zero, then: Consider $P=\{A\subseteq{R}\|m(A)=0\}$. Every totally ordered subset $(A_i)_{i\in{I}}$ of P has an upper bound $\bigcup_{i\in{I}}A_i$, which by your assertion is also in $P$. Then by Zorn's lemma, there is an element $B\in{P}$ such that no element in $P$ is strictly larger than $B$. But that is impossible, since obviously $B\neq{R}$, we can find $x\in{R\setminus{B}}$, and $B\cup{\{x\}}$ still have measure zero and is strictly larger than $B$, which is a contradiction. So the measure of $\bigcup_{i\in{I}}A_i$ may have positive measure, though I have not worked out an example yet...	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $(A_i)_{i\\in I}$ ($I$ is an index set) be a collection of real sets, all of measure zero, totally ordered with respect to inclusion, such that their union is measurable. \r\n\r\nIs it true that $\\bigcup_{i\\in I}A_i$ has measure zero?", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/6/e/6/6e6bc36705b067d4df27a4696bb66769b9608e5d.png\" class=\"latex\" alt=\"$(A_i)_{i\\in I}$\" style=\"vertical-align: -4px\" width=\"53\" height=\"18\" > <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/0/2/7/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" ></span> is an index set) be a collection of real sets, all of measure zero, totally ordered with respect to inclusion, such that their union is measurable.<br>\n<br>\nIs it true that <img src=\"//latex.artofproblemsolving.com/9/5/9/959291d3796de4cdb6036f9ea6dbc7df9d7a734a.png\" class=\"latex\" alt=\"$\\bigcup_{i\\in I}A_i$\" style=\"vertical-align: -21px\" width=\"41\" height=\"38\" > has measure zero?", "post_id": 560130, "post_number": 1, "post_time_unix": 1151504049, "post_time_utc": "2006-06-28 14:14:09 UTC", "thanks_received": 2, "user_id": 26, "username": "grobber" }, { "attachments": [], "content_bbcode": "[quote=\"grobber\"]Let $(A_i)_{i\\in I}$ ($I$ is an index set) be a collection of real sets, all of measure zero, totally ordered with respect to inclusion, such that their union is measurable. \n\nIs it true that $\\bigcup_{i\\in I}A_i$ has measure zero?[/quote]\r\n\r\nI have an idea, but I'm not sure whether it can be called a proof...\r\n\r\nIf the answer to your question is yes, that is, $\\bigcup_{i\\in I}A_i$ must have measure zero, then:\r\nConsider $P=\\{A\\subseteq{R}\|m(A)=0\\}$.\r\nEvery totally ordered subset $(A_i)_{i\\in{I}}$ of P has an upper bound $\\bigcup_{i\\in{I}}A_i$, which by your assertion is also in $P$.\r\nThen by Zorn's lemma, there is an element $B\\in{P}$ such that no element in $P$ is strictly larger than $B$.\r\nBut that is impossible, since obviously $B\\neq{R}$, we can find $x\\in{R\\setminus{B}}$, and $B\\cup{\\{x\\}}$ still have measure zero and is strictly larger than $B$, which is a contradiction.\r\nSo the measure of $\\bigcup_{i\\in{I}}A_i$ may have positive measure, though I have not worked out an example yet...", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">grobber wrote:</div>\n<div class=\"bbcode_quote_body\">Let <img src=\"//latex.artofproblemsolving.com/6/e/6/6e6bc36705b067d4df27a4696bb66769b9608e5d.png\" class=\"latex\" alt=\"$(A_i)_{i\\in I}$\" style=\"vertical-align: -4px\" width=\"53\" height=\"18\" > <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/0/2/7/027f4a11d6090f9eac0ce2488df6384dad1263ea.png\" class=\"latex\" alt=\"$I$\" width=\"9\" height=\"12\" ></span> is an index set) be a collection of real sets, all of measure zero, totally ordered with respect to inclusion, such that their union is measurable.<br>\n<br>\nIs it true that <img src=\"//latex.artofproblemsolving.com/9/5/9/959291d3796de4cdb6036f9ea6dbc7df9d7a734a.png\" class=\"latex\" alt=\"$\\bigcup_{i\\in I}A_i$\" style=\"vertical-align: -21px\" width=\"41\" height=\"38\" > has measure zero?</div>\n</div>\n<br>\nI have an idea, but I'm not sure whether it can be called a proof...<br>\n<br>\nIf the answer to your question is yes, that is, <img src=\"//latex.artofproblemsolving.com/9/5/9/959291d3796de4cdb6036f9ea6dbc7df9d7a734a.png\" class=\"latex\" alt=\"$\\bigcup_{i\\in I}A_i$\" style=\"vertical-align: -21px\" width=\"41\" height=\"38\" > must have measure zero, then:<br>\nConsider <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/8/7/c87b7cbd9bf0658c706bb4706ee32afe0be79e26.png\" class=\"latex\" alt=\"$P=\\{A\\subseteq{R}\|m(A)=0\\}$\" style=\"vertical-align: -4px\" width=\"189\" height=\"18\" >.</span><br>\nEvery totally ordered subset <img src=\"//latex.artofproblemsolving.com/9/6/8/968361969a178bee0aff518fa3d94842cb8e0d29.png\" class=\"latex\" alt=\"$(A_i)_{i\\in{I}}$\" style=\"vertical-align: -4px\" width=\"53\" height=\"18\" > of P has an upper bound <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/5/fa5ec6ed6879e54dc85ce3d5e404ecd95cd1c230.png\" class=\"latex\" alt=\"$\\bigcup_{i\\in{I}}A_i$\" style=\"vertical-align: -21px\" width=\"41\" height=\"38\" >,</span> which by your assertion is also in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >.</span><br>\nThen by Zorn's lemma, there is an element <img src=\"//latex.artofproblemsolving.com/2/8/3/28378e0c5d7cd8c484512842c6f40fbc26db5e76.png\" class=\"latex\" alt=\"$B\\in{P}$\" style=\"vertical-align: -1px\" width=\"51\" height=\"13\" > such that no element in <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > is strictly larger than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" >.</span><br>\nBut that is impossible, since obviously <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/6/2/162fe12d3f93f082895e3675ef2910f4667d871f.png\" class=\"latex\" alt=\"$B\\neq{R}$\" style=\"vertical-align: -4px\" width=\"53\" height=\"17\" >,</span> we can find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/f/e/0fe3e34f6cbd05b10d15315de6396f103cd965f0.png\" class=\"latex\" alt=\"$x\\in{R\\setminus{B}}$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" >,</span> and <img src=\"//latex.artofproblemsolving.com/b/4/a/b4a06ab22bd6d94a172876949aa0d41a6d30838b.png\" class=\"latex\" alt=\"$B\\cup{\\{x\\}}$\" style=\"vertical-align: -4px\" width=\"62\" height=\"18\" > still have measure zero and is strictly larger than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" >,</span> which is a contradiction.<br>\nSo the measure of <img src=\"//latex.artofproblemsolving.com/f/a/5/fa5ec6ed6879e54dc85ce3d5e404ecd95cd1c230.png\" class=\"latex\" alt=\"$\\bigcup_{i\\in{I}}A_i$\" style=\"vertical-align: -21px\" width=\"41\" height=\"38\" > may have positive measure, though I have not worked out an example yet...", "post_id": 560977, "post_number": 2, "post_time_unix": 1151549483, "post_time_utc": "2006-06-29 02:51:23 UTC", "thanks_received": 1, "user_id": 20595, "username": "AlvaroRecoba" } ], "source": null }	Let $(A_i)_{i\in I}$ be a collection of measurable subsets of $\mathbb{R}$, each of measure zero, totally ordered by inclusion, and such that $\bigcup_{i\in I}A_i$ is measurable. Is it true that $\bigcup_{i\in I}A_i$ has measure zero?	[ "/Mathematics/CalculusandAnalysis/MeasureTheory/LebesgueMeasure", "/Mathematics/CalculusandAnalysis/MeasureTheory/MeasurableSet", "/Mathematics/CalculusandAnalysis/MeasureTheory/Measure", "/Mathematics/CalculusandAnalysis/MeasureTheory/MeasureZero", "/Mathematics/FoundationsofMathematics/MathematicalProblems/UnsolvedProblems", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/AxiomofChoice", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/WellOrderingPrinciple", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/Zermelo-FraenkelAxioms", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/Zermelo-FraenkelSetTheory", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/Chain", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/OrderedSet", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/PartialOrder", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/PartiallyOrderedSet", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/TotalOrder", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/TotallyOrderedSet", "/Mathematics/FoundationsofMathematics/SetTheory/PartialOrders/ZornsLemma", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/IndexSet", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/Set", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/Subset", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/Superset" ]	Assuming every chain of null sets has null union yields a maximal null set, but any null set can be enlarged by a point while remaining null, giving a contradiction.	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aops_992673	I called it the reverse hockey stick indentity. Because it's the reflection of the hockey stick identy on pascals' triangle. alternative ly, just use n choose k = n choose n-k to get the hockey stick identity	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove that\r\n\r\n$ \\binom{n}{r} \\equal{} \\binom{n\\minus{}1}{r} \\plus{} \\binom{n\\minus{}2}{r\\minus{}1} \\plus{} \\binom{n \\minus{} 3}{r \\minus{} 2} \\plus{} ... \\plus{} \\binom{n\\minus{}r}{1}$.\r\n\r\nSource: own\r\n\r\nYay, it's math. Haven't posted math in a while. :P . This one isn't very hard, though. Is there any specific name for this identity?", "content_html": "Prove that<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/5/1/c5106ecbce98845ace50e69cc8a0cafdf3972670.png\" class=\"latex\" alt=\"$ \\binom{n}{r} = \\binom{n-1}{r} + \\binom{n-2}{r-1} + \\binom{n - 3}{r - 2} + ... + \\binom{n-r}{1}$\" style=\"vertical-align: -22px\" width=\"444\" height=\"53\" >.</span><br>\n<br>\nSource: own<br>\n<br>\nYay, it's math. Haven't posted math in a while. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /> . This one isn't very hard, though. Is there any specific name for this identity?", "post_id": 4405138, "post_number": 1, "post_time_unix": 1221349433, "post_time_utc": "2008-09-13 23:43:53 UTC", "thanks_received": 1, "user_id": 30252, "username": "Nerd_of_the_Ages" }, { "attachments": [], "content_bbcode": "I called it the reverse hockey stick indentity.\r\n\r\nBecause it's the reflection of the hockey stick identy on pascals' triangle.\r\n\r\nalternative ly, just use n choose k = n choose n-k to get the hockey stick identity", "content_html": "I called it the reverse hockey stick indentity.<br>\n<br>\nBecause it's the reflection of the hockey stick identy on pascals' triangle.<br>\n<br>\nalternative ly, just use n choose k = n choose n-k to get the hockey stick identity", "post_id": 4405139, "post_number": 2, "post_time_unix": 1221351301, "post_time_utc": "2008-09-14 00:15:01 UTC", "thanks_received": 1, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "Hehe, good point. Lol, I'm all like, hm my proof using repeated pascal's identity reminds me of the proof of the hockey stick identity. But i didn't realize it WAS the hockey-stick identity.", "content_html": "Hehe, good point. Lol, I'm all like, hm my proof using repeated pascal's identity reminds me of the proof of the hockey stick identity. But i didn't realize it WAS the hockey-stick identity.", "post_id": 4405140, "post_number": 3, "post_time_unix": 1221352553, "post_time_utc": "2008-09-14 00:35:53 UTC", "thanks_received": 1, "user_id": 30252, "username": "Nerd_of_the_Ages" }, { "attachments": [], "content_bbcode": "I found this by search the hockey stick identity on google lol", "content_html": "I found this by search the hockey stick identity on google lol", "post_id": 34796541, "post_number": 4, "post_time_unix": 1746924543, "post_time_utc": "2025-05-11 00:49:03 UTC", "thanks_received": 0, "user_id": 700379, "username": "orangebear" } ], "source": null }	Prove that \[ \binom{n}{r}=\binom{n-1}{r}+\binom{n-2}{r-1}+\binom{n-3}{r-2}+\cdots+\binom{n-r}{1}. \]	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]	Apply the symmetry ℓCk = ℓC(n-k) to turn the sum into the standard hockey‑stick identity.	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aops_992803	[hide="Complimentary Counting"] It's easier to count what's the probability there are no 1's, which is $ \left(\frac56\right)^{\infty}\equal{}0$. Therefore, the probability is 1? [/hide] [hide="Casework"] The probability on the first roll is 1/6. The next is 1/65/6. The next is 1/65/6*5/6. ... The total sum is: $ \frac16\left(\frac56\right)^0\plus{}\frac16\left(\frac56\right)^1\plus{}\frac16\left(\frac56\right)^2\plus{}\cdots\\ \equal{}\frac16\left(1\plus{}\frac56\plus{}\left(\frac56\right)^2\plus{}\cdots\right)\\ \equal{}\frac16(6)\\ \equal{}1$ [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "A kinda neat-ish problem (not that hard):\r\n\r\nGiven an n sided dice, approximate to the nearest thousandth the probability that at least one 1 is rolled after n rolls when n approaches infinity.\r\n\r\nThis should be done without an calculator. Sorry if that was badly worded.", "content_html": "A kinda neat-ish problem (not that hard):<br>\n<br>\nGiven an n sided dice, approximate to the nearest thousandth the probability that at least one 1 is rolled after n rolls when n approaches infinity.<br>\n<br>\nThis should be done without an calculator. Sorry if that was badly worded.", "post_id": 4405413, "post_number": 1, "post_time_unix": 1246595629, "post_time_utc": "2009-07-03 04:33:49 UTC", "thanks_received": 2, "user_id": 30252, "username": "Nerd_of_the_Ages" }, { "attachments": [], "content_bbcode": "[hide=\"Complimentary Counting\"]\nIt's easier to count what's the probability there are no 1's, which is $ \\left(\\frac56\\right)^{\\infty}\\equal{}0$. Therefore, the probability is 1?\n[/hide]\n\n[hide=\"Casework\"]\nThe probability on the first roll is 1/6. The next is 1/65/6. The next is 1/65/65/6. \n...\nThe total sum is:\n\n $ \\frac16\\left(\\frac56\\right)^0\\plus{}\\frac16\\left(\\frac56\\right)^1\\plus{}\\frac16\\left(\\frac56\\right)^2\\plus{}\\cdots\\\\\n\\equal{}\\frac16\\left(1\\plus{}\\frac56\\plus{}\\left(\\frac56\\right)^2\\plus{}\\cdots\\right)\\\\\n\\equal{}\\frac16(6)\\\\\n\\equal{}1$\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Complimentary Counting</a><div class=\"cmty-hide-content\" style=\"display:none\">It's easier to count what's the probability there are no 1's, which is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/d/c/9dc95f98ce91e0185db6be4b6910c499aaa442e5.png\" class=\"latex\" alt=\"$ \\left(\\frac56\\right)^{\\infty}=0$\" style=\"vertical-align: -17px\" width=\"86\" height=\"44\" >.</span> Therefore, the probability is 1?</div><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Casework</a><div class=\"cmty-hide-content\" style=\"display:none\">The probability on the first roll is 1/6. The next is 1/65/6. The next is 1/65/65/6.<br>\n...<br>\nThe total sum is:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/9/1/a91579817e0cacb777ba1e2af4b01a53c82773e7.png\" class=\"latex\" alt=\"$ \\frac16\\left(\\frac56\\right)^0+\\frac16\\left(\\frac56\\right)^1+\\frac16\\left(\\frac56\\right)^2+\\cdots\\\\\n=\\frac16\\left(1+\\frac56+\\left(\\frac56\\right)^2+\\cdots\\right)\\\\\n=\\frac16(6)\\\\\n=1$\" style=\"vertical-align: 0px\" width=\"274\" height=\"153\" ></div>", "post_id": 4405414, "post_number": 2, "post_time_unix": 1246597873, "post_time_utc": "2009-07-03 05:11:13 UTC", "thanks_received": 2, "user_id": 33079, "username": "dragon96" }, { "attachments": [], "content_bbcode": "You missed that the dice is n-sided.", "content_html": "You missed that the dice is n-sided.", "post_id": 4405415, "post_number": 3, "post_time_unix": 1246598142, "post_time_utc": "2009-07-03 05:15:42 UTC", "thanks_received": 2, "user_id": 30252, "username": "Nerd_of_the_Ages" }, { "attachments": [], "content_bbcode": "Oh pshh. I'll work on that later.", "content_html": "Oh pshh. I'll work on that later.", "post_id": 4405416, "post_number": 4, "post_time_unix": 1246603454, "post_time_utc": "2009-07-03 06:44:14 UTC", "thanks_received": 2, "user_id": 33079, "username": "dragon96" }, { "attachments": [], "content_bbcode": "[hide=\"Solution\"]\nUse complementary counting.\n\n$ 1 \\minus{} (\\frac{n\\minus{}1}{n})^{n}$\n\nDistribute out the denominator\n\n$ 1 \\minus{} (1 \\minus{} \\frac{1}{n})^{n}$\n\nSince $ \\lim_{n\\rightarrow\\infty}{(1 \\minus{} \\frac{1}{n})^{\\minus{}n}} \\equal{} e$,\n\n$ \\lim_{n\\rightarrow\\infty}{1\\minus{}(1 \\minus{} \\frac{1}{n})^{n}} \\equal{} 1 \\minus{} \\frac{1}{e}$\n\nThen assuming you know the first couple digits of e, just do the rest by hand.\nSo yeah, this problem is actually doable in 1 minute.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Solution</a><div class=\"cmty-hide-content\" style=\"display:none\">Use complementary counting.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/e/7/ae7410ed0a6ead18c32234e8815d1838c8926695.png\" class=\"latex\" alt=\"$ 1 - (\\frac{n-1}{n})^{n}$\" style=\"vertical-align: -12px\" width=\"99\" height=\"37\" ><br>\n<br>\nDistribute out the denominator<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/a/7/ba75de8e42d86003890e1cf73aa7094809b0ccc6.png\" class=\"latex\" alt=\"$ 1 - (1 - \\frac{1}{n})^{n}$\" style=\"vertical-align: -12px\" width=\"99\" height=\"37\" ><br>\n<br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/3/3/f337424298c30276b6bee8e4764fb7f49ff45534.png\" class=\"latex\" alt=\"$ \\lim_{n\\rightarrow\\infty}{(1 - \\frac{1}{n})^{-n}} = e$\" style=\"vertical-align: -12px\" width=\"149\" height=\"37\" >,</span><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/8/a/88ac5af27ce048e4e19fbccd327c3a9e78c08930.png\" class=\"latex\" alt=\"$ \\lim_{n\\rightarrow\\infty}{1-(1 - \\frac{1}{n})^{n}} = 1 - \\frac{1}{e}$\" style=\"vertical-align: -12px\" width=\"205\" height=\"37\" ><br>\n<br>\nThen assuming you know the first couple digits of e, just do the rest by hand.<br>\nSo yeah, this problem is actually doable in 1 minute.</div>", "post_id": 4405417, "post_number": 5, "post_time_unix": 1246643422, "post_time_utc": "2009-07-03 17:50:22 UTC", "thanks_received": 2, "user_id": 30252, "username": "Nerd_of_the_Ages" }, { "attachments": [], "content_bbcode": "[quote]approximate to the nearest thousandth[/quote]\r\n\r\nBut why this?", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">approximate to the nearest thousandth</div>\n</div>\n<br>\nBut why this?", "post_id": 4405418, "post_number": 6, "post_time_unix": 1246648290, "post_time_utc": "2009-07-03 19:11:30 UTC", "thanks_received": 2, "user_id": 35062, "username": "lifeisacircle" }, { "attachments": [], "content_bbcode": "It makes it basically impossible to just plug in a really large number for n, I guess.", "content_html": "It makes it basically impossible to just plug in a really large number for n, I guess.", "post_id": 4405419, "post_number": 7, "post_time_unix": 1246648684, "post_time_utc": "2009-07-03 19:18:04 UTC", "thanks_received": 2, "user_id": 30252, "username": "Nerd_of_the_Ages" } ], "source": null }	Given an n-sided die, approximate to the nearest thousandth the probability that at least one 1 is rolled after n independent rolls, in the limit as n → ∞.	[ "/Mathematics/DiscreteMathematics/Combinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/MultiplicationPrinciple", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace", "/Mathematics/ProbabilityandStatistics/Trials/BernoulliTrial", "/Mathematics/ProbabilityandStatistics/Trials/Event", "/Mathematics/ProbabilityandStatistics/Trials/Experiment", "/Mathematics/ProbabilityandStatistics/Trials/IndependentEvents", "/Mathematics/ProbabilityandStatistics/Trials/Trial" ]	Use the complement: probability of no 1’s is (5/6)^n, which tends to 0 as n→∞, so the desired probability tends to 1.	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aops_992815	$ x\plus{}y\plus{}z \equal{} xyz$, then $ \sum_{cyc}{\frac{1}{1\plus{}xy}}\le 3/4.$ $ 1/yz\plus{}1/zx\plus{}1/xy \equal{} 1$ $ 1/yz \equal{} a$ $ 1/zx \equal{} b$ $ 1/xy \equal{} c$ $ a\plus{}b\plus{}c \equal{} 1$ $ \sum\frac{1}{1\plus{}1/a}\le 3/4$ $ \sum\frac{a}{a\plus{}1}\le 3/4$ $ \sum1\minus{}\frac{1}{a\plus{}1}\le 3/4$ $ 3\minus{}3/4\equal{}9/4\ge\sum\frac{1}{a\plus{}1}$ but cauchy says $ (\sum\frac{1}{a\plus{}1})(\sum a\plus{}1)\equal{}(\sum\frac{1}{a\plus{}1})(4)\ge(1\plus{}1\plus{}1)^{3}\equal{}9$ so $ 9/4\ge\sum\frac{1}{a\plus{}1}$ and we are done what does b.1.1 mean easy?	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "yay... I just did B1.1 on MOP tests... if $ x\\plus{}y\\plus{}z\\equal{}xyz$, then $ \\sum_{cyc}{\\frac{1}{1\\plus{}xy}}\\le 3/4$.\r\n\r\nanother problem I can't do: show that if the four faces of a tetrahedron have the same area, then all four faces are congruent.", "content_html": "yay... I just did B1.1 on MOP tests... if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/4/4/144be6f29ba4a1e17a3e46235360f6e4bd893550.png\" class=\"latex\" alt=\"$ x+y+z=xyz$\" style=\"vertical-align: -3px\" width=\"126\" height=\"14\" >,</span> then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/f/f/cff9be820373bb47280781986ea263d94421f843.png\" class=\"latex\" alt=\"$ \\sum_{cyc}{\\frac{1}{1+xy}}\\le 3/4$\" style=\"vertical-align: -22px\" width=\"135\" height=\"47\" >.</span><br>\n<br>\nanother problem I can't do: show that if the four faces of a tetrahedron have the same area, then all four faces are congruent.", "post_id": 4405475, "post_number": 1, "post_time_unix": 1188383406, "post_time_utc": "2007-08-29 10:30:06 UTC", "thanks_received": 2, "user_id": 18909, "username": "not_trig" }, { "attachments": [], "content_bbcode": "$ x\\plus{}y\\plus{}z \\equal{} xyz$, then $ \\sum_{cyc}{\\frac{1}{1\\plus{}xy}}\\le 3/4.$\r\n$ 1/yz\\plus{}1/zx\\plus{}1/xy \\equal{} 1$\r\n$ 1/yz \\equal{} a$\r\n$ 1/zx \\equal{} b$\r\n$ 1/xy \\equal{} c$\r\n$ a\\plus{}b\\plus{}c \\equal{} 1$\r\n$ \\sum\\frac{1}{1\\plus{}1/a}\\le 3/4$\r\n$ \\sum\\frac{a}{a\\plus{}1}\\le 3/4$\r\n$ \\sum1\\minus{}\\frac{1}{a\\plus{}1}\\le 3/4$\r\n$ 3\\minus{}3/4\\equal{}9/4\\ge\\sum\\frac{1}{a\\plus{}1}$\r\nbut cauchy says\r\n$ (\\sum\\frac{1}{a\\plus{}1})(\\sum a\\plus{}1)\\equal{}(\\sum\\frac{1}{a\\plus{}1})(4)\\ge(1\\plus{}1\\plus{}1)^{3}\\equal{}9$\r\nso \r\n$ 9/4\\ge\\sum\\frac{1}{a\\plus{}1}$\r\nand we are done\r\nwhat does b.1.1 mean\r\neasy?", "content_html": "<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/4/0/54036e9b41ea7493e274ba836167ded097160ee7.png\" class=\"latex\" alt=\"$ x+y+z = xyz$\" style=\"vertical-align: -3px\" width=\"126\" height=\"14\" >,</span> then <img src=\"//latex.artofproblemsolving.com/6/9/0/6906936ad1e1fcaf4b131ba3c3bd47265c5d9940.png\" class=\"latex\" alt=\"$ \\sum_{cyc}{\\frac{1}{1+xy}}\\le 3/4.$\" style=\"vertical-align: -22px\" width=\"138\" height=\"47\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/a/0/6a0f6ac17ced75d563faf41bd243efc547f0e48e.png\" class=\"latex\" alt=\"$ 1/yz+1/zx+1/xy = 1$\" style=\"vertical-align: -4px\" width=\"189\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/8/e/98e7223c5410d9b9893f038d60a0ec767ec75103.png\" class=\"latex\" alt=\"$ 1/yz = a$\" style=\"vertical-align: -4px\" width=\"70\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a571c360356943fdcc8ba2f4faaff6d4c56b0a7.png\" class=\"latex\" alt=\"$ 1/zx = b$\" style=\"vertical-align: -4px\" width=\"70\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/9/9/3995458b4c3ec25aedf740fbc7ca73b0be8dc5fe.png\" class=\"latex\" alt=\"$ 1/xy = c$\" style=\"vertical-align: -4px\" width=\"70\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/3/9/939643886aa3018edb25d37722e2a205693c6e55.png\" class=\"latex\" alt=\"$ a+b+c = 1$\" style=\"vertical-align: -1px\" width=\"102\" height=\"14\" ><br>\n<img src=\"//latex.artofproblemsolving.com/d/0/9/d09ff052b6008a8623ce9c9152d3b56fb370aafa.png\" class=\"latex\" alt=\"$ \\sum\\frac{1}{1+1/a}\\le 3/4$\" style=\"vertical-align: -17px\" width=\"143\" height=\"41\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/5/7/c574570fa7342c2dd9cb5daec779a24b717dfd23.png\" class=\"latex\" alt=\"$ \\sum\\frac{a}{a+1}\\le 3/4$\" style=\"vertical-align: -14px\" width=\"125\" height=\"34\" ><br>\n<img src=\"//latex.artofproblemsolving.com/b/6/8/b6894e18196606590307d2dc2c3bfd5e1ad0776d.png\" class=\"latex\" alt=\"$ \\sum1-\\frac{1}{a+1}\\le 3/4$\" style=\"vertical-align: -14px\" width=\"156\" height=\"38\" ><br>\n<img src=\"//latex.artofproblemsolving.com/2/f/b/2fb33f363678ecdb6da72617084f653eef6c7fc5.png\" class=\"latex\" alt=\"$ 3-3/4=9/4\\ge\\sum\\frac{1}{a+1}$\" style=\"vertical-align: -14px\" width=\"206\" height=\"38\" ><br>\nbut cauchy says<br>\n<img src=\"//latex.artofproblemsolving.com/9/a/8/9a8d8ebec8bbb4f42d633a25fb188ca6348464b8.png\" class=\"latex\" alt=\"$ (\\sum\\frac{1}{a+1})(\\sum a+1)=(\\sum\\frac{1}{a+1})(4)\\ge(1+1+1)^{3}=9$\" style=\"vertical-align: -14px\" width=\"457\" height=\"38\" ><br>\nso<br>\n<img src=\"//latex.artofproblemsolving.com/d/c/f/dcf1378a9a5bb71fbd1e4864c17505adc8153324.png\" class=\"latex\" alt=\"$ 9/4\\ge\\sum\\frac{1}{a+1}$\" style=\"vertical-align: -14px\" width=\"123\" height=\"38\" ><br>\nand we are done<br>\nwhat does b.1.1 mean<br>\neasy?", "post_id": 4405476, "post_number": 2, "post_time_unix": 1188402921, "post_time_utc": "2007-08-29 15:55:21 UTC", "thanks_received": 1, "user_id": 19764, "username": "junggi" }, { "attachments": [], "content_bbcode": "pretty much\r\n\r\nI made same substitution \r\n\r\nthen\r\n\r\n(used some calculus to find this, but no calc in proof)\r\n\r\n$ 9a^{2}\\minus{}6a\\plus{}1\\ge0\\Leftrightarrow 16a\\ge(9a\\plus{}1)(a\\plus{}1)\\Leftrightarrow\\frac{a}{a\\plus{}1}\\le\\frac{1}{16}(9a\\plus{}1)$. Then, summing cyclically gives the required result.", "content_html": "pretty much<br>\n<br>\nI made same substitution<br>\n<br>\nthen<br>\n<br>\n(used some calculus to find this, but no calc in proof)<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/a/c/1acb0274545e0540e5e8b403fda2456360eb9112.png\" class=\"latex\" alt=\"$ 9a^{2}-6a+1\\ge0\\Leftrightarrow 16a\\ge(9a+1)(a+1)\\Leftrightarrow\\frac{a}{a+1}\\le\\frac{1}{16}(9a+1)$\" style=\"vertical-align: -14px\" width=\"513\" height=\"38\" >.</span> Then, summing cyclically gives the required result.", "post_id": 4405477, "post_number": 3, "post_time_unix": 1188408136, "post_time_utc": "2007-08-29 17:22:16 UTC", "thanks_received": 1, "user_id": 18909, "username": "not_trig" }, { "attachments": [], "content_bbcode": "haha i cross multiplied and a bunch of stuff canceled and I ended up with a few simple inequalities. it was really easy.", "content_html": "haha i cross multiplied and a bunch of stuff canceled and I ended up with a few simple inequalities. it was really easy.", "post_id": 4405478, "post_number": 4, "post_time_unix": 1188428584, "post_time_utc": "2007-08-29 23:03:04 UTC", "thanks_received": 1, "user_id": 26057, "username": "tjhance" }, { "attachments": [], "content_bbcode": "um\r\ndumbassing\r\nive been told to never dumbass while practicing", "content_html": "um<br>\ndumbassing<br>\nive been told to never dumbass while practicing", "post_id": 4405479, "post_number": 5, "post_time_unix": 1188525670, "post_time_utc": "2007-08-31 02:01:10 UTC", "thanks_received": 1, "user_id": 19764, "username": "junggi" }, { "attachments": [], "content_bbcode": "by whom... tis a very useful method", "content_html": "by whom... tis a very useful method", "post_id": 4405480, "post_number": 6, "post_time_unix": 1188558611, "post_time_utc": "2007-08-31 11:10:11 UTC", "thanks_received": 1, "user_id": 18909, "username": "not_trig" }, { "attachments": [], "content_bbcode": "hm it was in mildorfs packet, and numerous people in w00t have said so\r\nand it is quite a useless method if you want to improve\r\nsince it consists of \r\n1. exanding out everything\r\n2 do a few am gms or muirhead\r\ndone\r\n\r\nno\r\nits not good", "content_html": "hm it was in mildorfs packet, and numerous people in w00t have said so<br>\nand it is quite a useless method if you want to improve<br>\nsince it consists of<br>\n1. exanding out everything<br>\n2 do a few am gms or muirhead<br>\ndone<br>\n<br>\nno<br>\nits not good", "post_id": 4405481, "post_number": 7, "post_time_unix": 1188601287, "post_time_utc": "2007-08-31 23:01:27 UTC", "thanks_received": 1, "user_id": 19764, "username": "junggi" }, { "attachments": [], "content_bbcode": "aha but I didn't say it was good... it's USEFUL\r\n\r\nbtw and sometimes it requires schur, because AM-GM is like super-weak...", "content_html": "aha but I didn't say it was good... it's USEFUL<br>\n<br>\nbtw and sometimes it requires schur, because AM-GM is like super-weak...", "post_id": 4405482, "post_number": 8, "post_time_unix": 1188604104, "post_time_utc": "2007-08-31 23:48:24 UTC", "thanks_received": 1, "user_id": 18909, "username": "not_trig" }, { "attachments": [], "content_bbcode": "Try a tangent substitution.\r\n$ a\\equal{}\\tan A$, $ b\\equal{}\\tan B$, $ c\\equal{}\\tan C$ where $ A\\plus{}B\\plus{}C\\equal{}180$", "content_html": "Try a tangent substitution.<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/6/6/9667c6e2f198550b43fc2f58067cfa8c1bdbebc6.png\" class=\"latex\" alt=\"$ a=\\tan A$\" width=\"76\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/8/4/4841ad0cef34ab03c93062853ca5a70187508e66.png\" class=\"latex\" alt=\"$ b=\\tan B$\" width=\"75\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/6/1/2/6129ed94a29dcc0a39fde4dc0c8059dc7da33738.png\" class=\"latex\" alt=\"$ c=\\tan C$\" width=\"75\" height=\"12\" > where <img src=\"//latex.artofproblemsolving.com/c/4/6/c46d85fcdea66a021ec49ec529cc0b1342245578.png\" class=\"latex\" alt=\"$ A+B+C=180$\" style=\"vertical-align: -1px\" width=\"138\" height=\"14\" >", "post_id": 4405483, "post_number": 9, "post_time_unix": 1188652996, "post_time_utc": "2007-09-01 13:23:16 UTC", "thanks_received": 1, "user_id": 28075, "username": "diophantient" }, { "attachments": [], "content_bbcode": "wait did you substitute $ a\\equal{}\\frac{1}{xy}$... what? we have $ a\\plus{}b\\plus{}c\\equal{}1$ NOT arctan(a)+arctan(b)+arctan(c) = 1... like the conditions are not exactly equivalent...", "content_html": "wait did you substitute <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/d/1/ed13e94b31b67f4fef3dcab8b8b13083d3162bd7.png\" class=\"latex\" alt=\"$ a=\\frac{1}{xy}$\" style=\"vertical-align: -16px\" width=\"56\" height=\"40\" >.</span>.. what? we have <img src=\"//latex.artofproblemsolving.com/d/1/e/d1eaea3b37bd460d7824c3cb2fc63a2f6f97028c.png\" class=\"latex\" alt=\"$ a+b+c=1$\" style=\"vertical-align: -1px\" width=\"102\" height=\"14\" > NOT arctan(a)+arctan(b)+arctan(c) = 1... like the conditions are not exactly equivalent...", "post_id": 4405484, "post_number": 10, "post_time_unix": 1188656038, "post_time_utc": "2007-09-01 14:13:58 UTC", "thanks_received": 1, "user_id": 18909, "username": "not_trig" } ], "source": null }	yay... I just did B1.1 on MOP tests... if $ x\plus{}y\plus{}z\equal{}xyz$, then $ \sum_{cyc}{\frac{1}{1\plus{}xy}}\le 3/4$. another problem I can't do: show that if the four faces of a tetrahedron have the same area, then all four faces are congruent.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression", "/Mathematics/Algebra/AlgebraicIdentities", "/Mathematics/Algebra/Sums/Sum" ]	Rewrite the condition as a+b+c=1 via substitution and apply Cauchy‑Schwarz to bound the sum of 1/(a+1).	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aops_99291	[quote="t0rajir0u"]The answer changes depending on how the points are randomly selected.[/quote] Although it should have been specified, selecting a point "randomly" on a curve usually means you choose by arclength -- the point has a probability $p$ of landing on any arc which is $p$ of the total arclength.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "A semicircle has $O$ as its diameter midpoint and radius $1$.\r\n\r\nTwo points $P_1$ and $P_2$ are randomly and independently picked on the perimeter of the semicircle, including its diameter. Let $\\triangle$ denote the area of triangle $OP_1P_2$.\r\n\r\nWhat is the expected value of $\\triangle$?", "content_html": "A semicircle has <img src=\"//latex.artofproblemsolving.com/5/1/d/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" > as its diameter midpoint and radius <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >.</span><br>\n<br>\nTwo points <img src=\"//latex.artofproblemsolving.com/a/5/b/a5b0256a78d58b6d5f7a53400b19e7747d2983fa.png\" class=\"latex\" alt=\"$P_1$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/1/0/7/107c7c8e5a64d81ed24331e510b777b7d08345b7.png\" class=\"latex\" alt=\"$P_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > are randomly and independently picked on the perimeter of the semicircle, including its diameter. Let <img src=\"//latex.artofproblemsolving.com/0/0/9/009cf3eeb0ff3789cc057632947cadb200ab4663.png\" class=\"latex\" alt=\"$\\triangle$\" width=\"15\" height=\"13\" > denote the area of triangle <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/5/f550e836dea1bbbe5bc7025fd7eba729626bebf3.png\" class=\"latex\" alt=\"$OP_1P_2$\" style=\"vertical-align: -2px\" width=\"51\" height=\"15\" >.</span><br>\n<br>\nWhat is the expected value of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/0/9/009cf3eeb0ff3789cc057632947cadb200ab4663.png\" class=\"latex\" alt=\"$\\triangle$\" width=\"15\" height=\"13\" >?</span>", "post_id": 560606, "post_number": 1, "post_time_unix": 1151527323, "post_time_utc": "2006-06-28 20:42:03 UTC", "thanks_received": 2, "user_id": 18317, "username": "Centy" }, { "attachments": [], "content_bbcode": "What do you mean by \"expected value\"?", "content_html": "What do you mean by "expected value"?", "post_id": 560613, "post_number": 2, "post_time_unix": 1151527662, "post_time_utc": "2006-06-28 20:47:42 UTC", "thanks_received": 2, "user_id": 16759, "username": "rem" }, { "attachments": [], "content_bbcode": "this seems to need calculus, \r\n\r\nClearly the first point is independent of the second one, so we say that $P_1$ is fixed. The other point can be anywhere on the circle with uniform distribution in terms of arc length or angle. Since the circle is symmetric, consider only one half of the circle that is divided by diameter $OP_1$. The area of the triangle is:\r\n\r\n$\\[ \\frac{1}{2}\\sin\\theta$\r\n\r\nby the sine area formula where $0\\le \\theta \\le \\pi$.\r\n\r\nThen the expected value is just the average of the areas. We can easily do that using integrals.\r\n\r\n$\\frac{1}{b-a}\\int_{a}^bf(x) \\, dx$\r\n\r\n$\\frac{1}{2\\pi}\\int_0^\\pi \\sin x \\, dx$\r\n\r\n$\\frac{-\\cos\\pi+\\cos 0}{2\\pi}=\\frac{1}{\\pi}$", "content_html": "this seems to need calculus,<br>\n<br>\nClearly the first point is independent of the second one, so we say that <img src=\"//latex.artofproblemsolving.com/a/5/b/a5b0256a78d58b6d5f7a53400b19e7747d2983fa.png\" class=\"latex\" alt=\"$P_1$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > is fixed. The other point can be anywhere on the circle with uniform distribution in terms of arc length or angle. Since the circle is symmetric, consider only one half of the circle that is divided by diameter <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/3/6/936eecd1dc2e1f636c2f7353c42b0ab8a3ed3b9d.png\" class=\"latex\" alt=\"$OP_1$\" style=\"vertical-align: -2px\" width=\"31\" height=\"15\" >.</span> The area of the triangle is:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/5/4/a54b7b140fb81caa77e751446485ed7376f7bcaa.png\" class=\"latex\" alt=\"$ \\frac{1}{2}\\sin\\theta$\" style=\"vertical-align: -12px\" width=\"64\" height=\"37\" ><br>\n<br>\nby the sine area formula where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/1/c/61cbd9054c3cc57f5a74819435bcf8a513a351db.png\" class=\"latex\" alt=\"$0\\le \\theta \\le \\pi$\" style=\"vertical-align: -2px\" width=\"77\" height=\"15\" >.</span><br>\n<br>\nThen the expected value is just the average of the areas. We can easily do that using integrals.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/0/4/9043251f2e1ee49e7f91b0c93ff9ce1033f8f6c0.png\" class=\"latex\" alt=\"$\\frac{1}{b-a}\\int_{a}^bf(x) \\, dx$\" style=\"vertical-align: -16px\" width=\"132\" height=\"44\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/0/f/80fa90adabd23e6dcb5a13b1035149c13a099a31.png\" class=\"latex\" alt=\"$\\frac{1}{2\\pi}\\int_0^\\pi \\sin x \\, dx$\" style=\"vertical-align: -16px\" width=\"115\" height=\"41\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/d/e/9de0b7bc44e7426c128a6a5aaf354e26f7c6b4db.png\" class=\"latex\" alt=\"$\\frac{-\\cos\\pi+\\cos 0}{2\\pi}=\\frac{1}{\\pi}$\" style=\"vertical-align: -12px\" width=\"156\" height=\"37\" >", "post_id": 560632, "post_number": 3, "post_time_unix": 1151528534, "post_time_utc": "2006-06-28 21:02:14 UTC", "thanks_received": 1, "user_id": 10035, "username": "Altheman" }, { "attachments": [], "content_bbcode": "nice i didnt even think about using calculus...i was trying to use \"basic\" geometry and such (didnt work). :)", "content_html": "nice i didnt even think about using calculus...i was trying to use "basic" geometry and such (didnt work). <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 560928, "post_number": 4, "post_time_unix": 1151546539, "post_time_utc": "2006-06-29 02:02:19 UTC", "thanks_received": 1, "user_id": 10301, "username": "maokid7" }, { "attachments": [], "content_bbcode": "The answer changes depending on how the points are randomly selected. \r\n\r\nMethod 1: Select a random $x$-coordinate, then pick one of the two viable $y$-coordinates.\r\n\r\nMethod 2: Place the center of the circle at the origin of a polar coordinate system, and select a random $\\theta$. (Altheman's solution)\r\n\r\nYou can clearly see that these two methods give different distributions and hence a different answer.", "content_html": "The answer changes depending on how the points are randomly selected.<br>\n<br>\nMethod 1: Select a random <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" >-</span>coordinate, then pick one of the two viable <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" >-</span>coordinates.<br>\n<br>\nMethod 2: Place the center of the circle at the origin of a polar coordinate system, and select a random <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/2/e/52e8ed7a3ba22130ad3984eb2cd413406475a689.png\" class=\"latex\" alt=\"$\\theta$\" width=\"8\" height=\"13\" >.</span> (Altheman's solution)<br>\n<br>\nYou can clearly see that these two methods give different distributions and hence a different answer.", "post_id": 561000, "post_number": 5, "post_time_unix": 1151551971, "post_time_utc": "2006-06-29 03:32:51 UTC", "thanks_received": 1, "user_id": 14052, "username": "t0rajir0u" }, { "attachments": [], "content_bbcode": "Altheman: You forgot that the points can lie on the diameter.", "content_html": "Altheman: You forgot that the points can lie on the diameter.", "post_id": 561357, "post_number": 6, "post_time_unix": 1151587718, "post_time_utc": "2006-06-29 13:28:38 UTC", "thanks_received": 2, "user_id": 18317, "username": "Centy" }, { "attachments": [], "content_bbcode": "[quote=\"t0rajir0u\"]The answer changes depending on how the points are randomly selected.[/quote]\r\n\r\nAlthough it should have been specified, selecting a point \"randomly\" on a curve usually means you choose by arclength -- the point has a probability $p$ of landing on any arc which is $p$ of the total arclength.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">t0rajir0u wrote:</div>\n<div class=\"bbcode_quote_body\">The answer changes depending on how the points are randomly selected.</div>\n</div>\n<br>\nAlthough it should have been specified, selecting a point "randomly" on a curve usually means you choose by arclength -- the point has a probability <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > of landing on any arc which is <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > of the total arclength.", "post_id": 561504, "post_number": 7, "post_time_unix": 1151595159, "post_time_utc": "2006-06-29 15:32:39 UTC", "thanks_received": 2, "user_id": 1430, "username": "JBL" } ], "source": null }	A semicircle has center $O$, diameter midpoint $O$, and radius $1$. Two points $P_1$ and $P_2$ are chosen independently and uniformly on the perimeter of the semicircle, including its diameter. Let $\triangle$ denote the area of triangle $OP_1P_2$. What is the expected value of $\triangle$?	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Arcs/Arc", "/Mathematics/Geometry/PlaneGeometry/Arcs/Semicircle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Semicircle", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles/CircularTriangle", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine", "/Mathematics/ProbabilityandStatistics/Moments/Mean", "/Mathematics/ProbabilityandStatistics/Moments/PopulationMean", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/UniformDistribution" ]	Interpret the random points as being uniformly distributed along the semicircle’s arclength.	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aops_992999	I'm assuming the circle contains neither A nor B. Draw a tangent from A to the circle, and draw a tangent on the same side of AB from B to the circle. The path is along those two tangents and around the circle. Also, you seem to have little to say about Obama's victory :P	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Also, Algebraic Number Theory is pretty useful...\r\n\r\nBTW, here is a problem. We have two points A and B and a circle whose center lies on the line through the two points. Also, the center is between A and B. What is the shortest path from A to B that does not go through the interior of the circle?", "content_html": "Also, Algebraic Number Theory is pretty useful...<br>\n<br>\nBTW, here is a problem. We have two points A and B and a circle whose center lies on the line through the two points. Also, the center is between A and B. What is the shortest path from A to B that does not go through the interior of the circle?", "post_id": 4406240, "post_number": 1, "post_time_unix": 1225859147, "post_time_utc": "2008-11-05 04:25:47 UTC", "thanks_received": 1, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "I'm assuming the circle contains neither A nor B. Draw a tangent from A to the circle, and draw a tangent on the same side of AB from B to the circle. The path is along those two tangents and around the circle.\r\n\r\nAlso, you seem to have little to say about Obama's victory :P", "content_html": "I'm assuming the circle contains neither A nor B. Draw a tangent from A to the circle, and draw a tangent on the same side of AB from B to the circle. The path is along those two tangents and around the circle.<br>\n<br>\nAlso, you seem to have little to say about Obama's victory <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />", "post_id": 4406241, "post_number": 2, "post_time_unix": 1226022115, "post_time_utc": "2008-11-07 01:41:55 UTC", "thanks_received": 1, "user_id": 37564, "username": "zephyredx" } ], "source": null }	We have two points $A$ and $B$ and a circle whose center lies on the line through $A$ and $B$. The center lies between $A$ and $B$. What is the shortest path from $A$ to $B$ that does not go through the interior of the circle?	[ "/Mathematics/Geometry/Distance/Geodesic", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeometricConstruction", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/CircleTangentLine", "/Mathematics/Geometry/Points/Point" ]	Use the two external tangents from A and B and the minor arc between their touch points as the shortest admissible route.	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aops_99315	[hide="well"] the $f(xy)=f(x)+f(y)$ sorta made me look at $f(x)=\ln(g(x))$ and then I got $f'(x)=\frac{g'(x)}{g(x)}$ so $3g(e)=g'(e)$ from this i figured that $g(x)=e^{Ax}$ since then $g'(x)=Ae^{Ax}=Ag(x)$ [/hide] where did I go wrong?	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Given that $f(xy)=f(x)+f(y)$ for $x, y >0$ and that $f'(e)=3$ what is $f(x)$?\r\n[hide=\"here is what I got but am not 100% sure\"]\n$f(x)=\\ln(e^{3x})$\n[/hide]", "content_html": "Given that <img src=\"//latex.artofproblemsolving.com/2/a/a/2aa75726cabbdfe32044021eb50fed2b212116dd.png\" class=\"latex\" alt=\"$f(xy)=f(x)+f(y)$\" style=\"vertical-align: -4px\" width=\"160\" height=\"18\" > for <img src=\"//latex.artofproblemsolving.com/d/7/f/d7f6089c9ccfd5df7994f6a33ad53e1c648b07fa.png\" class=\"latex\" alt=\"$x, y >0$\" style=\"vertical-align: -3px\" width=\"61\" height=\"16\" > and that <img src=\"//latex.artofproblemsolving.com/f/6/3/f6361a181975f9f2c750121ce98a6728a53a237a.png\" class=\"latex\" alt=\"$f'(e)=3$\" style=\"vertical-align: -4px\" width=\"71\" height=\"18\" > what is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png\" class=\"latex\" alt=\"$f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" >?</span><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">here is what I got but am not 100% sure</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/8/2/a82671fac5ad9842aa36ff59a9208e7d3b69a530.png\" class=\"latex\" alt=\"$f(x)=\\ln(e^{3x})$\" style=\"vertical-align: -4px\" width=\"111\" height=\"19\" ></div>", "post_id": 560811, "post_number": 1, "post_time_unix": 1151540778, "post_time_utc": "2006-06-29 00:26:18 UTC", "thanks_received": 2, "user_id": 10301, "username": "maokid7" }, { "attachments": [], "content_bbcode": "Ummm... $\\ln(e^{3x})$ is just $3x.$ That's not the answer. What you really meant to say was ... ?", "content_html": "Ummm... <img src=\"//latex.artofproblemsolving.com/5/a/e/5ae61f58d07f19a12c3040c439d90318339c2e5d.png\" class=\"latex\" alt=\"$\\ln(e^{3x})$\" style=\"vertical-align: -4px\" width=\"51\" height=\"19\" > is just <img src=\"//latex.artofproblemsolving.com/0/9/5/095f951f73e4c1d8b43fc3ae03bc5ff9332d6705.png\" class=\"latex\" alt=\"$3x.$\" width=\"23\" height=\"12\" > That's not the answer. What you really meant to say was ... ?", "post_id": 560841, "post_number": 2, "post_time_unix": 1151542543, "post_time_utc": "2006-06-29 00:55:43 UTC", "thanks_received": 1, "user_id": 2948, "username": "Kent Merryfield" }, { "attachments": [], "content_bbcode": "[hide=\"well\"] the $f(xy)=f(x)+f(y)$ sorta made me look at $f(x)=\\ln(g(x))$ and then I got $f'(x)=\\frac{g'(x)}{g(x)}$ so $3g(e)=g'(e)$\nfrom this i figured that $g(x)=e^{Ax}$ since then $g'(x)=Ae^{Ax}=Ag(x)$\n[/hide]\r\nwhere did I go wrong?", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">well</a><div class=\"cmty-hide-content\" style=\"display:none\">the <img src=\"//latex.artofproblemsolving.com/2/a/a/2aa75726cabbdfe32044021eb50fed2b212116dd.png\" class=\"latex\" alt=\"$f(xy)=f(x)+f(y)$\" style=\"vertical-align: -4px\" width=\"160\" height=\"18\" > sorta made me look at <img src=\"//latex.artofproblemsolving.com/8/2/e/82ef229d8921867f38e402a7777ada999bb845b7.png\" class=\"latex\" alt=\"$f(x)=\\ln(g(x))$\" style=\"vertical-align: -4px\" width=\"121\" height=\"18\" > and then I got <img src=\"//latex.artofproblemsolving.com/1/8/8/188077c4280963bf881e0f3f3bbd1c5df1e43442.png\" class=\"latex\" alt=\"$f'(x)=\\frac{g'(x)}{g(x)}$\" style=\"vertical-align: -17px\" width=\"105\" height=\"43\" > so <img src=\"//latex.artofproblemsolving.com/b/d/3/bd30ef62a0270e46a926c7f20aeccad59b0de577.png\" class=\"latex\" alt=\"$3g(e)=g'(e)$\" style=\"vertical-align: -4px\" width=\"100\" height=\"18\" ><br>\nfrom this i figured that <img src=\"//latex.artofproblemsolving.com/d/1/1/d11fdcb5fe77ba22962be889917318e9740c443a.png\" class=\"latex\" alt=\"$g(x)=e^{Ax}$\" style=\"vertical-align: -4px\" width=\"83\" height=\"19\" > since then <img src=\"//latex.artofproblemsolving.com/d/6/6/d663195d139fde736aa543d41da00ea384434b56.png\" class=\"latex\" alt=\"$g'(x)=Ae^{Ax}=Ag(x)$\" style=\"vertical-align: -4px\" width=\"173\" height=\"19\" ></div><br>\nwhere did I go wrong?", "post_id": 560891, "post_number": 3, "post_time_unix": 1151544990, "post_time_utc": "2006-06-29 01:36:30 UTC", "thanks_received": 4, "user_id": 10301, "username": "maokid7" }, { "attachments": [], "content_bbcode": "3eln(x)", "content_html": "3eln(x)", "post_id": 560974, "post_number": 4, "post_time_unix": 1151548870, "post_time_utc": "2006-06-29 02:41:10 UTC", "thanks_received": 2, "user_id": 18124, "username": "PTynan89" }, { "attachments": [], "content_bbcode": "[quote=\"maokid7\"][hide=\"well\"] the $f(xy)=f(x)+f(y)$ sorta made me look at $f(x)=\\ln(g(x))$ and then I got $f'(x)=\\frac{g'(x)}{g(x)}$ so $3g(e)=g'(e)$\nfrom this i figured that $g(x)=e^{Ax}$ since then $g'(x)=Ae^{Ax}=Ag(x)$\n[/hide]\nwhere did I go wrong?[/quote]\r\n\r\nWell, $3g(x)=g'(x)$ holds when $x=e$ but it may not be true for other $x$.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">maokid7 wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">well</a><div class=\"cmty-hide-content\" style=\"display:none\">the <img src=\"//latex.artofproblemsolving.com/2/a/a/2aa75726cabbdfe32044021eb50fed2b212116dd.png\" class=\"latex\" alt=\"$f(xy)=f(x)+f(y)$\" style=\"vertical-align: -4px\" width=\"160\" height=\"18\" > sorta made me look at <img src=\"//latex.artofproblemsolving.com/8/2/e/82ef229d8921867f38e402a7777ada999bb845b7.png\" class=\"latex\" alt=\"$f(x)=\\ln(g(x))$\" style=\"vertical-align: -4px\" width=\"121\" height=\"18\" > and then I got <img src=\"//latex.artofproblemsolving.com/1/8/8/188077c4280963bf881e0f3f3bbd1c5df1e43442.png\" class=\"latex\" alt=\"$f'(x)=\\frac{g'(x)}{g(x)}$\" style=\"vertical-align: -17px\" width=\"105\" height=\"43\" > so <img src=\"//latex.artofproblemsolving.com/b/d/3/bd30ef62a0270e46a926c7f20aeccad59b0de577.png\" class=\"latex\" alt=\"$3g(e)=g'(e)$\" style=\"vertical-align: -4px\" width=\"100\" height=\"18\" ><br>\nfrom this i figured that <img src=\"//latex.artofproblemsolving.com/d/1/1/d11fdcb5fe77ba22962be889917318e9740c443a.png\" class=\"latex\" alt=\"$g(x)=e^{Ax}$\" style=\"vertical-align: -4px\" width=\"83\" height=\"19\" > since then <img src=\"//latex.artofproblemsolving.com/d/6/6/d663195d139fde736aa543d41da00ea384434b56.png\" class=\"latex\" alt=\"$g'(x)=Ae^{Ax}=Ag(x)$\" style=\"vertical-align: -4px\" width=\"173\" height=\"19\" ></div><br>\nwhere did I go wrong?</div>\n</div>\n<br>\nWell, <img src=\"//latex.artofproblemsolving.com/8/7/1/87193f348c033dee61ad5126f75b0f1c92d59811.png\" class=\"latex\" alt=\"$3g(x)=g'(x)$\" style=\"vertical-align: -4px\" width=\"104\" height=\"18\" > holds when <img src=\"//latex.artofproblemsolving.com/2/1/2/212f6aa15ffcb2c3d4a360521fb6fb1d01c2020b.png\" class=\"latex\" alt=\"$x=e$\" width=\"42\" height=\"8\" > but it may not be true for other <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" >.</span>", "post_id": 561008, "post_number": 5, "post_time_unix": 1151552416, "post_time_utc": "2006-06-29 03:40:16 UTC", "thanks_received": 2, "user_id": 20595, "username": "AlvaroRecoba" }, { "attachments": [], "content_bbcode": "[quote=\"maokid7\"]Given that $f(xy)=f(x)+f(y)$ for $x, y >0$ and that $f'(e)=3$ what is $f(x)$?\n[hide=\"here is what I got but am not 100% sure\"]\n$f(x)=\\ln(e^{3x})$\n[/hide][/quote]\r\n\r\n$f'(x)$\r\n$=\\lim_{z\\rightarrow{0}}\\frac{f(x+z)-f(x)}{z}$\r\n$=\\lim_{z\\rightarrow{0}}\\frac{f({y}(e+\\frac{z}{y}))-f({y}e)}{z}$\r\n (Here $y=\\frac{x}{e}$)\r\n$=\\lim_{z\\rightarrow{0}}\\frac{f(e+\\frac{z}{y})-f(e)}{{y}\\frac{z}{y}}$\r\n (Using $f(ab)=f(a)+f(b)$)\r\n$=\\frac{1}{y}f'(e)$\r\n$=\\frac{3e}{x}$\r\nThus $f'(x)=\\frac{3e}{x}$, and notice that $f(1)=0$,\r\nSo $f(x)=3e\\ln{x}$", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">maokid7 wrote:</div>\n<div class=\"bbcode_quote_body\">Given that <img src=\"//latex.artofproblemsolving.com/2/a/a/2aa75726cabbdfe32044021eb50fed2b212116dd.png\" class=\"latex\" alt=\"$f(xy)=f(x)+f(y)$\" style=\"vertical-align: -4px\" width=\"160\" height=\"18\" > for <img src=\"//latex.artofproblemsolving.com/d/7/f/d7f6089c9ccfd5df7994f6a33ad53e1c648b07fa.png\" class=\"latex\" alt=\"$x, y >0$\" style=\"vertical-align: -3px\" width=\"61\" height=\"16\" > and that <img src=\"//latex.artofproblemsolving.com/f/6/3/f6361a181975f9f2c750121ce98a6728a53a237a.png\" class=\"latex\" alt=\"$f'(e)=3$\" style=\"vertical-align: -4px\" width=\"71\" height=\"18\" > what is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png\" class=\"latex\" alt=\"$f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" >?</span><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">here is what I got but am not 100% sure</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/8/2/a82671fac5ad9842aa36ff59a9208e7d3b69a530.png\" class=\"latex\" alt=\"$f(x)=\\ln(e^{3x})$\" style=\"vertical-align: -4px\" width=\"111\" height=\"19\" ></div></div>\n</div>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/b/a/bbad379658bd32e9b6479bbd666b93a96e6b48ba.png\" class=\"latex\" alt=\"$f'(x)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/4/2/742fe918fdb75d7739c569c373490053f10f6c70.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f(x+z)-f(x)}{z}$\" style=\"vertical-align: -12px\" width=\"175\" height=\"38\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/1/4/c14a3d7de78d139f1571ff123988a5daf8b7cbea.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f({y}(e+\\frac{z}{y}))-f({y}e)}{z}$\" style=\"vertical-align: -12px\" width=\"207\" height=\"41\" ><br>\n(Here <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/b/c/8bcaf6c250f19a920e611e6829912cc6bd9bba3e.png\" class=\"latex\" alt=\"$y=\\frac{x}{e}$\" style=\"vertical-align: -12px\" width=\"46\" height=\"33\" >)</span><br>\n<img src=\"//latex.artofproblemsolving.com/0/2/1/02151ddefdb48c2c18b20e7cc2d7fcc1b5e1195f.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f(e+\\frac{z}{y})-f(e)}{{y}\\frac{z}{y}}$\" style=\"vertical-align: -20px\" width=\"174\" height=\"49\" ><br>\n(Using <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/c/1/3c13a0c03552f08e8c85b5c6888b4255d994cb91.png\" class=\"latex\" alt=\"$f(ab)=f(a)+f(b)$\" style=\"vertical-align: -4px\" width=\"155\" height=\"18\" >)</span><br>\n<img src=\"//latex.artofproblemsolving.com/5/9/7/597d899949860c65ed8888b535b403cfad292648.png\" class=\"latex\" alt=\"$=\\frac{1}{y}f'(e)$\" style=\"vertical-align: -16px\" width=\"69\" height=\"40\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/b/9/6b9c9b4af56a51f65e08a52f03c98f89bd85ac4b.png\" class=\"latex\" alt=\"$=\\frac{3e}{x}$\" style=\"vertical-align: -12px\" width=\"39\" height=\"37\" ><br>\nThus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/a/f/2af07383829f4787da5c6869a32f89061da41f60.png\" class=\"latex\" alt=\"$f'(x)=\\frac{3e}{x}$\" style=\"vertical-align: -12px\" width=\"84\" height=\"37\" >,</span> and notice that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/9/d/89d358367015e5ceeb10a72c78d0b627d586f3ac.png\" class=\"latex\" alt=\"$f(1)=0$\" style=\"vertical-align: -4px\" width=\"67\" height=\"18\" >,</span><br>\nSo <img src=\"//latex.artofproblemsolving.com/8/e/5/8e5be90a9f899f23af3a43ea8c3e180bb056734f.png\" class=\"latex\" alt=\"$f(x)=3e\\ln{x}$\" style=\"vertical-align: -4px\" width=\"108\" height=\"18\" >", "post_id": 561034, "post_number": 6, "post_time_unix": 1151554326, "post_time_utc": "2006-06-29 04:12:06 UTC", "thanks_received": 2, "user_id": 20595, "username": "AlvaroRecoba" }, { "attachments": [], "content_bbcode": "[quote=\"AlvaroRecoba\"][quote=\"maokid7\"]Given that $f(xy)=f(x)+f(y)$ for $x, y >0$ and that $f'(e)=3$ what is $f(x)$?\n[hide=\"here is what I got but am not 100% sure\"]\n$f(x)=\\ln(e^{3x})$\n[/hide][/quote]\n\n$f'(x)$\n$=\\lim_{z\\rightarrow{0}}\\frac{f(x+z)-f(x)}{z}$\n$=\\lim_{z\\rightarrow{0}}\\frac{f({y}(e+\\frac{z}{y}))-f({y}e)}{z}$\n (Here $y=\\frac{x}{e}$)\n$=\\lim_{z\\rightarrow{0}}\\frac{f(e+\\frac{z}{y})-f(e)}{{y}\\frac{z}{y}}$\n (Using $f(ab)=f(a)+f(b)$)\n$=\\frac{1}{y}f'(e)$\n$=\\frac{3e}{x}$\nThus $f'(x)=\\frac{3e}{x}$, and notice that $f(1)=0$,\nSo $f(x)=3e\\ln{x}$[/quote]\r\n I dont follow how you got $\\frac{1}{y}f'(e)$ could you please explain that.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">AlvaroRecoba wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">maokid7 wrote:</div>\n<div class=\"bbcode_quote_body\">Given that <img src=\"//latex.artofproblemsolving.com/2/a/a/2aa75726cabbdfe32044021eb50fed2b212116dd.png\" class=\"latex\" alt=\"$f(xy)=f(x)+f(y)$\" style=\"vertical-align: -4px\" width=\"160\" height=\"18\" > for <img src=\"//latex.artofproblemsolving.com/d/7/f/d7f6089c9ccfd5df7994f6a33ad53e1c648b07fa.png\" class=\"latex\" alt=\"$x, y >0$\" style=\"vertical-align: -3px\" width=\"61\" height=\"16\" > and that <img src=\"//latex.artofproblemsolving.com/f/6/3/f6361a181975f9f2c750121ce98a6728a53a237a.png\" class=\"latex\" alt=\"$f'(e)=3$\" style=\"vertical-align: -4px\" width=\"71\" height=\"18\" > what is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png\" class=\"latex\" alt=\"$f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" >?</span><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">here is what I got but am not 100% sure</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/8/2/a82671fac5ad9842aa36ff59a9208e7d3b69a530.png\" class=\"latex\" alt=\"$f(x)=\\ln(e^{3x})$\" style=\"vertical-align: -4px\" width=\"111\" height=\"19\" ></div></div>\n</div>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/b/a/bbad379658bd32e9b6479bbd666b93a96e6b48ba.png\" class=\"latex\" alt=\"$f'(x)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/4/2/742fe918fdb75d7739c569c373490053f10f6c70.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f(x+z)-f(x)}{z}$\" style=\"vertical-align: -12px\" width=\"175\" height=\"38\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/1/4/c14a3d7de78d139f1571ff123988a5daf8b7cbea.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f({y}(e+\\frac{z}{y}))-f({y}e)}{z}$\" style=\"vertical-align: -12px\" width=\"207\" height=\"41\" ><br>\n(Here <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/b/c/8bcaf6c250f19a920e611e6829912cc6bd9bba3e.png\" class=\"latex\" alt=\"$y=\\frac{x}{e}$\" style=\"vertical-align: -12px\" width=\"46\" height=\"33\" >)</span><br>\n<img src=\"//latex.artofproblemsolving.com/0/2/1/02151ddefdb48c2c18b20e7cc2d7fcc1b5e1195f.png\" class=\"latex\" alt=\"$=\\lim_{z\\rightarrow{0}}\\frac{f(e+\\frac{z}{y})-f(e)}{{y}\\frac{z}{y}}$\" style=\"vertical-align: -20px\" width=\"174\" height=\"49\" ><br>\n(Using <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/c/1/3c13a0c03552f08e8c85b5c6888b4255d994cb91.png\" class=\"latex\" alt=\"$f(ab)=f(a)+f(b)$\" style=\"vertical-align: -4px\" width=\"155\" height=\"18\" >)</span><br>\n<img src=\"//latex.artofproblemsolving.com/5/9/7/597d899949860c65ed8888b535b403cfad292648.png\" class=\"latex\" alt=\"$=\\frac{1}{y}f'(e)$\" style=\"vertical-align: -16px\" width=\"69\" height=\"40\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/b/9/6b9c9b4af56a51f65e08a52f03c98f89bd85ac4b.png\" class=\"latex\" alt=\"$=\\frac{3e}{x}$\" style=\"vertical-align: -12px\" width=\"39\" height=\"37\" ><br>\nThus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/a/f/2af07383829f4787da5c6869a32f89061da41f60.png\" class=\"latex\" alt=\"$f'(x)=\\frac{3e}{x}$\" style=\"vertical-align: -12px\" width=\"84\" height=\"37\" >,</span> and notice that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/9/d/89d358367015e5ceeb10a72c78d0b627d586f3ac.png\" class=\"latex\" alt=\"$f(1)=0$\" style=\"vertical-align: -4px\" width=\"67\" height=\"18\" >,</span><br>\nSo <img src=\"//latex.artofproblemsolving.com/8/e/5/8e5be90a9f899f23af3a43ea8c3e180bb056734f.png\" class=\"latex\" alt=\"$f(x)=3e\\ln{x}$\" style=\"vertical-align: -4px\" width=\"108\" height=\"18\" ></div>\n</div>\nI dont follow how you got <img src=\"//latex.artofproblemsolving.com/e/6/3/e637681159cc68c3a121ddfe6d8b45a1f33a676c.png\" class=\"latex\" alt=\"$\\frac{1}{y}f'(e)$\" style=\"vertical-align: -16px\" width=\"50\" height=\"40\" > could you please explain that.", "post_id": 563668, "post_number": 7, "post_time_unix": 1151764971, "post_time_utc": "2006-07-01 14:42:51 UTC", "thanks_received": 2, "user_id": 10301, "username": "maokid7" } ], "source": null }	Given f satisfies f(xy)=f(x)+f(y) for x,y>0 and f′(e)=3. Find f(x).	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicProperties", "/Mathematics/Algebra/GeneralAlgebra/AbstractAlgebra", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/LogarithmicDerivative", "/Mathematics/CalculusandAnalysis/Functions/AdditiveFunction", "/Mathematics/CalculusandAnalysis/Functions/DependentVariable", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/IndependentVariable", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/RealVariable", "/Mathematics/CalculusandAnalysis/Functions/Scalar-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/ScalarFunction", "/Mathematics/CalculusandAnalysis/Functions/Single-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis" ]	Recognize that the functional equation forces f to be a constant multiple of the natural logarithm.	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aops_993241	Ik heb een bewijs gevonden dat het idioot is om te zoeken naar een reekssom die nul is: Splits alle hyperharmonische reeksen met willekeurige +'en en -'en in twee gevallen: 1) De eerste term is positief ($ \plus{}1$ dus) Dan is $ \lim_{n\rightarrow\plus{}\infty}s_n\geq1\minus{}\left(\left(\sum_{n\equal{}1}^{\plus{}\infty}{\frac{1}{n^2}}\right)\minus{}1\right)\equal{}1\minus{}\left(\frac{\pi^2}{6}\minus{}1\right)\\\equal{}2\minus{}\frac{\pi^2}{6}\equal{}0,3550659331$ 2) De eerste term is negatief ($ \minus{}1$) Dan is $ \lim_{n\rightarrow\plus{}\infty}s_n\leq\minus{}1\plus{}\left(\left(\sum_{n\equal{}1}^{\plus{}\infty}{\frac{1}{n^2}}\right)\minus{}1\right)\equal{}\minus{}1\plus{}\left(\frac{\pi^2}{6}\minus{}1\right)\\\equal{}\minus{}2\plus{}\frac{\pi^2}{6}\equal{}\minus{}0,3550659331$ Dus $ s_n\in\left[\minus{}\frac{\pi^2}{6},\frac{\pi^2}{6}\right]\setminus \left]\minus{}2\plus{}\frac{\pi^2}{6},2\minus{}\frac{\pi^2}{6}\right[$ en nul zit daar niet bij.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Ik vraag me het volgende af:\r\n1) Kunnen we door in de reeks $ \\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac1{n^2}$ bepaalde $ \\plus{}$-tekens te vervangen door een $ \\minus{}$-teken een reekssom verkrijgen gelijk aan $ \\frac{\\pi^2}{23}$? (Het is evident dat dit niet lukt door slechts een eindig aantal + tekens te vervangen...)\r\n\r\n2) Kunnen we door in de reeks $ \\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac1{n^2}$ bepaalde $ \\plus{}$-tekens te vervangen door een $ \\minus{}$-teken een reekssom verkrijgen die gelijk is aan nul of willekeurig dicht bij nul ligt?", "content_html": "Ik vraag me het volgende af:<br>\n1) Kunnen we door in de reeks <img src=\"//latex.artofproblemsolving.com/5/c/9/5c932c158fb80b13f6cdeb9ee525b80621783786.png\" class=\"latex\" alt=\"$ \\sum_{n=1}^{+\\infty}\\frac1{n^2}$\" style=\"vertical-align: -20px\" width=\"49\" height=\"50\" > bepaalde <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/8/248fce2bc5a8fa499d9ebd6b6524084dffc86951.png\" class=\"latex\" alt=\"$ +$\" style=\"vertical-align: -1px\" width=\"13\" height=\"12\" >-</span>tekens te vervangen door een <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/c/8cc809405164c38b91adffd25eb9c2ab191d2144.png\" class=\"latex\" alt=\"$ -$\" style=\"vertical-align: 4px\" width=\"13\" height=\"1\" >-</span>teken een reekssom verkrijgen gelijk aan <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" >?</span> (Het is evident dat dit niet lukt door slechts een eindig aantal + tekens te vervangen...)<br>\n<br>\n2) Kunnen we door in de reeks <img src=\"//latex.artofproblemsolving.com/5/c/9/5c932c158fb80b13f6cdeb9ee525b80621783786.png\" class=\"latex\" alt=\"$ \\sum_{n=1}^{+\\infty}\\frac1{n^2}$\" style=\"vertical-align: -20px\" width=\"49\" height=\"50\" > bepaalde <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/8/248fce2bc5a8fa499d9ebd6b6524084dffc86951.png\" class=\"latex\" alt=\"$ +$\" style=\"vertical-align: -1px\" width=\"13\" height=\"12\" >-</span>tekens te vervangen door een <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/c/8cc809405164c38b91adffd25eb9c2ab191d2144.png\" class=\"latex\" alt=\"$ -$\" style=\"vertical-align: 4px\" width=\"13\" height=\"1\" >-</span>teken een reekssom verkrijgen die gelijk is aan nul of willekeurig dicht bij nul ligt?", "post_id": 4406732, "post_number": 1, "post_time_unix": 1192103485, "post_time_utc": "2007-10-11 11:51:25 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" }, { "attachments": [], "content_bbcode": "Ik heb een bewijs gevonden dat het idioot is om te zoeken naar een reekssom die nul is:\r\n\r\nSplits alle hyperharmonische reeksen met willekeurige +'en en -'en in twee gevallen:\r\n\r\n1) De eerste term is positief ($ \\plus{}1$ dus)\r\n\r\nDan is $ \\lim_{n\\rightarrow\\plus{}\\infty}s_n\\geq1\\minus{}\\left(\\left(\\sum_{n\\equal{}1}^{\\plus{}\\infty}{\\frac{1}{n^2}}\\right)\\minus{}1\\right)\\equal{}1\\minus{}\\left(\\frac{\\pi^2}{6}\\minus{}1\\right)\\\\\\equal{}2\\minus{}\\frac{\\pi^2}{6}\\equal{}0,3550659331$\r\n\r\n2) De eerste term is negatief ($ \\minus{}1$)\r\n\r\nDan is $ \\lim_{n\\rightarrow\\plus{}\\infty}s_n\\leq\\minus{}1\\plus{}\\left(\\left(\\sum_{n\\equal{}1}^{\\plus{}\\infty}{\\frac{1}{n^2}}\\right)\\minus{}1\\right)\\equal{}\\minus{}1\\plus{}\\left(\\frac{\\pi^2}{6}\\minus{}1\\right)\\\\\\equal{}\\minus{}2\\plus{}\\frac{\\pi^2}{6}\\equal{}\\minus{}0,3550659331$\r\n\r\nDus $ s_n\\in\\left[\\minus{}\\frac{\\pi^2}{6},\\frac{\\pi^2}{6}\\right]\\setminus \\left]\\minus{}2\\plus{}\\frac{\\pi^2}{6},2\\minus{}\\frac{\\pi^2}{6}\\right[$\r\nen nul zit daar niet bij.", "content_html": "Ik heb een bewijs gevonden dat het idioot is om te zoeken naar een reekssom die nul is:<br>\n<br>\nSplits alle hyperharmonische reeksen met willekeurige +'en en -'en in twee gevallen:<br>\n<br>\n1) De eerste term is positief <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/9/7/e/97e739d0c675e71fdda2adcca580a73752b66b35.png\" class=\"latex\" alt=\"$ +1$\" style=\"vertical-align: -1px\" width=\"22\" height=\"13\" ></span> dus)<br>\n<br>\nDan is <img src=\"//latex.artofproblemsolving.com/9/5/f/95fad1ddaad0853b7d80418b9f06c0c1a0335b01.png\" class=\"latex\" alt=\"$ \\lim_{n\\rightarrow+\\infty}s_n\\geq1-\\left(\\left(\\sum_{n=1}^{+\\infty}{\\frac{1}{n^2}}\\right)-1\\right)=1-\\left(\\frac{\\pi^2}{6}-1\\right)\\\\=2-\\frac{\\pi^2}{6}=0,3550659331$\" style=\"vertical-align: -12px\" width=\"391\" height=\"94\" ><br>\n<br>\n2) De eerste term is negatief <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/8/3/7/837d12377ff70746d353a495e0ac3cefb465208e.png\" class=\"latex\" alt=\"$ -1$\" style=\"vertical-align: 0px\" width=\"22\" height=\"12\" >)</span><br>\n<br>\nDan is <img src=\"//latex.artofproblemsolving.com/f/9/c/f9c7b869825bf404b6201a1066eb03c64eaa3b6c.png\" class=\"latex\" alt=\"$ \\lim_{n\\rightarrow+\\infty}s_n\\leq-1+\\left(\\left(\\sum_{n=1}^{+\\infty}{\\frac{1}{n^2}}\\right)-1\\right)=-1+\\left(\\frac{\\pi^2}{6}-1\\right)\\\\=-2+\\frac{\\pi^2}{6}=-0,3550659331$\" style=\"vertical-align: -12px\" width=\"419\" height=\"94\" ><br>\n<br>\nDus <img src=\"//latex.artofproblemsolving.com/8/9/a/89ae7dc206ed2b8e6d09b7b2aa6edf2cb48db5ce.png\" class=\"latex\" alt=\"$ s_n\\in\\left[-\\frac{\\pi^2}{6},\\frac{\\pi^2}{6}\\right]\\setminus \\left]-2+\\frac{\\pi^2}{6},2-\\frac{\\pi^2}{6}\\right[$\" style=\"vertical-align: -17px\" width=\"291\" height=\"43\" ><br>\nen nul zit daar niet bij.", "post_id": 4406733, "post_number": 2, "post_time_unix": 1192462259, "post_time_utc": "2007-10-15 15:30:59 UTC", "thanks_received": 1, "user_id": 28394, "username": "Goe.Pieter" }, { "attachments": [], "content_bbcode": "Dat is inderdaad een correct antwoord!\r\nBedankt, Pieter.", "content_html": "Dat is inderdaad een correct antwoord!<br>\nBedankt, Pieter.", "post_id": 4406734, "post_number": 3, "post_time_unix": 1192469637, "post_time_utc": "2007-10-15 17:33:57 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" }, { "attachments": [], "content_bbcode": "Voor $ \\frac{\\pi^2}{23}$ werkt deze methode niet doordat het net iets te groot is. Als uw geluksgetal het perfecte getal 28 was geweest was alles veel simpeler geweest.\r\nEen hyperharmonische reeks met reekssom $ \\frac{\\pi^2}{23}$ bestaat dus waarschijnlijk wel. Met ongeveer dezelfde methode als die hierboven kan je de eerste 5 termen met zekerheid vinden:\r\n\r\n$ \\plus{}1\\minus{}\\frac{1}{2^2}\\minus{}\\frac{1}{3^2}\\minus{}\\frac{1}{4^2}\\minus{}\\frac{1}{5^2}$\r\n\r\nWaarom dat zo is lijkt me logisch, maar toch teveel werk om helemaal uit te schrijven. Vanaf de zesde term valt voorlopig alle zekerheid weg, en zijn er misschien zelfs meerdere mogelijkheden.\r\nMaar als je nu voor elke term $ n$ apart gaat zien of $ s_n$ groter of kleiner dan $ \\frac{\\pi^2}{23}$ is en zo kiest tussen + en - moet je uiteindelijk toch op eender welk getal binnen het bereik kunnen komen?", "content_html": "Voor <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > werkt deze methode niet doordat het net iets te groot is. Als uw geluksgetal het perfecte getal 28 was geweest was alles veel simpeler geweest.<br>\nEen hyperharmonische reeks met reekssom <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > bestaat dus waarschijnlijk wel. Met ongeveer dezelfde methode als die hierboven kan je de eerste 5 termen met zekerheid vinden:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/5/4/a543c1ab6d88ce06a51909d54e3876610a7efea6.png\" class=\"latex\" alt=\"$ +1-\\frac{1}{2^2}-\\frac{1}{3^2}-\\frac{1}{4^2}-\\frac{1}{5^2}$\" style=\"vertical-align: -13px\" width=\"192\" height=\"37\" ><br>\n<br>\nWaarom dat zo is lijkt me logisch, maar toch teveel werk om helemaal uit te schrijven. Vanaf de zesde term valt voorlopig alle zekerheid weg, en zijn er misschien zelfs meerdere mogelijkheden.<br>\nMaar als je nu voor elke term <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > apart gaat zien of <img src=\"//latex.artofproblemsolving.com/f/7/2/f726505b9245dbd13b9e55343e07962ffa4b34c4.png\" class=\"latex\" alt=\"$ s_n$\" style=\"vertical-align: -2px\" width=\"16\" height=\"10\" > groter of kleiner dan <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > is en zo kiest tussen + en - moet je uiteindelijk toch op eender welk getal binnen het bereik kunnen komen?", "post_id": 4406735, "post_number": 4, "post_time_unix": 1192475419, "post_time_utc": "2007-10-15 19:10:19 UTC", "thanks_received": 1, "user_id": 28394, "username": "Goe.Pieter" }, { "attachments": [], "content_bbcode": "Ik heb ook de hoop opgegeven om dit met de hand te doen bij de term met $ \\mp\\frac{1}{7^2}$. Dat er meerdere mogelijkheden zijn is zeer aannemelijk (zie hiervoor een andere post). Misschien dat ik eens de computer inschakel om het saaie rekenwerk voor ons te doen, maar ik vrees dat het aantal mogelijkheden snel de spuigaten zal uitlopen...\r\n[quote]Maar als je nu voor elke term $ n$ apart gaat zien of $ s_n$ groter of kleiner dan $ \\frac{\\pi^2}{23}$ is en zo kiest tussen + en - moet je uiteindelijk toch op eender welk getal binnen het bereik kunnen komen?\n[/quote]\r\nAls je bedoelt dat we willekeurig dicht bij $ \\frac{\\pi^2}{23}$ kunnen geraken door telkens het volgende teken zó te kiezen dat we het dichtste bij $ \\frac{\\pi^2}{23}$ komen, dan lijkt me dat ook wel waar te zijn. Maar het is niet bewezen. Trouwens weten we ook niet precies wat het bereik is: misschien vallen er nog wel hier en daar kleine intervalletjes uit de boot...", "content_html": "Ik heb ook de hoop opgegeven om dit met de hand te doen bij de term met <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/9/0/c9085d70035227abd619c1b6b75aa907d9310db7.png\" class=\"latex\" alt=\"$ \\mp\\frac{1}{7^2}$\" style=\"vertical-align: -12px\" width=\"33\" height=\"37\" >.</span> Dat er meerdere mogelijkheden zijn is zeer aannemelijk (zie hiervoor een andere post). Misschien dat ik eens de computer inschakel om het saaie rekenwerk voor ons te doen, maar ik vrees dat het aantal mogelijkheden snel de spuigaten zal uitlopen...\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">Maar als je nu voor elke term <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > apart gaat zien of <img src=\"//latex.artofproblemsolving.com/f/7/2/f726505b9245dbd13b9e55343e07962ffa4b34c4.png\" class=\"latex\" alt=\"$ s_n$\" style=\"vertical-align: -2px\" width=\"16\" height=\"10\" > groter of kleiner dan <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > is en zo kiest tussen + en - moet je uiteindelijk toch op eender welk getal binnen het bereik kunnen komen?</div>\n</div>\nAls je bedoelt dat we willekeurig dicht bij <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > kunnen geraken door telkens het volgende teken zó te kiezen dat we het dichtste bij <img src=\"//latex.artofproblemsolving.com/0/4/b/04bb8e492c97fbf2a1c033c9ab97e78b14fdcbef.png\" class=\"latex\" alt=\"$ \\frac{\\pi^2}{23}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"39\" > komen, dan lijkt me dat ook wel waar te zijn. Maar het is niet bewezen. Trouwens weten we ook niet precies wat het bereik is: misschien vallen er nog wel hier en daar kleine intervalletjes uit de boot...", "post_id": 4406736, "post_number": 5, "post_time_unix": 1192561425, "post_time_utc": "2007-10-16 19:03:45 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" } ], "source": null }	Ik vraag me het volgende af: 1) Kunnen we door in de reeks $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2}}$ bepaalde plustekens te vervangen door een minteken een reekssom verkrijgen gelijk aan $\displaystyle \frac{\pi^{2}}{23}$? 2) Kunnen we door in de reeks $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2}}$ bepaalde plustekens te vervangen door een minteken een reekssom verkrijgen die gelijk is aan nul of willekeurig dicht bij nul ligt?	[ "/Mathematics/CalculusandAnalysis/Calculus/Limits/GreatestLowerBound", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Infimum", "/Mathematics/CalculusandAnalysis/Calculus/Limits/LeastUpperBound", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Supremum", "/Mathematics/CalculusandAnalysis/Series/Convergence", "/Mathematics/CalculusandAnalysis/Series/GeneralSeries" ]	Bound any signed version of the Basel series by the first term plus the total remaining absolute sum, showing it cannot approach zero.	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aops_993244	Juist! We weten immers dat: $ \frac1{n(n\plus{}1)}\equal{}\frac1n\minus{}<\frac1{n\plus{}1}$ De reeks is dus te schrijven als: $ \sum_{n\equal{}1}^{\plus{}\infty}\frac{x^n}{n}\minus{}\sum_{n\equal{}1}^{\plus{}\infty}\frac{x^n}{n\plus{}1}$ u weten we voor $ x\in]\minus{}1,1]$: $ \ln(1\plus{}x)\equal{}\sum_{n\equal{}1}^{\plus{}\infty}(\minus{}1)^{n\minus{}1}\frac{x^n}{n}$ waaruit volgt, voor $ x\in[\minus{}1,1[$: $ \ln(1\minus{}x)\equal{}\sum_{n\equal{}1}^{\plus{}\infty}(\minus{}1)^{n\minus{}1}\frac{(\minus{}x)^n}{n}\equal{}\sum_{n\equal{}1}^{\plus{}\infty}(\minus{}1)^{2n\minus{}1}\frac{x^n}{n}\equal{}\minus{}\sum_{n\equal{}1}^{\plus{}\infty}\frac{x^n}{n}$ Anderzijds is $ \sum_{n\equal{}1}^{\plus{}\infty}\frac{x^n}{n\plus{}1}\equal{}\frac1x\sum_{n\equal{}1}^{\plus{}\infty}\frac{x^{n\plus{}1}}{n\plus{}1}\equal{}\frac1x\cdot\left(\minus{}\ln(1\minus{}x)\minus{}x\right)$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "[b]oefening[/b]\r\nVan welke functie is dit de machtreeks:\r\n$ \\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^n}{n(n\\plus{}1)}$\r\nmet $ \|x\|<1$.", "content_html": "<b>oefening</b><br>\nVan welke functie is dit de machtreeks:<br>\n<img src=\"//latex.artofproblemsolving.com/8/4/4/8443d782531197f319ac2e8e44e5c77dd5321863.png\" class=\"latex\" alt=\"$ \\sum_{n=1}^{+\\infty}\\frac{x^n}{n(n+1)}$\" style=\"vertical-align: -20px\" width=\"98\" height=\"50\" ><br>\nmet <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/6/3/6638fed90d8bd2cfd2d01f9f42013522d74b346b.png\" class=\"latex\" alt=\"$ \|x\|<1$\" style=\"vertical-align: -4px\" width=\"52\" height=\"18\" >.</span>", "post_id": 4406740, "post_number": 1, "post_time_unix": 1192622901, "post_time_utc": "2007-10-17 12:08:21 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" }, { "attachments": [], "content_bbcode": "Ik denk dat het $ \\minus{}\\ln(1\\minus{}x)\\plus{}\\frac{\\ln(1\\minus{}x)}{x}\\plus{}1$ is.", "content_html": "Ik denk dat het <img src=\"//latex.artofproblemsolving.com/d/c/a/dca0231e4946f3170b5d1f42288a8bf8afb059f6.png\" class=\"latex\" alt=\"$ -\\ln(1-x)+\\frac{\\ln(1-x)}{x}+1$\" style=\"vertical-align: -12px\" width=\"215\" height=\"38\" > is.", "post_id": 4406741, "post_number": 2, "post_time_unix": 1192628592, "post_time_utc": "2007-10-17 13:43:12 UTC", "thanks_received": 1, "user_id": 28394, "username": "Goe.Pieter" }, { "attachments": [], "content_bbcode": "Juist!\r\nWe weten immers dat: $ \\frac1{n(n\\plus{}1)}\\equal{}\\frac1n\\minus{}<\\frac1{n\\plus{}1}$\r\nDe reeks is dus te schrijven als:\r\n$ \\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^n}{n}\\minus{}\\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^n}{n\\plus{}1}$\r\nu weten we voor $ x\\in]\\minus{}1,1]$:\r\n$ \\ln(1\\plus{}x)\\equal{}\\sum_{n\\equal{}1}^{\\plus{}\\infty}(\\minus{}1)^{n\\minus{}1}\\frac{x^n}{n}$\r\nwaaruit volgt, voor $ x\\in[\\minus{}1,1[$:\r\n$ \\ln(1\\minus{}x)\\equal{}\\sum_{n\\equal{}1}^{\\plus{}\\infty}(\\minus{}1)^{n\\minus{}1}\\frac{(\\minus{}x)^n}{n}\\equal{}\\sum_{n\\equal{}1}^{\\plus{}\\infty}(\\minus{}1)^{2n\\minus{}1}\\frac{x^n}{n}\\equal{}\\minus{}\\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^n}{n}$\r\nAnderzijds is\r\n$ \\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^n}{n\\plus{}1}\\equal{}\\frac1x\\sum_{n\\equal{}1}^{\\plus{}\\infty}\\frac{x^{n\\plus{}1}}{n\\plus{}1}\\equal{}\\frac1x\\cdot\\left(\\minus{}\\ln(1\\minus{}x)\\minus{}x\\right)$", "content_html": "Juist!<br>\nWe weten immers dat: <img src=\"//latex.artofproblemsolving.com/d/6/8/d68d466f005520da44bebc1a83fbacaaf889c662.png\" class=\"latex\" alt=\"$ \\frac1{n(n+1)}=\\frac1n-<\\frac1{n+1}$\" style=\"vertical-align: -17px\" width=\"193\" height=\"41\" ><br>\nDe reeks is dus te schrijven als:<br>\n<img src=\"//latex.artofproblemsolving.com/6/c/1/6c139eb7a18a179bb96ffc90d77ee0fba233fb38.png\" class=\"latex\" alt=\"$ \\sum_{n=1}^{+\\infty}\\frac{x^n}{n}-\\sum_{n=1}^{+\\infty}\\frac{x^n}{n+1}$\" style=\"vertical-align: -20px\" width=\"147\" height=\"50\" ><br>\nu weten we voor <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/2/f/e2f457c6c15324b7615196c36b561ca5909f29ed.png\" class=\"latex\" alt=\"$ x\\in]-1,1]$\" style=\"vertical-align: -5px\" width=\"84\" height=\"18\" >:</span><br>\n<img src=\"//latex.artofproblemsolving.com/e/f/2/ef2677064531d8a66b55c72793e2a69174b10734.png\" class=\"latex\" alt=\"$ \\ln(1+x)=\\sum_{n=1}^{+\\infty}(-1)^{n-1}\\frac{x^n}{n}$\" style=\"vertical-align: -20px\" width=\"206\" height=\"50\" ><br>\nwaaruit volgt, voor <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/2/9/a296b2cd3e6d3e50596dfb807f4b16deb81ff066.png\" class=\"latex\" alt=\"$ x\\in[-1,1[$\" style=\"vertical-align: -5px\" width=\"83\" height=\"18\" >:</span><br>\n<img src=\"//latex.artofproblemsolving.com/2/0/4/204a20ad7e0973b6f6a2111a073528a2608ab47f.png\" class=\"latex\" alt=\"$ \\ln(1-x)=\\sum_{n=1}^{+\\infty}(-1)^{n-1}\\frac{(-x)^n}{n}=\\sum_{n=1}^{+\\infty}(-1)^{2n-1}\\frac{x^n}{n}=-\\sum_{n=1}^{+\\infty}\\frac{x^n}{n}$\" style=\"vertical-align: -20px\" width=\"471\" height=\"50\" ><br>\nAnderzijds is<br>\n<img src=\"//latex.artofproblemsolving.com/b/f/d/bfd0f5f652e47952608d528ba45c0cd11190b435.png\" class=\"latex\" alt=\"$ \\sum_{n=1}^{+\\infty}\\frac{x^n}{n+1}=\\frac1x\\sum_{n=1}^{+\\infty}\\frac{x^{n+1}}{n+1}=\\frac1x\\cdot\\left(-\\ln(1-x)-x\\right)$\" style=\"vertical-align: -20px\" width=\"376\" height=\"50\" >", "post_id": 4406742, "post_number": 3, "post_time_unix": 1192658461, "post_time_utc": "2007-10-17 22:01:01 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" } ], "source": null }	Geef de functie waarvan de volgende machtreeks de ontwikkeling is: \[ \sum_{n=1}^{\infty}\frac{x^n}{n(n+1)},\qquad \|x\|<1. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/InfinitesimalAnalysis", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/InfinitesimalCalculus", "/Mathematics/CalculusandAnalysis/Functions/ElementaryFunction", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis", "/Mathematics/CalculusandAnalysis/Series/GeneralSeries", "/Mathematics/CalculusandAnalysis/Series/SeriesExpansions" ]	Rewrite 1/[n(n+1)] as 1/n – 1/(n+1) to express the series in terms of the known logarithmic series.	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aops_993247	Het antwoord is 23, en daar zijn drie redenen voor: [list]1) Wat zou het anders moeten zijn op een blog als deze? 2) De rekenmachine zegt het 3) Het bewijs waar de les fysica me tot inspireerde[/list]Voeg vooraan de factor $ \left(1 \plus{} \tan0\right)$ toe. Omdat $ \left(1 \plus{} \tan0\right) \equal{} 1$ verandert dat de uitkomst niet. Er zijn nu 46 factoren. Net als Gauss nemen we nu de eerste en de laatse factoren samen, dan de tweede en de voorlaatste, enz. Zo'n twee factoren geven dan: $ \left(1 \plus{} \tan a\right)\left(1 \plus{} \tan b\right) \equal{} \frac {\left(\cos a \plus{} \sin a\right)\left(\cos b \plus{} \sin b\right)}{\cos a\cdot\cos b}$ $ \equal{} \frac {\cos a\cdot\cos b \plus{} \cos a\cdot\sin b \plus{} \sin a\cdot\cos b \plus{} \sin a\cdot\sin b}{\cos a\cdot\cos b}$ $ \equal{} \frac {\sin\left(a \plus{} b\right) \plus{} \cos\left(a \minus{} b\right)}{\frac {\cos\left(a \plus{} b\right) \plus{} \cos\left(a \minus{} b\right)}{2}} \equal{} 2\cdot\frac {\frac {\sqrt {2}}{2} \plus{} \cos\left(a \minus{} b\right)}{\frac {\sqrt {2}}{2} \plus{} \cos\left(a \minus{} b\right)}$ (want $ ^{a \plus{} b \equal{} 45}$) $ \equal{} 2$ Omdat er 46 factoren zijn die we per 2 samen nemen blijven er 23 over. Dat resulteert dus in $ 2^{23}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Bepaal $ x$ in volgende uitdrukking (hoeken zijn in graden):\r\n$ (1\\plus{}\\tan1)(1\\plus{}\\tan2)(1\\plus{}\\tan3)\\cdots(1\\plus{}\\tan45)\\equal{}2^x$", "content_html": "Bepaal <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > in volgende uitdrukking (hoeken zijn in graden):<br>\n<img src=\"//latex.artofproblemsolving.com/e/1/8/e184bd4e4cb0902ecbf382f8f24289aab3b020a1.png\" class=\"latex\" alt=\"$ (1+\\tan1)(1+\\tan2)(1+\\tan3)\\cdots(1+\\tan45)=2^x$\" style=\"vertical-align: -4px\" width=\"412\" height=\"18\" >", "post_id": 4406747, "post_number": 1, "post_time_unix": 1193694815, "post_time_utc": "2007-10-29 21:53:35 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" }, { "attachments": [], "content_bbcode": "Het antwoord is 23, en daar zijn drie redenen voor:\r\n[list]1) Wat zou het anders moeten zijn op een blog als deze?\n2) De rekenmachine zegt het\n3) Het bewijs waar de les fysica me tot inspireerde[/list]Voeg vooraan de factor $ \\left(1 \\plus{} \\tan0\\right)$ toe. Omdat $ \\left(1 \\plus{} \\tan0\\right) \\equal{} 1$ verandert dat de uitkomst niet. Er zijn nu 46 factoren. Net als Gauss nemen we nu de eerste en de laatse factoren samen, dan de tweede en de voorlaatste, enz.\r\n\r\nZo'n twee factoren geven dan:\r\n\r\n$ \\left(1 \\plus{} \\tan a\\right)\\left(1 \\plus{} \\tan b\\right) \\equal{} \\frac {\\left(\\cos a \\plus{} \\sin a\\right)\\left(\\cos b \\plus{} \\sin b\\right)}{\\cos a\\cdot\\cos b}$\r\n\r\n$ \\equal{} \\frac {\\cos a\\cdot\\cos b \\plus{} \\cos a\\cdot\\sin b \\plus{} \\sin a\\cdot\\cos b \\plus{} \\sin a\\cdot\\sin b}{\\cos a\\cdot\\cos b}$\r\n\r\n$ \\equal{} \\frac {\\sin\\left(a \\plus{} b\\right) \\plus{} \\cos\\left(a \\minus{} b\\right)}{\\frac {\\cos\\left(a \\plus{} b\\right) \\plus{} \\cos\\left(a \\minus{} b\\right)}{2}} \\equal{} 2\\cdot\\frac {\\frac {\\sqrt {2}}{2} \\plus{} \\cos\\left(a \\minus{} b\\right)}{\\frac {\\sqrt {2}}{2} \\plus{} \\cos\\left(a \\minus{} b\\right)}$ (want $ ^{a \\plus{} b \\equal{} 45}$)\r\n$ \\equal{} 2$\r\n\r\nOmdat er 46 factoren zijn die we per 2 samen nemen blijven er 23 over. Dat resulteert dus in $ 2^{23}$", "content_html": "Het antwoord is 23, en daar zijn drie redenen voor:\n<ul class=\"bbcode_list\">\n1) Wat zou het anders moeten zijn op een blog als deze?<br>\n2) De rekenmachine zegt het<br>\n3) Het bewijs waar de les fysica me tot inspireerde</ul>\nVoeg vooraan de factor <img src=\"//latex.artofproblemsolving.com/2/a/a/2aad2ed2c0425e7241359a96215a0ae2522a9b7a.png\" class=\"latex\" alt=\"$ \\left(1 + \\tan0\\right)$\" style=\"vertical-align: -4px\" width=\"82\" height=\"18\" > toe. Omdat <img src=\"//latex.artofproblemsolving.com/b/7/c/b7cc742ed222a6a887c9fd34f2f5e0b13e2ee7aa.png\" class=\"latex\" alt=\"$ \\left(1 + \\tan0\\right) = 1$\" style=\"vertical-align: -4px\" width=\"116\" height=\"18\" > verandert dat de uitkomst niet. Er zijn nu 46 factoren. Net als Gauss nemen we nu de eerste en de laatse factoren samen, dan de tweede en de voorlaatste, enz.<br>\n<br>\nZo'n twee factoren geven dan:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/1/8/e18ae3ad27ca82656d002b3e1653304aab3d38ba.png\" class=\"latex\" alt=\"$ \\left(1 + \\tan a\\right)\\left(1 + \\tan b\\right) = \\frac {\\left(\\cos a + \\sin a\\right)\\left(\\cos b + \\sin b\\right)}{\\cos a\\cdot\\cos b}$\" style=\"vertical-align: -12px\" width=\"412\" height=\"38\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/d/4/dd439e03bf30dd52ec1572431a2f42f346c41532.png\" class=\"latex\" alt=\"$ = \\frac {\\cos a\\cdot\\cos b + \\cos a\\cdot\\sin b + \\sin a\\cdot\\cos b + \\sin a\\cdot\\sin b}{\\cos a\\cdot\\cos b}$\" style=\"vertical-align: -12px\" width=\"421\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/1/6/b162e2060afcc45eabd9d4b97309d93cf53265f9.png\" class=\"latex\" alt=\"$ = \\frac {\\sin\\left(a + b\\right) + \\cos\\left(a - b\\right)}{\\frac {\\cos\\left(a + b\\right) + \\cos\\left(a - b\\right)}{2}} = 2\\cdot\\frac {\\frac {\\sqrt {2}}{2} + \\cos\\left(a - b\\right)}{\\frac {\\sqrt {2}}{2} + \\cos\\left(a - b\\right)}$\" style=\"vertical-align: -22px\" width=\"379\" height=\"53\" > (want <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/e/0/4e0a2ae94a4f9366f605e34f0fde062b07048b18.png\" class=\"latex\" alt=\"$ ^{a + b = 45}$\" style=\"vertical-align: 6px\" width=\"45\" height=\"9\" >)</span><br>\n<img src=\"//latex.artofproblemsolving.com/d/c/8/dc84ca8d5a01635853d324802b42b9a0f7ec50f9.png\" class=\"latex\" alt=\"$ = 2$\" width=\"27\" height=\"12\" ><br>\n<br>\nOmdat er 46 factoren zijn die we per 2 samen nemen blijven er 23 over. Dat resulteert dus in <img src=\"//latex.artofproblemsolving.com/7/e/5/7e527fdc4d049e189c7d82b96b9b81c99bfdd9f5.png\" class=\"latex\" alt=\"$ 2^{23}$\" width=\"21\" height=\"15\" >", "post_id": 4406748, "post_number": 2, "post_time_unix": 1194624126, "post_time_utc": "2007-11-09 16:02:06 UTC", "thanks_received": 1, "user_id": 28394, "username": "Goe.Pieter" }, { "attachments": [], "content_bbcode": "Dat is inderdaad juist!", "content_html": "Dat is inderdaad juist!", "post_id": 4406749, "post_number": 3, "post_time_unix": 1194867372, "post_time_utc": "2007-11-12 11:36:12 UTC", "thanks_received": 1, "user_id": 21653, "username": "B23" } ], "source": null }	Bepaal $x$ in de volgende uitdrukking (hoeken zijn in graden): \[ (1+\tan 1^\circ)(1+\tan 2^\circ)(1+\tan 3^\circ)\cdots(1+\tan 45^\circ)=2^x. \]	[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Pair each factor (1+tan a) with (1+tan (45°‑a)) so that their product equals 2.	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aops_993340	um, factoring gives you; $ 3\sin{2\theta} \equal{} 2\sin{2\theta}\cos{2\theta}(2\cos{2\theta} \plus{} 1)$ so if you say that $ \cos{2\theta} \equal{} n$, you need to solve the quadratic equation: $ 2n(2n\plus{}1) \equal{} 3$ $ 4n^2 \plus{} 2n \minus{} 3\equal{}0$ $ \frac{2 \pm \sqrt{2^2\plus{}3\cdot 4}}{8}$ $ \cos{2\theta} \equal{} (\frac{\minus{}1}{2}, \frac{3}{4})$ since $ \cos^{\minus{}1}{\minus{}1/2} > \cos^{\minus{}1}{3\4}$, the answer is $ \cos^{\minus{}1}({3/4})/2 \approx 20.705^\circ$ huh? the maximum value however is $ 120/2 \equal{} 60^\circ$ which is a nicer number .	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Okay here's my attempt at a decent, if ugly, problem (jumbled a bunch of trig identities together hinthint, blah.):\r\n\r\nFind the lowest positive value of $ \\theta$:\r\n\\[ 2\\sin{4\\theta}\\cos{2\\theta}\\equal{}4\\sin{4\\theta}\\minus{}3\\sin{2\\theta}\r\n\\]", "content_html": "Okay here's my attempt at a decent, if ugly, problem (jumbled a bunch of trig identities together hinthint, blah.):<br>\n<br>\nFind the lowest positive value of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/d/c/cdc87059950c5386ad9d0c2c42c598418eda35d3.png\" class=\"latex\" alt=\"$ \\theta$\" width=\"8\" height=\"13\" >:</span><br>\n<img src=\"//latex.artofproblemsolving.com/4/0/5/4053471034d554ea7e34b08160a0e469f153adb7.png\" class=\"latexcenter\" alt=\"\\[ 2\\sin{4\\theta}\\cos{2\\theta}=4\\sin{4\\theta}-3\\sin{2\\theta}\n\\]\" width=\"260\" height=\"13\" >", "post_id": 4406985, "post_number": 1, "post_time_unix": 1216274934, "post_time_utc": "2008-07-17 06:08:54 UTC", "thanks_received": 1, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "um, factoring gives you;\r\n\r\n$ 3\\sin{2\\theta} \\equal{} 2\\sin{2\\theta}\\cos{2\\theta}(2\\cos{2\\theta} \\plus{} 1)$\r\n\r\nso if you say that $ \\cos{2\\theta} \\equal{} n$, you need to solve the quadratic equation:\r\n\r\n$ 2n(2n\\plus{}1) \\equal{} 3$\r\n\r\n$ 4n^2 \\plus{} 2n \\minus{} 3\\equal{}0$\r\n\r\n$ \\frac{2 \\pm \\sqrt{2^2\\plus{}3\\cdot 4}}{8}$\r\n\r\n$ \\cos{2\\theta} \\equal{} (\\frac{\\minus{}1}{2}, \\frac{3}{4})$\r\n\r\nsince $ \\cos^{\\minus{}1}{\\minus{}1/2} > \\cos^{\\minus{}1}{3\\4}$, the answer is $ \\cos^{\\minus{}1}({3/4})/2 \\approx 20.705^\\circ$ huh?\r\n\r\nthe maximum value however is $ 120/2 \\equal{} 60^\\circ$ which is a nicer number .", "content_html": "um, factoring gives you;<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/6/f/a/6faa99de1dac617cc72f3750ba7a7ad00574ae08.png\" class=\"latex\" alt=\"$ 3\\sin{2\\theta} = 2\\sin{2\\theta}\\cos{2\\theta}(2\\cos{2\\theta} + 1)$\" style=\"vertical-align: -4px\" width=\"285\" height=\"18\" ><br>\n<br>\nso if you say that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/e/d2e9d024c5026b58f97ba95b4026cd950bc65ba8.png\" class=\"latex\" alt=\"$ \\cos{2\\theta} = n$\" width=\"80\" height=\"13\" >,</span> you need to solve the quadratic equation:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/9/b/89b24615e16549f543dca1e60d4c0b013f9647a7.png\" class=\"latex\" alt=\"$ 2n(2n+1) = 3$\" style=\"vertical-align: -4px\" width=\"118\" height=\"18\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/3/7/f/37f573a7cdbb2c5a76ad8e9f98d8e12fc5da48cb.png\" class=\"latex\" alt=\"$ 4n^2 + 2n - 3=0$\" style=\"vertical-align: -1px\" width=\"133\" height=\"16\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/a/6/ea6294a8d4dae9c66b631f459feb0c515caf3594.png\" class=\"latex\" alt=\"$ \\frac{2 \\pm \\sqrt{2^2+3\\cdot 4}}{8}$\" style=\"vertical-align: -12px\" width=\"119\" height=\"40\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/b/b/abb1be766cb7da3454eabbbf6034c2cd6709fe9e.png\" class=\"latex\" alt=\"$ \\cos{2\\theta} = (\\frac{-1}{2}, \\frac{3}{4})$\" style=\"vertical-align: -13px\" width=\"130\" height=\"38\" ><br>\n<br>\nsince <span style=\"white-space:nowrap;\"><span class=\"aopscode-error aopscode-latex-error\">$ \\cos^{-1}{-1/2} > \\cos^{-1}{3\\4}$</span>,</span> the answer is <img src=\"//latex.artofproblemsolving.com/1/7/a/17a5adb90e3f2ac1c348602eaaf208961d7ac151.png\" class=\"latex\" alt=\"$ \\cos^{-1}({3/4})/2 \\approx 20.705^\\circ$\" style=\"vertical-align: -4px\" width=\"182\" height=\"19\" > huh?<br>\n<br>\nthe maximum value however is <img src=\"//latex.artofproblemsolving.com/a/8/4/a84402f5dadc4ff57d914e28761ddecc5cea6264.png\" class=\"latex\" alt=\"$ 120/2 = 60^\\circ$\" style=\"vertical-align: -4px\" width=\"93\" height=\"18\" > which is a nicer number .", "post_id": 4406986, "post_number": 2, "post_time_unix": 1216311418, "post_time_utc": "2008-07-17 16:16:58 UTC", "thanks_received": 1, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "uh oh, I made an arithmetic mistake while making the problem XD. edited.", "content_html": "uh oh, I made an arithmetic mistake while making the problem XD. edited.", "post_id": 4406987, "post_number": 3, "post_time_unix": 1216334524, "post_time_utc": "2008-07-17 22:42:04 UTC", "thanks_received": 1, "user_id": 27042, "username": "undefined117" } ], "source": null }	Find the lowest positive value of $\theta$ satisfying \[ 2\sin(4\theta)\cos(2\theta)=4\sin(4\theta)-3\sin(2\theta). \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial" ]	Factor the equation to isolate sin 2θ and obtain a quadratic equation in cos 2θ.	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aops_993345	[hide="Solution"]\[ \begin{align} y&\equal{}Cx^2\plus{}x\\ \cfrac{y\minus{}x}{x^2}&\equal{}C\\ yx^{\minus{}2}\minus{}x^{\minus{}1}&\equal{}C\\ \minus{}2yx^{\minus{}3}\plus{}x^{\minus{}2}\cfrac{dy}{dx}\plus{}x^{\minus{}2}&\equal{}0\\ \cfrac{1}{x^2}\cdot\cfrac{dy}{dx}&\equal{}\cfrac{2y}{x^3}\minus{}\cfrac{1}{x^2}\\ \cfrac{dy}{dx}&\equal{}\boxed{\cfrac{2y}{x}\minus{}1} \end{align}\][/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find the differential equation whose general solution is $ y\\equal{}Cx^2\\plus{}x$, where $ C$ is any constant.\r\n\r\nDWIT.", "content_html": "Find the differential equation whose general solution is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/7/30795da433b46d40b6f2ec1b39c59247a63335f3.png\" class=\"latex\" alt=\"$ y=Cx^2+x$\" style=\"vertical-align: -3px\" width=\"98\" height=\"18\" >,</span> where <img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" > is any constant.<br>\n<br>\nDWIT.", "post_id": 4407015, "post_number": 1, "post_time_unix": 1216803739, "post_time_utc": "2008-07-23 09:02:19 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "[hide=\"Solution\"]\\[ \\begin{align}\ny&\\equal{}Cx^2\\plus{}x\\\\\n\\cfrac{y\\minus{}x}{x^2}&\\equal{}C\\\\\nyx^{\\minus{}2}\\minus{}x^{\\minus{}1}&\\equal{}C\\\\\n\\minus{}2yx^{\\minus{}3}\\plus{}x^{\\minus{}2}\\cfrac{dy}{dx}\\plus{}x^{\\minus{}2}&\\equal{}0\\\\\n\\cfrac{1}{x^2}\\cdot\\cfrac{dy}{dx}&\\equal{}\\cfrac{2y}{x^3}\\minus{}\\cfrac{1}{x^2}\\\\\n\\cfrac{dy}{dx}&\\equal{}\\boxed{\\cfrac{2y}{x}\\minus{}1}\n\\end{align}\\][/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Solution</a><div class=\"cmty-hide-content\" style=\"display:none\"><pre class=\"aopscode-error aopscode-latex-error\">\\[ \\begin{align}\ny&=Cx^2+x\\\\\n\\cfrac{y-x}{x^2}&=C\\\\\nyx^{-2}-x^{-1}&=C\\\\\n-2yx^{-3}+x^{-2}\\cfrac{dy}{dx}+x^{-2}&=0\\\\\n\\cfrac{1}{x^2}\\cdot\\cfrac{dy}{dx}&=\\cfrac{2y}{x^3}-\\cfrac{1}{x^2}\\\\\n\\cfrac{dy}{dx}&=\\boxed{\\cfrac{2y}{x}-1}\n\\end{align}\\]</pre></div>", "post_id": 4407016, "post_number": 2, "post_time_unix": 1217119572, "post_time_utc": "2008-07-27 00:46:12 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" } ], "source": null }	Find the differential equation whose general solution is $y = Cx^2 + x$, where $C$ is an arbitrary constant.	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/ImplicitDifferentiation", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialEquations/DifferentialEquationSolving", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialEquations/OrdinaryDifferentialEquations", "/Mathematics/CalculusandAnalysis/DifferentialEquations/DifferentialEquationSolving/ODESolving", "/Mathematics/CalculusandAnalysis/DifferentialEquations/OrdinaryDifferentialEquations/First-OrderOrdinaryDifferentialEquation", "/Mathematics/CalculusandAnalysis/DifferentialEquations/OrdinaryDifferentialEquations/ODE", "/Mathematics/CalculusandAnalysis/DifferentialEquations/OrdinaryDifferentialEquations/OrdinaryDifferentialEquation" ]	Isolate the constant C from the general solution and differentiate to eliminate it, yielding the differential equation.	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aops_993346	Here's my solution: With algebraic intuition and experience (which I have to some degree :D ), we think to multiply the first and last terms on the left side of the equation and the second and third terms. This yields: $ \displaystyle (2 \plus{} x)(3 \plus{} x)(4 \plus{} x)(5 \plus{} x) \equal{} 2115\Rightarrow(x^2 \plus{} 7x \plus{} 10)(x^2 \plus{} 7x \plus{} 12) \equal{} 2115$ Now we let $ \displaystyle n \equal{} x^2 \plus{} 7x \plus{} 10$ to get: $ \displaystyle n(n \plus{} 2) \equal{} 2115\Rightarrow n^2 \plus{} 2n \minus{} 2115 \equal{} 0$. We take the prime factorization of $ \displaystyle 2115$, which is $ \displaystyle 3^25^147^1$. Then, we can quickly see how to factor the quadratic in $ \displaystyle n$ above: $ \displaystyle n^2 \plus{} 2n \minus{} 2115 \equal{} 0\Rightarrow(n \plus{} 47)(n \minus{} 45) \equal{} 0$. So, our solutions are $ \displaystyle n \equal{} \minus{} 47$ and $ \displaystyle n \equal{} 45$. We plug in both solutions of $ \displaystyle n$ in our definition of $ \displaystyle n$ ($ \displaystyle n \equal{} x^2 \plus{} 7x \plus{} 10$) and see what happens after rearranging: $ \begin{align}\displaystyle \minus{} 47 \equal{} x^2 \plus{} 7x \plus{} 10\Rightarrow x^2 \plus{} 7x \plus{} 57 \equal{} 0 \\ \displaystyle 45 \equal{} x^2 \plus{} 7x \plus{} 10\Rightarrow x^2 \plus{} 7x \minus{} 35 \equal{} 0\end{align}$ Obviously, we can see that the discriminant is negative in $ \displaystyle (1)$, so only non-real complex solutions will arise. We discard that equation. (Note that we are asked for the greatest value of $ \displaystyle x$; how can one non-real complex number be greater than another? It can't. Thus the non-real, complex solutions of $ \displaystyle x$ cannot be solutions because we won't be able to pick the "greatest" one.) The second equation seems alright, so we apply the quadratic formula (after seeing that trying to mentally factor is futile): $ \displaystyle x^2 \plus{} 7x \minus{} 35 \equal{} 0\Rightarrow x \equal{} \frac { \minus{} 7\pm3\sqrt {21}}{2}$ We want the largest $ \displaystyle x$ possible, so we choose the $ \displaystyle \plus{}$ sign in our $ \displaystyle \pm$ expression to get our final answer of: \[ \displaystyle \boxed{\frac { \minus{} 7 \plus{} 3\sqrt {21}}{2}} \]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find the largest $ x$ that satisfies the following condition:\r\n\r\n$ (2 \\plus{} x)(3 \\plus{} x)(4 \\plus{} x)(5 \\plus{} x) \\equal{}2115$", "content_html": "Find the largest <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > that satisfies the following condition:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/5/3/d/53daff4b9bfe3a8a4149c5e6accedb8bbe278670.png\" class=\"latex\" alt=\"$ (2 + x)(3 + x)(4 + x)(5 + x) =2115$\" style=\"vertical-align: -4px\" width=\"283\" height=\"18\" >", "post_id": 4407017, "post_number": 1, "post_time_unix": 1217035434, "post_time_utc": "2008-07-26 01:23:54 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "Here's my solution:\r\n\r\n\r\nWith algebraic intuition and experience (which I have to some degree :D ), we think to multiply the first and last terms on the left side of the equation and the second and third terms.\r\n\r\nThis yields:\r\n$ \\displaystyle (2 \\plus{} x)(3 \\plus{} x)(4 \\plus{} x)(5 \\plus{} x) \\equal{} 2115\\Rightarrow(x^2 \\plus{} 7x \\plus{} 10)(x^2 \\plus{} 7x \\plus{} 12) \\equal{} 2115$\r\n\r\nNow we let $ \\displaystyle n \\equal{} x^2 \\plus{} 7x \\plus{} 10$ to get:\r\n$ \\displaystyle n(n \\plus{} 2) \\equal{} 2115\\Rightarrow n^2 \\plus{} 2n \\minus{} 2115 \\equal{} 0$.\r\n\r\nWe take the prime factorization of $ \\displaystyle 2115$, which is $ \\displaystyle 3^25^147^1$. \r\n\r\nThen, we can quickly see how to factor the quadratic in $ \\displaystyle n$ above:\r\n$ \\displaystyle n^2 \\plus{} 2n \\minus{} 2115 \\equal{} 0\\Rightarrow(n \\plus{} 47)(n \\minus{} 45) \\equal{} 0$.\r\n\r\nSo, our solutions are $ \\displaystyle n \\equal{} \\minus{} 47$ and $ \\displaystyle n \\equal{} 45$.\r\n\r\nWe plug in both solutions of $ \\displaystyle n$ in our definition of $ \\displaystyle n$ ($ \\displaystyle n \\equal{} x^2 \\plus{} 7x \\plus{} 10$) and see what happens after rearranging:\r\n$ \\begin{align}\\displaystyle \\minus{} 47 \\equal{} x^2 \\plus{} 7x \\plus{} 10\\Rightarrow x^2 \\plus{} 7x \\plus{} 57 \\equal{} 0 \\\\\r\n\\displaystyle 45 \\equal{} x^2 \\plus{} 7x \\plus{} 10\\Rightarrow x^2 \\plus{} 7x \\minus{} 35 \\equal{} 0\\end{align}$\r\n\r\nObviously, we can see that the discriminant is negative in $ \\displaystyle (1)$, so only non-real complex solutions will arise. We discard that equation. (Note that we are asked for the greatest value of $ \\displaystyle x$; how can one non-real complex number be greater than another? It can't. Thus the non-real, complex solutions of $ \\displaystyle x$ cannot be solutions because we won't be able to pick the \"greatest\" one.)\r\n\r\nThe second equation seems alright, so we apply the quadratic formula (after seeing that trying to mentally factor is futile):\r\n$ \\displaystyle x^2 \\plus{} 7x \\minus{} 35 \\equal{} 0\\Rightarrow x \\equal{} \\frac { \\minus{} 7\\pm3\\sqrt {21}}{2}$\r\n\r\nWe want the largest $ \\displaystyle x$ possible, so we choose the $ \\displaystyle \\plus{}$ sign in our $ \\displaystyle \\pm$ expression to get our final answer of:\r\n\\[ \\displaystyle \\boxed{\\frac { \\minus{} 7 \\plus{} 3\\sqrt {21}}{2}}\r\n\\]", "content_html": "Here's my solution:<br>\n<br>\n<br>\nWith algebraic intuition and experience (which I have to some degree <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> ), we think to multiply the first and last terms on the left side of the equation and the second and third terms.<br>\n<br>\nThis yields:<br>\n<img src=\"//latex.artofproblemsolving.com/a/7/7/a776bc43af4ad16455ba9a5c23934adc1e5297ca.png\" class=\"latex\" alt=\"$ \\displaystyle (2 + x)(3 + x)(4 + x)(5 + x) = 2115\\Rightarrow(x^2 + 7x + 10)(x^2 + 7x + 12) = 2115$\" style=\"vertical-align: -4px\" width=\"600\" height=\"19\" ><br>\n<br>\nNow we let <img src=\"//latex.artofproblemsolving.com/7/b/6/7b629d5201e7bdc664968cb5c99c845e758a041b.png\" class=\"latex\" alt=\"$ \\displaystyle n = x^2 + 7x + 10$\" style=\"vertical-align: -1px\" width=\"134\" height=\"16\" > to get:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/d/0/dd0bd4b64dc702af08f460c6a6b4b1dc4e3cd4ec.png\" class=\"latex\" alt=\"$ \\displaystyle n(n + 2) = 2115\\Rightarrow n^2 + 2n - 2115 = 0$\" style=\"vertical-align: -4px\" width=\"308\" height=\"19\" >.</span><br>\n<br>\nWe take the prime factorization of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/9/b/19bd32fcc08cfcb79898c149a2185f5aefa2f442.png\" class=\"latex\" alt=\"$ \\displaystyle 2115$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" >,</span> which is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/6/8/3682edae1088970e484f8d71932a113bc62aa805.png\" class=\"latex\" alt=\"$ \\displaystyle 3^25^147^1$\" style=\"vertical-align: 0px\" width=\"56\" height=\"15\" >.</span><br>\n<br>\nThen, we can quickly see how to factor the quadratic in <img src=\"//latex.artofproblemsolving.com/c/8/c/c8c5ab8ad4b11fd8a04a68b20f4ab2b9f6cd54aa.png\" class=\"latex\" alt=\"$ \\displaystyle n$\" width=\"10\" height=\"8\" > above:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/c/d2cbbe14905f6f6950b235ed9c1d351b1911431f.png\" class=\"latex\" alt=\"$ \\displaystyle n^2 + 2n - 2115 = 0\\Rightarrow(n + 47)(n - 45) = 0$\" style=\"vertical-align: -4px\" width=\"344\" height=\"19\" >.</span><br>\n<br>\nSo, our solutions are <img src=\"//latex.artofproblemsolving.com/7/c/a/7ca2e80e6357503ecc369010e9642857fd3f38d8.png\" class=\"latex\" alt=\"$ \\displaystyle n = - 47$\" style=\"vertical-align: 0px\" width=\"67\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/f/b/1fb306c924c54b97e97efbcb34f20d58ba49a4ee.png\" class=\"latex\" alt=\"$ \\displaystyle n = 45$\" style=\"vertical-align: 0px\" width=\"52\" height=\"13\" >.</span><br>\n<br>\nWe plug in both solutions of <img src=\"//latex.artofproblemsolving.com/c/8/c/c8c5ab8ad4b11fd8a04a68b20f4ab2b9f6cd54aa.png\" class=\"latex\" alt=\"$ \\displaystyle n$\" width=\"10\" height=\"8\" > in our definition of <img src=\"//latex.artofproblemsolving.com/c/8/c/c8c5ab8ad4b11fd8a04a68b20f4ab2b9f6cd54aa.png\" class=\"latex\" alt=\"$ \\displaystyle n$\" width=\"10\" height=\"8\" > <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/7/b/6/7b629d5201e7bdc664968cb5c99c845e758a041b.png\" class=\"latex\" alt=\"$ \\displaystyle n = x^2 + 7x + 10$\" style=\"vertical-align: -1px\" width=\"134\" height=\"16\" >)</span> and see what happens after rearranging:<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align}\\displaystyle - 47 = x^2 + 7x + 10\\Rightarrow x^2 + 7x + 57 = 0 \\\\\n\\displaystyle 45 = x^2 + 7x + 10\\Rightarrow x^2 + 7x - 35 = 0\\end{align}$</span><br>\n<br>\nObviously, we can see that the discriminant is negative in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/8/30862d2648af36313659365cf48c703bb6508374.png\" class=\"latex\" alt=\"$ \\displaystyle (1)$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" >,</span> so only non-real complex solutions will arise. We discard that equation. (Note that we are asked for the greatest value of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/8/7/287ad620b3b594b40c92eb14c53e4b5c0b8a4e73.png\" class=\"latex\" alt=\"$ \\displaystyle x$\" width=\"10\" height=\"8\" >;</span> how can one non-real complex number be greater than another? It can't. Thus the non-real, complex solutions of <img src=\"//latex.artofproblemsolving.com/2/8/7/287ad620b3b594b40c92eb14c53e4b5c0b8a4e73.png\" class=\"latex\" alt=\"$ \\displaystyle x$\" width=\"10\" height=\"8\" > cannot be solutions because we won't be able to pick the "greatest" one.)<br>\n<br>\nThe second equation seems alright, so we apply the quadratic formula (after seeing that trying to mentally factor is futile):<br>\n<img src=\"//latex.artofproblemsolving.com/0/3/9/0397a82f8269d533698559ac29ca93ae3ba9c3e5.png\" class=\"latex\" alt=\"$ \\displaystyle x^2 + 7x - 35 = 0\\Rightarrow x = \\frac { - 7\\pm3\\sqrt {21}}{2}$\" style=\"vertical-align: -12px\" width=\"286\" height=\"41\" ><br>\n<br>\nWe want the largest <img src=\"//latex.artofproblemsolving.com/2/8/7/287ad620b3b594b40c92eb14c53e4b5c0b8a4e73.png\" class=\"latex\" alt=\"$ \\displaystyle x$\" width=\"10\" height=\"8\" > possible, so we choose the <img src=\"//latex.artofproblemsolving.com/a/1/b/a1b1b35cb2dcddff67b578e2fc36dc531dcc566b.png\" class=\"latex\" alt=\"$ \\displaystyle +$\" style=\"vertical-align: -1px\" width=\"13\" height=\"12\" > sign in our <img src=\"//latex.artofproblemsolving.com/9/9/3/99398392253efd8d8df5e1def698706eae829db0.png\" class=\"latex\" alt=\"$ \\displaystyle \\pm$\" style=\"vertical-align: 0px\" width=\"13\" height=\"12\" > expression to get our final answer of:<br>\n<img src=\"//latex.artofproblemsolving.com/1/2/7/1278345f59fd2c99573858ba5bc2de299563c8ab.png\" class=\"latexcenter\" alt=\"\\[ \\displaystyle \\boxed{\\frac { - 7 + 3\\sqrt {21}}{2}}\n\\]\" width=\"103\" height=\"53\" >", "post_id": 4407018, "post_number": 2, "post_time_unix": 1217040632, "post_time_utc": "2008-07-26 02:50:32 UTC", "thanks_received": 2, "user_id": 37350, "username": "MathAndKnowledge" }, { "attachments": [], "content_bbcode": "whatwhat\r\n\r\nlet's stop here:\r\n\\[ n(n \\plus{} 2) \\equal{} 2115 \\equal{} 45(45 \\plus{} 2)\r\n\\]\r\nbtw this is exactly what cognos did, so I get no credit :D\r\n\\[ x^2 \\plus{} 7x \\plus{} 10 \\equal{} 45\r\n\\]\r\nand you just solve that to get the same answer MathAnd got above :P", "content_html": "whatwhat<br>\n<br>\nlet's stop here:<br>\n<img src=\"//latex.artofproblemsolving.com/9/c/c/9cccda079e7e0f829278b25f7e4955bc923b9bbe.png\" class=\"latexcenter\" alt=\"\\[ n(n + 2) = 2115 = 45(45 + 2)\n\\]\" width=\"231\" height=\"18\" ><br>\nbtw this is exactly what cognos did, so I get no credit <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<img src=\"//latex.artofproblemsolving.com/5/f/5/5f502b224b6f9ebf53c99110624f6d1698227429.png\" class=\"latexcenter\" alt=\"\\[ x^2 + 7x + 10 = 45\n\\]\" width=\"141\" height=\"16\" ><br>\nand you just solve that to get the same answer MathAnd got above <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />", "post_id": 4407019, "post_number": 3, "post_time_unix": 1217074894, "post_time_utc": "2008-07-26 12:21:34 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "good job MathAnd and undefined!!!!! yeah i threw this problem out of the Mock AMC 12, it was too similar to a problem i saw somewhere. these were the makers of each problem we have done so far:\r\n\r\n10. AP [own]\r\n11. MP [sister]\r\n12. AP [own]\r\n13 AP [own]\r\n14. AP [own]\r\n15. AP [own]\r\n16. AP [own]\r\n17. MP [AMC 12]\r\n19. AP [own]\r\n20. MP [sister]\r\n21. AP [ARML]\r\n22. MP [sister]\r\n23. AP [own]\r\n24. AP [own]\r\n25. AP [Some-other-country Math olympiad (not USAMO)]\r\n \r\nwe don't have a number 18 yet. all of the problems are original but i was inspired by other contests as shown above.", "content_html": "good job MathAnd and undefined!!!!! yeah i threw this problem out of the Mock AMC 12, it was too similar to a problem i saw somewhere. these were the makers of each problem we have done so far:<br>\n<br>\n10. AP [own]<br>\n11. MP [sister]<br>\n12. AP [own]<br>\n13 AP [own]<br>\n14. AP [own]<br>\n15. AP [own]<br>\n16. AP [own]<br>\n17. MP [AMC 12]<br>\n19. AP [own]<br>\n20. MP [sister]<br>\n21. AP [ARML]<br>\n22. MP [sister]<br>\n23. AP [own]<br>\n24. AP [own]<br>\n25. AP [Some-other-country Math olympiad (not USAMO)]<br>\n<br>\nwe don't have a number 18 yet. all of the problems are original but i was inspired by other contests as shown above.", "post_id": 4407020, "post_number": 4, "post_time_unix": 1217098732, "post_time_utc": "2008-07-26 18:58:52 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" } ], "source": null }	Find the largest $x$ that satisfies the following condition: \[ (2+x)(3+x)(4+x)(5+x)=2115. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial" ]	Pair the outer and inner factors to form two quadratics, then substitute to reduce the equation to a simple quadratic in the new variable.	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aops_993347	It is not difficult (took me a minute :D ) to see that the given equation factors as: $ (x \plus{} y \minus{} 2)((x \minus{} y)^2 \plus{} (x \plus{} 2)(y \plus{} 2)) \equal{} 0$ Obviously, the first factor is a line on the coordinate plane, while the second factor is a single point. The equation of the line is $ x \plus{} y \minus{} 2 \equal{} 0$. The single point can be found quite simply. We can assume that the expressions $ (x \minus{} y)^2$ and $ (x \plus{} 2)(y \plus{} 2)$ must be $ 0$ in order to bring about our only solution; if it turns out that that is impossible, then we must find another way to get the solution. After some applications of the Trivial Inequality (no guesswork needed, hooray!), we find that $ x \equal{} y \equal{} \minus{} 2$. The formula for the shortest distance between a point and a line is: $ d \equal{} \frac {\|Ax \plus{} By \plus{} C\|}{\sqrt {A^2 \plus{} B^2}}$ We just plug in the numbers (where $ d$ is the shortest distance from the point to the line, $ A,B,$ and $ C$ are the coefficients of our linear equation and $ x$ and $ y$ are the coordinates for our single point), yielding: $ d \equal{} \boxed{3\sqrt {2}}$. Notice that this solution skipped a lot of the algebra and explanation because I was lazy :lol: . So don't be mad because it's not a full solution; it's not meant to be. I felt beastly after solving this probem; thanks, cognos599! I especially felt that feeling after factoring the given equation, without guessing the factorization (it's too hard to guess, anyways)! Now I shall do what any dignified mathematician does after he/she thinks that he/she accomplished something significant (when in reality they accomplished nothing significant): cackle. Cackle! ... Cackle! Cackle! ... Cackle!	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "The locus of all points $ (x,y)$ that satisfy the equation $ x^3 \\plus{} 6xy \\plus{} y^3 \\equal{} 8$ is a point and a line find the shortest distance from this point to the line. (wording is kinda messed up).", "content_html": "The locus of all points <img src=\"//latex.artofproblemsolving.com/2/8/c/28cb007e0f02b790009f3e599e9c1462a2d9f679.png\" class=\"latex\" alt=\"$ (x,y)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > that satisfy the equation <img src=\"//latex.artofproblemsolving.com/4/e/c/4ecb1ff4e302bc9d811bad9f9b3ed9999f580431.png\" class=\"latex\" alt=\"$ x^3 + 6xy + y^3 = 8$\" style=\"vertical-align: -3px\" width=\"141\" height=\"18\" > is a point and a line find the shortest distance from this point to the line. (wording is kinda messed up).", "post_id": 4407021, "post_number": 1, "post_time_unix": 1217099325, "post_time_utc": "2008-07-26 19:08:45 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "It is not difficult (took me a minute :D ) to see that the given equation factors as:\r\n$ (x \\plus{} y \\minus{} 2)((x \\minus{} y)^2 \\plus{} (x \\plus{} 2)(y \\plus{} 2)) \\equal{} 0$\r\n\r\nObviously, the first factor is a line on the coordinate plane, while the second factor is a single point.\r\n\r\nThe equation of the line is $ x \\plus{} y \\minus{} 2 \\equal{} 0$.\r\n\r\nThe single point can be found quite simply. We can assume that the expressions $ (x \\minus{} y)^2$ and $ (x \\plus{} 2)(y \\plus{} 2)$ must be $ 0$ in order to bring about our only solution; if it turns out that that is impossible, then we must find another way to get the solution. After some applications of the Trivial Inequality (no guesswork needed, hooray!), we find that $ x \\equal{} y \\equal{} \\minus{} 2$.\r\n\r\nThe formula for the shortest distance between a point and a line is:\r\n$ d \\equal{} \\frac {\|Ax \\plus{} By \\plus{} C\|}{\\sqrt {A^2 \\plus{} B^2}}$\r\n\r\nWe just plug in the numbers (where $ d$ is the shortest distance from the point to the line, $ A,B,$ and $ C$ are the coefficients of our linear equation and $ x$ and $ y$ are the coordinates for our single point), yielding:\r\n$ d \\equal{} \\boxed{3\\sqrt {2}}$.\r\n\r\nNotice that this solution skipped a lot of the algebra and explanation because I was lazy :lol: . So don't be mad because it's not a full solution; it's not meant to be.\r\n\r\nI felt beastly after solving this probem; thanks, cognos599! I especially felt that feeling after factoring the given equation, without guessing the factorization (it's too hard to guess, anyways)!\r\n\r\nNow I shall do what any dignified mathematician does after he/she thinks that he/she accomplished something significant (when in reality they accomplished nothing significant): cackle. Cackle! ... Cackle! Cackle! ... Cackle!", "content_html": "It is not difficult (took me a minute <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> ) to see that the given equation factors as:<br>\n<img src=\"//latex.artofproblemsolving.com/c/d/3/cd30c4293b23ad6042a300c96cc071ddae4c926e.png\" class=\"latex\" alt=\"$ (x + y - 2)((x - y)^2 + (x + 2)(y + 2)) = 0$\" style=\"vertical-align: -4px\" width=\"331\" height=\"19\" ><br>\n<br>\nObviously, the first factor is a line on the coordinate plane, while the second factor is a single point.<br>\n<br>\nThe equation of the line is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/8/1/0818564483cd200ce67202675b0f68cba488b6e7.png\" class=\"latex\" alt=\"$ x + y - 2 = 0$\" style=\"vertical-align: -3px\" width=\"106\" height=\"16\" >.</span><br>\n<br>\nThe single point can be found quite simply. We can assume that the expressions <img src=\"//latex.artofproblemsolving.com/5/e/9/5e9ade0a934979290831711c2310c2376aac54a2.png\" class=\"latex\" alt=\"$ (x - y)^2$\" style=\"vertical-align: -4px\" width=\"62\" height=\"19\" > and <img src=\"//latex.artofproblemsolving.com/f/5/8/f581fe52872803b313665c357fbd833c156cecd8.png\" class=\"latex\" alt=\"$ (x + 2)(y + 2)$\" style=\"vertical-align: -4px\" width=\"109\" height=\"18\" > must be <img src=\"//latex.artofproblemsolving.com/d/6/0/d6033a0bb0547c396d9c45af99df7a43af61eb69.png\" class=\"latex\" alt=\"$ 0$\" width=\"8\" height=\"12\" > in order to bring about our only solution; if it turns out that that is impossible, then we must find another way to get the solution. After some applications of the Trivial Inequality (no guesswork needed, hooray!), we find that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/0/d/00da69edbb3bf05e4cc96b12d24ad1052f1a1c9e.png\" class=\"latex\" alt=\"$ x = y = - 2$\" style=\"vertical-align: -3px\" width=\"91\" height=\"15\" >.</span><br>\n<br>\nThe formula for the shortest distance between a point and a line is:<br>\n<img src=\"//latex.artofproblemsolving.com/f/e/4/fe4579229948dc430c5a7d4a8c0a501c6eb5ef05.png\" class=\"latex\" alt=\"$ d = \\frac {\|Ax + By + C\|}{\\sqrt {A^2 + B^2}}$\" style=\"vertical-align: -17px\" width=\"153\" height=\"43\" ><br>\n<br>\nWe just plug in the numbers (where <img src=\"//latex.artofproblemsolving.com/3/8/e/38e0372252b796e84b1b18a897f413701331c9d6.png\" class=\"latex\" alt=\"$ d$\" width=\"9\" height=\"12\" > is the shortest distance from the point to the line, <img src=\"//latex.artofproblemsolving.com/0/9/e/09e8b6a2b93d13efbac517dbd4c7c9df5808f084.png\" class=\"latex\" alt=\"$ A,B,$\" style=\"vertical-align: -3px\" width=\"40\" height=\"16\" > and <img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" > are the coefficients of our linear equation and <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/8/9/b8959a2220db8bc60d06e50de97fb5e86756e0f8.png\" class=\"latex\" alt=\"$ y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > are the coordinates for our single point), yielding:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/6/46687afb358c7eb9d1108f0fc6a6f44878bb5d75.png\" class=\"latex\" alt=\"$ d = \\boxed{3\\sqrt {2}}$\" style=\"vertical-align: -7px\" width=\"78\" height=\"30\" >.</span><br>\n<br>\nNotice that this solution skipped a lot of the algebra and explanation because I was lazy <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /> . So don't be mad because it's not a full solution; it's not meant to be.<br>\n<br>\nI felt beastly after solving this probem; thanks, cognos599! I especially felt that feeling after factoring the given equation, without guessing the factorization (it's too hard to guess, anyways)!<br>\n<br>\nNow I shall do what any dignified mathematician does after he/she thinks that he/she accomplished something significant (when in reality they accomplished nothing significant): cackle. Cackle! ... Cackle! Cackle! ... Cackle!", "post_id": 4407022, "post_number": 2, "post_time_unix": 1217116993, "post_time_utc": "2008-07-27 00:03:13 UTC", "thanks_received": 2, "user_id": 37350, "username": "MathAndKnowledge" }, { "attachments": [], "content_bbcode": "huh\r\n\r\nNotice the equation is cyclic. This means that $ x$ and $ y$ are inversely related $ \\Rightarrow$ the line has a slope of $ \\minus{} 1$.\r\n\r\nReplacing $ y$ with $ x$ (since it's cyclic, this will give some solution pairs) we get $ 2x^3 \\plus{} 6x^2 \\equal{} 8$, which can be rearranged to $ x^3 \\plus{} 3x^2 \\minus{} 4 \\equal{} 0$.\r\n\r\nIts solutions are $ x \\equal{} 1$ and $ x \\equal{} \\minus{} 2$, which implies $ y \\equal{} 1$ and $ y \\equal{} \\minus{} 2$ at the respective x-values. They're positioned on the line $ y \\equal{} x$, which has a slope of $ 1$. This line and the solution line are perpendicular to each other. The solution line passes through one of these points, but I don't care which one.\r\n\r\nBecause the distance is just $ \\sqrt {2(1 \\minus{} ( \\minus{} 2))^2} \\equal{} \\boxed{3\\sqrt {2}}$ :P", "content_html": "huh<br>\n<br>\nNotice the equation is cyclic. This means that <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/8/9/b8959a2220db8bc60d06e50de97fb5e86756e0f8.png\" class=\"latex\" alt=\"$ y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > are inversely related <img src=\"//latex.artofproblemsolving.com/b/4/e/b4ec148589a55c44ce7b1ecc40660052598d0a7b.png\" class=\"latex\" alt=\"$ \\Rightarrow$\" style=\"vertical-align: 0px\" width=\"17\" height=\"10\" > the line has a slope of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/6/a/16a47c093112ae6bc4f1f00fccd49ff3c4ffc1a1.png\" class=\"latex\" alt=\"$ - 1$\" style=\"vertical-align: 0px\" width=\"22\" height=\"12\" >.</span><br>\n<br>\nReplacing <img src=\"//latex.artofproblemsolving.com/b/8/9/b8959a2220db8bc60d06e50de97fb5e86756e0f8.png\" class=\"latex\" alt=\"$ y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > with <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > (since it's cyclic, this will give some solution pairs) we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/2/2/422e2e3533ed6834c97deb4455dafcf3661abdcc.png\" class=\"latex\" alt=\"$ 2x^3 + 6x^2 = 8$\" style=\"vertical-align: -1px\" width=\"109\" height=\"16\" >,</span> which can be rearranged to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/1/f/31f7edaf8cf2da0f8e2f182b7c41b5106e153b82.png\" class=\"latex\" alt=\"$ x^3 + 3x^2 - 4 = 0$\" style=\"vertical-align: -1px\" width=\"131\" height=\"16\" >.</span><br>\n<br>\nIts solutions are <img src=\"//latex.artofproblemsolving.com/1/d/7/1d74df744e28e008410329b0edc01624c37d62e3.png\" class=\"latex\" alt=\"$ x = 1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/5/d/b5d103bd79dc1c03306775a5551797a66ca6f1da.png\" class=\"latex\" alt=\"$ x = - 2$\" width=\"57\" height=\"12\" >,</span> which implies <img src=\"//latex.artofproblemsolving.com/f/e/9/fe97047471451b679afa888d08d508a9f864cdfb.png\" class=\"latex\" alt=\"$ y = 1$\" style=\"vertical-align: -3px\" width=\"42\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/d/2/0/d202f5dd97f55da4f97f0b5c73296f04b308d102.png\" class=\"latex\" alt=\"$ y = - 2$\" style=\"vertical-align: -3px\" width=\"56\" height=\"15\" > at the respective x-values. They're positioned on the line <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/7/9/6797d933bb7d43ea7757b68d02a2ca08632bf1e4.png\" class=\"latex\" alt=\"$ y = x$\" style=\"vertical-align: -3px\" width=\"44\" height=\"11\" >,</span> which has a slope of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/9/0/39064bdd89b3dfa0626ca59d843d926ea072830b.png\" class=\"latex\" alt=\"$ 1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >.</span> This line and the solution line are perpendicular to each other. The solution line passes through one of these points, but I don't care which one.<br>\n<br>\nBecause the distance is just <img src=\"//latex.artofproblemsolving.com/8/4/d/84de422ce96e17fdab24c42e234e111d8cd109ef.png\" class=\"latex\" alt=\"$ \\sqrt {2(1 - ( - 2))^2} = \\boxed{3\\sqrt {2}}$\" style=\"vertical-align: -10px\" width=\"186\" height=\"33\" > <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />", "post_id": 4407023, "post_number": 3, "post_time_unix": 1217119153, "post_time_utc": "2008-07-27 00:39:13 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "if you do some rearranging, you notice that $ x^3 \\plus{} y^3 \\plus{} ( \\minus{} 2)^3 \\equal{} 3xy( \\minus{} 2)$ this reminds us of our algebraic identity:\r\n\r\n $ a^3 \\plus{} b^3 \\plus{} c^3 \\minus{} 3abc \\equal{} (a \\plus{} b \\plus{} c)(a^2 \\plus{} b^2 \\plus{} c^2 \\minus{} ab \\minus{} bc \\minus{} ac)$ \r\n\r\nbut since $ a^3 \\plus{} b^3 \\plus{} c^3 \\equal{} 3abc$, then we get the following equation:\r\n\r\n$ (x \\plus{} y \\minus{} 2)(x^2 \\plus{} y^2 \\plus{} ( \\minus{} 2)^2 \\minus{} xy \\plus{} 2x \\plus{} 2y) \\equal{} 0$ \r\n\r\nCase 1: $ x \\plus{} y \\minus{} 2 \\equal{} 0$, therefore\r\n\r\n$ x \\plus{} y \\equal{} 2$ , which gives us our line...\r\n\r\nCase 2: $ x^2 \\plus{} y^2 \\plus{} ( \\minus{} 2)^2 \\minus{} xy \\plus{} 2x \\plus{} 2y \\equal{} 0$\r\n\r\nwhich can be rewritten as $ \\frac {1}{2}((x \\minus{} y)^2 \\plus{} (x \\plus{} 2)^2 \\plus{} (y \\plus{} 2)^2 \\equal{} 0$, \r\n\r\nfor this to be $ 0$ each of the squares have to be equal to $ 0$, and the only point that satisfies that condition is:\r\n\r\n$ ( \\minus{} 2, \\minus{} 2)$, now we use the equation for the distance from a point to a line. \r\n\r\nthe distance from a point $ (p,q)$ to the line $ ax \\plus{} by \\equal{} c$ is $ \\frac {\|ap \\plus{} bq \\minus{} c\|}{\\sqrt {a \\plus{} 2 \\plus{} b^2}}$\r\n\r\napplying this: our point is $ ( \\minus{} 2, \\minus{} 2)$ and our line is $ x \\plus{} y \\equal{} 2$, so the answer is $ \\frac {6}{\\sqrt {2}} \\equal{} \\fbox{3\\sqrt {2}}$.\r\n\r\nso what number do you think this problem would have been if i had put it on the test? \r\n\r\n\r\nMathAnd, those are the kind of mistakes i am counting on you to make next year so i can beat you. :P you were exactly right until the end where you made a typo and wrote $ 3\\sqrt{3}$ instead of $ 3\\sqrt{2}$, that is the only chance i have of beating you...... :)", "content_html": "if you do some rearranging, you notice that <img src=\"//latex.artofproblemsolving.com/8/7/5/875cabcc35a8d9eb23feba5d7816e804e1ecec36.png\" class=\"latex\" alt=\"$ x^3 + y^3 + ( - 2)^3 = 3xy( - 2)$\" style=\"vertical-align: -4px\" width=\"213\" height=\"19\" > this reminds us of our algebraic identity:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/e/0/9e079ab07ee152d4289cfb720c0dbfd75250b88e.png\" class=\"latex\" alt=\"$ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ac)$\" style=\"vertical-align: -4px\" width=\"478\" height=\"19\" ><br>\n<br>\nbut since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/8/4/f84e5609aa59f31bba07d31053f9e86ac3fe3ce7.png\" class=\"latex\" alt=\"$ a^3 + b^3 + c^3 = 3abc$\" style=\"vertical-align: -1px\" width=\"151\" height=\"16\" >,</span> then we get the following equation:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/2/c/72cbe5280df6b0fe4af3c4cefe4d8517a5a788f9.png\" class=\"latex\" alt=\"$ (x + y - 2)(x^2 + y^2 + ( - 2)^2 - xy + 2x + 2y) = 0$\" style=\"vertical-align: -4px\" width=\"383\" height=\"19\" ><br>\n<br>\nCase 1: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/8/1/0818564483cd200ce67202675b0f68cba488b6e7.png\" class=\"latex\" alt=\"$ x + y - 2 = 0$\" style=\"vertical-align: -3px\" width=\"106\" height=\"16\" >,</span> therefore<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/3/a/e3a1ded3484b3f21494bd75a2edd71da03fb6748.png\" class=\"latex\" alt=\"$ x + y = 2$\" style=\"vertical-align: -3px\" width=\"75\" height=\"15\" > , which gives us our line...<br>\n<br>\nCase 2: <img src=\"//latex.artofproblemsolving.com/6/0/2/60221cc6561cb5013116c178ea6ad52ef4341b8b.png\" class=\"latex\" alt=\"$ x^2 + y^2 + ( - 2)^2 - xy + 2x + 2y = 0$\" style=\"vertical-align: -4px\" width=\"281\" height=\"19\" ><br>\n<br>\nwhich can be rewritten as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/8/8c8a54145e47140cc514c14e89650eaa5c5547a2.png\" class=\"latex\" alt=\"$ \\frac {1}{2}((x - y)^2 + (x + 2)^2 + (y + 2)^2 = 0$\" style=\"vertical-align: -12px\" width=\"287\" height=\"37\" >,</span><br>\n<br>\nfor this to be <img src=\"//latex.artofproblemsolving.com/d/6/0/d6033a0bb0547c396d9c45af99df7a43af61eb69.png\" class=\"latex\" alt=\"$ 0$\" width=\"8\" height=\"12\" > each of the squares have to be equal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/6/0/d6033a0bb0547c396d9c45af99df7a43af61eb69.png\" class=\"latex\" alt=\"$ 0$\" width=\"8\" height=\"12\" >,</span> and the only point that satisfies that condition is:<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/e/b/9eb2b26480c536f68eff7a884ab00a0e080b230a.png\" class=\"latex\" alt=\"$ ( - 2, - 2)$\" style=\"vertical-align: -4px\" width=\"67\" height=\"18\" >,</span> now we use the equation for the distance from a point to a line.<br>\n<br>\nthe distance from a point <img src=\"//latex.artofproblemsolving.com/e/9/6/e961747226cbee361450fc3a30acb2a6d7c60dfe.png\" class=\"latex\" alt=\"$ (p,q)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > to the line <img src=\"//latex.artofproblemsolving.com/4/7/4/47468ab4b8f924a3317302d594fcbb8a0318e3de.png\" class=\"latex\" alt=\"$ ax + by = c$\" style=\"vertical-align: -3px\" width=\"92\" height=\"16\" > is <img src=\"//latex.artofproblemsolving.com/8/4/c/84cfe96e99b06d25bfdc97924b11e9c39471247b.png\" class=\"latex\" alt=\"$ \\frac {\|ap + bq - c\|}{\\sqrt {a + 2 + b^2}}$\" style=\"vertical-align: -17px\" width=\"100\" height=\"43\" ><br>\n<br>\napplying this: our point is <img src=\"//latex.artofproblemsolving.com/9/e/b/9eb2b26480c536f68eff7a884ab00a0e080b230a.png\" class=\"latex\" alt=\"$ ( - 2, - 2)$\" style=\"vertical-align: -4px\" width=\"67\" height=\"18\" > and our line is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/a/e3a1ded3484b3f21494bd75a2edd71da03fb6748.png\" class=\"latex\" alt=\"$ x + y = 2$\" style=\"vertical-align: -3px\" width=\"75\" height=\"15\" >,</span> so the answer is <span style=\"white-space:nowrap;\"><span class=\"aopscode-error aopscode-latex-error\">$ \\frac {6}{\\sqrt {2}} = \\fbox{3\\sqrt {2}}$</span>.</span><br>\n<br>\nso what number do you think this problem would have been if i had put it on the test?<br>\n<br>\n<br>\nMathAnd, those are the kind of mistakes i am counting on you to make next year so i can beat you. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /> you were exactly right until the end where you made a typo and wrote <img src=\"//latex.artofproblemsolving.com/6/f/b/6fb60f62e647bf01bcd2efed29226f67a3d5d253.png\" class=\"latex\" alt=\"$ 3\\sqrt{3}$\" style=\"vertical-align: -1px\" width=\"33\" height=\"18\" > instead of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/d/b/0db5ceb3dbfaa203f8fdd0adf470ddcfead87ea1.png\" class=\"latex\" alt=\"$ 3\\sqrt{2}$\" style=\"vertical-align: -1px\" width=\"33\" height=\"18\" >,</span> that is the only chance i have of beating you...... <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 4407024, "post_number": 4, "post_time_unix": 1217128353, "post_time_utc": "2008-07-27 03:12:33 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "meh.. it's a little trivial, though I guess I like it this way :D otherwise, I'd fail.\r\n\r\nIt could be a #15.\r\n\r\naghhhh I need sleep.", "content_html": "meh.. it's a little trivial, though I guess I like it this way <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> otherwise, I'd fail.<br>\n<br>\nIt could be a #15.<br>\n<br>\naghhhh I need sleep.", "post_id": 4407025, "post_number": 5, "post_time_unix": 1217129922, "post_time_utc": "2008-07-27 03:38:42 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" } ], "source": null }	Find the shortest distance between the point and the line that form the locus of all points $(x,y)$ satisfying \[ x^{3}+6xy+y^{3}=8. \]	[ "/Mathematics/Algebra/AlgebraicCurves/AlgebraicCurve", "/Mathematics/Algebra/AlgebraicCurves/AlgebraicGeometry", "/Mathematics/Algebra/AlgebraicCurves/CubicCurve", "/Mathematics/Algebra/AlgebraicCurves/PolynomialCurve", "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/CubicEquation", "/Mathematics/Algebra/AlgebraicGeometry/AbstractAlgebraicCurves/Acnode", "/Mathematics/Algebra/AlgebraicGeometry/AbstractAlgebraicCurves/AlgebraicCurve", "/Mathematics/Algebra/AlgebraicGeometry/AbstractAlgebraicCurves/IsolatedPoint", "/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry", "/Mathematics/Geometry/CoordinateGeometry/Cartesian", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/CoordinateGeometry/CartesianPlane", "/Mathematics/Geometry/CoordinateGeometry/CoordinatePlane", "/Mathematics/Geometry/Distance/Locus", "/Mathematics/Geometry/Distance/Point-LineDistance2-Dimensional", "/Mathematics/Geometry/LineGeometry/Lines/Line", "/Mathematics/Geometry/LineGeometry/Lines/Point-LineDistance2-Dimensional", "/Mathematics/Geometry/LineGeometry/Lines/StraightLine", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves/AlgebraicCurves", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves/GeneralPlaneCurves", "/Mathematics/Geometry/PlaneGeometry/PlaneCurves/ImplicitCurves", "/Mathematics/Geometry/Points/Locus", "/Mathematics/Geometry/Points/PlanarDistance", "/Mathematics/Geometry/Points/Point" ]	Factor the cubic equation to isolate a linear factor (the line) and a zero-dimensional factor (the single point).	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aops_993349	er \[ (10a \plus{} b) \minus{} (10c \plus{} d) \equal{} (10b \plus{} a) \minus{} (10d \plus{} c) \] \[ a \minus{} b \equal{} c \minus{} d \] which means the difference of the two digits is the same for both numbers. Then some clever, confusing counting stuff that I don't want to do at the moment. SPAZ	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "What is the probability that two randomly selected distinct two-digit integers $ ab$ and $ cd$ (where $ a$ and $ b$ are digits not the product $ a\\times b$.) have the property that $ ab \\minus{} cd \\equal{} ba \\minus{} dc$? (Assume the digit reverse of 20 is not possible for both of the reversed numbers have to be two-digit as well).", "content_html": "What is the probability that two randomly selected distinct two-digit integers <img src=\"//latex.artofproblemsolving.com/d/1/5/d15a5fdc286fe50ee5854e45c979373baacdd841.png\" class=\"latex\" alt=\"$ ab$\" width=\"17\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/e/c/d/ecd44d6224319b906f4c1500ebaddb0fbc5044c0.png\" class=\"latex\" alt=\"$ cd$\" width=\"17\" height=\"12\" > (where <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > are digits not the product <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/9/f/59fefe43a9933933261f5cc7f9ae2e9091c9a077.png\" class=\"latex\" alt=\"$ a\\times b$\" width=\"40\" height=\"12\" >.</span>) have the property that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/b/9/5b9ea02da00703fee21e62922facc5cd1d96aa59.png\" class=\"latex\" alt=\"$ ab - cd = ba - dc$\" width=\"138\" height=\"12\" >?</span> (Assume the digit reverse of 20 is not possible for both of the reversed numbers have to be two-digit as well).", "post_id": 4407028, "post_number": 1, "post_time_unix": 1217467708, "post_time_utc": "2008-07-31 01:28:28 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "er\r\n\\[ (10a \\plus{} b) \\minus{} (10c \\plus{} d) \\equal{} (10b \\plus{} a) \\minus{} (10d \\plus{} c)\r\n\\]\r\n\r\n\\[ a \\minus{} b \\equal{} c \\minus{} d\r\n\\]\r\nwhich means the difference of the two digits is the same for both numbers.\r\n\r\nThen some clever, confusing counting stuff that I don't want to do at the moment. SPAZ", "content_html": "er<br>\n<img src=\"//latex.artofproblemsolving.com/4/f/c/4fce20f488361f1e89fbbf3311c48a816bec07b9.png\" class=\"latexcenter\" alt=\"\\[ (10a + b) - (10c + d) = (10b + a) - (10d + c)\n\\]\" width=\"353\" height=\"18\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/e/5/9e5ec82252c13dadf3e86999f4c989457d4e25ba.png\" class=\"latexcenter\" alt=\"\\[ a - b = c - d\n\\]\" width=\"103\" height=\"12\" ><br>\nwhich means the difference of the two digits is the same for both numbers.<br>\n<br>\nThen some clever, confusing counting stuff that I don't want to do at the moment. SPAZ", "post_id": 4407029, "post_number": 2, "post_time_unix": 1217470092, "post_time_utc": "2008-07-31 02:08:12 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "Blargh, ur right, there is no ez way of doing it..... that is why my sister rejected it. :( the answer ends up being like $ \\frac{1}{10}$ or $ \\frac{1}{20}$.....unless i did it wrong.", "content_html": "Blargh, ur right, there is no ez way of doing it..... that is why my sister rejected it. <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /> the answer ends up being like <img src=\"//latex.artofproblemsolving.com/4/8/5/485ac0d99da346057dd01831d2c3d0b2f64144c9.png\" class=\"latex\" alt=\"$ \\frac{1}{10}$\" style=\"vertical-align: -13px\" width=\"20\" height=\"37\" > or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/2/1/621f8e9d5dd547652cfb6fa271e2b3dedfbf8505.png\" class=\"latex\" alt=\"$ \\frac{1}{20}$\" style=\"vertical-align: -12px\" width=\"20\" height=\"37\" >.</span>....unless i did it wrong.", "post_id": 4407030, "post_number": 3, "post_time_unix": 1217518683, "post_time_utc": "2008-07-31 15:38:03 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "okey total number of combinations of two two-digit numbers: $ \\cfrac{81\\cdot{80}}{2}\\equal{}3240...(1)$\r\n\r\ndifference is 0 ----> $ \\cfrac{9\\cdot{8}}{2}\\equal{}36...(2)$\r\n\r\ndifference is 1, 2, ... 8:\r\n\r\n$ \\cfrac{8\\cdot7}{2}\\plus{}\\cfrac{7\\cdot6}{2}\\plus{}\\cdots\\plus{}\\cfrac{2\\cdot1}{2}$\r\n\r\nand we have to multiply that by 2 to account for differences of -1, -2, ... -8\r\n\r\n$ \\equal{}2\\sum_{k\\equal{}1}^7\\cfrac{k(k\\plus{}1)}{2}\\equal{}\\sum_{k\\equal{}1}^7k^2\\plus{}\\sum_{k\\equal{}1}^7k\\equal{}\\cfrac{7\\cdot8\\cdot15}{6}\\plus{}28\\equal{}168...(3)$\r\n\r\nCombining (1), (2), and (3), we get $ \\cfrac{36\\plus{}168}{3240}\\equal{}\\boxed{\\cfrac{17}{270}}$\r\n\r\nI MADE A MISTAKE", "content_html": "okey total number of combinations of two two-digit numbers: <img src=\"//latex.artofproblemsolving.com/a/b/a/abab14279dd120e39defab49f530d389330a36c2.png\" class=\"latex\" alt=\"$ \\cfrac{81\\cdot{80}}{2}=3240...(1)$\" style=\"vertical-align: -12px\" width=\"148\" height=\"38\" ><br>\n<br>\ndifference is 0 ----> <img src=\"//latex.artofproblemsolving.com/6/8/f/68fc1e81792840b39ff840650abac875f5be77de.png\" class=\"latex\" alt=\"$ \\cfrac{9\\cdot{8}}{2}=36...(2)$\" style=\"vertical-align: -12px\" width=\"112\" height=\"38\" ><br>\n<br>\ndifference is 1, 2, ... 8:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/5/1/e/51eccf8e479ec32e2e938c97a8d83837e7f3e528.png\" class=\"latex\" alt=\"$ \\cfrac{8\\cdot7}{2}+\\cfrac{7\\cdot6}{2}+\\cdots+\\cfrac{2\\cdot1}{2}$\" style=\"vertical-align: -12px\" width=\"188\" height=\"38\" ><br>\n<br>\nand we have to multiply that by 2 to account for differences of -1, -2, ... -8<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/4/e/74e903088cd0a40f6d63ab17ce9db6afd424d03f.png\" class=\"latex\" alt=\"$ =2\\sum_{k=1}^7\\cfrac{k(k+1)}{2}=\\sum_{k=1}^7k^2+\\sum_{k=1}^7k=\\cfrac{7\\cdot8\\cdot15}{6}+28=168...(3)$\" style=\"vertical-align: -20px\" width=\"476\" height=\"50\" ><br>\n<br>\nCombining (1), (2), and (3), we get <img src=\"//latex.artofproblemsolving.com/c/8/4/c845983f846c67fdbf0a0318dd60af2cc4a85d89.png\" class=\"latex\" alt=\"$ \\cfrac{36+168}{3240}=\\boxed{\\cfrac{17}{270}}$\" style=\"vertical-align: -18px\" width=\"135\" height=\"51\" ><br>\n<br>\nI MADE A MISTAKE", "post_id": 4407031, "post_number": 4, "post_time_unix": 1217551685, "post_time_utc": "2008-08-01 00:48:05 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" } ], "source": null }	What is the probability that two randomly selected distinct two-digit integers $ab$ and $cd$ (where $a$ and $b$ are digits, not the product $a\times b$) have the property \[ ab - cd = ba - dc? \] Assume digit reversals must also be two-digit numbers (so the reversal of 20 is not allowed).	[ "/Mathematics/Algebra/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/RateProblems", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/MultiplicationPrinciple", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Simplify the equation to show the digit differences are equal: a − b = c − d	[ -0.00013947486877441406, -0.019195556640625, 0.03900146484375, -0.027557373046875, -0.0003399848937988281, 0.00751495361328125, 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aops_993354	One must find $ 2005^{2005} (\mod{7})$ Notice that by Euler's formula, $ 2005^{6} \equiv 1 (\mod{7})$ therefore $ 2005^{2004} \equiv 1 (\mod{7})$ $ 2005^{2005} \equiv 2005 (\mod{7})$ $ 2005 \equiv 3 (\mod{7})$ So the day of the week that corresponds to 3 is $ \boxed{\mbox{Thursday}}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "What day of the week is $ 2005^{2005}$ days after Monday? (This problem is from the MMPC competition, my cousin showed it to me). If you saw my previous post before i deleted, it, that is a big hint.", "content_html": "What day of the week is <img src=\"//latex.artofproblemsolving.com/e/2/3/e2309d7bfb9ae6cf64db296db4d2b71fccc84570.png\" class=\"latex\" alt=\"$ 2005^{2005}$\" width=\"61\" height=\"15\" > days after Monday? (This problem is from the MMPC competition, my cousin showed it to me). If you saw my previous post before i deleted, it, that is a big hint.", "post_id": 4407050, "post_number": 1, "post_time_unix": 1220472890, "post_time_utc": "2008-09-03 20:14:50 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "One must find $ 2005^{2005} (\\mod{7})$\r\n\r\nNotice that by Euler's formula, $ 2005^{6} \\equiv 1 (\\mod{7})$\r\n\r\ntherefore $ 2005^{2004} \\equiv 1 (\\mod{7})$\r\n\r\n$ 2005^{2005} \\equiv 2005 (\\mod{7})$\r\n\r\n$ 2005 \\equiv 3 (\\mod{7})$\r\n\r\nSo the day of the week that corresponds to 3 is $ \\boxed{\\mbox{Thursday}}$", "content_html": "One must find <img src=\"//latex.artofproblemsolving.com/0/a/c/0ac5163c0e10c2919d6dbb494ed48c6e82802bbc.png\" class=\"latex\" alt=\"$ 2005^{2005} (\\mod{7})$\" style=\"vertical-align: -4px\" width=\"138\" height=\"19\" ><br>\n<br>\nNotice that by Euler's formula, <img src=\"//latex.artofproblemsolving.com/7/b/3/7b33b853912e879287fdd855d4ce42f20aabeb40.png\" class=\"latex\" alt=\"$ 2005^{6} \\equiv 1 (\\mod{7})$\" style=\"vertical-align: -4px\" width=\"151\" height=\"19\" ><br>\n<br>\ntherefore <img src=\"//latex.artofproblemsolving.com/2/2/2/222603d1a6e5f2834ca59e7ec5c58fc99b32183e.png\" class=\"latex\" alt=\"$ 2005^{2004} \\equiv 1 (\\mod{7})$\" style=\"vertical-align: -4px\" width=\"171\" height=\"19\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/5/2/d5255c49b85d8cec398f6973a1bd5308a5dd6354.png\" class=\"latex\" alt=\"$ 2005^{2005} \\equiv 2005 (\\mod{7})$\" style=\"vertical-align: -4px\" width=\"199\" height=\"19\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/5/b/d5b018c79d7366d49adef3dbb4c67c28a5359d7c.png\" class=\"latex\" alt=\"$ 2005 \\equiv 3 (\\mod{7})$\" style=\"vertical-align: -4px\" width=\"144\" height=\"18\" ><br>\n<br>\nSo the day of the week that corresponds to 3 is <img src=\"//latex.artofproblemsolving.com/1/a/2/1a23e99fe7106a64d1a113fac1c885f39242a5be.png\" class=\"latex\" alt=\"$ \\boxed{\\mbox{Thursday}}$\" style=\"vertical-align: -9px\" width=\"80\" height=\"26\" >", "post_id": 4407051, "post_number": 2, "post_time_unix": 1220489486, "post_time_utc": "2008-09-04 00:51:26 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "This Euler's formula thing sounds pretty useful.", "content_html": "This Euler's formula thing sounds pretty useful.", "post_id": 4407052, "post_number": 3, "post_time_unix": 1220540204, "post_time_utc": "2008-09-04 14:56:44 UTC", "thanks_received": 2, "user_id": 27042, "username": "undefined117" }, { "attachments": [], "content_bbcode": "Um, correct me if I'm wrong, but don't you mean Fermat's Little Theorem? Euler's generalization only applies when the modular base isn't prime, in which case you raise the number to the euler's totient function of that nonprime rather than raising it to one less than the nonprime. :wink:", "content_html": "Um, correct me if I'm wrong, but don't you mean Fermat's Little Theorem? Euler's generalization only applies when the modular base isn't prime, in which case you raise the number to the euler's totient function of that nonprime rather than raising it to one less than the nonprime. <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />", "post_id": 4407053, "post_number": 4, "post_time_unix": 1220665730, "post_time_utc": "2008-09-06 01:48:50 UTC", "thanks_received": 2, "user_id": 29190, "username": "Math Geek" } ], "source": null }	What day of the week is $2005^{2005}$ days after Monday?	[ "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/ClockArithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/EulersTotientTheorem", "/Mathematics/NumberTheory/Congruences/FermatsLittleTheorem", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Use Euler's theorem to reduce the large exponent modulo 7, the length of the week cycle.	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aops_99360	[quote="Ignite168"]Baphomet has 45 Yuan (元). If 1 Euro (€) is pegged at 1.25 US Dollars ($\$$) and 1 € is pegged at 10 元, what is the amount of money in 元s Baphomet will have left over once he buys an item costing $\$$3.25?[/quote] [hide]$\$$3.25 is the same as 26 元. Therefore, Baphomet will have $45-26=\boxed{19}$ 元 left.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Baphomet has 45 Yuan (元). If 1 Euro (€) is pegged at 1.25 US Dollars ($\\$$) and 1 € is pegged at 10 元, what is the amount of money in 元s Baphomet will have left over once he buys an item costing $\\$$3.25?", "content_html": "Baphomet has 45 Yuan (元). If 1 Euro (€) is pegged at 1.25 US Dollars <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >)</span> and 1 € is pegged at 10 元, what is the amount of money in 元s Baphomet will have left over once he buys an item costing <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >3</span>.25?", "post_id": 561058, "post_number": 1, "post_time_unix": 1151557983, "post_time_utc": "2006-06-29 05:13:03 UTC", "thanks_received": 2, "user_id": 17514, "username": "Ignite168" }, { "attachments": [], "content_bbcode": "[quote=\"Ignite168\"]Baphomet has 45 Yuan (元). If 1 Euro (€) is pegged at 1.25 US Dollars ($\\$$) and 1 € is pegged at 10 元, what is the amount of money in 元s Baphomet will have left over once he buys an item costing $\\$$3.25?[/quote]\r\n\r\n[hide]$\\$$3.25 is the same as 26 元. Therefore, Baphomet will have $45-26=\\boxed{19}$ 元 left.[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Ignite168 wrote:</div>\n<div class=\"bbcode_quote_body\">Baphomet has 45 Yuan (元). If 1 Euro (€) is pegged at 1.25 US Dollars <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >)</span> and 1 € is pegged at 10 元, what is the amount of money in 元s Baphomet will have left over once he buys an item costing <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >3</span>.25?</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/7/347d5d8ccef965291225560dfe843b447c59955e.png\" class=\"latex\" alt=\"$\\$$\" style=\"vertical-align: -1px\" width=\"8\" height=\"14\" >3</span>.25 is the same as 26 元. Therefore, Baphomet will have <img src=\"//latex.artofproblemsolving.com/3/d/5/3d5e6bcf39ca8f083eb47c9943f61a600eef60ea.png\" class=\"latex\" alt=\"$45-26=\\boxed{19}$\" style=\"vertical-align: -5px\" width=\"112\" height=\"23\" > 元 left.</div>", "post_id": 561320, "post_number": 2, "post_time_unix": 1151585717, "post_time_utc": "2006-06-29 12:55:17 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[hide]I got 19.\n\nConvert 3.25 dollars into euros to get 26\n\n45-26 =19[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">I got 19.<br>\n<br>\nConvert 3.25 dollars into euros to get 26<br>\n<br>\n45-26 =19</div>", "post_id": 562217, "post_number": 3, "post_time_unix": 1151631097, "post_time_utc": "2006-06-30 01:31:37 UTC", "thanks_received": 2, "user_id": 8131, "username": "math92" }, { "attachments": [], "content_bbcode": "Your solutions are not thorough enough, how did you know how many dollars equaled a Yuan? There's no exchange rate for that in the problem.", "content_html": "Your solutions are not thorough enough, how did you know how many dollars equaled a Yuan? There's no exchange rate for that in the problem.", "post_id": 562303, "post_number": 4, "post_time_unix": 1151633972, "post_time_utc": "2006-06-30 02:19:32 UTC", "thanks_received": 2, "user_id": 17514, "username": "Ignite168" }, { "attachments": [], "content_bbcode": "[hide=\"A thorough solution.\"]\nWe take fractions.\n\n$\\frac{euro}{yuan}=10$\n\n$\\frac{euro}{dollar}=1.25$\n\n$\\frac{dollar}{yuan}=\\frac{euro}{yuan}\\frac{dollar}{euro}=10\\frac{1}{1.25}=\\frac{2}{.25}=\\frac{2}{\\frac{1}{4}}=8$\n\nTherefore, there are 8 yuan in one dollar.\nTherefore, there are $83.25=24+2=26$? in 3.25.\nTherefore, he has $45-26=\\boxed{19}$? left.\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">A thorough solution.</a><div class=\"cmty-hide-content\" style=\"display:none\">We take fractions.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/6/f/86feb50b99b3145a67b25e3055ff75a108161845.png\" class=\"latex\" alt=\"$\\frac{euro}{yuan}=10$\" style=\"vertical-align: -16px\" width=\"86\" height=\"36\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/8/3/78376d55366a5d67588ba88f36ccf50bdb639668.png\" class=\"latex\" alt=\"$\\frac{euro}{dollar}=1.25$\" style=\"vertical-align: -12px\" width=\"108\" height=\"33\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/6/c/7/6c74d78c3c2729714d804cabfa4086fb833e4755.png\" class=\"latex\" alt=\"$\\frac{dollar}{yuan}=\\frac{euro}{yuan}\\frac{dollar}{euro}=10\\frac{1}{1.25}=\\frac{2}{.25}=\\frac{2}{\\frac{1}{4}}=8$\" style=\"vertical-align: -19px\" width=\"409\" height=\"44\" ><br>\n<br>\nTherefore, there are 8 yuan in one dollar.<br>\nTherefore, there are <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/8/8/6889642b4f88e38a80ab8593a8f8e5f6893ce668.png\" class=\"latex\" alt=\"$83.25=24+2=26$\" style=\"vertical-align: -1px\" width=\"174\" height=\"14\" >?</span> in 3.25.<br>\nTherefore, he has <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/d/5/3d5e6bcf39ca8f083eb47c9943f61a600eef60ea.png\" class=\"latex\" alt=\"$45-26=\\boxed{19}$\" style=\"vertical-align: -5px\" width=\"112\" height=\"23\" >?</span> left.</div>", "post_id": 566242, "post_number": 5, "post_time_unix": 1152044772, "post_time_utc": "2006-07-04 20:26:12 UTC", "thanks_received": 1, "user_id": 15223, "username": "1=2" } ], "source": null }	Baphomet has $45$ yuan (元). If $1$ euro (€) is pegged at $1.25$ US dollars ($\$$) and $1$ € is pegged at $10$ 元, what is the amount of money in 元 that Baphomet will have left over once he buys an item costing $\$3.25$?	[ "/Mathematics/AppliedMathematics" ]	Convert the dollar amount to yuan using the two given euro exchange rates and then subtract.	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aops_99366	[quote="inom"]$sin3z-cos3z=\sqrt\frac{3}{2}$ Solve for z?[/quote] [hide] Squar both sides and simple algebraic computation yields: $-\frac{1}{2}=2\cos(3z)\sin(3z)$. This is equal to $-\frac{1}{2}=\sin(6z)$.. This is pretty much it... :D [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "$sin3z-cos3z=\\sqrt\\frac{3}{2}$\r\nSolve for z?", "content_html": "<img src=\"//latex.artofproblemsolving.com/5/2/e/52ea84d746ac3779f0e9d42cfd5c1610736e2311.png\" class=\"latex\" alt=\"$sin3z-cos3z=\\sqrt\\frac{3}{2}$\" style=\"vertical-align: -14px\" width=\"165\" height=\"43\" ><br>\nSolve for z?", "post_id": 561070, "post_number": 1, "post_time_unix": 1151558980, "post_time_utc": "2006-06-29 05:29:40 UTC", "thanks_received": 2, "user_id": 20532, "username": "inom" }, { "attachments": [], "content_bbcode": "[quote=\"inom\"]$sin3z-cos3z=\\sqrt\\frac{3}{2}$\nSolve for z?[/quote]\r\n[hide]\nSquar both sides and simple algebraic computation yields:\n$-\\frac{1}{2}=2\\cos(3z)\\sin(3z)$.\nThis is equal to $-\\frac{1}{2}=\\sin(6z)$..\nThis is pretty much it... :D [/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">inom wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/5/2/e/52ea84d746ac3779f0e9d42cfd5c1610736e2311.png\" class=\"latex\" alt=\"$sin3z-cos3z=\\sqrt\\frac{3}{2}$\" style=\"vertical-align: -14px\" width=\"165\" height=\"43\" ><br>\nSolve for z?</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Squar both sides and simple algebraic computation yields:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/0/0/20045937d9cd40a8ef70a0e0b68709ac901abdd2.png\" class=\"latex\" alt=\"$-\\frac{1}{2}=2\\cos(3z)\\sin(3z)$\" style=\"vertical-align: -12px\" width=\"176\" height=\"37\" >.</span><br>\nThis is equal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/1/9218eacce16f052d9ec76056f53a66f2712d1c99.png\" class=\"latex\" alt=\"$-\\frac{1}{2}=\\sin(6z)$\" style=\"vertical-align: -12px\" width=\"104\" height=\"37\" >.</span>.<br>\nThis is pretty much it... <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div>", "post_id": 561150, "post_number": 2, "post_time_unix": 1151566259, "post_time_utc": "2006-06-29 07:30:59 UTC", "thanks_received": 2, "user_id": 9197, "username": "kimby_102" }, { "attachments": [], "content_bbcode": "[hide]Let $3z = x$\nAfter squaring both sides we get $1 - 2\\sin x\\cos x = \\frac{3}{2} \\Rightarrow 2\\sin x\\cos x = -\\frac{1}{2} \\Rightarrow \\sin 2x = -\\frac{1}{2}$.\nHence $z = \\frac{11}{36}\\pi + k\\frac{\\pi}{3}$. [/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let <img src=\"//latex.artofproblemsolving.com/0/e/2/0e273539260225515d64116566b243617df83726.png\" class=\"latex\" alt=\"$3z = x$\" width=\"52\" height=\"12\" ><br>\nAfter squaring both sides we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/4/1/7414b1aed0c0e394f57ef8d43e6328bb1bfce7ab.png\" class=\"latex\" alt=\"$1 - 2\\sin x\\cos x = \\frac{3}{2} \\Rightarrow 2\\sin x\\cos x = -\\frac{1}{2} \\Rightarrow \\sin 2x = -\\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"448\" height=\"37\" >.</span><br>\nHence <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/7/4/3746ad9d7763b5a90b15e32f4583b328c0f33594.png\" class=\"latex\" alt=\"$z = \\frac{11}{36}\\pi + k\\frac{\\pi}{3}$\" style=\"vertical-align: -12px\" width=\"112\" height=\"37\" >.</span></div>", "post_id": 561159, "post_number": 3, "post_time_unix": 1151566575, "post_time_utc": "2006-06-29 07:36:15 UTC", "thanks_received": 2, "user_id": 5839, "username": "Andreas" }, { "attachments": [], "content_bbcode": "[hide]\nsquare both sides to get $1-\\sin{6z} = \\frac{3}{2}$. then we get $6z = \\frac{(7\\pm12k)\\pi}{6}, \\frac{(11 \\pm 12k)\\pi}{6}$, divide by 6 to get $z = \\frac{(7\\pm12k)\\pi}{36}, \\frac{(11 \\pm 12k)\\pi}{36}$\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">square both sides to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/3/0/63077b0a7547704988e6cee2f4ccea1d01de362a.png\" class=\"latex\" alt=\"$1-\\sin{6z} = \\frac{3}{2}$\" style=\"vertical-align: -12px\" width=\"110\" height=\"37\" >.</span> then we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/5/8456ed4637f8d743c29729b240b868f2d07a2c01.png\" class=\"latex\" alt=\"$6z = \\frac{(7\\pm12k)\\pi}{6}, \\frac{(11 \\pm 12k)\\pi}{6}$\" style=\"vertical-align: -12px\" width=\"235\" height=\"38\" >,</span> divide by 6 to get <img src=\"//latex.artofproblemsolving.com/0/0/b/00b8bbf15123804b6bc316801cb2153059c40c18.png\" class=\"latex\" alt=\"$z = \\frac{(7\\pm12k)\\pi}{36}, \\frac{(11 \\pm 12k)\\pi}{36}$\" style=\"vertical-align: -12px\" width=\"226\" height=\"38\" ></div>", "post_id": 561179, "post_number": 4, "post_time_unix": 1151567489, "post_time_utc": "2006-06-29 07:51:29 UTC", "thanks_received": 2, "user_id": 10705, "username": "pkerichang" }, { "attachments": [], "content_bbcode": "Hint\r\n[hide]$z_1=35^\\circ+(120^\\circ)k$\n$z_2=55\\circ+(120^\\circ)k$[/hide]", "content_html": "Hint<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/1/5/e/15ee0ca191caa897895e17abe5fc780bf9ae53f2.png\" class=\"latex\" alt=\"$z_1=35^\\circ+(120^\\circ)k$\" style=\"vertical-align: -4px\" width=\"146\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/3/d/13d2285c92385a077b64d72b580263a0e47b7088.png\" class=\"latex\" alt=\"$z_2=55\\circ+(120^\\circ)k$\" style=\"vertical-align: -4px\" width=\"148\" height=\"18\" ></div>", "post_id": 561186, "post_number": 5, "post_time_unix": 1151567821, "post_time_utc": "2006-06-29 07:57:01 UTC", "thanks_received": 2, "user_id": 20532, "username": "inom" } ], "source": null }	Solve for $z$: \[ \sin 3z - \cos 3z = \sqrt{\frac{3}{2}}. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity" ]	Square the equation and apply the identity (sin A – cos A)² = 1 – sin 2A to obtain a simple sine equation.	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aops_99372	[quote="inom"]$cos^4(2x)+6cos^2(2x)=\frac{25}{16}$ Solve for x?[/quote] [hide]this is pretty straight-forward.. let $\cos^2(2x)$ be $a$, then the given equation becomes: $a^2+6a-\frac{25}{16}=0$. Multiply $16$ on the equation, we get: $16a^2+96a-25=0$. This is factored into: $(4a-1)(4a+25)=0$.. from here it 's easy to find solutions to the problem.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "$cos^4(2x)+6cos^2(2x)=\\frac{25}{16}$\r\nSolve for x?", "content_html": "<img src=\"//latex.artofproblemsolving.com/c/8/a/c8ab5838439b8ee0514f60b63671c0b428471495.png\" class=\"latex\" alt=\"$cos^4(2x)+6cos^2(2x)=\\frac{25}{16}$\" style=\"vertical-align: -13px\" width=\"208\" height=\"38\" ><br>\nSolve for x?", "post_id": 561078, "post_number": 1, "post_time_unix": 1151559555, "post_time_utc": "2006-06-29 05:39:15 UTC", "thanks_received": 2, "user_id": 20532, "username": "inom" }, { "attachments": [], "content_bbcode": "[quote=\"inom\"]$cos^4(2x)+6cos^2(2x)=\\frac{25}{16}$\nSolve for x?[/quote]\r\n[hide]this is pretty straight-forward..\nlet $\\cos^2(2x)$ be $a$, then the given equation becomes:\n$a^2+6a-\\frac{25}{16}=0$.\nMultiply $16$ on the equation, we get:\n$16a^2+96a-25=0$. \nThis is factored into:\n$(4a-1)(4a+25)=0$.. \nfrom here it 's easy to find solutions to the problem.[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">inom wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/c/8/a/c8ab5838439b8ee0514f60b63671c0b428471495.png\" class=\"latex\" alt=\"$cos^4(2x)+6cos^2(2x)=\\frac{25}{16}$\" style=\"vertical-align: -13px\" width=\"208\" height=\"38\" ><br>\nSolve for x?</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">this is pretty straight-forward..<br>\nlet <img src=\"//latex.artofproblemsolving.com/7/8/2/782e8d44d2bf9700ca53abbecbba154acaf33f23.png\" class=\"latex\" alt=\"$\\cos^2(2x)$\" style=\"vertical-align: -4px\" width=\"64\" height=\"19\" > be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" >,</span> then the given equation becomes:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/7/dc76bcfb73bce834b7f12190c171b52fb514f14d.png\" class=\"latex\" alt=\"$a^2+6a-\\frac{25}{16}=0$\" style=\"vertical-align: -13px\" width=\"135\" height=\"38\" >.</span><br>\nMultiply <img src=\"//latex.artofproblemsolving.com/9/a/5/9a5b4928c8fe50ce3c2428da3bee3505e891b788.png\" class=\"latex\" alt=\"$16$\" style=\"vertical-align: 0px\" width=\"17\" height=\"13\" > on the equation, we get:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/e/eae3b8a12cb323dfe388a081a79086d2521d7d26.png\" class=\"latex\" alt=\"$16a^2+96a-25=0$\" style=\"vertical-align: -1px\" width=\"158\" height=\"16\" >.</span><br>\nThis is factored into:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/b/e0bf948dd5960aba662599216d554b7dce3aaee0.png\" class=\"latex\" alt=\"$(4a-1)(4a+25)=0$\" style=\"vertical-align: -4px\" width=\"170\" height=\"18\" >.</span>.<br>\nfrom here it 's easy to find solutions to the problem.</div>", "post_id": 561144, "post_number": 2, "post_time_unix": 1151566025, "post_time_utc": "2006-06-29 07:27:05 UTC", "thanks_received": 2, "user_id": 9197, "username": "kimby_102" }, { "attachments": [], "content_bbcode": "[hide]Let $2x = \\alpha$ and $\\cos^2 \\alpha = a$.\n$16a^2 + 96a - 25 = 0 \\Rightarrow \\cos^2 \\alpha = -\\frac{25}{4} \\vee \\cos^2 \\alpha = \\frac{1}{4}$.\nHence $\\cos \\alpha = \\pm \\frac{1}{2} \\Rightarrow x = \\frac{\\pi}{6} + k\\pi \\vee x = \\frac{\\pi}{3} + k\\pi$.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let <img src=\"//latex.artofproblemsolving.com/4/c/e/4ce6703790866f7486376611010324062ba54bc6.png\" class=\"latex\" alt=\"$2x = \\alpha$\" width=\"54\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/8/2/08254388ec6279a7bbf7d4273c971503945ec522.png\" class=\"latex\" alt=\"$\\cos^2 \\alpha = a$\" width=\"80\" height=\"15\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/7/3/673a2b5d9e5e565d1a47b495f5e83c14f08d9405.png\" class=\"latex\" alt=\"$16a^2 + 96a - 25 = 0 \\Rightarrow \\cos^2 \\alpha = -\\frac{25}{4} \\vee \\cos^2 \\alpha = \\frac{1}{4}$\" style=\"vertical-align: -13px\" width=\"397\" height=\"38\" >.</span><br>\nHence <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/d/6adaa8d65ccb01d97c76547b2ffe7b354d31251f.png\" class=\"latex\" alt=\"$\\cos \\alpha = \\pm \\frac{1}{2} \\Rightarrow x = \\frac{\\pi}{6} + k\\pi \\vee x = \\frac{\\pi}{3} + k\\pi$\" style=\"vertical-align: -12px\" width=\"324\" height=\"37\" >.</span></div>", "post_id": 561145, "post_number": 3, "post_time_unix": 1151566065, "post_time_utc": "2006-06-29 07:27:45 UTC", "thanks_received": 2, "user_id": 5839, "username": "Andreas" }, { "attachments": [], "content_bbcode": "I believe the solutions for $x$ are\r\n\r\n $\\frac{\\pi}{6}+\\frac{k\\pi}{2}, \\frac{\\pi}{3}+\\frac{k\\pi}{2}$", "content_html": "I believe the solutions for <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > are<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/5/0/b503bd05e343c8bc9137c5e16ce2b91094187c6f.png\" class=\"latex\" alt=\"$\\frac{\\pi}{6}+\\frac{k\\pi}{2}, \\frac{\\pi}{3}+\\frac{k\\pi}{2}$\" style=\"vertical-align: -12px\" width=\"131\" height=\"37\" >", "post_id": 564595, "post_number": 4, "post_time_unix": 1151862419, "post_time_utc": "2006-07-02 17:46:59 UTC", "thanks_received": 2, "user_id": 12699, "username": "jhcreinhardt" } ], "source": null }	Solve for $x$: \[ \cos^4(2x)+6\cos^2(2x)=\frac{25}{16}. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial" ]	Introduce a variable for cos²(2x) to reduce the equation to a quadratic and solve for that variable.	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aops_99390	(2) It's obvious that there's no three points collinear (otherwise we get $k\geq 2>2\sin 70^{\circ}$). Also, we've the following [b]Lemma[/b]: For any triangle $ABC$, $\frac{BC}{\min \{ BA,AC\} }\geq 2\sin \frac{A}{2}$. Suppose that $k\leq 2\sin 70^{\circ}$. Let $M$ is the smallest distance of two points among $A,B,C,D,E$. Let $AC\cap BD=T$. WLOG that $E$ is in the interior of $\triangle{ABT}$. By [b]Lemma[/b], we get $\angle{AEB},\angle{BED}\leq 140^{\circ}\implies \angle{AED}\geq 80^{\circ}$. By (1), we get that $\angle{ABC},\angle{BAD}\leq 70^{\circ}$. WLOG $\angle{BCD}\geq 110^{\circ}$. In triangle $BCE$, we've $\frac{BC}{\min \{ BE,EC\} }\geq 2\sin 40^{\circ}\implies BC\geq 2M\sin 40^{\circ}$. By law of cosines, $BD^2=BC^2+CD^2-2BC\times CD\times \cos (\angle{BCD})\geq BC^2+CD^2+2BC\times CD\times \sin 70^{\circ}\geq M^2(4\sin^2 40^{\circ} +1+4\sin 40^{\circ} \sin 70^{\circ})$. Hence, $BD^2\geq 4M^2\sin^2 70^{\circ}\implies BD\geq 2M\sin 70^{\circ}\implies 2\sin 70^{\circ} \geq k\geq 2\sin 70^{\circ}$. We get that if $k\leq 2\sin 70^{\circ}$, then $k=2\sin 70^{\circ}$. This gives $k\geq 2\sin 70^{\circ}$. For the equality case, we get that all equalities in the above bounds holds. This's not hard to verify that it happens when $ABCD$ is an isosceles trapezoid with $\angle{ABC}=\angle{BAD}=70^{\circ}$ and $E$ is the point in quadrilateral $ABCD$ that $DEC$ is an equilateral triangle.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "(1) $D$ is an arbitary point in $\\triangle{ABC}$. Prove that:\r\n\\[ \\frac{BC}{\\min{AD,BD,CD}} \\geq \\{ \\begin{array}{c} \\displaystyle 2\\sin{A}, \\ \\angle{A}< 90^o \\\\ \\\\ 2, \\ \\angle{A} \\geq 90^o \\end{array} \\]\r\n(2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \\geq 2\\sin{70^o}$. Can equality be achieved?", "content_html": "(1) <img src=\"//latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is an arbitary point in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/f/8/1f828f10cad9235a2c72c34369f5fd4a1179e579.png\" class=\"latex\" alt=\"$\\triangle{ABC}$\" width=\"58\" height=\"13\" >.</span> Prove that:<br>\n<img src=\"//latex.artofproblemsolving.com/d/b/c/dbc154a2aefacceb2126682f05c1cb54d7522cc1.png\" class=\"latexcenter\" alt=\"\\[ \\frac{BC}{\\min{AD,BD,CD}} \\geq \\{ \\begin{array}{c} \\displaystyle 2\\sin{A}, \\ \\angle{A}< 90^o \\\\ \\\\ 2, \\ \\angle{A} \\geq 90^o \\end{array} \\]\" width=\"322\" height=\"56\" ><br>\n(2<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" ></span> is an arbitary point in convex quadrilateral <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/9/e/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" >.</span> Denote <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > the ratio of the largest and least distances of any two points among <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" >.</span> Prove that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/1/3/6139f7950bacac81730816000c86500da24c5a15.png\" class=\"latex\" alt=\"$k \\geq 2\\sin{70^o}$\" style=\"vertical-align: -2px\" width=\"96\" height=\"15\" >.</span> Can equality be achieved?", "post_id": 561140, "post_number": 1, "post_time_unix": 1151565877, "post_time_utc": "2006-06-29 07:24:37 UTC", "thanks_received": 2, "user_id": 6601, "username": "shobber" }, { "attachments": [], "content_bbcode": "1): We have:\r\nIf $\\angle{A} <90^o$ then:\r\n$\\frac{BC}{\\min{AD,BD,CD}} \\geq \\frac{BC}{R}$, where $R$ is circumcentre.\r\n$R=\\frac{BC}{2sinA}$. So:\r\n$\\frac{BC}{\\min{AD,BD,CD}} \\geq \\frac{BC}{R}$\r\n$\\geq 2sinA$.\r\nIf $\\angle {A} \\geq 90^o$ then $BC > AC,AB$, and:\r\nif $D$ is midpt of $BC$, then $minAD,BD,CD \\leq \\frac{BC}{2}$ and so $\\frac{BC}{\\min{AD,BD,CD}} \\geq 2$.\r\nLet $M$ be midpoint of $BC$.\r\nif $D$ is inside $\\triangle AMB$, $minAD,BD \\le BM$, and if $D$ is inside $\\triangle AMC$, $minAD,CD \\le CM$. Therfore:\r\n$minAD,BD,CD \\le CM= \\frac{BC}{2}$.So:\r\n$\\frac{BC}{\\min{AD,BD,CD}} \\geq 2$\r\nQED", "content_html": "1): We have:<br>\nIf <img src=\"//latex.artofproblemsolving.com/f/d/4/fd4dde215579a8c1b2361a82a904cac6e4880d9e.png\" class=\"latex\" alt=\"$\\angle{A} <90^o$\" style=\"vertical-align: 0px\" width=\"75\" height=\"13\" > then:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/d/9/7d91fe8f337d8c7e9cbe26b1c29a704d3ab60d28.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min{AD,BD,CD}} \\geq \\frac{BC}{R}$\" style=\"vertical-align: -16px\" width=\"198\" height=\"41\" >,</span> where <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > is circumcentre.<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/2/7/b27bb6de890b52770e094a878fb32472852b987f.png\" class=\"latex\" alt=\"$R=\\frac{BC}{2sinA}$\" style=\"vertical-align: -12px\" width=\"88\" height=\"37\" >.</span> So:<br>\n<img src=\"//latex.artofproblemsolving.com/7/d/9/7d91fe8f337d8c7e9cbe26b1c29a704d3ab60d28.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min{AD,BD,CD}} \\geq \\frac{BC}{R}$\" style=\"vertical-align: -16px\" width=\"198\" height=\"41\" ><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/f/6/df6a3b23e0405ab457ab60b25aeae8ee4a9dd5a6.png\" class=\"latex\" alt=\"$\\geq 2sinA$\" style=\"vertical-align: -2px\" width=\"67\" height=\"15\" >.</span><br>\nIf <img src=\"//latex.artofproblemsolving.com/d/e/7/de7570e8e576b5f4fab76ee16cd1953624e540b7.png\" class=\"latex\" alt=\"$\\angle {A} \\geq 90^o$\" style=\"vertical-align: -2px\" width=\"75\" height=\"15\" > then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/8/8/38860cc6862c0958de4d5e4dd215ac6c79d8c82a.png\" class=\"latex\" alt=\"$BC > AC,AB$\" style=\"vertical-align: -3px\" width=\"115\" height=\"16\" >,</span> and:<br>\nif <img src=\"//latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is midpt of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >,</span> then <img src=\"//latex.artofproblemsolving.com/6/2/3/623387e650bbfbe205509767c5cb88fb520c3534.png\" class=\"latex\" alt=\"$minAD,BD,CD \\leq \\frac{BC}{2}$\" style=\"vertical-align: -12px\" width=\"194\" height=\"37\" > and so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/0/f/50f535fc9664aaefeed7069ed12d82da00411cc7.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min{AD,BD,CD}} \\geq 2$\" style=\"vertical-align: -16px\" width=\"175\" height=\"41\" >.</span><br>\nLet <img src=\"//latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > be midpoint of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" >.</span><br>\nif <img src=\"//latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is inside <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/9/d/69dec09caa3fc15dd235bbb46718b3b8dd3ca725.png\" class=\"latex\" alt=\"$\\triangle AMB$\" width=\"63\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/f/6/df6e466513a85590734f51808449f226dd1b3123.png\" class=\"latex\" alt=\"$minAD,BD \\le BM$\" style=\"vertical-align: -3px\" width=\"158\" height=\"16\" >,</span> and if <img src=\"//latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png\" class=\"latex\" alt=\"$D$\" width=\"15\" height=\"12\" > is inside <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/3/3431b35dcd263cac56e5c8a8aed399ba0ce6fce2.png\" class=\"latex\" alt=\"$\\triangle AMC$\" width=\"63\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/c/b/9cba19c0146945ba001740f1395140532b6cee56.png\" class=\"latex\" alt=\"$minAD,CD \\le CM$\" style=\"vertical-align: -3px\" width=\"158\" height=\"16\" >.</span> Therfore:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/8/b/58bca666a3bbce604c62349ae33372940af8e47c.png\" class=\"latex\" alt=\"$minAD,BD,CD \\le CM= \\frac{BC}{2}$\" style=\"vertical-align: -12px\" width=\"252\" height=\"37\" >.</span>So:<br>\n<img src=\"//latex.artofproblemsolving.com/5/0/f/50f535fc9664aaefeed7069ed12d82da00411cc7.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min{AD,BD,CD}} \\geq 2$\" style=\"vertical-align: -16px\" width=\"175\" height=\"41\" ><br>\nQED", "post_id": 561439, "post_number": 2, "post_time_unix": 1151592201, "post_time_utc": "2006-06-29 14:43:21 UTC", "thanks_received": 1, "user_id": 16759, "username": "rem" }, { "attachments": [], "content_bbcode": "(2)\nIt's obvious that there's no three points collinear (otherwise we get $k\\geq 2>2\\sin 70^{\\circ}$).\nAlso, we've the following [b]Lemma[/b]: For any triangle $ABC$, $\\frac{BC}{\\min \\{ BA,AC\\} }\\geq 2\\sin \\frac{A}{2}$.\nSuppose that $k\\leq 2\\sin 70^{\\circ}$. Let $M$ is the smallest distance of two points among $A,B,C,D,E$.\nLet $AC\\cap BD=T$. WLOG that $E$ is in the interior of $\\triangle{ABT}$.\nBy [b]Lemma[/b], we get $\\angle{AEB},\\angle{BED}\\leq 140^{\\circ}\\implies \\angle{AED}\\geq 80^{\\circ}$.\nBy (1), we get that $\\angle{ABC},\\angle{BAD}\\leq 70^{\\circ}$. WLOG $\\angle{BCD}\\geq 110^{\\circ}$.\nIn triangle $BCE$, we've $\\frac{BC}{\\min \\{ BE,EC\\} }\\geq 2\\sin 40^{\\circ}\\implies BC\\geq 2M\\sin 40^{\\circ}$.\nBy law of cosines, $BD^2=BC^2+CD^2-2BC\\times CD\\times \\cos (\\angle{BCD})\\geq BC^2+CD^2+2BC\\times CD\\times \\sin 70^{\\circ}\\geq M^2(4\\sin^2 40^{\\circ} +1+4\\sin 40^{\\circ} \\sin 70^{\\circ})$.\nHence, $BD^2\\geq 4M^2\\sin^2 70^{\\circ}\\implies BD\\geq 2M\\sin 70^{\\circ}\\implies 2\\sin 70^{\\circ} \\geq k\\geq 2\\sin 70^{\\circ}$.\nWe get that if $k\\leq 2\\sin 70^{\\circ}$, then $k=2\\sin 70^{\\circ}$. This gives $k\\geq 2\\sin 70^{\\circ}$.\n\nFor the equality case, we get that all equalities in the above bounds holds.\nThis's not hard to verify that it happens when $ABCD$ is an isosceles trapezoid with $\\angle{ABC}=\\angle{BAD}=70^{\\circ}$ and $E$ is the point in quadrilateral $ABCD$ that $DEC$ is an equilateral triangle.", "content_html": "(2)<br>\nIt's obvious that there's no three points collinear (otherwise we get <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/9/6/b/96bee62f9d1a605daf535d08ae55874a6efa7147.png\" class=\"latex\" alt=\"$k\\geq 2>2\\sin 70^{\\circ}$\" style=\"vertical-align: -2px\" width=\"129\" height=\"15\" >)</span>.<br>\nAlso, we've the following <b>Lemma</b>: For any triangle <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" >,</span> <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/8/9/0/890dd979798c60f2e306256199b4e9896ca77b3b.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min \\{ BA,AC\\} }\\geq 2\\sin \\frac{A}{2}$\" style=\"vertical-align: -17px\" width=\"194\" height=\"42\" >.</span><br>\nSuppose that <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/b/8/0/b80fe15e6e9b5cff6723dcc0cb407a89485dae62.png\" class=\"latex\" alt=\"$k\\leq 2\\sin 70^{\\circ}$\" style=\"vertical-align: -2px\" width=\"96\" height=\"15\" >.</span> Let <img src=\"//latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is the smallest distance of two points among <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/9/5/8/9580b15ecb95e817f4cc6603baaa03ed71254759.png\" class=\"latex\" alt=\"$A,B,C,D,E$\" style=\"vertical-align: -3px\" width=\"103\" height=\"16\" >.</span><br>\nLet <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/4/5/d/45da4c517e0842c92c9b0b6017cdf942f2955d89.png\" class=\"latex\" alt=\"$AC\\cap BD=T$\" style=\"vertical-align: 0px\" width=\"115\" height=\"13\" >.</span> WLOG that <img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > is in the interior of <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/5/6/3/563544d0dbd47310a75678c831e3d30c58ce6cb0.png\" class=\"latex\" alt=\"$\\triangle{ABT}$\" width=\"57\" height=\"13\" >.</span><br>\nBy <b>Lemma</b>, we get <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/c/6/0/c600fbe2dfdca47f383304c39640fb761270c115.png\" class=\"latex\" alt=\"$\\angle{AEB},\\angle{BED}\\leq 140^{\\circ}\\implies \\angle{AED}\\geq 80^{\\circ}$\" style=\"vertical-align: -3px\" width=\"334\" height=\"16\" >.</span><br>\nBy (1), we get that <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/8/5/b/85b365b0742c64bbe178bbfb8bfae3d545d77320.png\" class=\"latex\" alt=\"$\\angle{ABC},\\angle{BAD}\\leq 70^{\\circ}$\" style=\"vertical-align: -3px\" width=\"168\" height=\"16\" >.</span> WLOG <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/7/a/1/7a10a6a7884f9d8ad76094b7216d81f2e6c4c6f8.png\" class=\"latex\" alt=\"$\\angle{BCD}\\geq 110^{\\circ}$\" style=\"vertical-align: -2px\" width=\"114\" height=\"15\" >.</span><br>\nIn triangle <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/9/1/2/91275aa1f85be8f39ea3ae3cef68f10100fc664b.png\" class=\"latex\" alt=\"$BCE$\" width=\"43\" height=\"12\" >,</span> we've <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/6/9/a/69a4c6f00db76b43265e3757ab2e9498a6a92afc.png\" class=\"latex\" alt=\"$\\frac{BC}{\\min \\{ BE,EC\\} }\\geq 2\\sin 40^{\\circ}\\implies BC\\geq 2M\\sin 40^{\\circ}$\" style=\"vertical-align: -17px\" width=\"390\" height=\"42\" >.</span><br>\nBy law of cosines, <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/d/8/4/d8412631b70d8c6721c967a5e70307e3b2c4d640.png\" class=\"latex\" alt=\"$BD^2=BC^2+CD^2-2BC\\times CD\\times \\cos (\\angle{BCD})\\geq BC^2+CD^2+2BC\\times CD\\times \\sin 70^{\\circ}\\geq M^2(4\\sin^2 40^{\\circ} +1+4\\sin 40^{\\circ} \\sin 70^{\\circ})$\" style=\"vertical-align: -4px\" width=\"1000\" height=\"20\" >.</span><br>\nHence, <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/2/7/3/273971e88f815a696c1eb09ed532af39c3e1caf4.png\" class=\"latex\" alt=\"$BD^2\\geq 4M^2\\sin^2 70^{\\circ}\\implies BD\\geq 2M\\sin 70^{\\circ}\\implies 2\\sin 70^{\\circ} \\geq k\\geq 2\\sin 70^{\\circ}$\" style=\"vertical-align: -2px\" width=\"579\" height=\"18\" >.</span><br>\nWe get that if <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/b/8/0/b80fe15e6e9b5cff6723dcc0cb407a89485dae62.png\" class=\"latex\" alt=\"$k\\leq 2\\sin 70^{\\circ}$\" style=\"vertical-align: -2px\" width=\"96\" height=\"15\" >,</span> then <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/f/4/a/f4ae8e7eb4f3032614eb12e9913776617ad2db27.png\" class=\"latex\" alt=\"$k=2\\sin 70^{\\circ}$\" width=\"96\" height=\"13\" >.</span> This gives <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/a/4/5/a45ac2a46aedafffbd75a477bcf3bac31e4f710b.png\" class=\"latex\" alt=\"$k\\geq 2\\sin 70^{\\circ}$\" style=\"vertical-align: -2px\" width=\"96\" height=\"15\" >.</span><br>\n<br>\nFor the equality case, we get that all equalities in the above bounds holds.<br>\nThis's not hard to verify that it happens when <img src=\"//latex.artofproblemsolving.com/f/9/e/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > is an isosceles trapezoid with <img src=\"//latex.artofproblemsolving.com/a/4/d/a4d4003045518fd91d283312bcedd96a4815f875.png\" class=\"latex\" alt=\"$\\angle{ABC}=\\angle{BAD}=70^{\\circ}$\" width=\"184\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > is the point in quadrilateral <img src=\"//latex.artofproblemsolving.com/f/9/e/f9efaf9474c5658c4089523e2aff4e11488f8603.png\" class=\"latex\" alt=\"$ABCD$\" width=\"57\" height=\"13\" > that <img src=\"//latex.artofproblemsolving.com/b/7/a/b7a9191aefcf2eb237d1999486083ed78f6d515f.png\" class=\"latex\" alt=\"$DEC$\" width=\"44\" height=\"12\" > is an equilateral triangle.", "post_id": 9598204, "post_number": 3, "post_time_unix": 1514531032, "post_time_utc": "2017-12-29 07:03:52 UTC", "thanks_received": 3, "user_id": 243405, "username": "ThE-dArK-lOrD" }, { "attachments": [], "content_bbcode": "Different solution for $(2)$\n\n$(2)$ First of all, if $E$ lies on $AC$ or $BD$, then in fact we can get a bound of $2 > 2 \\sin 70.$ Hence, assume from now on that $E$ is in the strict interior of $\\triangle BCD$ and $\\triangle BCA$, WLOG. We can also observe that if any triangle consisting of three of $A, B, C, D, E$ has an obtuse angle which is greater than $140$, we are done by Law of Sines within that triangle.\n\nBy part $(1)$, if any of $\\angle BAC, \\angle ABC, \\angle ACB, \\angle BDC, \\angle DCB, \\angle DBC$ are greater than or equal to $70$ degrees, we are done. Hence, suppose that they all have measures in the range $(40, 70].$ Let $\\theta_1 = \\angle ABC, \\theta_2 = \\angle DCB.$ Observe that if $\\angle BAD \\le 70$ then $\\angle ADC \\ge 150$ and we're done. Analogously, we're also done if $\\angle ADC \\le 70.$ Therefore, WLOG assume that $\\angle BAD, \\angle CDA >70.$ Now, observe that either $\\angle BAD \\le 180 - \\theta_1$ or $\\angle ADC \\le 180 - \\theta_2.$ By symmetry, we will consider just the former. We then have that $\\angle BAD \\in [70, 180 - \\theta_1]$ and so as $\\theta_1 \\le 70$, we've $\\sin \\angle BAD \\ge \\sin \\theta_1.$ Therefore, $\\frac{BD}{AD} \\ge \\frac{\\sin \\theta_1}{\\sin (\\theta_1 - 40)},$ where we also used $\\angle ABD = \\angle ABC - \\angle DBC \\le \\theta_1 - 40.$ It's easily checked that for $\\theta_1 \\in (40, 70],$ $\\frac{\\sin \\theta_1}{\\sin (\\theta_1 - 40)} \\ge \\frac{\\sin 70}{\\sin 30} = 2 \\sin 70,$ and so we're done.\n\n$\\square$ ", "content_html": "Different solution for <img src=\"//latex.artofproblemsolving.com/6/2/f/62f62a19f0425d8e6121ca0c1cfd3ce96e43c9c5.png\" class=\"latex\" alt=\"$(2)$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/6/2/f/62f62a19f0425d8e6121ca0c1cfd3ce96e43c9c5.png\" class=\"latex\" alt=\"$(2)$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" > First of all, if <img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > lies on <img src=\"//latex.artofproblemsolving.com/a/1/7/a179ff2638e4799cadd820db205c2beff6299ce9.png\" class=\"latex\" alt=\"$AC$\" width=\"27\" height=\"13\" > or <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/8/9/f/89fff82bb65d0215e49c8c91cb7c553da52205e2.png\" class=\"latex\" alt=\"$BD$\" width=\"29\" height=\"12\" >,</span> then in fact we can get a bound of <img src=\"//latex.artofproblemsolving.com/e/2/2/e226c05574b652e9ed7743a5e47383656ddfd5c3.png\" class=\"latex\" alt=\"$2 > 2 \\sin 70.$\" style=\"vertical-align: 0px\" width=\"92\" height=\"13\" > Hence, assume from now on that <img src=\"//latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png\" class=\"latex\" alt=\"$E$\" width=\"14\" height=\"12\" > is in the strict interior of <img src=\"//latex.artofproblemsolving.com/1/2/f/12f372757d55c2d07adc720932c6d6e1396aa389.png\" class=\"latex\" alt=\"$\\triangle BCD$\" width=\"60\" height=\"13\" > and <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/7/9/c/79c35bcb734d7ec239f0e2939d436cca06f71806.png\" class=\"latex\" alt=\"$\\triangle BCA$\" width=\"58\" height=\"13\" >,</span> WLOG. We can also observe that if any triangle consisting of three of <img src=\"//latex.artofproblemsolving.com/6/3/5/63568d5821ffdfe52fbdbe448e607084e73f9457.png\" class=\"latex\" alt=\"$A, B, C, D, E$\" style=\"vertical-align: -3px\" width=\"103\" height=\"16\" > has an obtuse angle which is greater than <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/0/b/5/0b51850ccd3440214d4ccfbc14d29b2a31bfda55.png\" class=\"latex\" alt=\"$140$\" style=\"vertical-align: 0px\" width=\"26\" height=\"13\" >,</span> we are done by Law of Sines within that triangle.<br>\n<br>\nBy part <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/0/d/9/0d9c3b1c142cbd3f8217dabc160c57e4750df241.png\" class=\"latex\" alt=\"$(1)$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" >,</span> if any of <img src=\"//latex.artofproblemsolving.com/0/4/5/0459aad7ece47d3493080b2b7b28ef8fe2b60807.png\" class=\"latex\" alt=\"$\\angle BAC, \\angle ABC, \\angle ACB, \\angle BDC, \\angle DCB, \\angle DBC$\" style=\"vertical-align: -3px\" width=\"375\" height=\"16\" > are greater than or equal to <img src=\"//latex.artofproblemsolving.com/a/a/9/aa94edfb9739dcc66dad90f7dd5c148f44a57e2c.png\" class=\"latex\" alt=\"$70$\" width=\"17\" height=\"12\" > degrees, we are done. Hence, suppose that they all have measures in the range <img src=\"//latex.artofproblemsolving.com/6/4/c/64cd78696c9bc2ad5ae0af4e870ec737ad02b096.png\" class=\"latex\" alt=\"$(40, 70].$\" style=\"vertical-align: -5px\" width=\"60\" height=\"18\" > Let <img src=\"//latex.artofproblemsolving.com/3/6/4/3644fc0f8e68a10e1d04d001e443b97949215bf3.png\" class=\"latex\" alt=\"$\\theta_1 = \\angle ABC, \\theta_2 = \\angle DCB.$\" style=\"vertical-align: -3px\" width=\"204\" height=\"16\" > Observe that if <img src=\"//latex.artofproblemsolving.com/0/0/5/005bfbe620571d10ceedde4928e0b63159a5c01f.png\" class=\"latex\" alt=\"$\\angle BAD \\le 70$\" style=\"vertical-align: -2px\" width=\"98\" height=\"15\" > then <img src=\"//latex.artofproblemsolving.com/b/9/9/b998c88f02f36e3f8c83ab8f5fa558eec34afd48.png\" class=\"latex\" alt=\"$\\angle ADC \\ge 150$\" style=\"vertical-align: -2px\" width=\"107\" height=\"15\" > and we're done. Analogously, we're also done if <img src=\"//latex.artofproblemsolving.com/d/0/3/d0314d95b0186700e6710dbc68cb2ffa9b4a0c26.png\" class=\"latex\" alt=\"$\\angle ADC \\le 70.$\" style=\"vertical-align: -2px\" width=\"102\" height=\"15\" > Therefore, WLOG assume that <img src=\"//latex.artofproblemsolving.com/0/b/a/0badd51b8f7e58bb8818b6ddf2fec34e2edd9f82.png\" class=\"latex\" alt=\"$\\angle BAD, \\angle CDA >70.$\" style=\"vertical-align: -3px\" width=\"167\" height=\"16\" > Now, observe that either <img src=\"//latex.artofproblemsolving.com/a/2/1/a219401ea757e3c5a16b94c7ceadb67e0c8ac348.png\" class=\"latex\" alt=\"$\\angle BAD \\le 180 - \\theta_1$\" style=\"vertical-align: -2px\" width=\"144\" height=\"15\" > or <img src=\"//latex.artofproblemsolving.com/c/5/5/c554b4667f2136a9b17a2d9f7f3a1f8dc60eb3da.png\" class=\"latex\" alt=\"$\\angle ADC \\le 180 - \\theta_2.$\" style=\"vertical-align: -2px\" width=\"149\" height=\"15\" > By symmetry, we will consider just the former. We then have that <img src=\"//latex.artofproblemsolving.com/4/2/7/427f18863e2d87608c9311f629fcd5faf6da5d12.png\" class=\"latex\" alt=\"$\\angle BAD \\in [70, 180 - \\theta_1]$\" style=\"vertical-align: -5px\" width=\"178\" height=\"18\" > and so as <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/c/2/9/c29b8c1f8e137f241b0a32b85713f2528869a030.png\" class=\"latex\" alt=\"$\\theta_1 \\le 70$\" style=\"vertical-align: -2px\" width=\"57\" height=\"15\" >,</span> we've <img src=\"//latex.artofproblemsolving.com/d/7/b/d7bf8c8617d04fca892cdaa47b5a37635fbd306f.png\" class=\"latex\" alt=\"$\\sin \\angle BAD \\ge \\sin \\theta_1.$\" style=\"vertical-align: -2px\" width=\"151\" height=\"15\" > Therefore, <img src=\"//latex.artofproblemsolving.com/d/d/b/ddbaf93a0d7b99f802e5cbc9592afcc266aeb6ef.png\" class=\"latex\" alt=\"$\\frac{BD}{AD} \\ge \\frac{\\sin \\theta_1}{\\sin (\\theta_1 - 40)},$\" style=\"vertical-align: -17px\" width=\"159\" height=\"42\" > where we also used <img src=\"//latex.artofproblemsolving.com/b/9/f/b9f1e350a324b6ea682ad362339b9a261cd4e2e6.png\" class=\"latex\" alt=\"$\\angle ABD = \\angle ABC - \\angle DBC \\le \\theta_1 - 40.$\" style=\"vertical-align: -2px\" width=\"300\" height=\"15\" > It's easily checked that for <img src=\"//latex.artofproblemsolving.com/9/2/4/924a3597b0c5737eb235ebfa2e50fca848da92f4.png\" class=\"latex\" alt=\"$\\theta_1 \\in (40, 70],$\" style=\"vertical-align: -5px\" width=\"98\" height=\"18\" > <img src=\"//latex.artofproblemsolving.com/5/c/1/5c1e5d293b0c6e29281dfd37769aacab93aab15a.png\" class=\"latex\" alt=\"$\\frac{\\sin \\theta_1}{\\sin (\\theta_1 - 40)} \\ge \\frac{\\sin 70}{\\sin 30} = 2 \\sin 70,$\" style=\"vertical-align: -17px\" width=\"252\" height=\"42\" > and so we're done.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/6/c/b6cf65b1f2fbdf388e3daeff9b96b34d3399d777.png\" class=\"latex\" alt=\"$\\square$\" width=\"13\" height=\"12\" >", "post_id": 12971362, "post_number": 4, "post_time_unix": 1566078204, "post_time_utc": "2019-08-17 21:43:24 UTC", "thanks_received": 1, "user_id": 426228, "username": "Pathological" } ], "source": null }	(1) Let $D$ be an arbitrary point in $\triangle ABC$. Prove that \[ \frac{BC}{\min\{AD,BD,CD\}}\ge \begin{cases} 2\sin A,&\angle A<90^\circ,\\[6pt] 2,&\angle A\ge90^\circ. \end{cases} \] (2) Let $E$ be an arbitrary point in the convex quadrilateral $ABCD$. Let $k$ be the ratio of the largest and smallest distances among the six pairwise distances between $A,B,C,D,E$. Prove that $k\ge2\sin70^\circ$. Can equality be achieved?	[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/GeometricInequalities/HingeTheorem", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/IsoscelesTrapezium", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/IsoscelesTrapezoid", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Quadrilateral", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/Geometry/Trigonometry/Angles/AcuteAngle", "/Mathematics/Geometry/Trigonometry/Angles/Angle", "/Mathematics/Geometry/Trigonometry/Angles/ObtuseAngle", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/LawofCosines", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine", "/Mathematics/Geometry/Trigonometry/TrigonometricInequalities" ]	Use the triangle side‑to‑minimum‑adjacent‑side inequality (2 sin (A/2)) to bound angles, then apply the law of cosines on a large angle to force a long distance.	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aops_99394	What do you mean by "the finite lattice", exactly? Is it just a finite subset of the unit lattice in the plane? Anyway, construct a bipartite graph $G$ as follows: its vertices are the rows and columns of points, and its bipartition is $R,C$, where $R$ represents the set of rows, and $C$ the set of columns. Two vertices $r\in R,c\in C$ are connected iff there is a point of $S$ at the intersection between $r$ and $c$. Notice that $A$ is a [i]maximal matching[/i] of $G$, i.e. the maximal cardinality of a set of edges such that no two share a vertex, while $B$ is a [i]minimal vertex cover[/i] of $G$: a set of vertices of $G$ with minimal cardinality containing at least one endpoint from every edge. The inequality (in fact, it's an equality, because the reversed inequality is easy to prove) is precisely Konig's Theorem: in a bipartite graph, the cardinality of a maximal matching equals the cardinality of a minimal vertex cover. P.S. This was much better suited for the "Combinatorics" section, I think :).	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\\in S$, $x \\in B$ or $y \\in B$ holds. Prove that $\|A\| \\geq \|B\|$.", "content_html": "Given <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > be the finite lattice (with integer coordinate) set in the <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" >-</span>plane. <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > is the subset of <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > with most elements such that the line connecting any two points in <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > is not parallel to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" >-</span>axis or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" >-</span>axis. <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > is the subset of integer with least elements such that for any <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/4/e/54e9a31f8cc970df6495f352c70671f28753b508.png\" class=\"latex\" alt=\"$(x,y)\\in S$\" style=\"vertical-align: -4px\" width=\"76\" height=\"18\" >,</span> <img src=\"//latex.artofproblemsolving.com/6/b/d/6bd8fed2102b508ab36c0339647907260bc95dc2.png\" class=\"latex\" alt=\"$x \\in B$\" style=\"vertical-align: -1px\" width=\"46\" height=\"13\" > or <img src=\"//latex.artofproblemsolving.com/8/7/d/87d4ac9ed225894e281d8f52654fd22fe87e6336.png\" class=\"latex\" alt=\"$y \\in B$\" style=\"vertical-align: -3px\" width=\"46\" height=\"16\" > holds. Prove that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/d/2/cd2eb3a5ddf6bcf98cc8a9bc91cee57d25642af7.png\" class=\"latex\" alt=\"$\|A\| \\geq \|B\|$\" style=\"vertical-align: -4px\" width=\"70\" height=\"18\" >.</span>", "post_id": 561167, "post_number": 1, "post_time_unix": 1151567015, "post_time_utc": "2006-06-29 07:43:35 UTC", "thanks_received": 3, "user_id": 6601, "username": "shobber" }, { "attachments": [], "content_bbcode": "What do you mean by \"the finite lattice\", exactly? Is it just a finite subset of the unit lattice in the plane?\r\n\r\nAnyway, construct a bipartite graph $G$ as follows: its vertices are the rows and columns of points, and its bipartition is $R,C$, where $R$ represents the set of rows, and $C$ the set of columns. Two vertices $r\\in R,c\\in C$ are connected iff there is a point of $S$ at the intersection between $r$ and $c$.\r\n\r\nNotice that $A$ is a [i]maximal matching[/i] of $G$, i.e. the maximal cardinality of a set of edges such that no two share a vertex, while $B$ is a [i]minimal vertex cover[/i] of $G$: a set of vertices of $G$ with minimal cardinality containing at least one endpoint from every edge. \r\n\r\nThe inequality (in fact, it's an equality, because the reversed inequality is easy to prove) is precisely Konig's Theorem: in a bipartite graph, the cardinality of a maximal matching equals the cardinality of a minimal vertex cover.\r\n\r\n\r\nP.S.\r\n\r\nThis was much better suited for the \"Combinatorics\" section, I think :).", "content_html": "What do you mean by "the finite lattice", exactly? Is it just a finite subset of the unit lattice in the plane?<br>\n<br>\nAnyway, construct a bipartite graph <img src=\"//latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > as follows: its vertices are the rows and columns of points, and its bipartition is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/4/e/d4e1ed70ec950a5792330f281b4ec1a3ea4210d4.png\" class=\"latex\" alt=\"$R,C$\" style=\"vertical-align: -3px\" width=\"36\" height=\"16\" >,</span> where <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > represents the set of rows, and <img src=\"//latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png\" class=\"latex\" alt=\"$C$\" width=\"14\" height=\"12\" > the set of columns. Two vertices <img src=\"//latex.artofproblemsolving.com/4/e/e/4eeb0da1ab6a8da114f8d806ca9c1db6afa60f06.png\" class=\"latex\" alt=\"$r\\in R,c\\in C$\" style=\"vertical-align: -3px\" width=\"97\" height=\"16\" > are connected iff there is a point of <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > at the intersection between <img src=\"//latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png\" class=\"latex\" alt=\"$r$\" width=\"8\" height=\"8\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png\" class=\"latex\" alt=\"$c$\" width=\"8\" height=\"8\" >.</span><br>\n<br>\nNotice that <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > is a <i>maximal matching</i> of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" >,</span> i.e. the maximal cardinality of a set of edges such that no two share a vertex, while <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > is a <i>minimal vertex cover</i> of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" >:</span> a set of vertices of <img src=\"//latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > with minimal cardinality containing at least one endpoint from every edge.<br>\n<br>\nThe inequality (in fact, it's an equality, because the reversed inequality is easy to prove) is precisely Konig's Theorem: in a bipartite graph, the cardinality of a maximal matching equals the cardinality of a minimal vertex cover.<br>\n<br>\n<br>\nP.S.<br>\n<br>\nThis was much better suited for the "Combinatorics" section, I think <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />.", "post_id": 561242, "post_number": 2, "post_time_unix": 1151572141, "post_time_utc": "2006-06-29 09:09:01 UTC", "thanks_received": 2, "user_id": 26, "username": "grobber" }, { "attachments": [], "content_bbcode": "A little problem,as an element in B can mean two vertexs,the problem is still a inequality.", "content_html": "A little problem,as an element in B can mean two vertexs,the problem is still a inequality.", "post_id": 7147718, "post_number": 3, "post_time_unix": 1477224242, "post_time_utc": "2016-10-23 12:04:02 UTC", "thanks_received": 2, "user_id": 300496, "username": "P-H-David-Clarence" } ], "source": null }	Let $S$ be a finite lattice set in the $xy$-plane (points with integer coordinates). Let $A\subseteq S$ be a subset of maximum size such that the line connecting any two distinct points in $A$ is not parallel to the $x$-axis or the $y$-axis. Let $B$ be a subset of integers of minimum size such that for every $(x,y)\in S$, either $x\in B$ or $y\in B$. Prove that $\|A\|\ge \|B\|$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/Covers", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GraphTheory/GeneralGraphTheory", "/Mathematics/DiscreteMathematics/GraphTheory/Matchings", "/Mathematics/DiscreteMathematics/GraphTheory/SimpleGraphs", "/Mathematics/DiscreteMathematics/GraphTheory/VertexCovers" ]	Translate rows and columns into a bipartite graph so that A becomes a maximum matching and B a minimum vertex cover, then apply König's theorem.	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aops_99396	If I am not mistaken this is extraordinarily easy for a Chinese TST. :huh: We have to show: (1) $\forall X,Y \subseteq S: f(X \cup Y) + f(X \cap Y) \le f(X) + f(Y)$ $\Leftrightarrow$ (2) $g_a$ is decreasing for any $a$. For (1) $\Rightarrow$ (2) note that it suffices to show that $g_a(X) \ge g_a(X \cup \{b\})$ for all $a \neq b \in S$, $X \subset S \setminus \{a,b\}$. This however follows directly from (1) applied to the sets $X \cup \{a\}$ and $X \cup \{b\}$. For (2) $\Rightarrow$ (1) we use induction on the number $n$ of elements of $X \cup Y$. If $n = 0$ (1) holds trivially. (1) also is trivial if $X=Y$. So w.l.o.g. there is an $a \in S$ such that $a \in X$, $a \notin Y$. By induction hypothesis (1) holds for the sets $Y$ and $X' : = X \setminus \{a\}$. Hence it suffices to show that $f(X') - f(X' \cup Y) \le f(X) - f(X \cup Y)$. But this is the same as $g_a(X' \cup Y) \le g_a(X')$, which holds by (2).	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\\textsl{monotonic decreasing}$ if when $X \\subseteq Y\\subseteq S$, then $f(X) \\geq f(Y)$ holds. Prove that: $f(X \\cup Y)+f(X \\cap Y ) \\leq f(X)+ f(Y)$ for $X, Y \\subseteq S$ if and only if $g(X)=f(X \\cup \\{ a \\}) - f(X)$ is a $\\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \\setminus \\{a\\}}$ of set $S \\setminus \\{a\\}$ for any $a \\in S$.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > be a finite set. <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > is a function defined on the subset-group <img src=\"//latex.artofproblemsolving.com/4/7/5/4759f99bb09ab92276fa04f740b863b509e772a9.png\" class=\"latex\" alt=\"$2^S$\" width=\"17\" height=\"15\" > of set <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" >.</span> <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > is called <img src=\"//latex.artofproblemsolving.com/1/3/2/1325d5766c3a5951b7a105781a02aaaee9478ea9.png\" class=\"latex\" alt=\"$\\textsl{monotonic decreasing}$\" style=\"vertical-align: -3px\" width=\"157\" height=\"15\" > if when <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/f/b/9fba27d4371fc70d7377df70fabeba135db4f145.png\" class=\"latex\" alt=\"$X \\subseteq Y\\subseteq S$\" style=\"vertical-align: -2px\" width=\"91\" height=\"15\" >,</span> then <img src=\"//latex.artofproblemsolving.com/f/6/2/f629153f10f9458156fc536f722c8f860ae8d111.png\" class=\"latex\" alt=\"$f(X) \\geq f(Y)$\" style=\"vertical-align: -4px\" width=\"104\" height=\"18\" > holds. Prove that: <img src=\"//latex.artofproblemsolving.com/c/a/2/ca262ba17fac7b27db6e8fa7c6bfc2f423e0aa8e.png\" class=\"latex\" alt=\"$f(X \\cup Y)+f(X \\cap Y ) \\leq f(X)+ f(Y)$\" style=\"vertical-align: -4px\" width=\"301\" height=\"18\" > for <img src=\"//latex.artofproblemsolving.com/d/b/9/db995a1522f9b8582fd1a63ebcb38c503c93ef5a.png\" class=\"latex\" alt=\"$X, Y \\subseteq S$\" style=\"vertical-align: -3px\" width=\"74\" height=\"16\" > if and only if <img src=\"//latex.artofproblemsolving.com/3/4/0/340205bba92ced594fb03df1e6391a0cd561d017.png\" class=\"latex\" alt=\"$g(X)=f(X \\cup \\{ a \\}) - f(X)$\" style=\"vertical-align: -4px\" width=\"216\" height=\"18\" > is a <img src=\"//latex.artofproblemsolving.com/1/3/2/1325d5766c3a5951b7a105781a02aaaee9478ea9.png\" class=\"latex\" alt=\"$\\textsl{monotonic decreasing}$\" style=\"vertical-align: -3px\" width=\"157\" height=\"15\" > funnction on the subset-group <img src=\"//latex.artofproblemsolving.com/b/1/0/b10fb348df7c8fb4e1c99f2cd640e29f8d7dd348.png\" class=\"latex\" alt=\"$2^{S \\setminus \\{a\\}}$\" width=\"45\" height=\"16\" > of set <img src=\"//latex.artofproblemsolving.com/3/6/3/36346ef72d55396701e461fea667309d4686f883.png\" class=\"latex\" alt=\"$S \\setminus \\{a\\}$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" > for any <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/2/bb24ceea9f976239608d778a6f0bf736a9f8019c.png\" class=\"latex\" alt=\"$a \\in S$\" style=\"vertical-align: -1px\" width=\"44\" height=\"13\" >.</span>", "post_id": 561176, "post_number": 1, "post_time_unix": 1151567358, "post_time_utc": "2006-06-29 07:49:18 UTC", "thanks_received": 2, "user_id": 6601, "username": "shobber" }, { "attachments": [], "content_bbcode": "If I am not mistaken this is extraordinarily easy for a Chinese TST. :huh: \r\n\r\nWe have to show: \r\n(1) $\\forall X,Y \\subseteq S: f(X \\cup Y) + f(X \\cap Y) \\le f(X) + f(Y)$ $\\Leftrightarrow$\r\n(2) $g_a$ is decreasing for any $a$. \r\n\r\nFor (1) $\\Rightarrow$ (2) note that it suffices to show that \r\n$g_a(X) \\ge g_a(X \\cup \\{b\\})$ for all $a \\neq b \\in S$, $X \\subset S \\setminus \\{a,b\\}$. \r\nThis however follows directly from (1) applied to the sets $X \\cup \\{a\\}$ and \r\n$X \\cup \\{b\\}$. \r\n\r\nFor (2) $\\Rightarrow$ (1) we use induction on the number $n$ of elements of $X \\cup Y$. \r\nIf $n = 0$ (1) holds trivially. (1) also is trivial if $X=Y$. So w.l.o.g. there is an $a \\in S$\r\nsuch that $a \\in X$, $a \\notin Y$. By induction hypothesis (1) holds for \r\nthe sets $Y$ and $X' : = X \\setminus \\{a\\}$. Hence it suffices to show that \r\n$f(X') - f(X' \\cup Y) \\le f(X) - f(X \\cup Y)$. \r\nBut this is the same as $g_a(X' \\cup Y) \\le g_a(X')$, which holds by (2).", "content_html": "If I am not mistaken this is extraordinarily easy for a Chinese TST. <img src=\"/assets/images/smilies/huh.gif\" width=\"20\" height=\"20\" alt=\":huh:\" title=\":huh:\" class=\"bbcode_smiley\" /><br>\n<br>\nWe have to show:<br>\n(1) <img src=\"//latex.artofproblemsolving.com/c/9/2/c9250cd95674c934adc2c862302cd65c91e0f5e3.png\" class=\"latex\" alt=\"$\\forall X,Y \\subseteq S: f(X \\cup Y) + f(X \\cap Y) \\le f(X) + f(Y)$\" style=\"vertical-align: -4px\" width=\"400\" height=\"18\" > <img src=\"//latex.artofproblemsolving.com/a/6/e/a6ec0db174f3b705df19623976bf4b88ff502f3a.png\" class=\"latex\" alt=\"$\\Leftrightarrow$\" style=\"vertical-align: 0px\" width=\"18\" height=\"10\" ><br>\n(2) <img src=\"//latex.artofproblemsolving.com/5/7/1/5710fbb7370455fb81823eda1c925cede8a81d86.png\" class=\"latex\" alt=\"$g_a$\" style=\"vertical-align: -3px\" width=\"15\" height=\"11\" > is decreasing for any <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" >.</span><br>\n<br>\nFor (1) <img src=\"//latex.artofproblemsolving.com/8/4/3/843622567e12d686e4e2f94ddfcb444e8ecde0d2.png\" class=\"latex\" alt=\"$\\Rightarrow$\" style=\"vertical-align: 0px\" width=\"17\" height=\"10\" > (2) note that it suffices to show that<br>\n<img src=\"//latex.artofproblemsolving.com/7/5/f/75f3ce0774a80fb0325e256566eea0db3a123f47.png\" class=\"latex\" alt=\"$g_a(X) \\ge g_a(X \\cup \\{b\\})$\" style=\"vertical-align: -4px\" width=\"163\" height=\"18\" > for all <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/0/8/208adac0d5fcb1039d9aa9c9cc522af0e01319cb.png\" class=\"latex\" alt=\"$a \\neq b \\in S$\" style=\"vertical-align: -4px\" width=\"76\" height=\"17\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/f/c/3fcbe428246599f9f99e676b8225ff38b89a2c72.png\" class=\"latex\" alt=\"$X \\subset S \\setminus \\{a,b\\}$\" style=\"vertical-align: -4px\" width=\"113\" height=\"18\" >.</span><br>\nThis however follows directly from (1) applied to the sets <img src=\"//latex.artofproblemsolving.com/9/7/e/97ea156ece9d0ecab8f8e16472305e5c2912929b.png\" class=\"latex\" alt=\"$X \\cup \\{a\\}$\" style=\"vertical-align: -4px\" width=\"63\" height=\"18\" > and<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/b/24b2e9e4437ca60d7c98f167131ca2b2834d76ec.png\" class=\"latex\" alt=\"$X \\cup \\{b\\}$\" style=\"vertical-align: -4px\" width=\"61\" height=\"18\" >.</span><br>\n<br>\nFor (2) <img src=\"//latex.artofproblemsolving.com/8/4/3/843622567e12d686e4e2f94ddfcb444e8ecde0d2.png\" class=\"latex\" alt=\"$\\Rightarrow$\" style=\"vertical-align: 0px\" width=\"17\" height=\"10\" > (1) we use induction on the number <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > of elements of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/6/c/a6cdbe37fa68d1fd550d5b249b72e32832d00f30.png\" class=\"latex\" alt=\"$X \\cup Y$\" style=\"vertical-align: 0px\" width=\"50\" height=\"13\" >.</span><br>\nIf <img src=\"//latex.artofproblemsolving.com/2/d/6/2d6fc266463b33c8b1c079a50909f3fd03ed5ffd.png\" class=\"latex\" alt=\"$n = 0$\" width=\"43\" height=\"12\" > (1) holds trivially. (1) also is trivial if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/d/a/4da586db5670d15a8fd95a1cb334d535f4335a05.png\" class=\"latex\" alt=\"$X=Y$\" width=\"54\" height=\"12\" >.</span> So w.l.o.g. there is an <img src=\"//latex.artofproblemsolving.com/b/b/2/bb24ceea9f976239608d778a6f0bf736a9f8019c.png\" class=\"latex\" alt=\"$a \\in S$\" style=\"vertical-align: -1px\" width=\"44\" height=\"13\" ><br>\nsuch that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/9/6/396150ce5dfce53dced188c0bd97d98304b70c3b.png\" class=\"latex\" alt=\"$a \\in X$\" style=\"vertical-align: -1px\" width=\"47\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/9/bd901aa25ff8011a8b5c0cd6d6e5b787819a9afe.png\" class=\"latex\" alt=\"$a \\notin Y$\" style=\"vertical-align: -4px\" width=\"46\" height=\"18\" >.</span> By induction hypothesis (1) holds for<br>\nthe sets <img src=\"//latex.artofproblemsolving.com/c/e/5/ce58e4af225c93d08606c26554caaa5ae32edeba.png\" class=\"latex\" alt=\"$Y$\" width=\"14\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/e/c/1ec0f663f838f7c54a881aafd5b3b979b5f32268.png\" class=\"latex\" alt=\"$X' : = X \\setminus \\{a\\}$\" style=\"vertical-align: -4px\" width=\"110\" height=\"18\" >.</span> Hence it suffices to show that<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/b/bdbc82697ee9304f20db00c5d7325e81571a7291.png\" class=\"latex\" alt=\"$f(X') - f(X' \\cup Y) \\le f(X) - f(X \\cup Y)$\" style=\"vertical-align: -4px\" width=\"312\" height=\"18\" >.</span><br>\nBut this is the same as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/e/1/be19f719e27e035d0f018a3f9d347e6268a1afe8.png\" class=\"latex\" alt=\"$g_a(X' \\cup Y) \\le g_a(X')$\" style=\"vertical-align: -4px\" width=\"161\" height=\"18\" >,</span> which holds by (2).", "post_id": 561360, "post_number": 2, "post_time_unix": 1151587848, "post_time_utc": "2006-06-29 13:30:48 UTC", "thanks_received": 2, "user_id": 4420, "username": "solyaris" }, { "attachments": [], "content_bbcode": "A curious note: Such an $f$ is called a [url=https://en.wikipedia.org/wiki/Submodular_set_function]submodular set function[/url], and at the forefront of discrete optimization theory these days. ", "content_html": "A curious note: Such an <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > is called a <a href=\"https://en.wikipedia.org/wiki/Submodular_set_function\" class=\"bbcode_url\" target=\"_blank\" rel=\"nofollow\">submodular set function</a>, and at the forefront of discrete optimization theory these days.", "post_id": 12938985, "post_number": 3, "post_time_unix": 1565644577, "post_time_utc": "2019-08-12 21:16:17 UTC", "thanks_received": 2, "user_id": 64049, "username": "grupyorum" } ], "source": null }	Let S be a finite set. Let f: 2^S → ℝ. f is called monotonic decreasing if X ⊆ Y ⊆ S implies f(X) ≥ f(Y). Prove that for all X, Y ⊆ S, f(X ∪ Y) + f(X ∩ Y) ≤ f(X) + f(Y) if and only if for every a ∈ S the function g_a: 2^{S\setminus\{a\}} → ℝ defined by g_a(X) = f(X ∪ {a}) − f(X) is monotonic decreasing on 2^{S\setminus\{a\}} (i.e. for X ⊆ Y ⊆ S\setminus\{a\} we have g_a(X) ≥ g_a(Y)).	[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]	Relate the submodular inequality to decreasing marginal gains by testing it on singleton extensions and then use induction on the union size.	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aops_99399	[hide]$3^{3x - 3y} \cdot 3^{2x + 2y} \cdot 5^{x + y} = 3^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}$ $3^{5x - y} \cdot 5^{x + y} = 3^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} \Longrightarrow x = \frac{1}{6}$[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find the value of the real number $x$ satisfying the equation,\r\n\r\n$(27)^{x-y}\\times(45)^{x+y}=\\sqrt{15}$", "content_html": "Find the value of the real number <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > satisfying the equation,<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/c/7/7/c77903168b46145378f063af299bd2b0a0fa6ca4.png\" class=\"latex\" alt=\"$(27)^{x-y}\\times(45)^{x+y}=\\sqrt{15}$\" style=\"vertical-align: -4px\" width=\"196\" height=\"21\" >", "post_id": 561201, "post_number": 1, "post_time_unix": 1151569249, "post_time_utc": "2006-06-29 08:20:49 UTC", "thanks_received": 2, "user_id": 20652, "username": "Nekruzjon_eko" }, { "attachments": [], "content_bbcode": "Rewrite the equation as $3^{3x-3y}3^{2x+2y}5^{x+y}=3^{5x-y}5^{x+y}=\\sqrt{15}$. Noting that $\\sqrt{15}=3^{.5}5^{.5}$ we have $5x-y=.5$ and $x+y=.5$. Then $x=\\frac{1}{6}$.", "content_html": "Rewrite the equation as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/5/b/95b6afa8e537c1ce7bd90920edc98bedddf37648.png\" class=\"latex\" alt=\"$3^{3x-3y}3^{2x+2y}5^{x+y}=3^{5x-y}5^{x+y}=\\sqrt{15}$\" style=\"vertical-align: -1px\" width=\"289\" height=\"18\" >.</span> Noting that <img src=\"//latex.artofproblemsolving.com/b/4/4/b4423b9c33f6d6a3e4ee7a803ad103014482e865.png\" class=\"latex\" alt=\"$\\sqrt{15}=3^{.5}5^{.5}$\" style=\"vertical-align: -1px\" width=\"97\" height=\"18\" > we have <img src=\"//latex.artofproblemsolving.com/f/7/6/f7692a54f9f4f6d2d3765b33dc99c8a0aa2fd5b9.png\" class=\"latex\" alt=\"$5x-y=.5$\" style=\"vertical-align: -3px\" width=\"89\" height=\"16\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/6/7/86793fa2970412f410e9ce27088683b78e9a9c98.png\" class=\"latex\" alt=\"$x+y=.5$\" style=\"vertical-align: -3px\" width=\"80\" height=\"16\" >.</span> Then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/a/4/7a40c9c10b928a26c578d04585c8c5596327a8b0.png\" class=\"latex\" alt=\"$x=\\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"46\" height=\"37\" >.</span>", "post_id": 561227, "post_number": 2, "post_time_unix": 1151571146, "post_time_utc": "2006-06-29 08:52:26 UTC", "thanks_received": 1, "user_id": 7685, "username": "drunner2007" }, { "attachments": [], "content_bbcode": "[hide]$3^{3x - 3y} \\cdot 3^{2x + 2y} \\cdot 5^{x + y} = 3^{\\frac{1}{2}} \\cdot 5^{\\frac{1}{2}}$\n$3^{5x - y} \\cdot 5^{x + y} = 3^{\\frac{1}{2}} \\cdot 5^{\\frac{1}{2}} \\Longrightarrow x = \\frac{1}{6}$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/8/8/b/88b617c7cecb1bb48a69ed8a722139574ae3502f.png\" class=\"latex\" alt=\"$3^{3x - 3y} \\cdot 3^{2x + 2y} \\cdot 5^{x + y} = 3^{\\frac{1}{2}} \\cdot 5^{\\frac{1}{2}}$\" width=\"231\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/e/a/fea3c70072b121f2d0b559ac5d1e1e8b8039b6c1.png\" class=\"latex\" alt=\"$3^{5x - y} \\cdot 5^{x + y} = 3^{\\frac{1}{2}} \\cdot 5^{\\frac{1}{2}} \\Longrightarrow x = \\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"251\" height=\"37\" ></div>", "post_id": 561229, "post_number": 3, "post_time_unix": 1151571192, "post_time_utc": "2006-06-29 08:53:12 UTC", "thanks_received": 3, "user_id": 5839, "username": "Andreas" } ], "source": null }	Find the value of the real number $x$ satisfying the equation \[ 27^{\,x-y}\cdot 45^{\,x+y}=\sqrt{15}. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression", "/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations/Exponentiation", "/Mathematics/Algebra/Products/Product" ]	Rewrite all numbers as powers of the same primes and then equate the exponents.	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aops_99401	AYMANE's post doesn't make much sense to me either: if $g$ has a primitive, it doesn't follow that $g^2$ has a primitive and if $f^2$ has a primitive, then it doesn't follow in general that $f$ has a primitive. The solution I know runs as follows. Let $G$ and $H$ be the primitives of $g$ and $h$ respectively. Let $F(x)=G(x)\sin x+H(x)\cos x$. Then $F$ is differentiable and $F'(x)=f(x)+[G(x)\cos x-H(x)\sin x]$. But the function in brackets is continuous and, therefore, has an antiderivative. It remains to subtract this antiderivative from $F$ to get an antiderivative of $f$.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $f,g,h: \\mathbb{R}\\to\\mathbb{R},\\ g(x)=f(x)\\sin(x), \\ h(x)=f(x)\\cos(x)$ for any real $x$. If $g,h$ have primitives on $\\mathbb{R}$ show that $f$ has too.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/b/9/3/b9328f00dd56757a960b5b42bcd2ed685d1336c5.png\" class=\"latex\" alt=\"$f,g,h: \\mathbb{R}\\to\\mathbb{R},\\ g(x)=f(x)\\sin(x), \\ h(x)=f(x)\\cos(x)$\" style=\"vertical-align: -4px\" width=\"430\" height=\"18\" > for any real <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" >.</span> If <img src=\"//latex.artofproblemsolving.com/8/1/0/810ddff7f8e93a1de60193c1f463e1e2e977c9e3.png\" class=\"latex\" alt=\"$g,h$\" style=\"vertical-align: -3px\" width=\"27\" height=\"16\" > have primitives on <img src=\"//latex.artofproblemsolving.com/9/0/6/906fddead65545297c74bbc2ca83cd01f3f9ecc9.png\" class=\"latex\" alt=\"$\\mathbb{R}$\" width=\"12\" height=\"12\" > show that <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > has too.", "post_id": 561215, "post_number": 1, "post_time_unix": 1151570259, "post_time_utc": "2006-06-29 08:37:39 UTC", "thanks_received": 2, "user_id": 7271, "username": "Slizzel" }, { "attachments": [], "content_bbcode": "1)$f(x)^2=g(x)^2+h(x)^2$\r\n2)$(\\int{f(x)dx})^2\\leq \\int{f^2(x)dx}$", "content_html": "1<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/c/8/2/c8216ca94459829526d932ad9400651a09fa4137.png\" class=\"latex\" alt=\"$f(x)^2=g(x)^2+h(x)^2$\" style=\"vertical-align: -4px\" width=\"171\" height=\"19\" ></span><br>\n2<span style=\"white-space:nowrap;\">)<img src=\"//latex.artofproblemsolving.com/4/a/5/4a59783be38e8746841b0ae7e5ccea87777a4d1b.png\" class=\"latex\" alt=\"$(\\int{f(x)dx})^2\\leq \\int{f^2(x)dx}$\" style=\"vertical-align: -15px\" width=\"205\" height=\"39\" ></span>", "post_id": 561259, "post_number": 2, "post_time_unix": 1151573584, "post_time_utc": "2006-06-29 09:33:04 UTC", "thanks_received": 2, "user_id": 7340, "username": "AYMANE" }, { "attachments": [], "content_bbcode": "you say that if $f^2$ has primitives so does $f$? Your answer (2) isn't clear enough to since my knowledge about primitives are small..", "content_html": "you say that if <img src=\"//latex.artofproblemsolving.com/7/5/6/7569ac9e2a50191b339e733c3e526f8152486504.png\" class=\"latex\" alt=\"$f^2$\" style=\"vertical-align: -3px\" width=\"17\" height=\"18\" > has primitives so does <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" >?</span> Your answer (2) isn't clear enough to since my knowledge about primitives are small..", "post_id": 561354, "post_number": 3, "post_time_unix": 1151587556, "post_time_utc": "2006-06-29 13:25:56 UTC", "thanks_received": 2, "user_id": 7271, "username": "Slizzel" }, { "attachments": [], "content_bbcode": "AYMANE's post doesn't make much sense to me either: if $g$ has a primitive, it doesn't follow that $g^2$ has a primitive and if $f^2$ has a primitive, then it doesn't follow in general that $f$ has a primitive. The solution I know runs as follows. Let $G$ and $H$ be the primitives of $g$ and $h$ respectively. Let $F(x)=G(x)\\sin x+H(x)\\cos x$. Then $F$ is differentiable and $F'(x)=f(x)+[G(x)\\cos x-H(x)\\sin x]$. But the function in brackets is continuous and, therefore, has an antiderivative. It remains to subtract this antiderivative from $F$ to get an antiderivative of $f$.", "content_html": "AYMANE's post doesn't make much sense to me either: if <img src=\"//latex.artofproblemsolving.com/3/1/1/311cabda3a9b09f0dde217303ca9d1cd9201dcf6.png\" class=\"latex\" alt=\"$g$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > has a primitive, it doesn't follow that <img src=\"//latex.artofproblemsolving.com/e/b/f/ebfa63393ddea2b33dea39cfd7ed861ccc00b8b2.png\" class=\"latex\" alt=\"$g^2$\" style=\"vertical-align: -3px\" width=\"15\" height=\"18\" > has a primitive and if <img src=\"//latex.artofproblemsolving.com/7/5/6/7569ac9e2a50191b339e733c3e526f8152486504.png\" class=\"latex\" alt=\"$f^2$\" style=\"vertical-align: -3px\" width=\"17\" height=\"18\" > has a primitive, then it doesn't follow in general that <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > has a primitive. The solution I know runs as follows. Let <img src=\"//latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png\" class=\"latex\" alt=\"$G$\" width=\"14\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/b/1/9/b1902d279ba37d60bdce4e0e987b7cd19d48974e.png\" class=\"latex\" alt=\"$H$\" width=\"16\" height=\"12\" > be the primitives of <img src=\"//latex.artofproblemsolving.com/3/1/1/311cabda3a9b09f0dde217303ca9d1cd9201dcf6.png\" class=\"latex\" alt=\"$g$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > and <img src=\"//latex.artofproblemsolving.com/8/1/8/8189a5b5a0917b8c93350827be4038af1839139d.png\" class=\"latex\" alt=\"$h$\" width=\"10\" height=\"12\" > respectively. Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/d/4/fd4c001498aa2ff6b077717bfe0c4d70439dd0d9.png\" class=\"latex\" alt=\"$F(x)=G(x)\\sin x+H(x)\\cos x$\" style=\"vertical-align: -4px\" width=\"244\" height=\"18\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/a/0/5/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" > is differentiable and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/7/c/a7cc65cbb02b461e00ff3daf6209fa1ea4aafef9.png\" class=\"latex\" alt=\"$F'(x)=f(x)+[G(x)\\cos x-H(x)\\sin x]$\" style=\"vertical-align: -5px\" width=\"315\" height=\"19\" >.</span> But the function in brackets is continuous and, therefore, has an antiderivative. It remains to subtract this antiderivative from <img src=\"//latex.artofproblemsolving.com/a/0/5/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" > to get an antiderivative of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" >.</span>", "post_id": 561575, "post_number": 4, "post_time_unix": 1151598361, "post_time_utc": "2006-06-29 16:26:01 UTC", "thanks_received": 2, "user_id": 6542, "username": "fedja" }, { "attachments": [], "content_bbcode": "That's better. Taking a function $F(x)$ like you did isn't so obiouvsly for me. Maybe the problem is classical so there is an explination :)", "content_html": "That's better. Taking a function <img src=\"//latex.artofproblemsolving.com/d/e/8/de8fdaa56ea5a8940d4bba79faa9e3dea0f6bf7c.png\" class=\"latex\" alt=\"$F(x)$\" style=\"vertical-align: -4px\" width=\"37\" height=\"18\" > like you did isn't so obiouvsly for me. Maybe the problem is classical so there is an explination <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 563423, "post_number": 5, "post_time_unix": 1151738780, "post_time_utc": "2006-07-01 07:26:20 UTC", "thanks_received": 2, "user_id": 7271, "username": "Slizzel" }, { "attachments": [], "content_bbcode": "maybe i'm wrong, but:\r\n\r\nwe know (edit: i was wrong)\r\n\r\n:)", "content_html": "maybe i'm wrong, but:<br>\n<br>\nwe know (edit: i was wrong)<br>\n<br>\n<img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 565966, "post_number": 6, "post_time_unix": 1152021265, "post_time_utc": "2006-07-04 13:54:25 UTC", "thanks_received": 2, "user_id": 19204, "username": "vl4d" }, { "attachments": [], "content_bbcode": "You forgot $f'(x)$ coming from the chain rule (which may fail to exist, by the way) ;)", "content_html": "You forgot <img src=\"//latex.artofproblemsolving.com/b/b/a/bbad379658bd32e9b6479bbd666b93a96e6b48ba.png\" class=\"latex\" alt=\"$f'(x)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > coming from the chain rule (which may fail to exist, by the way) <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 565972, "post_number": 7, "post_time_unix": 1152021499, "post_time_utc": "2006-07-04 13:58:19 UTC", "thanks_received": 2, "user_id": 6542, "username": "fedja" }, { "attachments": [], "content_bbcode": "damn! it's true", "content_html": "damn! it's true", "post_id": 565973, "post_number": 8, "post_time_unix": 1152021689, "post_time_utc": "2006-07-04 14:01:29 UTC", "thanks_received": 2, "user_id": 19204, "username": "vl4d" } ], "source": null }	Let $f,g,h:\mathbb{R}\to\mathbb{R}$ with \[ g(x)=f(x)\sin x,\qquad h(x)=f(x)\cos x\quad\text{for all }x\in\mathbb{R}. \] If $g$ and $h$ have antiderivatives on $\mathbb{R}$, prove that $f$ has an antiderivative on $\mathbb{R}$.	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiable", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstFundamentalTheoremofCalculus", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FundamentalTheoremsofCalculus", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/ProductRule", "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/IndefiniteIntegrals", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/RealVariable", "/Mathematics/CalculusandAnalysis/Functions/Single-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis" ]	Combine the primitives of g and h with sin x and cos x to create a function whose derivative equals f plus a continuous term.	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aops_994021	So we have ln(21.00410^3)-ln(21.00210^3) and all of everything dies except ln(1.004)-ln(1.002), which is equal to ln(1.004/1.002) which is equal to ln(1+1/501). The Taylor series (1/x+1/2x^2+1/3x^3...) may be rewritten as, for x=1+y, y(1-y(1/2-y(1/3-y...))), and so we have ln(1+1/501) as 1/501(1-1/501(1/2-1/1503), or about 1/501(1-1/1000), or .002 times about 1, so its .002.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Approximate to the nearest thousandth (no calculator):\r\n$ \\ln(2008) \\minus{} \\ln(2004)$", "content_html": "Approximate to the nearest thousandth (no calculator):<br>\n<img src=\"//latex.artofproblemsolving.com/0/4/d/04d69a042c0343345d76d073d1ade6c800c569f3.png\" class=\"latex\" alt=\"$ \\ln(2008) - \\ln(2004)$\" style=\"vertical-align: -4px\" width=\"151\" height=\"18\" >", "post_id": 4409274, "post_number": 1, "post_time_unix": 1207359594, "post_time_utc": "2008-04-05 01:39:54 UTC", "thanks_received": 2, "user_id": 24504, "username": "Twiz" }, { "attachments": [], "content_bbcode": "So we have ln(21.00410^3)-ln(21.00210^3) and all of everything dies except ln(1.004)-ln(1.002), which is equal to ln(1.004/1.002) which is equal to ln(1+1/501). The Taylor series (1/x+1/2x^2+1/3x^3...) may be rewritten as, for x=1+y, y(1-y(1/2-y(1/3-y...))), and so we have ln(1+1/501) as 1/501(1-1/501(1/2-1/1503), or about 1/501(1-1/1000), or .002 times about 1, so its .002.", "content_html": "So we have ln(21.00410^3)-ln(21.00210^3) and all of everything dies except ln(1.004)-ln(1.002), which is equal to ln(1.004/1.002) which is equal to ln(1+1/501). The Taylor series (1/x+1/2x^2+1/3x^3...) may be rewritten as, for x=1+y, y(1-y(1/2-y(1/3-y...))), and so we have ln(1+1/501) as 1/501(1-1/501(1/2-1/1503), or about 1/501(1-1/1000), or .002 times about 1, so its .002.", "post_id": 4409275, "post_number": 2, "post_time_unix": 1207363351, "post_time_utc": "2008-04-05 02:42:31 UTC", "thanks_received": 2, "user_id": 2112, "username": "solafidefarms" }, { "attachments": [], "content_bbcode": "My solution was less rigorous and more complicated :maybe: : \r\n[hide]Clearly $ \\displaystyle\\lim_{x\\to\\infty}\\ln[(1 \\plus{} \\frac {1}{x})^x] \\equal{} 1$\nSince $ \\ln[(1 \\plus{} \\frac {1}{x})^x] \\equal{} x\\ln(1 \\plus{} \\frac {1}{x})\\approx{1}$ for large values of $ x$, \n$ \\ln(1 \\plus{} \\frac {1}{x}) \\approx{\\frac {1}{x}}$ for such values.\nRewrite the given value as $ \\ln(1 \\plus{} \\frac {1}{2007}) \\plus{} \\ln(1 \\plus{} \\frac {1}{2006}) \\plus{} \\ln(1 \\plus{} \\frac {1}{2005}) \\plus{} \\ln(1 \\plus{} \\frac {1}{2004})$, and since these numbers are large (i.e. large enough such that $ (1 \\plus{} \\frac {1}{x})^x \\approx{e}$, the given is about equal to $ \\frac {1}{2007} \\plus{} \\frac {1}{2006} \\plus{} \\frac {1}{2005} \\plus{} \\frac {1}{2004}\\approx{\\frac {4}{2000}} \\equal{} .002$[/hide]\r\nBy the way Billy - I know you know LaTeX... ;)", "content_html": "My solution was less rigorous and more complicated <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" /> :<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Clearly <img src=\"//latex.artofproblemsolving.com/1/4/9/1491b4eda4c125bf0b71e3a21d4364f19caf6d9a.png\" class=\"latex\" alt=\"$ \\displaystyle\\lim_{x\\to\\infty}\\ln[(1 + \\frac {1}{x})^x] = 1$\" style=\"vertical-align: -12px\" width=\"162\" height=\"37\" ><br>\nSince <img src=\"//latex.artofproblemsolving.com/1/0/8/10851cd887cc3f3fb291274f43d95c01be92321b.png\" class=\"latex\" alt=\"$ \\ln[(1 + \\frac {1}{x})^x] = x\\ln(1 + \\frac {1}{x})\\approx{1}$\" style=\"vertical-align: -12px\" width=\"238\" height=\"37\" > for large values of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" >,</span><br>\n<img src=\"//latex.artofproblemsolving.com/d/1/5/d15c4c4332eb72567cdf893dabd415763118d018.png\" class=\"latex\" alt=\"$ \\ln(1 + \\frac {1}{x}) \\approx{\\frac {1}{x}}$\" style=\"vertical-align: -12px\" width=\"111\" height=\"37\" > for such values.<br>\nRewrite the given value as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/d/bad88b4655c7478aeccbdb7a58983170abd78463.png\" class=\"latex\" alt=\"$ \\ln(1 + \\frac {1}{2007}) + \\ln(1 + \\frac {1}{2006}) + \\ln(1 + \\frac {1}{2005}) + \\ln(1 + \\frac {1}{2004})$\" style=\"vertical-align: -13px\" width=\"469\" height=\"37\" >,</span> and since these numbers are large (i.e. large enough such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/f/1/1f12c7a09aed078fa7195ab43d0435039fab86dc.png\" class=\"latex\" alt=\"$ (1 + \\frac {1}{x})^x \\approx{e}$\" style=\"vertical-align: -12px\" width=\"100\" height=\"37\" >,</span> the given is about equal to <img src=\"//latex.artofproblemsolving.com/e/b/6/eb643087f2330973c1080bd3540cbb3c521b79b2.png\" class=\"latex\" alt=\"$ \\frac {1}{2007} + \\frac {1}{2006} + \\frac {1}{2005} + \\frac {1}{2004}\\approx{\\frac {4}{2000}} = .002$\" style=\"vertical-align: -13px\" width=\"348\" height=\"37\" ></div><br>\nBy the way Billy - I know you know LaTeX... <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 4409276, "post_number": 3, "post_time_unix": 1207370260, "post_time_utc": "2008-04-05 04:37:40 UTC", "thanks_received": 2, "user_id": 24504, "username": "Twiz" }, { "attachments": [], "content_bbcode": "How about this?\r\n\\[ \\ln 2008 \\minus{} \\ln 2004 \\equal{} \\int_{2004}^{2008} \\frac{dx}{x} \\approx \\int_{2004}^{2008} \\frac{dx}{2000} \\equal{} 2/1000. \\]", "content_html": "How about this?<br>\n<img src=\"//latex.artofproblemsolving.com/b/8/e/b8ea3c681184beaad30754ca9943fe337bc94ea8.png\" class=\"latexcenter\" alt=\"\\[ \\ln 2008 - \\ln 2004 = \\int_{2004}^{2008} \\frac{dx}{x} \\approx \\int_{2004}^{2008} \\frac{dx}{2000} = 2/1000. \\]\" width=\"421\" height=\"43\" >", "post_id": 4409277, "post_number": 4, "post_time_unix": 1207699124, "post_time_utc": "2008-04-08 23:58:44 UTC", "thanks_received": 2, "user_id": 13517, "username": "Boy Soprano II" }, { "attachments": [], "content_bbcode": "Okay, you win. I think all three of the solutions are somewhat similar in principle though.", "content_html": "Okay, you win. I think all three of the solutions are somewhat similar in principle though.", "post_id": 4409278, "post_number": 5, "post_time_unix": 1207703977, "post_time_utc": "2008-04-09 01:19:37 UTC", "thanks_received": 2, "user_id": 24504, "username": "Twiz" } ], "source": null }	Approximate to the nearest thousandth (no calculator): \[ \ln(2008) - \ln(2004) \]	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/LogarithmicDerivative", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Maclaurin-CauchyTheorem", "/Mathematics/CalculusandAnalysis/Functions/ElementaryFunction", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/Single-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/TranscendentalFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction", "/Mathematics/CalculusandAnalysis/Series/SeriesExpansions" ]	Rewrite the difference as ln of a ratio close to 1 and approximate ln(1+ε) using the first term of its Taylor series.	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aops_99404	Suppose first that some vertex of our polygon $\mathcal P$ is not adjacent to any diameters (diagonals or sides of length $1$). Then pick any diagonal having this vertex as endpoint, and move the vertex slightly, so as to elongate the diagonal. Clearly, if the elongation is small enough, all diagonals with our vertex as endpoint stay smaller than $1$, but the area increases strictly, and this contradicts the maximality property of $\mathcal P$. We have thus shown that every vertex of $\mathcal P$. There are now two cases: either (a) every vertex is an endpoint of at least two diameters, or (b) there is a vertex which is an endpoint of exactly one diameter. (a) Notice that every two diameters must intersect (in an endpoint or otherwise). We’ll cal this assertion $()$. Suppose there are vertices $x,y,z$ such that $xy,xz$ are diameters, but $y,z$ are not consecutive vertices of $\mathcal P$. Then, if $t$ is a vertex between $y$ and $z$ and such that $yz$ separates $x$ and $t$, then one of the (at least two) diameters through $t$ would be disjoint from one of the two diameters $xy,xz$, and this contradicts $()$. This proves that in the situation we’re considering now, every vertex is an endpoint of [i]exactly[/i] two diameters, and, moreover, that whenever $xy,xz$ are diameters, $y,z$ are consecutive vertices of $\mathcal P$. It easily follows from here (just play around with a drawing) that $\mathcal P$ has an odd number of vertices, and this contradicts the fact that it actually has $2000$ vertices. Hence, (a) cannot occur. (b) Let $xy$ be the only diameter having $y$ as endpoint, and let $u,v$ be vertices immediately to the left and right of $y$. If $xy$ is not orthogonal to $uv$, we can rotate $y$ around $x$ slightly such that (i) $xy=1$ throughout the process, (ii) all other distances between vertices of $\mathcal P$ and $y$ stay $<1$ (or $\le 1$; it doesn’t matter), and (iii) the angle ($<\frac\pi 2$) between $xy$ and $uv$ increases, which is easily seen to make the area of $\mathcal P$ increase strictly. This contradicts the maximality of the area of $\mathcal P$, so we have what we wanted: the diagonals (well, $xy$ could be a side and not a diagonal in the strictest sense, but I assume this is allowed, i.e. by “diagonals” you mean “sides or diagonals”? :?) $xy$ and $uv$ must be orthogonal.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": ":o the convex 2000 gon M satisfies following prp. the max. dist. bet. 2 vertices equal to 1.its known among all convex 2000 gons with same prp'\r\nM has max. area. show some 2 diagonals of M are perpendicular.\r\nprp mans property", "content_html": "<img src=\"/assets/images/smilies/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" class=\"bbcode_smiley\" /> the convex 2000 gon M satisfies following prp. the max. dist. bet. 2 vertices equal to 1.its known among all convex 2000 gons with same prp'<br>\nM has max. area. show some 2 diagonals of M are perpendicular.<br>\nprp mans property", "post_id": 561224, "post_number": 1, "post_time_unix": 1151570851, "post_time_utc": "2006-06-29 08:47:31 UTC", "thanks_received": 2, "user_id": 20415, "username": "who" }, { "attachments": [], "content_bbcode": "Suppose first that some vertex of our polygon $\\mathcal P$ is not adjacent to any diameters (diagonals or sides of length $1$). Then pick any diagonal having this vertex as endpoint, and move the vertex slightly, so as to elongate the diagonal. Clearly, if the elongation is small enough, all diagonals with our vertex as endpoint stay smaller than $1$, but the area increases strictly, and this contradicts the maximality property of $\\mathcal P$. We have thus shown that every vertex of $\\mathcal P$.\r\n\r\nThere are now two cases: either (a) every vertex is an endpoint of at least two diameters, or (b) there is a vertex which is an endpoint of exactly one diameter.\r\n\r\n(a)\r\n\r\nNotice that every two diameters must intersect (in an endpoint or otherwise). We’ll cal this assertion $()$. \r\n\r\nSuppose there are vertices $x,y,z$ such that $xy,xz$ are diameters, but $y,z$ are not consecutive vertices of $\\mathcal P$. Then, if $t$ is a vertex between $y$ and $z$ and such that $yz$ separates $x$ and $t$, then one of the (at least two) diameters through $t$ would be disjoint from one of the two diameters $xy,xz$, and this contradicts $()$. This proves that in the situation we’re considering now, every vertex is an endpoint of [i]exactly[/i] two diameters, and, moreover, that whenever $xy,xz$ are diameters, $y,z$ are consecutive vertices of $\\mathcal P$. It easily follows from here (just play around with a drawing) that $\\mathcal P$ has an odd number of vertices, and this contradicts the fact that it actually has $2000$ vertices.\r\n\r\nHence, (a) cannot occur.\r\n\r\n(b)\r\n\r\nLet $xy$ be the only diameter having $y$ as endpoint, and let $u,v$ be vertices immediately to the left and right of $y$. If $xy$ is not orthogonal to $uv$, we can rotate $y$ around $x$ slightly such that (i) $xy=1$ throughout the process, (ii) all other distances between vertices of $\\mathcal P$ and $y$ stay $<1$ (or $\\le 1$; it doesn’t matter), and (iii) the angle ($<\\frac\\pi 2$) between $xy$ and $uv$ increases, which is easily seen to make the area of $\\mathcal P$ increase strictly. This contradicts the maximality of the area of $\\mathcal P$, so we have what we wanted: the diagonals (well, $xy$ could be a side and not a diagonal in the strictest sense, but I assume this is allowed, i.e. by “diagonals” you mean “sides or diagonals”? :?) $xy$ and $uv$ must be orthogonal.", "content_html": "Suppose first that some vertex of our polygon <img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" > is not adjacent to any diameters (diagonals or sides of length <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >)</span>. Then pick any diagonal having this vertex as endpoint, and move the vertex slightly, so as to elongate the diagonal. Clearly, if the elongation is small enough, all diagonals with our vertex as endpoint stay smaller than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> but the area increases strictly, and this contradicts the maximality property of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" >.</span> We have thus shown that every vertex of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" >.</span><br>\n<br>\nThere are now two cases: either (a) every vertex is an endpoint of at least two diameters, or (b) there is a vertex which is an endpoint of exactly one diameter.<br>\n<br>\n(a)<br>\n<br>\nNotice that every two diameters must intersect (in an endpoint or otherwise). We’ll cal this assertion <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/d/0/9d02e715e7f7195675151570530c5b62ba7b2744.png\" class=\"latex\" alt=\"$()$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" >.</span><br>\n<br>\nSuppose there are vertices <img src=\"//latex.artofproblemsolving.com/a/c/c/accc80fdf164cef264f56a82b6f9f6add320fe05.png\" class=\"latex\" alt=\"$x,y,z$\" style=\"vertical-align: -3px\" width=\"44\" height=\"11\" > such that <img src=\"//latex.artofproblemsolving.com/8/0/4/804178873f17bd8bb4e0712649dcdf0dd04c1ec8.png\" class=\"latex\" alt=\"$xy,xz$\" style=\"vertical-align: -3px\" width=\"47\" height=\"11\" > are diameters, but <img src=\"//latex.artofproblemsolving.com/6/f/5/6f51d57011f1c7cc6b8e9035808e3936ce524817.png\" class=\"latex\" alt=\"$y,z$\" style=\"vertical-align: -3px\" width=\"26\" height=\"11\" > are not consecutive vertices of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" >.</span> Then, if <img src=\"//latex.artofproblemsolving.com/e/0/d/e0d2bf360290fd61d1c1557e763f2622363b3d35.png\" class=\"latex\" alt=\"$t$\" width=\"6\" height=\"11\" > is a vertex between <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > and <img src=\"//latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png\" class=\"latex\" alt=\"$z$\" width=\"8\" height=\"8\" > and such that <img src=\"//latex.artofproblemsolving.com/5/d/3/5d379b13c2338ee58a27252096b2544c9e8e37c7.png\" class=\"latex\" alt=\"$yz$\" style=\"vertical-align: -3px\" width=\"18\" height=\"11\" > separates <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/d/e0d2bf360290fd61d1c1557e763f2622363b3d35.png\" class=\"latex\" alt=\"$t$\" width=\"6\" height=\"11\" >,</span> then one of the (at least two) diameters through <img src=\"//latex.artofproblemsolving.com/e/0/d/e0d2bf360290fd61d1c1557e763f2622363b3d35.png\" class=\"latex\" alt=\"$t$\" width=\"6\" height=\"11\" > would be disjoint from one of the two diameters <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/0/4/804178873f17bd8bb4e0712649dcdf0dd04c1ec8.png\" class=\"latex\" alt=\"$xy,xz$\" style=\"vertical-align: -3px\" width=\"47\" height=\"11\" >,</span> and this contradicts <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/d/0/9d02e715e7f7195675151570530c5b62ba7b2744.png\" class=\"latex\" alt=\"$()$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" >.</span> This proves that in the situation we’re considering now, every vertex is an endpoint of <i>exactly</i> two diameters, and, moreover, that whenever <img src=\"//latex.artofproblemsolving.com/8/0/4/804178873f17bd8bb4e0712649dcdf0dd04c1ec8.png\" class=\"latex\" alt=\"$xy,xz$\" style=\"vertical-align: -3px\" width=\"47\" height=\"11\" > are diameters, <img src=\"//latex.artofproblemsolving.com/6/f/5/6f51d57011f1c7cc6b8e9035808e3936ce524817.png\" class=\"latex\" alt=\"$y,z$\" style=\"vertical-align: -3px\" width=\"26\" height=\"11\" > are consecutive vertices of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" >.</span> It easily follows from here (just play around with a drawing) that <img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" > has an odd number of vertices, and this contradicts the fact that it actually has <img src=\"//latex.artofproblemsolving.com/0/5/f/05f1ffc8a6829f25b0b0366087a70dfd4e06fef6.png\" class=\"latex\" alt=\"$2000$\" width=\"35\" height=\"12\" > vertices.<br>\n<br>\nHence, (a) cannot occur.<br>\n<br>\n(b)<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" > be the only diameter having <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > as endpoint, and let <img src=\"//latex.artofproblemsolving.com/5/f/0/5f03710a9ffbf0ecba4ae631d6f3bd28ff7f48f2.png\" class=\"latex\" alt=\"$u,v$\" style=\"vertical-align: -3px\" width=\"27\" height=\"11\" > be vertices immediately to the left and right of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" >.</span> If <img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" > is not orthogonal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/f/4af6ae0ada20981b5d1ae4b17676e2cc12be86fe.png\" class=\"latex\" alt=\"$uv$\" width=\"19\" height=\"8\" >,</span> we can rotate <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > around <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > slightly such that (i) <img src=\"//latex.artofproblemsolving.com/f/a/2/fa22a3b6fdd8321ce93427ac874f16557593d663.png\" class=\"latex\" alt=\"$xy=1$\" style=\"vertical-align: -3px\" width=\"52\" height=\"15\" > throughout the process, (ii) all other distances between vertices of <img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" > and <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > stay <img src=\"//latex.artofproblemsolving.com/f/b/7/fb7ad445c31287ef502477d7f3fa24a3850110ab.png\" class=\"latex\" alt=\"$<1$\" style=\"vertical-align: 0px\" width=\"27\" height=\"12\" > (or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/d/1/8d1d3e9b91fd51e596ff2f380746034a1d58aa60.png\" class=\"latex\" alt=\"$\\le 1$\" style=\"vertical-align: -2px\" width=\"27\" height=\"14\" >;</span> it doesn’t matter), and (iii) the angle <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/8/e/3/8e3c459febcc0b392410ef0bfa7ac57dc74caad0.png\" class=\"latex\" alt=\"$<\\frac\\pi 2$\" style=\"vertical-align: -12px\" width=\"32\" height=\"33\" >)</span> between <img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" > and <img src=\"//latex.artofproblemsolving.com/4/a/f/4af6ae0ada20981b5d1ae4b17676e2cc12be86fe.png\" class=\"latex\" alt=\"$uv$\" width=\"19\" height=\"8\" > increases, which is easily seen to make the area of <img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" > increase strictly. This contradicts the maximality of the area of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/e/17e179805931e8a5004cd6ac6b3ad3a9b5020876.png\" class=\"latex\" alt=\"$\\mathcal P$\" style=\"vertical-align: -1px\" width=\"13\" height=\"14\" >,</span> so we have what we wanted: the diagonals (well, <img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" > could be a side and not a diagonal in the strictest sense, but I assume this is allowed, i.e. by “diagonals” you mean “sides or diagonals”? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" />) <img src=\"//latex.artofproblemsolving.com/3/f/7/3f79c3676b62fb47243e0d357c87115b6f3395e4.png\" class=\"latex\" alt=\"$xy$\" style=\"vertical-align: -3px\" width=\"19\" height=\"11\" > and <img src=\"//latex.artofproblemsolving.com/4/a/f/4af6ae0ada20981b5d1ae4b17676e2cc12be86fe.png\" class=\"latex\" alt=\"$uv$\" width=\"19\" height=\"8\" > must be orthogonal.", "post_id": 562442, "post_number": 2, "post_time_unix": 1151650409, "post_time_utc": "2006-06-30 06:53:29 UTC", "thanks_received": 3, "user_id": 26, "username": "grobber" } ], "source": null }	Let M be a convex 2000-gon with the property that the maximal distance between any two vertices equals 1. Among all convex 2000-gons with this property, M has maximal area. Show that some two diagonals of M are perpendicular.	[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/Area", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/ConvexPolygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/PolygonArea", "/Mathematics/Geometry/PlaneGeometry/Polygons/PolygonDiameter" ]	Move a vertex while keeping its longest incident segment (a diameter) length 1; area increases unless that segment is perpendicular to the adjacent side.	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aops_99409	[quote="Nekruzjon_eko"]If $x=\sqrt{27: \sqrt{27: \sqrt{27: ...}}}$ and ${y=\sqrt[3]{{9\sqrt[3]{9\sqrt[3]9}}}}...$ then find the quotient $\frac{x}{y}$[/quote] [hide]Since this is infinite we can assume some things. Substituting x into the radical we get $x=\sqrt{\frac{27}{x}}$, so $x=3$ $y=\sqrt[3]{9y}$, so $y=3$. So $\frac{x}{y}=1$. $1$[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "If $x=\\sqrt{27: \\sqrt{27: \\sqrt{27: ...}}}$ and\r\n\r\n${y=\\sqrt[3]{{9\\sqrt[3]{9\\sqrt[3]9}}}}...$\r\n\r\nthen find the quotient $\\frac{x}{y}$", "content_html": "If <img src=\"//latex.artofproblemsolving.com/a/0/3/a03bf13307bb06a2f9c0b655b28c539a1f95155c.png\" class=\"latex\" alt=\"$x=\\sqrt{27: \\sqrt{27: \\sqrt{27: ...}}}$\" style=\"vertical-align: -9px\" width=\"201\" height=\"43\" > and<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/b/3/db3e119e2cefbed9106b860427d963d64c9ffeca.png\" class=\"latex\" alt=\"${y=\\sqrt[3]{{9\\sqrt[3]{9\\sqrt[3]9}}}}...$\" style=\"vertical-align: -9px\" width=\"127\" height=\"43\" ><br>\n<br>\nthen find the quotient <img src=\"//latex.artofproblemsolving.com/e/3/2/e32f96b310bbe6cd6981b6c1da83fe7905ea3116.png\" class=\"latex\" alt=\"$\\frac{x}{y}$\" style=\"vertical-align: -16px\" width=\"12\" height=\"36\" >", "post_id": 561249, "post_number": 1, "post_time_unix": 1151572868, "post_time_utc": "2006-06-29 09:21:08 UTC", "thanks_received": 4, "user_id": 20652, "username": "Nekruzjon_eko" }, { "attachments": [], "content_bbcode": "$x=3$\r\n$y=3$\r\n$\\frac{x}{y}=1$", "content_html": "<img src=\"//latex.artofproblemsolving.com/b/0/2/b022dc8ef6d33931fc1ae3d27df3ea0a81b52936.png\" class=\"latex\" alt=\"$x=3$\" width=\"43\" height=\"12\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/2/b/a2b0e1404de4eca35c42876bcc83707b753e990e.png\" class=\"latex\" alt=\"$y=3$\" style=\"vertical-align: -3px\" width=\"42\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/b/0/1b0a25df91937164e14a2c6445eeb41895e2eb3f.png\" class=\"latex\" alt=\"$\\frac{x}{y}=1$\" style=\"vertical-align: -16px\" width=\"46\" height=\"36\" >", "post_id": 561278, "post_number": 2, "post_time_unix": 1151579402, "post_time_utc": "2006-06-29 11:10:02 UTC", "thanks_received": 1, "user_id": 20532, "username": "inom" }, { "attachments": [], "content_bbcode": "[hide] $x=\\sqrt{\\frac{27}{x}} \\longrightarrow x^2=\\frac{27}{x} \\longrightarrow x^3=27$. $x=\\boxed{3}$.\n\n$y=\\sqrt[3]{9y} \\longrightarrow y^3=9y \\longrightarrow y^2=9$. $y=\\boxed{3}$. [/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/8/e/98e56439a6004f28159c0db037a448c1d4af7aa1.png\" class=\"latex\" alt=\"$x=\\sqrt{\\frac{27}{x}} \\longrightarrow x^2=\\frac{27}{x} \\longrightarrow x^3=27$\" style=\"vertical-align: -14px\" width=\"277\" height=\"43\" >.</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/4/e/a4e99993cfca0d5682cae52e896c5577d370338e.png\" class=\"latex\" alt=\"$x=\\boxed{3}$\" style=\"vertical-align: -5px\" width=\"55\" height=\"23\" >.</span><br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/5/c6502a073c065e9b7854eb4b6b34fe97383659b4.png\" class=\"latex\" alt=\"$y=\\sqrt[3]{9y} \\longrightarrow y^3=9y \\longrightarrow y^2=9$\" style=\"vertical-align: -5px\" width=\"259\" height=\"22\" >.</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/0/0/1000088800e46025499661bf49f98073083c2422.png\" class=\"latex\" alt=\"$y=\\boxed{3}$\" style=\"vertical-align: -5px\" width=\"54\" height=\"23\" >.</span></div>", "post_id": 561773, "post_number": 3, "post_time_unix": 1151607196, "post_time_utc": "2006-06-29 18:53:16 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "[quote=\"Nekruzjon_eko\"]If $x=\\sqrt{27: \\sqrt{27: \\sqrt{27: ...}}}$ and\n\n${y=\\sqrt[3]{{9\\sqrt[3]{9\\sqrt[3]9}}}}...$\n\nthen find the quotient $\\frac{x}{y}$[/quote]\r\n\r\n[hide]Since this is infinite we can assume some things. Substituting x into the radical we get\n\n$x=\\sqrt{\\frac{27}{x}}$, so $x=3$\n\n$y=\\sqrt[3]{9y}$, so $y=3$.\n\nSo $\\frac{x}{y}=1$.\n\n$1$[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Nekruzjon_eko wrote:</div>\n<div class=\"bbcode_quote_body\">If <img src=\"//latex.artofproblemsolving.com/a/0/3/a03bf13307bb06a2f9c0b655b28c539a1f95155c.png\" class=\"latex\" alt=\"$x=\\sqrt{27: \\sqrt{27: \\sqrt{27: ...}}}$\" style=\"vertical-align: -9px\" width=\"201\" height=\"43\" > and<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/b/3/db3e119e2cefbed9106b860427d963d64c9ffeca.png\" class=\"latex\" alt=\"${y=\\sqrt[3]{{9\\sqrt[3]{9\\sqrt[3]9}}}}...$\" style=\"vertical-align: -9px\" width=\"127\" height=\"43\" ><br>\n<br>\nthen find the quotient <img src=\"//latex.artofproblemsolving.com/e/3/2/e32f96b310bbe6cd6981b6c1da83fe7905ea3116.png\" class=\"latex\" alt=\"$\\frac{x}{y}$\" style=\"vertical-align: -16px\" width=\"12\" height=\"36\" ></div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Since this is infinite we can assume some things. Substituting x into the radical we get<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/b/eeb7622a0f236b11f0ec51a961d2b3856efc5259.png\" class=\"latex\" alt=\"$x=\\sqrt{\\frac{27}{x}}$\" style=\"vertical-align: -14px\" width=\"75\" height=\"43\" >,</span> so <img src=\"//latex.artofproblemsolving.com/b/0/2/b022dc8ef6d33931fc1ae3d27df3ea0a81b52936.png\" class=\"latex\" alt=\"$x=3$\" width=\"43\" height=\"12\" ><br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/4/3/b43a95334380617aea8b98c2e156bb34cf17e053.png\" class=\"latex\" alt=\"$y=\\sqrt[3]{9y}$\" style=\"vertical-align: -5px\" width=\"71\" height=\"22\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/2/b/a2b0e1404de4eca35c42876bcc83707b753e990e.png\" class=\"latex\" alt=\"$y=3$\" style=\"vertical-align: -3px\" width=\"42\" height=\"16\" >.</span><br>\n<br>\nSo <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/b/0/1b0a25df91937164e14a2c6445eeb41895e2eb3f.png\" class=\"latex\" alt=\"$\\frac{x}{y}=1$\" style=\"vertical-align: -16px\" width=\"46\" height=\"36\" >.</span><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" ></div>", "post_id": 561918, "post_number": 4, "post_time_unix": 1151614740, "post_time_utc": "2006-06-29 20:59:00 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "you can actually find it without find the values of each one! try", "content_html": "you can actually find it without find the values of each one! try", "post_id": 562419, "post_number": 5, "post_time_unix": 1151646517, "post_time_utc": "2006-06-30 05:48:37 UTC", "thanks_received": 2, "user_id": 19927, "username": "srulikbd" }, { "attachments": [], "content_bbcode": "[quote=\"srulikbd\"]you can actually find it without find the values of each one! try[/quote]\r\nWe have:\r\n$\\frac{x}{y}=\\frac{\\sqrt{27: \\sqrt{27: \\sqrt{27: \\dots}}}}{\\sqrt[3]{9\\sqrt[3]{\\sqrt[3]{9\\dots}}}}=\\frac{\\sqrt{27}: \\sqrt[4]{27}\\cdot\\sqrt[8]{27}: \\dots}{\\sqrt[3]{9}\\cdot\\sqrt[9]{9}\\cdot\\sqrt[27]{9}\\dots}=\\frac{3^\\frac32\\cdot3^\\frac38\\cdot3^\\frac3{32}\\cdot\\dots}{(3^\\frac23\\cdot3^\\frac29\\cdot3^\\frac2{27}\\cdot\\dots)\\cdot(3^\\frac34\\cdot3^\\frac3{16}\\cdot\\dots)}=\\frac{3^\\frac{\\frac32}{1-\\frac14}}{3^{\\frac{\\frac23}{1-\\frac13}+\\frac{\\frac34}{1-\\frac14}}}=1$ :starwars: :spider:", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">srulikbd wrote:</div>\n<div class=\"bbcode_quote_body\">you can actually find it without find the values of each one! try</div>\n</div>\nWe have:<br>\n<img src=\"//latex.artofproblemsolving.com/2/e/2/2e2f97f72c15f95cbca727ccbfa5d1abe0addd05.png\" class=\"latex\" alt=\"$\\frac{x}{y}=\\frac{\\sqrt{27: \\sqrt{27: \\sqrt{27: \\dots}}}}{\\sqrt[3]{9\\sqrt[3]{\\sqrt[3]{9\\dots}}}}=\\frac{\\sqrt{27}: \\sqrt[4]{27}\\cdot\\sqrt[8]{27}: \\dots}{\\sqrt[3]{9}\\cdot\\sqrt[9]{9}\\cdot\\sqrt[27]{9}\\dots}=\\frac{3^\\frac32\\cdot3^\\frac38\\cdot3^\\frac3{32}\\cdot\\dots}{(3^\\frac23\\cdot3^\\frac29\\cdot3^\\frac2{27}\\cdot\\dots)\\cdot(3^\\frac34\\cdot3^\\frac3{16}\\cdot\\dots)}=\\frac{3^\\frac{\\frac32}{1-\\frac14}}{3^{\\frac{\\frac23}{1-\\frac13}+\\frac{\\frac34}{1-\\frac14}}}=1$\" style=\"vertical-align: -31px\" width=\"838\" height=\"73\" > <img src=\"/assets/images/smilies/starwars.gif\" width=\"100\" height=\"40\" alt=\":starwars:\" title=\":starwars:\" class=\"bbcode_smiley\" /> :spider:", "post_id": 562572, "post_number": 6, "post_time_unix": 1151671644, "post_time_utc": "2006-06-30 12:47:24 UTC", "thanks_received": 2, "user_id": 19904, "username": "xime" }, { "attachments": [], "content_bbcode": "So why do the nested expressions for $x$ and $y$ converge? People have only shown that if they converge they are equal to 3. It is my greatest pet peeve with the Intermediate forums that there often appears expressions defined by limits but that no one considers convergerence.\r\n\r\nIt is also the fault of problem constructors when they include problems on competitions that cannot be expected to be solved rigorously by the intended audience.", "content_html": "So why do the nested expressions for <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > converge? People have only shown that if they converge they are equal to 3. It is my greatest pet peeve with the Intermediate forums that there often appears expressions defined by limits but that no one considers convergerence.<br>\n<br>\nIt is also the fault of problem constructors when they include problems on competitions that cannot be expected to be solved rigorously by the intended audience.", "post_id": 562606, "post_number": 7, "post_time_unix": 1151674121, "post_time_utc": "2006-06-30 13:28:41 UTC", "thanks_received": 2, "user_id": 11334, "username": "Kalle" } ], "source": null }	If \[ x=\sqrt{27:\sqrt{27:\sqrt{27:\cdots}}} \] and \[ y=\sqrt[3]{9\sqrt[3]{9\sqrt[3]{9\cdots}}}, \] find the quotient $\dfrac{x}{y}$.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/CubicEquation", "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Set the infinite nested radical equal to a variable and solve the resulting self‑referential equation.	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aops_99421	1. $\left( \frac{35}{36} \right)^3$, pentru ca-s $36$ de cazuri, numai unul e bun. 2. $\frac{9 \cdot 8!}{10!}$, pentru ca bilele $1,2,3,4,7,8,9,10$ se aseaza mai intai la intamplare, apoi se pun $5,6$ intr-unul dintre cele noua spatii disponibile: $- 1 - 2 - 3 - 4 - 7 - 8 - 9 - 10 -$. Sper sa fie bine.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Recunosc ca am o varsta cam inaintata ptr.matematica si probabil de aceea m-am si impotmolit. Am doua probleme si va rog foarte mult sa nu ma ocoliti.\"Se arunca doua zaruri de 3 ori. Care este probabilitatea ca dubla 6,6 sa apara cel putin o data?\" si \"Zece bile numerotate de la 1 la 10 se aseaza la intamplare una dupa alta intr-un sir. Care este probabilitatea ca dupa bila nr.5 sa urmeze bila cu nr.6?\"\r\nLe-am rezolvat dar nu sunt sigura pe rezultat : 1-(25/36)la puterea a treia (doar paranteza ridicata la putere) si 9x8!/10! sau 9!8/10! (nu sunt sigura care). Ma poate ajuta cineva?", "content_html": "Recunosc ca am o varsta cam inaintata ptr.matematica si probabil de aceea m-am si impotmolit. Am doua probleme si va rog foarte mult sa nu ma ocoliti."Se arunca doua zaruri de 3 ori. Care este probabilitatea ca dubla 6,6 sa apara cel putin o data?" si "Zece bile numerotate de la 1 la 10 se aseaza la intamplare una dupa alta intr-un sir. Care este probabilitatea ca dupa bila nr.5 sa urmeze bila cu nr.6?"<br>\nLe-am rezolvat dar nu sunt sigura pe rezultat : 1-(25/36)la puterea a treia (doar paranteza ridicata la putere) si 9x8!/10! sau 9!8/10! (nu sunt sigura care). Ma poate ajuta cineva?", "post_id": 561285, "post_number": 1, "post_time_unix": 1151581582, "post_time_utc": "2006-06-29 11:46:22 UTC", "thanks_received": 1, "user_id": 20675, "username": "mirelah" }, { "attachments": [], "content_bbcode": "1. $\\left( \\frac{35}{36} \\right)^3$, pentru ca-s $36$ de cazuri, numai unul e bun.\r\n\r\n2. $\\frac{9 \\cdot 8!}{10!}$, pentru ca bilele $1,2,3,4,7,8,9,10$ se aseaza mai intai la intamplare, apoi se pun $5,6$ intr-unul dintre cele noua spatii disponibile: $- 1 - 2 - 3 - 4 - 7 - 8 - 9 - 10 -$. Sper sa fie bine.", "content_html": "1. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/0/2/9020ded76050ba67a4d9b89ac95d4ce037fad968.png\" class=\"latex\" alt=\"$\\left( \\frac{35}{36} \\right)^3$\" style=\"vertical-align: -17px\" width=\"54\" height=\"46\" >,</span> pentru ca-s <img src=\"//latex.artofproblemsolving.com/c/e/5/ce5d3cbb9ab0a992ed503c2d4499e4f4af2f9d65.png\" class=\"latex\" alt=\"$36$\" width=\"17\" height=\"12\" > de cazuri, numai unul e bun.<br>\n<br>\n2. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/b/2/0b24569fcdccde781bfba7f66a7ef8cabe13169b.png\" class=\"latex\" alt=\"$\\frac{9 \\cdot 8!}{10!}$\" style=\"vertical-align: -13px\" width=\"38\" height=\"38\" >,</span> pentru ca bilele <img src=\"//latex.artofproblemsolving.com/6/d/9/6d9af839c818954b0bab53ed7db46e7f37b695e5.png\" class=\"latex\" alt=\"$1,2,3,4,7,8,9,10$\" style=\"vertical-align: -3px\" width=\"137\" height=\"16\" > se aseaza mai intai la intamplare, apoi se pun <img src=\"//latex.artofproblemsolving.com/4/7/8/478f8b6c52d81d14c2aee0fdbf79b955410bc2c3.png\" class=\"latex\" alt=\"$5,6$\" style=\"vertical-align: -3px\" width=\"25\" height=\"16\" > intr-unul dintre cele noua spatii disponibile: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/8/d/b8d7b392149d8de7d2d3dd8baf397d9aa5f745d9.png\" class=\"latex\" alt=\"$- 1 - 2 - 3 - 4 - 7 - 8 - 9 - 10 -$\" style=\"vertical-align: 0px\" width=\"264\" height=\"13\" >.</span> Sper sa fie bine.", "post_id": 561404, "post_number": 2, "post_time_unix": 1151590320, "post_time_utc": "2006-06-29 14:12:00 UTC", "thanks_received": 2, "user_id": 6551, "username": "perfect_radio" }, { "attachments": [], "content_bbcode": "Cu raspunsul la intrebarea doi sunt de acord, dar nu sunt lamurita la prima. In manualul de cls.a X a este o poblema asemanatoare doar ca se arunca un zar de doua ori iar raspunsul este [b]1-(5/6)³[/b]. Dupa cum spui tu ar trebui sa fie doar[b] (5/6)³[/b].", "content_html": "Cu raspunsul la intrebarea doi sunt de acord, dar nu sunt lamurita la prima. In manualul de cls.a X a este o poblema asemanatoare doar ca se arunca un zar de doua ori iar raspunsul este <b>1-(5/6)³</b>. Dupa cum spui tu ar trebui sa fie doar<b> (5/6)³</b>.", "post_id": 561441, "post_number": 3, "post_time_unix": 1151592230, "post_time_utc": "2006-06-29 14:43:50 UTC", "thanks_received": 1, "user_id": 20675, "username": "mirelah" }, { "attachments": [], "content_bbcode": "ca sa apara cel putin o data inseamna: apare la una din cele 3 aruncari sau la doua din cele 3 aruncari sau la toate trei. Daca notam cu A evenimentul sa nu apara dubla 6,6 atunci avem da calculat nonA, adica P(nonA)=1-P(A)=1-$(\\frac{35}{36})^3$", "content_html": "ca sa apara cel putin o data inseamna: apare la una din cele 3 aruncari sau la doua din cele 3 aruncari sau la toate trei. Daca notam cu A evenimentul sa nu apara dubla 6,6 atunci avem da calculat nonA, adica P(nonA)=1-P(A)=1<span style=\"white-space:nowrap;\">-<img src=\"//latex.artofproblemsolving.com/2/4/1/2419a39af2f909ba7e852dd9681e69d7b963ceec.png\" class=\"latex\" alt=\"$(\\frac{35}{36})^3$\" style=\"vertical-align: -12px\" width=\"42\" height=\"37\" ></span>", "post_id": 561480, "post_number": 4, "post_time_unix": 1151594155, "post_time_utc": "2006-06-29 15:15:55 UTC", "thanks_received": 2, "user_id": 7334, "username": "sweetonect" } ], "source": null }	Se aruncă două zaruri de 3 ori. Care este probabilitatea ca dubla 6,6 să apară cel puțin o dată? Zece bile numerotate de la 1 la 10 se așază aleatoriu una după alta într-un șir. Care este probabilitatea ca după bila nr. 5 să urmeze bila nr. 6?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/MultiplicationPrinciple", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Use the complement rule for at least one occurrence and count positions by fixing the order of 5 then 6 among remaining slots.	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aops_99429	We need to assume that $x,y$ are coprime or something similar. When both have some other common divisors, we just forget them and thus assume $x,y$ coprime. Let $q \neq p$ be a odd prime divisor of $x^2-xy+\frac{p+1}{4}y^2=\left(x-\frac{y}{2}\right)^2+\frac{p}{4}y^2$, so by coprimality especially $q\nmid y$. Then $-p \equiv \left(\frac{2x-y}{y}\right)^2 \mod q$. Thus $1=\left( \frac{-p}{q} \right) = \left( \frac{q}{p} \right)$, thus $q$ is a square $\mod p$. Similar we can treat $q=2$ and get the same. Since squares are closed under multiplication, we cannot reach $-1 \mod p$ (by multiplication of the prime divisors) which is a nonsquare.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $p$ denote a prime number of the form $4k+3$ and $x,y$ be positive integers. Prove that $x^2-xy+\\frac{p+1}{4}y^2$ does not have any divisor of the form $kp-1$.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > denote a prime number of the form <img src=\"//latex.artofproblemsolving.com/c/d/f/cdf059b98920b73904f66dd65eccb9b62c9bc184.png\" class=\"latex\" alt=\"$4k+3$\" style=\"vertical-align: -1px\" width=\"50\" height=\"14\" > and <img src=\"//latex.artofproblemsolving.com/b/6/4/b6400c6fe3f1ed70e5afd387a9686cdc3fa09531.png\" class=\"latex\" alt=\"$x,y$\" style=\"vertical-align: -3px\" width=\"27\" height=\"11\" > be positive integers. Prove that <img src=\"//latex.artofproblemsolving.com/5/8/f/58fcc8c79f73caf8e95d6f67ca843e8f84d452c0.png\" class=\"latex\" alt=\"$x^2-xy+\\frac{p+1}{4}y^2$\" style=\"vertical-align: -13px\" width=\"142\" height=\"37\" > does not have any divisor of the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/a/0/1a018f104fe4c7e35495f0200abb6e42981695d5.png\" class=\"latex\" alt=\"$kp-1$\" style=\"vertical-align: -3px\" width=\"49\" height=\"16\" >.</span>", "post_id": 561326, "post_number": 1, "post_time_unix": 1151586014, "post_time_utc": "2006-06-29 13:00:14 UTC", "thanks_received": 2, "user_id": 5633, "username": "warut_suk" }, { "attachments": [], "content_bbcode": "We need to assume that $x,y$ are coprime or something similar.\r\nWhen both have some other common divisors, we just forget them and thus assume $x,y$ coprime.\r\nLet $q \\neq p$ be a odd prime divisor of $x^2-xy+\\frac{p+1}{4}y^2=\\left(x-\\frac{y}{2}\\right)^2+\\frac{p}{4}y^2$, so by coprimality especially $q\\nmid y$.\r\nThen $-p \\equiv \\left(\\frac{2x-y}{y}\\right)^2 \\mod q$.\r\nThus $1=\\left( \\frac{-p}{q} \\right) = \\left( \\frac{q}{p} \\right)$, thus $q$ is a square $\\mod p$.\r\nSimilar we can treat $q=2$ and get the same.\r\nSince squares are closed under multiplication, we cannot reach $-1 \\mod p$ (by multiplication of the prime divisors) which is a nonsquare.", "content_html": "We need to assume that <img src=\"//latex.artofproblemsolving.com/b/6/4/b6400c6fe3f1ed70e5afd387a9686cdc3fa09531.png\" class=\"latex\" alt=\"$x,y$\" style=\"vertical-align: -3px\" width=\"27\" height=\"11\" > are coprime or something similar.<br>\nWhen both have some other common divisors, we just forget them and thus assume <img src=\"//latex.artofproblemsolving.com/b/6/4/b6400c6fe3f1ed70e5afd387a9686cdc3fa09531.png\" class=\"latex\" alt=\"$x,y$\" style=\"vertical-align: -3px\" width=\"27\" height=\"11\" > coprime.<br>\nLet <img src=\"//latex.artofproblemsolving.com/f/d/c/fdcf9d8edc9044747ecaa431f9ab62ddda2732a4.png\" class=\"latex\" alt=\"$q \\neq p$\" style=\"vertical-align: -4px\" width=\"42\" height=\"17\" > be a odd prime divisor of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/c/47c64f94d9014850ca767911fa77fa9c3de0fbe3.png\" class=\"latex\" alt=\"$x^2-xy+\\frac{p+1}{4}y^2=\\left(x-\\frac{y}{2}\\right)^2+\\frac{p}{4}y^2$\" style=\"vertical-align: -13px\" width=\"293\" height=\"37\" >,</span> so by coprimality especially <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/9/c/b9caefe1bff848148f4e0cb192b186a96f9b5469.png\" class=\"latex\" alt=\"$q\\nmid y$\" style=\"vertical-align: -4px\" width=\"33\" height=\"18\" >.</span><br>\nThen <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/5/0/95068ad5e7db15617a3b375f7535d5cb0ab1c6ab.png\" class=\"latex\" alt=\"$-p \\equiv \\left(\\frac{2x-y}{y}\\right)^2 \\mod q$\" style=\"vertical-align: -17px\" width=\"201\" height=\"46\" >.</span><br>\nThus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/c/a/0ca8448819ea23e9a3734e2fbe17c24d59e142b5.png\" class=\"latex\" alt=\"$1=\\left( \\frac{-p}{q} \\right) = \\left( \\frac{q}{p} \\right)$\" style=\"vertical-align: -17px\" width=\"146\" height=\"43\" >,</span> thus <img src=\"//latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png\" class=\"latex\" alt=\"$q$\" style=\"vertical-align: -3px\" width=\"8\" height=\"11\" > is a square <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/f/9/0f965eb9407014fe0e4fc671eed0acbfd84eddb2.png\" class=\"latex\" alt=\"$\\mod p$\" style=\"vertical-align: -3px\" width=\"62\" height=\"16\" >.</span><br>\nSimilar we can treat <img src=\"//latex.artofproblemsolving.com/2/2/4/2249017892bda0490d89aa6d997b953b070b1ba1.png\" class=\"latex\" alt=\"$q=2$\" style=\"vertical-align: -3px\" width=\"41\" height=\"15\" > and get the same.<br>\nSince squares are closed under multiplication, we cannot reach <img src=\"//latex.artofproblemsolving.com/d/5/8/d58da14210e75da8ad55188ccd1035a1f9b144b2.png\" class=\"latex\" alt=\"$-1 \\mod p$\" style=\"vertical-align: -3px\" width=\"85\" height=\"16\" > (by multiplication of the prime divisors) which is a nonsquare.", "post_id": 561395, "post_number": 2, "post_time_unix": 1151589940, "post_time_utc": "2006-06-29 14:05:40 UTC", "thanks_received": 1, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "I use excactly the same method.", "content_html": "I use excactly the same method.", "post_id": 562245, "post_number": 3, "post_time_unix": 1151631556, "post_time_utc": "2006-06-30 01:39:16 UTC", "thanks_received": 1, "user_id": 14130, "username": "Hawk Tiger" } ], "source": null }	Let $p$ be a prime of the form $4k+3$ and $x,y$ be positive integers. Prove that \[ x^2-xy+\frac{p+1}{4}y^2 \] has no divisor of the form $kp-1$ (with integer $k$).	[ "/Mathematics/NumberTheory/Divisors/Coprime", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/RelativelyPrime", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/PrimeNumbers/PrimeRepresentations", "/Mathematics/NumberTheory/ReciprocityTheorems/EulersQuadraticResidueTheorem", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticNonresidue", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticReciprocityTheorem", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticResidue", "/Mathematics/NumberTheory/ReciprocityTheorems/ReciprocityTheorem" ]	Show every prime divisor of the form is a quadratic residue modulo p, so their product cannot be congruent to −1 (mod p).	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aops_994298	[b]Problem[/b]: $ \int_{ \minus{} \infty}^\infty e^{ \minus{} x^2}dx$ Well... if we remember the probability density function $ \frac 1{\sqrt {2\pi}}e^{ \minus{} \frac {x^2}2}$, then we should remember that $ \int_{ \minus{} \infty}^\infty \frac 1{\sqrt {2\pi}}e^{ \minus{} \frac {x^2}2}dx \equal{} 1$ Breaking it apart, then... $ \begin{align} \frac 1{\sqrt {2\pi}} \int_{ \minus{} \infty}^\infty e^{ \minus{} \frac {x^2}2}dx & \equal{} 1 \\ \int_{ \minus{} \infty}^\infty e^{ \minus{} \frac {x^2}2}dx & \equal{} \sqrt {2\pi} \\ \int_{ \minus{} \infty}^\infty e^{ \minus{} x^2}dx & \equal{} \sqrt {\pi} \end{align}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Lets have a math problem so I am not posting spam.\r\n\r\nI just found this cool problem and wanted to see if you remember integrals from last year at camp. \r\n(I should learn Latex but I haven't so this will be hard to read) \r\n\r\nEvaluate the integral of (e^(-x^2)) from negative infinity to infinity.\r\n\r\nIf you know Latex could you plz post it in Latex.\r\n\r\nIf you get the question I will give you a smiley. :D :lol: :) :P", "content_html": "Lets have a math problem so I am not posting spam.<br>\n<br>\nI just found this cool problem and wanted to see if you remember integrals from last year at camp.<br>\n(I should learn Latex but I haven't so this will be hard to read)<br>\n<br>\nEvaluate the integral of (e^(-x^2)) from negative infinity to infinity.<br>\n<br>\nIf you know Latex could you plz post it in Latex.<br>\n<br>\nIf you get the question I will give you a smiley. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />", "post_id": 4410040, "post_number": 1, "post_time_unix": 1215051803, "post_time_utc": "2008-07-03 02:23:23 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "[b]Problem[/b]: $ \\int_{ \\minus{} \\infty}^\\infty e^{ \\minus{} x^2}dx$\r\n\r\nWell... if we remember the probability density function $ \\frac 1{\\sqrt {2\\pi}}e^{ \\minus{} \\frac {x^2}2}$, then we should remember that \r\n$ \\int_{ \\minus{} \\infty}^\\infty \\frac 1{\\sqrt {2\\pi}}e^{ \\minus{} \\frac {x^2}2}dx \\equal{} 1$\r\n\r\nBreaking it apart, then...\r\n\r\n$ \\begin{align} \\frac 1{\\sqrt {2\\pi}} \\int_{ \\minus{} \\infty}^\\infty e^{ \\minus{} \\frac {x^2}2}dx & \\equal{} 1 \\\\\r\n\\int_{ \\minus{} \\infty}^\\infty e^{ \\minus{} \\frac {x^2}2}dx & \\equal{} \\sqrt {2\\pi} \\\\\r\n\\int_{ \\minus{} \\infty}^\\infty e^{ \\minus{} x^2}dx & \\equal{} \\sqrt {\\pi} \\end{align}$", "content_html": "<b>Problem</b>: <img src=\"//latex.artofproblemsolving.com/7/8/d/78d283869e5ea37518b7cc5871ff62be7cc0a6e7.png\" class=\"latex\" alt=\"$ \\int_{ - \\infty}^\\infty e^{ - x^2}dx$\" style=\"vertical-align: -16px\" width=\"90\" height=\"42\" ><br>\n<br>\nWell... if we remember the probability density function <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/6/a/f6aa980d9363e75582e3fd669fabc582ed8de631.png\" class=\"latex\" alt=\"$ \\frac 1{\\sqrt {2\\pi}}e^{ - \\frac {x^2}2}$\" style=\"vertical-align: -17px\" width=\"73\" height=\"41\" >,</span> then we should remember that<br>\n<img src=\"//latex.artofproblemsolving.com/9/b/2/9b20225703bab5dc857297320f65679d5f73635f.png\" class=\"latex\" alt=\"$ \\int_{ - \\infty}^\\infty \\frac 1{\\sqrt {2\\pi}}e^{ - \\frac {x^2}2}dx = 1$\" style=\"vertical-align: -17px\" width=\"165\" height=\"42\" ><br>\n<br>\nBreaking it apart, then...<br>\n<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align} \\frac 1{\\sqrt {2\\pi}} \\int_{ - \\infty}^\\infty e^{ - \\frac {x^2}2}dx & = 1 \\\\\n\\int_{ - \\infty}^\\infty e^{ - \\frac {x^2}2}dx & = \\sqrt {2\\pi} \\\\\n\\int_{ - \\infty}^\\infty e^{ - x^2}dx & = \\sqrt {\\pi} \\end{align}$</span>", "post_id": 4410041, "post_number": 2, "post_time_unix": 1215098497, "post_time_utc": "2008-07-03 15:21:37 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "Good Job!!!!!\r\nNow you get to pick your smiley.\r\n\r\nAlso if anyone can solve it without the probability density function I will give you all four smiley(s)?.\r\n(I would give you more but when trying to post all the smiley on (^_^)'s blog it said i could only have a maximum of 10 or 4) \r\nWhat is the plural of smiley? Cause under smileys i get a red line.", "content_html": "Good Job!!!!!<br>\nNow you get to pick your smiley.<br>\n<br>\nAlso if anyone can solve it without the probability density function I will give you all four smiley(s)?.<br>\n(I would give you more but when trying to post all the smiley on (^_^)'s blog it said i could only have a maximum of 10 or 4)<br>\nWhat is the plural of smiley? Cause under smileys i get a red line.", "post_id": 4410042, "post_number": 3, "post_time_unix": 1215104958, "post_time_utc": "2008-07-03 17:09:18 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" } ], "source": null }	Evaluate the integral \[ \int_{-\infty}^{\infty} e^{-x^{2}}\,dx. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/DefiniteIntegrals", "/Mathematics/CalculusandAnalysis/SpecialFunctions/Exponentials", "/Mathematics/CalculusandAnalysis/SpecialFunctions/GammaFunctions", "/Mathematics/CalculusandAnalysis/SpecialFunctions/NamedIntegrals", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilityDensityFunction", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/GaussianDistribution", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/NormalDistribution", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/NormalDistributionFunction", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/StandardNormalDistribution" ]	Recognize the integral as a scaled Gaussian integral and use the known value of the standard normal integral.	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aops_994316	this is equivalent to $ \binom{2007}{1}\plus{}\binom{2007}{2}\plus{}\binom{2007}{3}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ $ \binom{2007}{2}\plus{}\binom{2007}{3}\plus{}\binom{2007}{4}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ $ \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots$ $ \binom{2007}{2006}\plus{}\binom{2007}{2007}\plus{}$ $ \binom{2007}{2007}$ let this equal S S can be rewritten as $ \binom{2007}{2006}\plus{}\binom{2007}{2005}\plus{}\binom{2007}{2004}\plus{} \cdots \plus{}\binom{2007}{0}\plus{}$ $ \binom{2007}{2005}\plus{}\binom{2007}{2004}\plus{}\binom{2007}{2003}\plus{} \cdots \plus{}\binom{2007}{0}\plus{}$ $ \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots$ $ \binom{2007}{0}\plus{}\binom{2007}{0}\plus{}$ $ \binom{2007}{0}$ or $ \binom{2007}{0}$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}$ $ \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2006}\plus{}$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ we add S to itself $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ $ \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ $ \binom{2007}{0}\plus{}\binom{2007}{1}\plus{}\binom{2007}{2}\plus{} \cdots \plus{}\binom{2007}{2007}\plus{}$ (2007 rows) or $ 2S \equal{} 2007 \cdot 2^{2007}$ $ S \equal{} 2007 \cdot 2^{2006}$ which is congruent modulo 25 to... $ 7 \cdot 2^6 \equal{} 448$ which is congruent modulo 25 to 23! and the number added on is congruent to 0 modulo 25 so this does not change the remainder.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find the remainder when\r\n\\[ 1 \\cdot {2007 \\choose 1} \\plus{} 2\\cdot{2007 \\choose 2} \\plus{} 3\\cdot{2007 \\choose 3} \\plus{} \\dots \\plus{} 2007\\cdot{2007 \\choose 2007} \\plus{} 20938430427459084096849528975\r\n\\]\r\nis divided by 25.", "content_html": "Find the remainder when<br>\n<img src=\"//latex.artofproblemsolving.com/c/f/c/cfcafa712a42592ffecfd0b85292bf832a7d93fd.png\" class=\"latexcenter\" alt=\"\\[ 1 \\cdot {2007 \\choose 1} + 2\\cdot{2007 \\choose 2} + 3\\cdot{2007 \\choose 3} + \\dots + 2007\\cdot{2007 \\choose 2007} + 20938430427459084096849528975\n\\]\" width=\"694\" height=\"53\" ><br>\nis divided by 25.", "post_id": 4410125, "post_number": 1, "post_time_unix": 1215376975, "post_time_utc": "2008-07-06 20:42:55 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" }, { "attachments": [], "content_bbcode": "nice edit stevenmeow", "content_html": "nice edit stevenmeow", "post_id": 4410126, "post_number": 2, "post_time_unix": 1215381212, "post_time_utc": "2008-07-06 21:53:32 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" }, { "attachments": [], "content_bbcode": "this is equivalent to \r\n$ \\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{}\\binom{2007}{3}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\binom{2007}{2}\\plus{}\\binom{2007}{3}\\plus{}\\binom{2007}{4}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\r\n$ \\binom{2007}{2006}\\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\binom{2007}{2007}$\r\nlet this equal S\r\n\r\nS can be rewritten as\r\n$ \\binom{2007}{2006}\\plus{}\\binom{2007}{2005}\\plus{}\\binom{2007}{2004}\\plus{} \\cdots \\plus{}\\binom{2007}{0}\\plus{}$\r\n$ \\binom{2007}{2005}\\plus{}\\binom{2007}{2004}\\plus{}\\binom{2007}{2003}\\plus{} \\cdots \\plus{}\\binom{2007}{0}\\plus{}$\r\n$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{0}\\plus{}$\r\n$ \\binom{2007}{0}$\r\n\r\nor\r\n$ \\binom{2007}{0}$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}$\r\n$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2006}\\plus{}$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n\r\nwe add S to itself\r\n\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n$ \\binom{2007}{0}\\plus{}\\binom{2007}{1}\\plus{}\\binom{2007}{2}\\plus{} \\cdots \\plus{}\\binom{2007}{2007}\\plus{}$\r\n(2007 rows)\r\n\r\nor \r\n$ 2S \\equal{} 2007 \\cdot 2^{2007}$\r\n$ S \\equal{} 2007 \\cdot 2^{2006}$\r\nwhich is congruent modulo 25 to...\r\n$ 7 \\cdot 2^6 \\equal{} 448$\r\nwhich is congruent modulo 25 to 23!\r\nand the number added on is congruent to 0 modulo 25 so this does not change the remainder.", "content_html": "this is equivalent to<br>\n<img src=\"//latex.artofproblemsolving.com/8/0/0/80061c39e51557913b2c2097c4f62b69662a2ebf.png\" class=\"latex\" alt=\"$ \\binom{2007}{1}+\\binom{2007}{2}+\\binom{2007}{3}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/9/1/c912a0be5be42a31522acc14a3799c2cbd212ae4.png\" class=\"latex\" alt=\"$ \\binom{2007}{2}+\\binom{2007}{3}+\\binom{2007}{4}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/d/a/1dab0366e97f60e00bd0cd064bbb3367bcaecdb6.png\" class=\"latex\" alt=\"$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\" width=\"488\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/6/a/a6a9ae6b0a1e1d3b97bce6a44e3a6d739ca7b6fa.png\" class=\"latex\" alt=\"$ \\binom{2007}{2006}+\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/1/8/f1853afc5b9cd64c17d2a529d8918104f7e62c77.png\" class=\"latex\" alt=\"$ \\binom{2007}{2007}$\" style=\"vertical-align: -22px\" width=\"60\" height=\"53\" ><br>\nlet this equal S<br>\n<br>\nS can be rewritten as<br>\n<img src=\"//latex.artofproblemsolving.com/d/0/7/d07e49997eb3f5bf2b4a0990a42fdfef064e348b.png\" class=\"latex\" alt=\"$ \\binom{2007}{2006}+\\binom{2007}{2005}+\\binom{2007}{2004}+ \\cdots +\\binom{2007}{0}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/b/f/3/bf32e763289caa4bd2d56f684f28c5cea9f6d2d3.png\" class=\"latex\" alt=\"$ \\binom{2007}{2005}+\\binom{2007}{2004}+\\binom{2007}{2003}+ \\cdots +\\binom{2007}{0}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/d/a/1dab0366e97f60e00bd0cd064bbb3367bcaecdb6.png\" class=\"latex\" alt=\"$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\" width=\"488\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/4/2/a/42a632935eeb597dbef8ea2551a9bb683ed1e370.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{0}+$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/a/0/6a066e2afae9d8a7b74c3e13c92cc5ca4dfb5e2a.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}$\" style=\"vertical-align: -22px\" width=\"60\" height=\"53\" ><br>\n<br>\nor<br>\n<img src=\"//latex.artofproblemsolving.com/6/a/0/6a066e2afae9d8a7b74c3e13c92cc5ca4dfb5e2a.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}$\" style=\"vertical-align: -22px\" width=\"60\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/4/c/64cf0a285a161e8e63a4a147bdf4fae81cbf2658.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/d/a/1dab0366e97f60e00bd0cd064bbb3367bcaecdb6.png\" class=\"latex\" alt=\"$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\" width=\"488\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/d/0/6d035f07af81200915d9ba55cd35ff040cee99de.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2006}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a540786c74053779daa567ecaf3450091f02922.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<br>\nwe add S to itself<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a540786c74053779daa567ecaf3450091f02922.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a540786c74053779daa567ecaf3450091f02922.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/d/a/1dab0366e97f60e00bd0cd064bbb3367bcaecdb6.png\" class=\"latex\" alt=\"$ \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots \\vdots$\" width=\"488\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a540786c74053779daa567ecaf3450091f02922.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/a/5/9a540786c74053779daa567ecaf3450091f02922.png\" class=\"latex\" alt=\"$ \\binom{2007}{0}+\\binom{2007}{1}+\\binom{2007}{2}+ \\cdots +\\binom{2007}{2007}+$\" style=\"vertical-align: -22px\" width=\"381\" height=\"53\" ><br>\n(2007 rows)<br>\n<br>\nor<br>\n<img src=\"//latex.artofproblemsolving.com/0/f/a/0fa0cc38902e8d296b1f5d37dc886ed0373f6938.png\" class=\"latex\" alt=\"$ 2S = 2007 \\cdot 2^{2007}$\" width=\"130\" height=\"15\" ><br>\n<img src=\"//latex.artofproblemsolving.com/4/6/1/4614b0ed00cdaced02952566e2379d6cf9b1bc86.png\" class=\"latex\" alt=\"$ S = 2007 \\cdot 2^{2006}$\" width=\"120\" height=\"15\" ><br>\nwhich is congruent modulo 25 to...<br>\n<img src=\"//latex.artofproblemsolving.com/f/c/6/fc64cd4ce69d004bfb78fb81bf0c958f246aa2f0.png\" class=\"latex\" alt=\"$ 7 \\cdot 2^6 = 448$\" style=\"vertical-align: 0px\" width=\"89\" height=\"15\" ><br>\nwhich is congruent modulo 25 to 23!<br>\nand the number added on is congruent to 0 modulo 25 so this does not change the remainder.", "post_id": 4410127, "post_number": 3, "post_time_unix": 1215381682, "post_time_utc": "2008-07-06 22:01:22 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "correct!\r\n\r\n[hide=\"easier on the eyes\"]\nLet $ S\\equal{}1 \\cdot {2007 \\choose 1} \\plus{} 2\\cdot{2007 \\choose 2} \\plus{} 3\\cdot{2007 \\choose 3} \\plus{} \\dots \\plus{} 2007\\cdot{2007 \\choose 2007}$\n\nUsing the identity $ {n \\choose k}\\equal{}{n \\choose {n\\minus{}k}}$, we see that\n\n$ S\\equal{}2007\\left({2007 \\choose 0}\\plus{}{2007 \\choose 1}\\plus{}\\dots \\plus{}{2007 \\choose 1003} \\right)\\equal{}2007\\cdot \\frac{2^{2007}}{2}\\equal{}2007 \\cdot 2^{2006} \\equiv 23 \\pmod {25}$[/hide]", "content_html": "correct!<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">easier on the eyes</a><div class=\"cmty-hide-content\" style=\"display:none\">Let <img src=\"//latex.artofproblemsolving.com/3/7/9/379bc40e6bdf485d7f5e7598d338f8e495da06ec.png\" class=\"latex\" alt=\"$ S=1 \\cdot {2007 \\choose 1} + 2\\cdot{2007 \\choose 2} + 3\\cdot{2007 \\choose 3} + \\dots + 2007\\cdot{2007 \\choose 2007}$\" style=\"vertical-align: -22px\" width=\"516\" height=\"53\" ><br>\n<br>\nUsing the identity <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/b/3/ab3e70d3b30aee5faf2808633876c45a72fa0cdf.png\" class=\"latex\" alt=\"$ {n \\choose k}={n \\choose {n-k}}$\" style=\"vertical-align: -22px\" width=\"130\" height=\"53\" >,</span> we see that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/4/4/6/4468bee6877fe477b78ec775927f8cce3044923d.png\" class=\"latex\" alt=\"$ S=2007\\left({2007 \\choose 0}+{2007 \\choose 1}+\\dots +{2007 \\choose 1003} \\right)=2007\\cdot \\frac{2^{2007}}{2}=2007 \\cdot 2^{2006} \\equiv 23 \\pmod {25}$\" style=\"vertical-align: -22px\" width=\"731\" height=\"53\" ></div>", "post_id": 4410128, "post_number": 4, "post_time_unix": 1215382169, "post_time_utc": "2008-07-06 22:09:29 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" } ], "source": null }	Find the remainder when \[ 1\binom{2007}{1}+2\binom{2007}{2}+3\binom{2007}{3}+\dots+2007\binom{2007}{2007}+20938430427459084096849528975 \] is divided by $25$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Rewrite the weighted sum ℓk\binom{n}{k} as n\cdot2^{n-1} using the binomial identity.	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aops_994333	disproof: we know f(5)=55 f(55)=5555 f(555)=555555 and f(5555)=55555555 we set f(x)=ax^2+bx+c therefore we have a system of 4 equations in 3 variables, which we wish to prove is inconsistent 55=25a+5b+c 5555=3025a+55b+c 555555=308025a+555b+c 55555555=30858025a+5555b+c we subtrace the first equation from the other 3 giving us 5500=3000a+50b 555500=308000a+550b 55555500=30858000a+5550b we subtract 11 times the first from the second equation and 111 times the first from the third equation 495000=275000a ==> 495=275a 54945000=30525000a ==> 54945=30525a simplifiying, a must equal both 9/5 and 9/5 (rewind soundtrack marking mistake plays) ugh i guess there is a quadratic ehh :lol: time to turn this disproof into a proof so a=9/5 and b=2 and c=0 so our quadratic is f(x)=1.8x^2+2x we verify this for our first 4 terms and we wonder, does this work for all terms? we have the number 5555.....55555 (n 5's) and this is equivalent to 5/9(10^n-1) f(x) is equal to 5/9(10^n-1)^2+2*5/9(10^n-1)=5/9(10^(2n))-(2)5/9(10^n)+5/9+(2)5/9(10^n)-10/9 = 5/9(10^(2n))-5/9 =5/9(10^(2n)-1) which is what our function is supposed to do... so there is a function Q.E.D.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove or disprove: There is a quadratic polynomial $ f(x)$with rational coefficients such that, if $ n$ is a positive integer consisting of the digit 5 repeated $ k$ times, then $ f(n)$ consists of the digit 5 repeated $ 2k$ times. For example $ f(555) \\equal{} 555555$\r\n\r\nTINYTIM LOOK @ MY DISPROOF IT IS SO AWESOME", "content_html": "Prove or disprove: There is a quadratic polynomial <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/3/1/93151cc6bf9af56ebcb1b6e891401f06a07aefc8.png\" class=\"latex\" alt=\"$ f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" >w</span>ith rational coefficients such that, if <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > is a positive integer consisting of the digit 5 repeated <img src=\"//latex.artofproblemsolving.com/d/1/0/d10af8b3fc779f307fe5c87020e433b00a41801d.png\" class=\"latex\" alt=\"$ k$\" width=\"9\" height=\"12\" > times, then <img src=\"//latex.artofproblemsolving.com/2/4/7/247c5c4a02e9c1587ae084fc06cd45fc44368bdb.png\" class=\"latex\" alt=\"$ f(n)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" > consists of the digit 5 repeated <img src=\"//latex.artofproblemsolving.com/8/d/0/8d05756257f91d34512dea175c7f86ea118ff789.png\" class=\"latex\" alt=\"$ 2k$\" width=\"18\" height=\"12\" > times. For example <img src=\"//latex.artofproblemsolving.com/8/e/1/8e118c749c0e8559121f5f16fc1a9ecae141c521.png\" class=\"latex\" alt=\"$ f(555) = 555555$\" style=\"vertical-align: -4px\" width=\"130\" height=\"18\" ><br>\n<br>\nTINYTIM LOOK @ MY DISPROOF IT IS SO AWESOME", "post_id": 4410214, "post_number": 1, "post_time_unix": 1215549041, "post_time_utc": "2008-07-08 20:30:41 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "disproof:\r\nwe know f(5)=55 f(55)=5555 f(555)=555555 and f(5555)=55555555\r\nwe set f(x)=ax^2+bx+c\r\ntherefore we have a system of 4 equations in 3 variables, which we wish to prove is inconsistent\r\n\r\n55=25a+5b+c\r\n5555=3025a+55b+c\r\n555555=308025a+555b+c\r\n55555555=30858025a+5555b+c\r\nwe subtrace the first equation from the other 3 giving us\r\n\r\n5500=3000a+50b\r\n555500=308000a+550b\r\n55555500=30858000a+5550b\r\n\r\nwe subtract 11 times the first from the second equation and 111 times the first from the third equation\r\n\r\n495000=275000a ==> 495=275a\r\n54945000=30525000a ==> 54945=30525a\r\nsimplifiying, a must equal both 9/5 and 9/5 (rewind soundtrack marking mistake plays)\r\nugh i guess there is a quadratic\r\nehh :lol: time to turn this disproof into a proof\r\n\r\nso a=9/5 and b=2 and c=0 so our quadratic is f(x)=1.8x^2+2x\r\nwe verify this for our first 4 terms and we wonder, does this work for all terms?\r\n we have the number 5555.....55555 (n 5's) and this is equivalent to 5/9(10^n-1)\r\nf(x) is equal to 5/9(10^n-1)^2+25/9(10^n-1)=5/9(10^(2n))-(2)5/9(10^n)+5/9+(2)5/9(10^n)-10/9\r\n= 5/9(10^(2n))-5/9\r\n=5/9(10^(2n)-1)\r\nwhich is what our function is supposed to do... so there is a function Q.E.D.", "content_html": "disproof:<br>\nwe know f(5)=55 f(55)=5555 f(555)=555555 and f(5555)=55555555<br>\nwe set f(x)=ax^2+bx+c<br>\ntherefore we have a system of 4 equations in 3 variables, which we wish to prove is inconsistent<br>\n<br>\n55=25a+5b+c<br>\n5555=3025a+55b+c<br>\n555555=308025a+555b+c<br>\n55555555=30858025a+5555b+c<br>\nwe subtrace the first equation from the other 3 giving us<br>\n<br>\n5500=3000a+50b<br>\n555500=308000a+550b<br>\n55555500=30858000a+5550b<br>\n<br>\nwe subtract 11 times the first from the second equation and 111 times the first from the third equation<br>\n<br>\n495000=275000a ==> 495=275a<br>\n54945000=30525000a ==> 54945=30525a<br>\nsimplifiying, a must equal both 9/5 and 9/5 (rewind soundtrack marking mistake plays)<br>\nugh i guess there is a quadratic<br>\nehh <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /> time to turn this disproof into a proof<br>\n<br>\nso a=9/5 and b=2 and c=0 so our quadratic is f(x)=1.8x^2+2x<br>\nwe verify this for our first 4 terms and we wonder, does this work for all terms?<br>\nwe have the number 5555.....55555 (n 5's) and this is equivalent to 5/9(10^n-1)<br>\nf(x) is equal to 5/9(10^n-1)^2+25/9(10^n-1)=5/9(10^(2n))-(2)5/9(10^n)+5/9+(2)5/9(10^n)-10/9<br>\n= 5/9(10^(2n))-5/9<br>\n=5/9(10^(2n)-1)<br>\nwhich is what our function is supposed to do... so there is a function Q.E.D.", "post_id": 4410215, "post_number": 2, "post_time_unix": 1215550586, "post_time_utc": "2008-07-08 20:56:26 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "YAY! Correct!!! :rotfl: :rotfl: :rotfl:", "content_html": "YAY! Correct!!! <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" />", "post_id": 4410216, "post_number": 3, "post_time_unix": 1215554207, "post_time_utc": "2008-07-08 21:56:47 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "i totally need to write my proofs like this at USAMO...\r\n\r\nproblem:\r\nprove or disprove blah is true\r\nwe try to prove blah is true by contradiction and assuming blah is false\r\nuh-oh blah IS false, ok time to turn proof into disproof\r\nblah is false Q.E.D.", "content_html": "i totally need to write my proofs like this at USAMO...<br>\n<br>\nproblem:<br>\nprove or disprove blah is true<br>\nwe try to prove blah is true by contradiction and assuming blah is false<br>\nuh-oh blah IS false, ok time to turn proof into disproof<br>\nblah is false Q.E.D.", "post_id": 4410217, "post_number": 4, "post_time_unix": 1215554573, "post_time_utc": "2008-07-08 22:02:53 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" } ], "source": null }	Prove or disprove: There is a quadratic polynomial $f(x)$ with rational coefficients such that, if $n$ is a positive integer consisting of the digit 5 repeated $k$ times, then $f(n)$ consists of the digit 5 repeated $2k$ times. For example $f(555)=555555$.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialFunction", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/Algebra/Polynomials/Variable", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/RationalNumbers/Digit", "/Mathematics/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/NumberTheory/Sequences/PolynomialSequence" ]	Express a number of k repeated 5's as (5/9)(10^k‑1) and pick a quadratic that transforms it into (5/9)(10^{2k}‑1).	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aops_994342	YAY!! another recurrence problem!! Let $ P_n$ denote the probability you will have $ n$ points. Then $ P_n \equal{} \frac {1}{2}(P_{n \minus{} 1} \plus{} P_{n \minus{} 2})$. We know $ P_1 \equal{} \frac {1}{2}$ and $ P_2\equal{}\frac{3}{4}.$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "If you flip heads of a fair coin you get 2 points and if you flip tails you get 1 point. What is the probability that at some point you will have 1337 points?\r\n\r\nPut the the answer in the form of (a+bc^d) where a, b, c ,d are rational numbers.", "content_html": "If you flip heads of a fair coin you get 2 points and if you flip tails you get 1 point. What is the probability that at some point you will have 1337 points?<br>\n<br>\nPut the the answer in the form of (a+bc^d) where a, b, c ,d are rational numbers.", "post_id": 4410255, "post_number": 1, "post_time_unix": 1215634219, "post_time_utc": "2008-07-09 20:10:19 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "YAY!! another recurrence problem!!\r\n\r\nLet $ P_n$ denote the probability you will have $ n$ points. Then $ P_n \\equal{} \\frac {1}{2}(P_{n \\minus{} 1} \\plus{} P_{n \\minus{} 2})$. We know $ P_1 \\equal{} \\frac {1}{2}$ and $ P_2\\equal{}\\frac{3}{4}.$", "content_html": "YAY!! another recurrence problem!!<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/f/6/c/f6c19c7a065e856f8d3196d4a3b02b745094ddbc.png\" class=\"latex\" alt=\"$ P_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > denote the probability you will have <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > points. Then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/1/e/61e6fbd1840404eee960a540c2af246e61aceeab.png\" class=\"latex\" alt=\"$ P_n = \\frac {1}{2}(P_{n - 1} + P_{n - 2})$\" style=\"vertical-align: -12px\" width=\"168\" height=\"37\" >.</span> We know <img src=\"//latex.artofproblemsolving.com/c/3/2/c32fc5e84025bb99b81dec90c401af405522818a.png\" class=\"latex\" alt=\"$ P_1 = \\frac {1}{2}$\" style=\"vertical-align: -12px\" width=\"55\" height=\"37\" > and <img src=\"//latex.artofproblemsolving.com/3/c/a/3cae5ad79350556cdeebfdee78949c86d85c3532.png\" class=\"latex\" alt=\"$ P_2=\\frac{3}{4}.$\" style=\"vertical-align: -13px\" width=\"60\" height=\"38\" >", "post_id": 4410256, "post_number": 2, "post_time_unix": 1215634978, "post_time_utc": "2008-07-09 20:22:58 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "$ P_2 \\neq \\frac{1}{2}$\r\n\r\n$ P_1\\equal{}\\frac{1}{2}, P_2\\equal{}\\frac{1}{2}\\plus{}\\frac{1}{2}\\cdot\\frac{1}{2}$", "content_html": "<img src=\"//latex.artofproblemsolving.com/7/6/d/76d30049b120e2e439c73ddb1d57b60bcd212fa3.png\" class=\"latex\" alt=\"$ P_2 \\neq \\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"55\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/3/6/2362fa26315765e9273be4fd4e5c89d634c6a8a9.png\" class=\"latex\" alt=\"$ P_1=\\frac{1}{2}, P_2=\\frac{1}{2}+\\frac{1}{2}\\cdot\\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"181\" height=\"37\" >", "post_id": 4410257, "post_number": 3, "post_time_unix": 1215635370, "post_time_utc": "2008-07-09 20:29:30 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" }, { "attachments": [], "content_bbcode": "I don't get why he wants us to put it in the form of $ a\\plus{}bc^d$", "content_html": "I don't get why he wants us to put it in the form of <img src=\"//latex.artofproblemsolving.com/3/6/e/36e326322c91dd0f0499bfc6bb96471b057b5dca.png\" class=\"latex\" alt=\"$ a+bc^d$\" style=\"vertical-align: -1px\" width=\"55\" height=\"17\" >", "post_id": 4410258, "post_number": 4, "post_time_unix": 1215635865, "post_time_utc": "2008-07-09 20:37:45 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "I left out 1337 my bad.", "content_html": "I left out 1337 my bad.", "post_id": 4410259, "post_number": 5, "post_time_unix": 1215635912, "post_time_utc": "2008-07-09 20:38:32 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "ohh i get it\r\n\r\nIs it $ \\frac{1}{2}\\plus{}\\frac{2^{1334}\\plus{}1}{2^{1337}}$?", "content_html": "ohh i get it<br>\n<br>\nIs it <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/a/f/7af1fa00fab6ea32b8f41e7194ec5294cee2e685.png\" class=\"latex\" alt=\"$ \\frac{1}{2}+\\frac{2^{1334}+1}{2^{1337}}$\" style=\"vertical-align: -12px\" width=\"105\" height=\"39\" >?</span>", "post_id": 4410260, "post_number": 6, "post_time_unix": 1215636376, "post_time_utc": "2008-07-09 20:46:16 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "No not quite", "content_html": "No not quite", "post_id": 4410261, "post_number": 7, "post_time_unix": 1215636606, "post_time_utc": "2008-07-09 20:50:06 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "$ \\frac{5}{8}\\plus{}\\left(\\frac{1}{2}\\right)^{1337}$", "content_html": "<img src=\"//latex.artofproblemsolving.com/0/4/5/045bbecc8de342fcc3b73752aa97eb46df58ea7a.png\" class=\"latex\" alt=\"$ \\frac{5}{8}+\\left(\\frac{1}{2}\\right)^{1337}$\" style=\"vertical-align: -17px\" width=\"100\" height=\"46\" >", "post_id": 4410262, "post_number": 8, "post_time_unix": 1215636760, "post_time_utc": "2008-07-09 20:52:40 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "[hide]Think sum of geometric sequences[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Think sum of geometric sequences</div>", "post_id": 4410263, "post_number": 9, "post_time_unix": 1215638464, "post_time_utc": "2008-07-09 21:21:04 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "I FINALLY GOT IT!!!!!!!!!\r\n\r\n$ \\frac {2}{3} \\minus{} \\frac {1}{3}\\left(\\frac {1}{2}\\right)^{1337}$\r\n\r\nyou wasted an hour of my life. anyway, very nice problem.", "content_html": "I FINALLY GOT IT!!!!!!!!!<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/4/5/045b3929ca229b2aae67a0e61f3c6e90ab9da4dc.png\" class=\"latex\" alt=\"$ \\frac {2}{3} - \\frac {1}{3}\\left(\\frac {1}{2}\\right)^{1337}$\" style=\"vertical-align: -17px\" width=\"116\" height=\"46\" ><br>\n<br>\nyou wasted an hour of my life. anyway, very nice problem.", "post_id": 4410264, "post_number": 10, "post_time_unix": 1215641474, "post_time_utc": "2008-07-09 22:11:14 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" }, { "attachments": [], "content_bbcode": "Nice\r\n\r\nYup that is the answer :D", "content_html": "Nice<br>\n<br>\nYup that is the answer <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 4410265, "post_number": 11, "post_time_unix": 1215643702, "post_time_utc": "2008-07-09 22:48:22 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" } ], "source": null }	If you flip heads of a fair coin you get $2$ points and if you flip tails you get $1$ point. What is the probability that at some point you will have $1337$ points? Put the answer in the form $(a+b\cdot c^{d})$ where $a,b,c,d$ are rational numbers.	[ "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence", "/Mathematics/ProbabilityandStatistics/Probability/CoinFlipping", "/Mathematics/ProbabilityandStatistics/Probability/CoinTossing", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]	Apply first-step analysis to obtain a linear recurrence for the hitting probability.	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aops_994346	Well see here... Define the sequence $ P_n$ as the probability of getting no consecutive heads in $ n$ flips. Now, $ \begin{align} P_0 & \equal{} 1 \\ P_1 & \equal{} 1 \\ P_2 & \equal{} \frac 34 \\ P_3 & \equal{} \frac 58 \\ P_4 & \equal{} \frac 12 \\ \end{align}$ Now, we ponder. Hmm... (:D)... OK, recursion wins here. If the first flip is heads, the next flip must be tails and remaining probability is the probability of getting no consecutive heads in $ n \minus{} 2$ flips. If the first filp is tails, then the remaining probability is the probability of getting no consecutive heads in $ n \minus{} 1$ flips. Equation time! $ \begin{align}P_n & \equal{} \frac 12\left(\frac 12\left(P_{n \minus{} 2}\right)\right) \plus{} \frac 12\left(P_{n \minus{} 1}\right) \\ & \equal{} \frac 14P_{n \minus{} 2} \plus{} \frac 12P_{n \minus{} 1}\end{align}$ Now according to my calculator... the answer is: (drumroll) [hide]$ \boxed {\frac 9{64}}$[/hide] Is recursion like your favorite subject or something? :D Edit: I'm also noticing something here... FIBONACCI NUMBERS!!! See, the numerators are going $ 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \dots$ when unsimplified and the denominators are going $ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \dots$. So... my conjecture is: $ P_n \equal{} \frac {F_{n \plus{} 1}}{2^n}$ Next problem: Prove it.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "If you flip a coin 10 times what is the probability that no two consecutive flips are both heads?", "content_html": "If you flip a coin 10 times what is the probability that no two consecutive flips are both heads?", "post_id": 4410280, "post_number": 1, "post_time_unix": 1215647931, "post_time_utc": "2008-07-09 23:58:51 UTC", "thanks_received": 1, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "Well see here...\r\n\r\nDefine the sequence $ P_n$ as the probability of getting no consecutive heads in $ n$ flips. Now,\r\n\r\n$ \\begin{align} P_0 & \\equal{} 1 \\\\\r\nP_1 & \\equal{} 1 \\\\\r\nP_2 & \\equal{} \\frac 34 \\\\\r\nP_3 & \\equal{} \\frac 58 \\\\\r\nP_4 & \\equal{} \\frac 12 \\\\\r\n\\end{align}$\r\n\r\nNow, we ponder. Hmm... (:D)... OK, recursion wins here.\r\n\r\nIf the first flip is heads, the next flip must be tails and remaining probability is the probability of getting no consecutive heads in $ n \\minus{} 2$ flips.\r\n\r\nIf the first filp is tails, then the remaining probability is the probability of getting no consecutive heads in $ n \\minus{} 1$ flips.\r\n\r\nEquation time!\r\n$ \\begin{align}P_n & \\equal{} \\frac 12\\left(\\frac 12\\left(P_{n \\minus{} 2}\\right)\\right) \\plus{} \\frac 12\\left(P_{n \\minus{} 1}\\right) \\\\\r\n& \\equal{} \\frac 14P_{n \\minus{} 2} \\plus{} \\frac 12P_{n \\minus{} 1}\\end{align}$\r\n\r\nNow according to my calculator... the answer is:\r\n\r\n(drumroll)\r\n\r\n[hide]$ \\boxed {\\frac 9{64}}$[/hide]\r\n\r\nIs recursion like your favorite subject or something? :D\r\n\r\nEdit: I'm also noticing something here... FIBONACCI NUMBERS!!! See, the numerators are going $ 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \\dots$ when unsimplified and the denominators are going $ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \\dots$. So... my conjecture is:\r\n\r\n$ P_n \\equal{} \\frac {F_{n \\plus{} 1}}{2^n}$\r\n\r\nNext problem: Prove it.", "content_html": "Well see here...<br>\n<br>\nDefine the sequence <img src=\"//latex.artofproblemsolving.com/f/6/c/f6c19c7a065e856f8d3196d4a3b02b745094ddbc.png\" class=\"latex\" alt=\"$ P_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > as the probability of getting no consecutive heads in <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > flips. Now,<br>\n<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align} P_0 & = 1 \\\\\nP_1 & = 1 \\\\\nP_2 & = \\frac 34 \\\\\nP_3 & = \\frac 58 \\\\\nP_4 & = \\frac 12 \\\\\n\\end{align}$</span><br>\n<br>\nNow, we ponder. Hmm... (<img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />)... OK, recursion wins here.<br>\n<br>\nIf the first flip is heads, the next flip must be tails and remaining probability is the probability of getting no consecutive heads in <img src=\"//latex.artofproblemsolving.com/7/4/c/74cb539f5a34b578494e936625b2e35bc81b454f.png\" class=\"latex\" alt=\"$ n - 2$\" width=\"41\" height=\"12\" > flips.<br>\n<br>\nIf the first filp is tails, then the remaining probability is the probability of getting no consecutive heads in <img src=\"//latex.artofproblemsolving.com/4/f/9/4f937580dc77cb10ce0fca8c7499d80177966a15.png\" class=\"latex\" alt=\"$ n - 1$\" style=\"vertical-align: 0px\" width=\"41\" height=\"12\" > flips.<br>\n<br>\nEquation time!<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align}P_n & = \\frac 12\\left(\\frac 12\\left(P_{n - 2}\\right)\\right) + \\frac 12\\left(P_{n - 1}\\right) \\\\\n& = \\frac 14P_{n - 2} + \\frac 12P_{n - 1}\\end{align}$</span><br>\n<br>\nNow according to my calculator... the answer is:<br>\n<br>\n(drumroll)<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/3/9/8/398a55b647b66b5107866f4262c2a3a5d36d246c.png\" class=\"latex\" alt=\"$ \\boxed {\\frac 9{64}}$\" style=\"vertical-align: -18px\" width=\"34\" height=\"48\" ></div><br>\n<br>\nIs recursion like your favorite subject or something? <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\nEdit: I'm also noticing something here... FIBONACCI NUMBERS!!! See, the numerators are going <img src=\"//latex.artofproblemsolving.com/b/0/f/b0ff688a0d75ff59b3e4dcb12e8013e9de77600d.png\" class=\"latex\" alt=\"$ 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \\dots$\" style=\"vertical-align: -3px\" width=\"272\" height=\"16\" > when unsimplified and the denominators are going <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/5/d/a5dd950c6db3571efc8de12c75e2cd2ba6dca200.png\" class=\"latex\" alt=\"$ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, \\dots$\" style=\"vertical-align: -3px\" width=\"317\" height=\"16\" >.</span> So... my conjecture is:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/8/3/983ecac83f2482daf6df333576e842981414f3e3.png\" class=\"latex\" alt=\"$ P_n = \\frac {F_{n + 1}}{2^n}$\" style=\"vertical-align: -12px\" width=\"85\" height=\"37\" ><br>\n<br>\nNext problem: Prove it.", "post_id": 4410281, "post_number": 2, "post_time_unix": 1215649475, "post_time_utc": "2008-07-10 00:24:35 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "Correct :D", "content_html": "Correct <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 4410282, "post_number": 3, "post_time_unix": 1215652173, "post_time_utc": "2008-07-10 01:09:33 UTC", "thanks_received": 1, "user_id": 20380, "username": "budi713" } ], "source": null }	If you flip a coin 10 times, what is the probability that no two consecutive flips are both heads?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence", "/Mathematics/ProbabilityandStatistics/Probability/CoinFlipping", "/Mathematics/ProbabilityandStatistics/Probability/CoinTossing" ]	Condition on the first flip to derive a recurrence for the probability of no consecutive heads.	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aops_994348	I got 60/13 too. Basically, you do blah, blah and blah to find that its is the area of a triangle with vertices $ (154,10)$, $ (120,24)$, and $ (144,60)$. Then you can easily shoelace(or shoestring w/e) it and get that the area is 780. But then because of blah you must divide by 169.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Given $ \\Delta ABC$ with $ AB \\equal{} 5$, $ AC \\equal{} 12$, and $ BC \\equal{} 13$, construct the altitude to side $ BC$. Construct the incircles of the two smaller triangles formed. Find the area of the triangle formed by point A and the two incenters. Sorry for the bad wording...", "content_html": "Given <img src=\"//latex.artofproblemsolving.com/6/a/4/6a4d7c60de90a8227f9a09b610eede2cb00b20eb.png\" class=\"latex\" alt=\"$ \\Delta ABC$\" width=\"57\" height=\"13\" > with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/9/079284195066f81d3a969192f7adcebcd0969e87.png\" class=\"latex\" alt=\"$ AB = 5$\" width=\"61\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/4/9/a49dd3be71624afcd7f0a96b501fed3d8f561d28.png\" class=\"latex\" alt=\"$ AC = 12$\" style=\"vertical-align: 0px\" width=\"69\" height=\"13\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/e/46e738c1a638bbb057e2946994e4603023a53cc0.png\" class=\"latex\" alt=\"$ BC = 13$\" style=\"vertical-align: 0px\" width=\"70\" height=\"13\" >,</span> construct the altitude to side <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/4/e3427a9945d3b1de6a7c0b6862f3228415ea8bca.png\" class=\"latex\" alt=\"$ BC$\" width=\"28\" height=\"12\" >.</span> Construct the incircles of the two smaller triangles formed. Find the area of the triangle formed by point A and the two incenters. Sorry for the bad wording...", "post_id": 4410284, "post_number": 1, "post_time_unix": 1215654378, "post_time_utc": "2008-07-10 01:46:18 UTC", "thanks_received": 1, "user_id": 37259, "username": "math154" }, { "attachments": [], "content_bbcode": "60/13 maybe", "content_html": "60/13 maybe", "post_id": 4410285, "post_number": 2, "post_time_unix": 1215656833, "post_time_utc": "2008-07-10 02:27:13 UTC", "thanks_received": 1, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "I got 60/13 too.\r\n\r\nBasically, you do blah, blah and blah to find that its is the area of a triangle with vertices $ (154,10)$, $ (120,24)$, and $ (144,60)$. Then you can easily shoelace(or shoestring w/e) it and get that the area is 780. But then because of blah you must divide by 169.", "content_html": "I got 60/13 too.<br>\n<br>\nBasically, you do blah, blah and blah to find that its is the area of a triangle with vertices <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/d/f/ddfe6993308cc4e977e7c254084908ca631c8dd2.png\" class=\"latex\" alt=\"$ (154,10)$\" style=\"vertical-align: -4px\" width=\"66\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/b/9/9b93c3b5152bfa864e569927b88051a393ab0d9b.png\" class=\"latex\" alt=\"$ (120,24)$\" style=\"vertical-align: -4px\" width=\"66\" height=\"18\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/1/3/61325a0ef1b0c5554dfdaa0bfc6cc79f2e153422.png\" class=\"latex\" alt=\"$ (144,60)$\" style=\"vertical-align: -4px\" width=\"66\" height=\"18\" >.</span> Then you can easily shoelace(or shoestring w/e) it and get that the area is 780. But then because of blah you must divide by 169.", "post_id": 4410286, "post_number": 3, "post_time_unix": 1215708184, "post_time_utc": "2008-07-10 16:43:04 UTC", "thanks_received": 1, "user_id": 39364, "username": "RunpengFAILS" } ], "source": null }	Given triangle $ABC$ with $AB=5$, $AC=12$, and $BC=13$. Construct the altitude from $A$ to side $BC$. Construct the incircles of the two smaller triangles formed by this altitude (i.e., the incircles of the right triangles into which the altitude divides $\triangle ABC$). Find the area of the triangle formed by vertex $A$ and the two incenters.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeometricConstruction", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Incenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/Cevians", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles", "/Mathematics/Geometry/PlaneGeometry/Triangles/TrianglePoints" ]	Locate the two incenters using the right‑triangle inradius formula and then compute the area of triangle A‑I₁‑I₂ with the shoelace method.	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aops_994355	the solutions are really ugly, but ill post the general idea Let $ a\equal{}\sqrt{x\plus{}1},b\equal{}\sqrt{y\minus{}1}$ Then $ a\plus{}b\equal{}60$, $ a^2\plus{}b^2\equal{}3344$ Square the first and subtract, $ ab\equal{}128$ Now you make a polynomial with roots of $ a$ and $ b$ $ t^2\minus{}60t\plus{}128\equal{}0$ Solve for t and then find $ x,y$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "This time i am early\r\n\r\nFind all real solutions so that $ x$ and $ y$ satisfy both $ \\sqrt {x \\plus{} 1} \\plus{} \\sqrt {y \\minus{} 1} \\equal{} 66 \\minus{} 6$ and $ x \\plus{} y \\plus{} 1 \\equal{} 1337 \\plus{} 2008$.", "content_html": "This time i am early<br>\n<br>\nFind all real solutions so that <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/8/9/b8959a2220db8bc60d06e50de97fb5e86756e0f8.png\" class=\"latex\" alt=\"$ y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > satisfy both <img src=\"//latex.artofproblemsolving.com/5/2/7/5277d49bc37cf7ee985d2f686684410f1bff80eb.png\" class=\"latex\" alt=\"$ \\sqrt {x + 1} + \\sqrt {y - 1} = 66 - 6$\" style=\"vertical-align: -5px\" width=\"211\" height=\"22\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/a/7/1a7475928df901986cc43cd3df8d3ab5db524eec.png\" class=\"latex\" alt=\"$ x + y + 1 = 1337 + 2008$\" style=\"vertical-align: -3px\" width=\"192\" height=\"16\" >.</span>", "post_id": 4410308, "post_number": 1, "post_time_unix": 1215729071, "post_time_utc": "2008-07-10 22:31:11 UTC", "thanks_received": 2, "user_id": 20380, "username": "budi713" }, { "attachments": [], "content_bbcode": "the solutions are really ugly, but ill post the general idea\r\n\r\nLet $ a\\equal{}\\sqrt{x\\plus{}1},b\\equal{}\\sqrt{y\\minus{}1}$\r\n\r\nThen $ a\\plus{}b\\equal{}60$, $ a^2\\plus{}b^2\\equal{}3344$\r\n\r\nSquare the first and subtract, $ ab\\equal{}128$\r\n\r\nNow you make a polynomial with roots of $ a$ and $ b$ $ t^2\\minus{}60t\\plus{}128\\equal{}0$\r\n\r\nSolve for t and then find $ x,y$", "content_html": "the solutions are really ugly, but ill post the general idea<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/9/f/0/9f076b0c2fc08836afe9939c0d9d1dfc76d80300.png\" class=\"latex\" alt=\"$ a=\\sqrt{x+1},b=\\sqrt{y-1}$\" style=\"vertical-align: -5px\" width=\"189\" height=\"22\" ><br>\n<br>\nThen <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/9/6/696dc3b9c84b750ff5129996fe5a22be558b0a88.png\" class=\"latex\" alt=\"$ a+b=60$\" style=\"vertical-align: -1px\" width=\"81\" height=\"14\" >,</span> <img src=\"//latex.artofproblemsolving.com/5/a/4/5a43a48e0e3d033c1e8bcb97675427685a5512a4.png\" class=\"latex\" alt=\"$ a^2+b^2=3344$\" style=\"vertical-align: -1px\" width=\"115\" height=\"16\" ><br>\n<br>\nSquare the first and subtract, <img src=\"//latex.artofproblemsolving.com/5/6/1/5615537bab7745d6b5168e54559fe6d091fb78c5.png\" class=\"latex\" alt=\"$ ab=128$\" style=\"vertical-align: 0px\" width=\"68\" height=\"13\" ><br>\n<br>\nNow you make a polynomial with roots of <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > <img src=\"//latex.artofproblemsolving.com/8/1/2/812cf9b05a8b59525ab19a76c359cd00340b3971.png\" class=\"latex\" alt=\"$ t^2-60t+128=0$\" style=\"vertical-align: -1px\" width=\"143\" height=\"16\" ><br>\n<br>\nSolve for t and then find <img src=\"//latex.artofproblemsolving.com/5/5/a/55a5b72276af86fdc51bbcd3686f01af40983101.png\" class=\"latex\" alt=\"$ x,y$\" style=\"vertical-align: -3px\" width=\"27\" height=\"11\" >", "post_id": 4410309, "post_number": 2, "post_time_unix": 1215935058, "post_time_utc": "2008-07-13 07:44:18 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" } ], "source": null }	Find all real solutions (x, y) satisfying \[ \sqrt{x+1}+\sqrt{y-1}=66-6 \] and \[ x+y+1=1337+2008. \]	[ "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial" ]	Introduce a = √(x+1), b = √(y-1) and use (a+b)^2 = a^2 + b^2 + 2ab to find ab, then solve the resulting quadratic for a and b.	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aops_994397	Yes you can. Just memorize one formula: The volume of a [url=http://mathworld.wolfram.com/SphericalSegment.html]spherical segment[/url] is given by \[ V \equal{} \frac 16 \pih(3a^2 \plus{} 3b^2 \plus{} h^2) \] where $ a$ and $ b$ are the two "radii" and $ h$ is the height of the segment. :D :D :D	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "2 parralel planes cut a sphere into 3 regions of equal volume find the ratio of the distance between them to the radius of the sphere\r\ni think i would know how to do this using calculus...", "content_html": "2 parralel planes cut a sphere into 3 regions of equal volume find the ratio of the distance between them to the radius of the sphere<br>\ni think i would know how to do this using calculus...", "post_id": 4410463, "post_number": 1, "post_time_unix": 1217465520, "post_time_utc": "2008-07-31 00:52:00 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "Yes you can. Just memorize one formula:\r\n\r\nThe volume of a [url=http://mathworld.wolfram.com/SphericalSegment.html]spherical segment[/url] is given by\r\n\\[ V \\equal{} \\frac 16 \\pih(3a^2 \\plus{} 3b^2 \\plus{} h^2)\r\n\\]\r\n\r\nwhere $ a$ and $ b$ are the two \"radii\" and $ h$ is the height of the segment.\r\n\r\n :D :D :D", "content_html": "Yes you can. Just memorize one formula:<br>\n<br>\nThe volume of a <a href=\"http://mathworld.wolfram.com/SphericalSegment.html\" class=\"bbcode_url\" target=\"_blank\">spherical segment</a> is given by<br>\n<pre class=\"aopscode-error aopscode-latex-error\">\\[ V = \\frac 16 \\pih(3a^2 + 3b^2 + h^2)\n\\]</pre><br>\n<br>\nwhere <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > are the two "radii" and <img src=\"//latex.artofproblemsolving.com/2/5/6/2568a6b6e1ee432e4cfaae0c22987046bc23038a.png\" class=\"latex\" alt=\"$ h$\" width=\"10\" height=\"12\" > is the height of the segment.<br>\n<br>\n<img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 4410464, "post_number": 2, "post_time_unix": 1217466107, "post_time_utc": "2008-07-31 01:01:47 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "And you derive that formula from calculus! :D\r\nWell if there was a non calc solution that would be a great question to put on a middle school test.", "content_html": "And you derive that formula from calculus! <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\nWell if there was a non calc solution that would be a great question to put on a middle school test.", "post_id": 4410465, "post_number": 3, "post_time_unix": 1217466189, "post_time_utc": "2008-07-31 01:03:09 UTC", "thanks_received": 2, "user_id": 40979, "username": "CA Math" }, { "attachments": [], "content_bbcode": "i am thinking about the upper region for a non calc soluion\r\nthe upper region is the subtraction of a conical region from the 3-d equivalent of a circular sector (is this called a spherical sector?).\r\nwe set this equal to 1/3 of the sphere and were done\r\n\r\nthe part im not sure of is finding the volume of the sector", "content_html": "i am thinking about the upper region for a non calc soluion<br>\nthe upper region is the subtraction of a conical region from the 3-d equivalent of a circular sector (is this called a spherical sector?).<br>\nwe set this equal to 1/3 of the sphere and were done<br>\n<br>\nthe part im not sure of is finding the volume of the sector", "post_id": 4410466, "post_number": 4, "post_time_unix": 1217472295, "post_time_utc": "2008-07-31 02:44:55 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "lol you spelled parallel wrong........", "content_html": "lol you spelled parallel wrong........", "post_id": 4410467, "post_number": 5, "post_time_unix": 1217518982, "post_time_utc": "2008-07-31 15:43:02 UTC", "thanks_received": 2, "user_id": 27938, "username": "cognos599" }, { "attachments": [], "content_bbcode": "ya usamo graders take off 8 poitns for that you know", "content_html": "ya usamo graders take off 8 poitns for that you know", "post_id": 4410468, "post_number": 6, "post_time_unix": 1217523211, "post_time_utc": "2008-07-31 16:53:31 UTC", "thanks_received": 2, "user_id": 37259, "username": "math154" }, { "attachments": [], "content_bbcode": "you know what? i sometimes write my d's as g's and vice versa, so parralel is mild compared to that", "content_html": "you know what? i sometimes write my d's as g's and vice versa, so parralel is mild compared to that", "post_id": 4410469, "post_number": 7, "post_time_unix": 1217530240, "post_time_utc": "2008-07-31 18:50:40 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "just use integration...\r\nwill be faster than trying to do without calc...", "content_html": "just use integration...<br>\nwill be faster than trying to do without calc...", "post_id": 4410470, "post_number": 8, "post_time_unix": 1217620082, "post_time_utc": "2008-08-01 19:48:02 UTC", "thanks_received": 2, "user_id": 36435, "username": "Poincare" } ], "source": null }	Two parallel planes cut a sphere into three regions of equal volume. Find the ratio of the distance between the outer two planes to the radius of the sphere.	[ "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals", "/Mathematics/Geometry/SolidGeometry/Spheres/Sphere", "/Mathematics/Geometry/SolidGeometry/Spheres/SphericalSegment", "/Mathematics/Geometry/SolidGeometry/Volume" ]	Apply the spherical segment volume formula to relate the cut heights to the equal third‑volume condition.	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aops_994401	$ R_{19} \equal{} \left(\frac{10^{19} \minus{} 1}{9}\right)$ is said to be prime (see[url=http://en.wikipedia.org/wiki/repunit]repunits[/url]). (Notice that all repdigits greater than 11, that are not repunits, are never prime; furthermore, it is necessary, though not sufficient, that the number of "repetitions" of 1 be prime for a certain repunit to be prime.)	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "prove or disprove:\r\nthere are only 5 repdigit primes in base 10\r\n\r\nrepdigit means consisting on only 1 digit such as 6666666\r\nthe 5 repdigit primes for sure are 2,3,5,7,11; the question asks if there are more", "content_html": "prove or disprove:<br>\nthere are only 5 repdigit primes in base 10<br>\n<br>\nrepdigit means consisting on only 1 digit such as 6666666<br>\nthe 5 repdigit primes for sure are 2,3,5,7,11; the question asks if there are more", "post_id": 4410489, "post_number": 1, "post_time_unix": 1217635578, "post_time_utc": "2008-08-02 00:06:18 UTC", "thanks_received": 1, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "$ R_{19} \\equal{} \\left(\\frac{10^{19} \\minus{} 1}{9}\\right)$ is said to be prime (see[url=http://en.wikipedia.org/wiki/repunit]repunits[/url]).\r\n\r\n(Notice that all repdigits greater than 11, that are not repunits, are never prime; furthermore, it is necessary, though not sufficient, that the number of \"repetitions\" of 1 be prime for a certain repunit to be prime.)", "content_html": "<img src=\"//latex.artofproblemsolving.com/d/b/e/dbed6869dc02afe8f45b7ccdb3819b6aeb99f4e4.png\" class=\"latex\" alt=\"$ R_{19} = \\left(\\frac{10^{19} - 1}{9}\\right)$\" style=\"vertical-align: -17px\" width=\"142\" height=\"43\" > is said to be prime (see<a href=\"http://en.wikipedia.org/wiki/repunit\" class=\"bbcode_url\" target=\"_blank\">repunits</a>).<br>\n<br>\n(Notice that all repdigits greater than 11, that are not repunits, are never prime; furthermore, it is necessary, though not sufficient, that the number of "repetitions" of 1 be prime for a certain repunit to be prime.)", "post_id": 4410490, "post_number": 2, "post_time_unix": 1217662207, "post_time_utc": "2008-08-02 07:30:07 UTC", "thanks_received": 1, "user_id": 42058, "username": "metafor" }, { "attachments": [], "content_bbcode": "lol the best prime factorizer i found said 1111111111111111111=5555555555555556002", "content_html": "lol the best prime factorizer i found said 1111111111111111111=5555555555555556002", "post_id": 4410491, "post_number": 3, "post_time_unix": 1217692798, "post_time_utc": "2008-08-02 15:59:58 UTC", "thanks_received": 1, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "Get a new program. XD", "content_html": "Get a new program. XD", "post_id": 4410492, "post_number": 4, "post_time_unix": 1217735557, "post_time_utc": "2008-08-03 03:52:37 UTC", "thanks_received": 1, "user_id": 42058, "username": "metafor" } ], "source": null }	Prove or disprove: There are only five repdigit primes in base 10. A repdigit is a number consisting of only one repeated decimal digit (for example, 6666666). The five repdigit primes are 2, 3, 5, 7, and 11; determine whether any other repdigit primes exist.	[ "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberSequences", "/Mathematics/RecreationalMathematics" ]	Express any repdigit > 11 as a digit multiplied by a repunit, proving it’s composite unless it is a repunit, and note repunit primes require a prime number of digits.	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aops_994413	how about sum(x) where sum(x)=the sum of the first x natural numbers btw $ \frac {((\sqrt9)!)!}{9} \minus{} 9 \minus{} \sqrt9 \equal{} 68$ $ \frac {((\sqrt9)!)!}{9} \minus{} 9 \minus{} (\sqrt9)! \equal{} 65$ $ 9(\sqrt9)! \plus{} 9 \plus{} \sqrt9 \equal{} 66$ $ ((\sqrt9)\minus{}.\overline9)^{(\sqrt9)!}\plus{}\sqrt9$ beasted!	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "okay so like using 4 9's make the numbers 65,66,67,68\r\n\r\nYou can use like nearly everything\r\n\r\n$ \\plus{} , \\minus{} , /, , \\sqrt {\\text{blah}}, !,\\ \\overline{.9} \\text{\\ rofl}, \\text{exponents}$\r\n\r\nGood luck have fun\r\n\r\nALL ON ARIZONA!!! (civil war)\r\n\r\nARIZONA?! what about wyoming, they are next on your list\r\nthey were already at 9\r\nMISSOURI GOT OWNED", "content_html": "okay so like using 4 9's make the numbers 65,66,67,68<br>\n<br>\nYou can use like nearly everything<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/2/c/e2ce38bbb08e2c301a205bc46ff5361e01764fea.png\" class=\"latex\" alt=\"$ + , - , /, , \\sqrt {\\text{blah}}, !,\\ \\overline{.9} \\text{\\ rofl}, \\text{exponents}$\" style=\"vertical-align: -4px\" width=\"276\" height=\"21\" ><br>\n<br>\nGood luck have fun<br>\n<br>\nALL ON ARIZONA!!! (civil war)<br>\n<br>\nARIZONA?! what about wyoming, they are next on your list<br>\nthey were already at 9<br>\nMISSOURI GOT OWNED", "post_id": 4410541, "post_number": 1, "post_time_unix": 1218146563, "post_time_utc": "2008-08-07 22:02:43 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" }, { "attachments": [], "content_bbcode": "how about sum(x) where sum(x)=the sum of the first x natural numbers\r\n\r\nbtw\r\n$ \\frac {((\\sqrt9)!)!}{9} \\minus{} 9 \\minus{} \\sqrt9 \\equal{} 68$\r\n$ \\frac {((\\sqrt9)!)!}{9} \\minus{} 9 \\minus{} (\\sqrt9)! \\equal{} 65$\r\n$ 9(\\sqrt9)! \\plus{} 9 \\plus{} \\sqrt9 \\equal{} 66$\r\n$ ((\\sqrt9)\\minus{}.\\overline9)^{(\\sqrt9)!}\\plus{}\\sqrt9$\r\n\r\n\r\nbeasted!", "content_html": "how about sum(x) where sum(x)=the sum of the first x natural numbers<br>\n<br>\nbtw<br>\n<img src=\"//latex.artofproblemsolving.com/1/4/1/1413743d3b66698e03ae6f2fb77267667741e878.png\" class=\"latex\" alt=\"$ \\frac {((\\sqrt9)!)!}{9} - 9 - \\sqrt9 = 68$\" style=\"vertical-align: -12px\" width=\"186\" height=\"41\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/d/2/fd2ba099612697b8ba1411b88e68549b7f1c5359.png\" class=\"latex\" alt=\"$ \\frac {((\\sqrt9)!)!}{9} - 9 - (\\sqrt9)! = 65$\" style=\"vertical-align: -12px\" width=\"205\" height=\"41\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/5/6/7565c8d80608ace5ad04eff6da0c5905faae5fb3.png\" class=\"latex\" alt=\"$ 9(\\sqrt9)! + 9 + \\sqrt9 = 66$\" style=\"vertical-align: -4px\" width=\"172\" height=\"21\" ><br>\n<img src=\"//latex.artofproblemsolving.com/d/5/6/d56464f6c26141076f71013e11cae914458204be.png\" class=\"latex\" alt=\"$ ((\\sqrt9)-.\\overline9)^{(\\sqrt9)!}+\\sqrt9$\" style=\"vertical-align: -4px\" width=\"168\" height=\"22\" ><br>\n<br>\n<br>\nbeasted!", "post_id": 4410542, "post_number": 2, "post_time_unix": 1218151885, "post_time_utc": "2008-08-07 23:31:25 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "stevenmeow is a beast. This has been a public shaming of anyone not named stevenmeow, thank you.", "content_html": "stevenmeow is a beast. This has been a public shaming of anyone not named stevenmeow, thank you.", "post_id": 4410543, "post_number": 3, "post_time_unix": 1218154078, "post_time_utc": "2008-08-08 00:07:58 UTC", "thanks_received": 2, "user_id": 18870, "username": "stupidityismygam" }, { "attachments": [], "content_bbcode": "stupidityismygame shamed himself. This has been a public shaming of stupidityismygame. Gracias.", "content_html": "stupidityismygame shamed himself. This has been a public shaming of stupidityismygame. Gracias.", "post_id": 4410544, "post_number": 4, "post_time_unix": 1218154912, "post_time_utc": "2008-08-08 00:21:52 UTC", "thanks_received": 2, "user_id": 39364, "username": "RunpengFAILS" } ], "source": null }	Using four 9's, form the numbers 65, 66, 67, and 68. You may use the operations +, −, ×, ÷, exponents, factorial, square roots, decimal repetition (e.g. $\overline{.9}$), and parentheses. Provide one expression for each target number, each expression using exactly four 9s.	[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Use √9 = 3, apply factorial twice to obtain 6! = 720, divide by 9 to get 80, then add or subtract 9, √9 or .̅9 (which equals 1) to produce 65‑68.	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aops_994702	i dont think it is integral... oh well if $ a^3\plus{}b^3\equal{}(a\plus{}b)^3$, then $ 0\equal{}3ab^2\plus{}3ba^2 \implies (a\plus{}b)ab\equal{}0$ so either a, b, or a+b is 0 i consider this solved	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "What is the sum of all x’s such that the equation \r\n\r\n(2x+1)^3+(3x)^3 = (5x+1)^3\r\n\r\nis true? Express your answer as a common fraction.", "content_html": "What is the sum of all x’s such that the equation<br>\n<br>\n(2x+1)^3+(3x)^3 = (5x+1)^3<br>\n<br>\nis true? Express your answer as a common fraction.", "post_id": 4411027, "post_number": 1, "post_time_unix": 1225234869, "post_time_utc": "2008-10-28 23:01:09 UTC", "thanks_received": 2, "user_id": 40979, "username": "CA Math" }, { "attachments": [], "content_bbcode": "FLT :!: :!: :!:", "content_html": "FLT <img src=\"/assets/images/smilies/exclaim.gif\" width=\"19\" height=\"19\" alt=\":!:\" title=\":!:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/exclaim.gif\" width=\"19\" height=\"19\" alt=\":!:\" title=\":!:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/exclaim.gif\" width=\"19\" height=\"19\" alt=\":!:\" title=\":!:\" class=\"bbcode_smiley\" />", "post_id": 4411028, "post_number": 2, "post_time_unix": 1225236093, "post_time_utc": "2008-10-28 23:21:33 UTC", "thanks_received": 2, "user_id": 40880, "username": "leoxnlin" }, { "attachments": [], "content_bbcode": "i dont think it is integral...\r\noh well\r\nif $ a^3\\plus{}b^3\\equal{}(a\\plus{}b)^3$, then $ 0\\equal{}3ab^2\\plus{}3ba^2 \\implies (a\\plus{}b)ab\\equal{}0$\r\nso either a, b, or a+b is 0\r\ni consider this solved", "content_html": "i dont think it is integral...<br>\noh well<br>\nif <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/6/e/06e279a34b7c169f2be90660464dca5572360c90.png\" class=\"latex\" alt=\"$ a^3+b^3=(a+b)^3$\" style=\"vertical-align: -4px\" width=\"138\" height=\"19\" >,</span> then <img src=\"//latex.artofproblemsolving.com/6/5/0/6500de64d3c613b1238df88764a2d284036c0de1.png\" class=\"latex\" alt=\"$ 0=3ab^2+3ba^2 \\implies (a+b)ab=0$\" style=\"vertical-align: -4px\" width=\"277\" height=\"19\" ><br>\nso either a, b, or a+b is 0<br>\ni consider this solved", "post_id": 4411029, "post_number": 3, "post_time_unix": 1225239650, "post_time_utc": "2008-10-29 00:20:50 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" } ], "source": null }	What is the sum of all $x$ such that the equation \[ (2x+1)^3+(3x)^3=(5x+1)^3 \] is true? Express your answer as a common fraction.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/CubicEquation", "/Mathematics/Algebra/Polynomials/CubicEquation", "/Mathematics/Algebra/Polynomials/CubicPolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/Algebra/Polynomials/Variable" ]	Use the fact that a^3+b^3=(a+b)^3 implies a·b·(a+b)=0, then apply it to a=2x+1 and b=3x.	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aops_994744	Ok so umm I don't think $ \frac{20 \pi}{3}$ Is the right answer because if you think about it, the area from the top (looking down from a point at, say, (0,0,10) is going to be $ 4 \pi$ (circle with radius 2) and same for the bottom, so the surface area should be greater than $ 8 \pi$...umm actually, i'll go sit down with a crayon and a whiteboard and think about it some more.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "find the surface area of the solid $ 4x^2 \\plus{} 4y^2 \\plus{} z^2 \\le 4$\r\nok my approach\r\na logical (but not necessarily correct) approach is to derive (derivate?) the formula for the volume\r\nso we want the surface area of a sphere vertically distorted by a factor of 2\r\nlets try derivating $ \\frac {4\\pi}{3}r^2 \\cdot 2r$ in a speeciale way\r\n$ \\lim_{h \\rightarrow 0} \\frac {4\\pi\\left((x \\plus{} h)^2(2x \\plus{} h) \\minus{} x^22x\\right)}{3h}$\r\nwhich yields $ \\frac {4\\pi}{3}5x^2$\r\nso for a smaller radius of length 1, the surface area is $ \\frac {20\\pi}{3}$\r\nwhich makes me doubt myself\r\nif noone solves this, im posting this on the tx private forum", "content_html": "find the surface area of the solid <img src=\"//latex.artofproblemsolving.com/6/2/d/62d6171d3c873b669b2980279734ce34c3d7f316.png\" class=\"latex\" alt=\"$ 4x^2 + 4y^2 + z^2 \\le 4$\" style=\"vertical-align: -3px\" width=\"147\" height=\"18\" ><br>\nok my approach<br>\na logical (but not necessarily correct) approach is to derive (derivate?) the formula for the volume<br>\nso we want the surface area of a sphere vertically distorted by a factor of 2<br>\nlets try derivating <img src=\"//latex.artofproblemsolving.com/0/6/7/0679ab5f07b96205b3c5105b4ee741fe43e623ed.png\" class=\"latex\" alt=\"$ \\frac {4\\pi}{3}r^2 \\cdot 2r$\" style=\"vertical-align: -12px\" width=\"70\" height=\"37\" > in a speeciale way<br>\n<img src=\"//latex.artofproblemsolving.com/b/1/2/b121c088616e1c2c837c3f6c038e54784a404a69.png\" class=\"latex\" alt=\"$ \\lim_{h \\rightarrow 0} \\frac {4\\pi\\left((x + h)^2(2x + h) - x^22x\\right)}{3h}$\" style=\"vertical-align: -12px\" width=\"260\" height=\"39\" ><br>\nwhich yields <img src=\"//latex.artofproblemsolving.com/b/1/3/b138dc43a6c30ca320f38d5aea57b7c1bf9454ef.png\" class=\"latex\" alt=\"$ \\frac {4\\pi}{3}5x^2$\" style=\"vertical-align: -12px\" width=\"49\" height=\"37\" ><br>\nso for a smaller radius of length 1, the surface area is <img src=\"//latex.artofproblemsolving.com/e/3/7/e3798d52452a2c9238c4b6dd34f53275f88c22c9.png\" class=\"latex\" alt=\"$ \\frac {20\\pi}{3}$\" style=\"vertical-align: -12px\" width=\"31\" height=\"37\" ><br>\nwhich makes me doubt myself<br>\nif noone solves this, im posting this on the tx private forum", "post_id": 4411133, "post_number": 1, "post_time_unix": 1226634059, "post_time_utc": "2008-11-14 03:40:59 UTC", "thanks_received": 1, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "derivate can only be a noun or an adjective, so it should be derive.\r\n\r\nI'm still trying to think of how to solve this...I haven't had much experience in surface area.", "content_html": "derivate can only be a noun or an adjective, so it should be derive.<br>\n<br>\nI'm still trying to think of how to solve this...I haven't had much experience in surface area.", "post_id": 4411134, "post_number": 2, "post_time_unix": 1226701665, "post_time_utc": "2008-11-14 22:27:45 UTC", "thanks_received": 1, "user_id": 40979, "username": "CA Math" }, { "attachments": [], "content_bbcode": "Ok so umm I don't think $ \\frac{20 \\pi}{3}$ Is the right answer because if you think about it, the area from the top (looking down from a point at, say, (0,0,10) is going to be $ 4 \\pi$ (circle with radius 2) and same for the bottom, so the surface area should be greater than $ 8 \\pi$...umm actually, i'll go sit down with a crayon and a whiteboard and think about it some more.", "content_html": "Ok so umm I don't think <img src=\"//latex.artofproblemsolving.com/1/b/0/1b076f4b5517c44fba7cca0c45311c3a6134bf32.png\" class=\"latex\" alt=\"$ \\frac{20 \\pi}{3}$\" style=\"vertical-align: -12px\" width=\"31\" height=\"37\" > Is the right answer because if you think about it, the area from the top (looking down from a point at, say, (0,0,10) is going to be <img src=\"//latex.artofproblemsolving.com/4/0/b/40b36805fb540d97906db18b2639f510b3ec9d33.png\" class=\"latex\" alt=\"$ 4 \\pi$\" style=\"vertical-align: 0px\" width=\"19\" height=\"12\" > (circle with radius 2) and same for the bottom, so the surface area should be greater than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/2/7/7278b0ae7d7c09a185d84a3a976a2c08c9d6c5b0.png\" class=\"latex\" alt=\"$ 8 \\pi$\" width=\"19\" height=\"12\" >.</span>..umm actually, i'll go sit down with a crayon and a whiteboard and think about it some more.", "post_id": 4411135, "post_number": 3, "post_time_unix": 1226702274, "post_time_utc": "2008-11-14 22:37:54 UTC", "thanks_received": 1, "user_id": 40979, "username": "CA Math" }, { "attachments": [], "content_bbcode": "wrong! its two circles both with radius 1 so the area is t least 2 pi not 8 pi", "content_html": "wrong! its two circles both with radius 1 so the area is t least 2 pi not 8 pi", "post_id": 4411136, "post_number": 4, "post_time_unix": 1226703646, "post_time_utc": "2008-11-14 23:00:46 UTC", "thanks_received": 1, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "\"Differentiate\"", "content_html": ""Differentiate"", "post_id": 4411137, "post_number": 5, "post_time_unix": 1227219907, "post_time_utc": "2008-11-20 22:25:07 UTC", "thanks_received": 1, "user_id": 45289, "username": "dysfunctionalequations" } ], "source": null }	Find the surface area of the solid \[ 4x^2+4y^2+z^2\le 4. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/DefiniteIntegrals", "/Mathematics/CalculusandAnalysis/Calculus/MultivariableCalculus/MultipleIntegral", "/Mathematics/CalculusandAnalysis/Calculus/MultivariableCalculus/MultivariateCalculus", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry", "/Mathematics/Geometry/SolidGeometry/Ellipsoids/Ellipsoid", "/Mathematics/Geometry/SolidGeometry/Ellipsoids/ProlateSpheroid", "/Mathematics/Geometry/SolidGeometry/Ellipsoids/Spheroid", "/Mathematics/Geometry/SolidGeometry/Spheroids/ProlateSpheroid", "/Mathematics/Geometry/Surfaces/AlgebraicSurfaces/AlgebraicSurface", "/Mathematics/Geometry/Surfaces/AlgebraicSurfaces/QuadraticSurface", "/Mathematics/Geometry/Surfaces/AlgebraicSurfaces/Quadric", "/Mathematics/Geometry/Surfaces/ClosedSurfaces/Ellipsoid", "/Mathematics/Geometry/Surfaces/ClosedSurfaces/Spheroid", "/Mathematics/Geometry/Surfaces/SurfacesofRevolution/ProlateSpheroid" ]	Recognize the region as a spheroid (ellipsoid of revolution) and apply the known surface‑area formula for an ellipsoid with two equal radii.	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aops_994850	this is harder than ithought... ( <----> these doesn't count) $ \because \text{Binomial Thm.}$ $ \therefore \lim_{k \rightarrow \infty} \left(1\plus{}k^{\minus{}1} \right)^k$ $ \equal{}\lim_{k \rightarrow \infty} \sum^{\infty}_{a\equal{}0} \binom{k}{a} k^{\minus{}a}$ $ \equal{}\lim_{k \rightarrow \infty} \sum^{\infty}_{a\equal{}0} \frac{\binom{k}{a}}{k^a}$ $ \equal{}1\plus{} \lim_{k \rightarrow \infty} \sum^{\infty}_{a\equal{}1} \frac{\binom{k}{a}}{k^a}$ $ \because$ $ \because \lim_{k \rightarrow \infty} \frac{\binom{k}{a}}{k^a}\equal{}\lim_{k \rightarrow \infty} \frac{\frac{\prod^{k}_{b\equal{}k\minus{}a\plus{}1} b}{a!}}{k^a}\equal{}\lim_{k \rightarrow \infty} \frac{\prod^{k}_{b\equal{}k\minus{}a\plus{}1} \frac{b}{k}}{a!} \equal{} \left( a! \right)^{\minus{}1}$ $ \therefore 1\plus{} \lim_{k \rightarrow \infty} \sum^{\infty}_{a\equal{}1} \frac{\binom{k}{a}}{k^a}\equal{}\sum^{\infty}_{a\equal{}0} \left(a! \right)^{\minus{}1}\equal{}\sum^{\infty}_{k\equal{}0} \left( a! \right)^{\minus{}1}$ $ \therefore \lim_{k \rightarrow \infty} \left(1\plus{}k^{\minus{}1} \right)^k\equal{}\sum^{\infty}_{k\equal{}0} \left( a! \right)^{\minus{}1}$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "uhh yeah\r\nso first, pythag verified (almost) an idea for solving the problem below is as follows: (on the post b4 this)\r\nthen, i solved a normal PoTD\r\nthen, i emailed 3 ppl on gaggle\r\nthen, i found this person's phone number\r\nthen, i won a few FTW games\r\n(this is probably not in order)\r\ntime for some math content?\r\nprove the following equality without using the letter \"e\"\r\n$ \\sum^{\\infty}_{k \\equal{} 0} \\left( k! \\right)^{ \\minus{} 1} \\equal{} \\lim_{k \\rightarrow \\infty} \\left( 1 \\plus{} k^{ \\minus{} 1} \\right)^k$\r\n\r\nNO TYPOES ARE ALLOWED\r\nexpecially if it's really obvious, like\r\nbinomial thirum kills this", "content_html": "uhh yeah<br>\nso first, pythag verified (almost) an idea for solving the problem below is as follows: (on the post b4 this)<br>\nthen, i solved a normal PoTD<br>\nthen, i emailed 3 ppl on gaggle<br>\nthen, i found this person's phone number<br>\nthen, i won a few FTW games<br>\n(this is probably not in order)<br>\ntime for some math content?<br>\nprove the following equality without using the letter "e"<br>\n<img src=\"//latex.artofproblemsolving.com/c/1/4/c14a4b80564ffc7c2316a250735ff7a17ad3b1c1.png\" class=\"latex\" alt=\"$ \\sum^{\\infty}_{k = 0} \\left( k! \\right)^{ - 1} = \\lim_{k \\rightarrow \\infty} \\left( 1 + k^{ - 1} \\right)^k$\" style=\"vertical-align: -20px\" width=\"219\" height=\"48\" ><br>\n<br>\nNO TYPOES ARE ALLOWED<br>\nexpecially if it's really obvious, like<br>\nbinomial thirum kills this", "post_id": 4411399, "post_number": 1, "post_time_unix": 1230692033, "post_time_utc": "2008-12-31 02:53:53 UTC", "thanks_received": 1, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "You forgot tag again :huh:", "content_html": "You forgot tag again <img src=\"/assets/images/smilies/huh.gif\" width=\"20\" height=\"20\" alt=\":huh:\" title=\":huh:\" class=\"bbcode_smiley\" />", "post_id": 4411400, "post_number": 2, "post_time_unix": 1230692252, "post_time_utc": "2008-12-31 02:57:32 UTC", "thanks_received": 1, "user_id": 37710, "username": "shtsxc12" }, { "attachments": [], "content_bbcode": "binom theorem kills this.", "content_html": "binom theorem kills this.", "post_id": 4411401, "post_number": 3, "post_time_unix": 1230693635, "post_time_utc": "2008-12-31 03:20:35 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "incorrectt proof\r\nthe word \"theorem\" contains the letter \"e\" multiple times.\r\ntry again.\r\ntytia", "content_html": "incorrectt proof<br>\nthe word "theorem" contains the letter "e" multiple times.<br>\ntry again.<br>\ntytia", "post_id": 4411402, "post_number": 4, "post_time_unix": 1230693809, "post_time_utc": "2008-12-31 03:23:29 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "binom thorum kills this.\r\n\r\nQ. M.D.\r\n\r\nhalmostsymber.", "content_html": "binom thorum kills this.<br>\n<br>\nQ. M.D.<br>\n<br>\nhalmostsymber.", "post_id": 4411403, "post_number": 5, "post_time_unix": 1230694627, "post_time_utc": "2008-12-31 03:37:07 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "Full solution:\r\n\r\nLook at pythag011's proof.", "content_html": "Full solution:<br>\n<br>\nLook at pythag011's proof.", "post_id": 4411404, "post_number": 6, "post_time_unix": 1230734935, "post_time_utc": "2008-12-31 14:48:55 UTC", "thanks_received": 2, "user_id": 45289, "username": "dysfunctionalequations" }, { "attachments": [], "content_bbcode": "this is harder than ithought... ( <----> these doesn't count)\r\n\r\n$ \\because \\text{Binomial Thm.}$\r\n$ \\therefore \\lim_{k \\rightarrow \\infty} \\left(1\\plus{}k^{\\minus{}1} \\right)^k$\r\n$ \\equal{}\\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a\\equal{}0} \\binom{k}{a} k^{\\minus{}a}$\r\n$ \\equal{}\\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a\\equal{}0} \\frac{\\binom{k}{a}}{k^a}$\r\n$ \\equal{}1\\plus{} \\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a\\equal{}1} \\frac{\\binom{k}{a}}{k^a}$\r\n\r\n$ \\because$\r\n$ \\because \\lim_{k \\rightarrow \\infty} \\frac{\\binom{k}{a}}{k^a}\\equal{}\\lim_{k \\rightarrow \\infty} \\frac{\\frac{\\prod^{k}_{b\\equal{}k\\minus{}a\\plus{}1} b}{a!}}{k^a}\\equal{}\\lim_{k \\rightarrow \\infty} \\frac{\\prod^{k}_{b\\equal{}k\\minus{}a\\plus{}1} \\frac{b}{k}}{a!} \\equal{} \\left( a! \\right)^{\\minus{}1}$\r\n$ \\therefore 1\\plus{} \\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a\\equal{}1} \\frac{\\binom{k}{a}}{k^a}\\equal{}\\sum^{\\infty}_{a\\equal{}0} \\left(a! \\right)^{\\minus{}1}\\equal{}\\sum^{\\infty}_{k\\equal{}0} \\left( a! \\right)^{\\minus{}1}$\r\n\r\n$ \\therefore \\lim_{k \\rightarrow \\infty} \\left(1\\plus{}k^{\\minus{}1} \\right)^k\\equal{}\\sum^{\\infty}_{k\\equal{}0} \\left( a! \\right)^{\\minus{}1}$", "content_html": "this is harder than ithought... ( <----> these doesn't count)<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/9/0/b/90ba09f69e67b8794a0dd644ea8c483286b36a99.png\" class=\"latex\" alt=\"$ \\because \\text{Binomial Thm.}$\" style=\"vertical-align: -1px\" width=\"126\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/9/4/3/9438e022e1d2079e564378dc5020abf4910bc98a.png\" class=\"latex\" alt=\"$ \\therefore \\lim_{k \\rightarrow \\infty} \\left(1+k^{-1} \\right)^k$\" style=\"vertical-align: -11px\" width=\"136\" height=\"30\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/7/4/c74144b4c4d87058a35f8d964540bcb2ed6ed559.png\" class=\"latex\" alt=\"$ =\\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a=0} \\binom{k}{a} k^{-a}$\" style=\"vertical-align: -22px\" width=\"150\" height=\"53\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/e/0/7e065ff18a5cce5be214f861f462644eaa53bbb2.png\" class=\"latex\" alt=\"$ =\\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a=0} \\frac{\\binom{k}{a}}{k^a}$\" style=\"vertical-align: -20px\" width=\"110\" height=\"50\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/0/9/509c28eefa1cb6accd2f0673e541f91c9dbb004d.png\" class=\"latex\" alt=\"$ =1+ \\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a=1} \\frac{\\binom{k}{a}}{k^a}$\" style=\"vertical-align: -20px\" width=\"142\" height=\"50\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/7/1/f71a3ac923b457c97fd63fe89a60eeb954017bc0.png\" class=\"latex\" alt=\"$ \\because$\" style=\"vertical-align: -1px\" width=\"11\" height=\"10\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/0/1/c015f2684a8f4f71773f951ab6522aa638275ab8.png\" class=\"latex\" alt=\"$ \\because \\lim_{k \\rightarrow \\infty} \\frac{\\binom{k}{a}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\frac{\\prod^{k}_{b=k-a+1} b}{a!}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\prod^{k}_{b=k-a+1} \\frac{b}{k}}{a!} = \\left( a! \\right)^{-1}$\" style=\"vertical-align: -12px\" width=\"445\" height=\"47\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/c/a/3cae2bbdc14dfa165fad2d7910aa741f352b09aa.png\" class=\"latex\" alt=\"$ \\therefore 1+ \\lim_{k \\rightarrow \\infty} \\sum^{\\infty}_{a=1} \\frac{\\binom{k}{a}}{k^a}=\\sum^{\\infty}_{a=0} \\left(a! \\right)^{-1}=\\sum^{\\infty}_{k=0} \\left( a! \\right)^{-1}$\" style=\"vertical-align: -20px\" width=\"338\" height=\"50\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/1/0/d106f048c2367092bfd5dc4c64bee4f4ee7e8d52.png\" class=\"latex\" alt=\"$ \\therefore \\lim_{k \\rightarrow \\infty} \\left(1+k^{-1} \\right)^k=\\sum^{\\infty}_{k=0} \\left( a! \\right)^{-1}$\" style=\"vertical-align: -20px\" width=\"235\" height=\"48\" >", "post_id": 4411405, "post_number": 7, "post_time_unix": 1230754840, "post_time_utc": "2008-12-31 20:20:40 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "\\b[b][size=150]e[/size][/b]caus[b][size=200]e[/size][/b] \\lim_{k \\rightarrow \\infty} \\frac{\\binom{k}{a}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\frac{\\prod^{k}_{b=k-a+1} b}{a!}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\prod^{k}_{b=k-a+1} \\frac{b}{k}}{a!} = \\left( a! \\right)^{-1}\r\n\r\nFail.", "content_html": "\\b<b><span class=\"bbfont-one-five\">e</span></b>caus<b><span class=\"bbfont-double\">e</span></b> \\lim_{k \\rightarrow \\infty} \\frac{\\binom{k}{a}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\frac{\\prod^{k}_{b=k-a+1} b}{a!}}{k^a}=\\lim_{k \\rightarrow \\infty} \\frac{\\prod^{k}_{b=k-a+1} \\frac{b}{k}}{a!} = \\left( a! \\right)^{-1}<br>\n<br>\nFail.", "post_id": 4411406, "post_number": 8, "post_time_unix": 1231455082, "post_time_utc": "2009-01-08 22:51:22 UTC", "thanks_received": 2, "user_id": 45289, "username": "dysfunctionalequations" } ], "source": null }	Prove the following equality without using the letter "e": \[ \sum_{k=0}^{\infty}\frac{1}{k!}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/InfinitesimalAnalysis", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/InfinitesimalCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis", "/Mathematics/CalculusandAnalysis/Series/Convergence", "/Mathematics/CalculusandAnalysis/Series/GeneralSeries", "/Mathematics/CalculusandAnalysis/Series/SeriesExpansions", "/Mathematics/FoundationsofMathematics/MathematicalProblems/SolvedProblems/PowerSeries", "/Mathematics/FoundationsofMathematics/TheoremProving/FlawedProofs/Fallacy", "/Mathematics/FoundationsofMathematics/TheoremProving/Proofs/ElementaryProof", "/Mathematics/FoundationsofMathematics/TheoremProving/Proofs/Proof" ]	Expand (1+1/n)^n with the binomial theorem and observe that each term ↔ \frac{\binom{n}{k}}{n^k} tends to 1/k! as n\to\infty.	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aops_994903	Introduce an n+1th element. The probability that this new element is greater than max{first n elements} is equal to 1-max{first n elements}. So if we average over all sets then the average of max{first n elements}=1-P(the n+1th element is the greatest one), which is n/n+1.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "In terms of n, find the expected value of the maximum element of a list of n random numbers.\r\nIn this case, a ransom number is a randomly chosen number ranging from 0 to 1\r\nMy guess is $ \\frac {n}{n \\plus{} 1}$\r\nFor example, if n=0, the maximum element in an empty list is cough zero.\r\nIf n=1, the maximum element is just that random number which has expected value .5\r\ni used pretty intense calculus to find n=2\r\nlet's go!\r\nSuppose each random number is limited to .5 or 1.\r\nThe expected value for the maximum is $ \\frac78$\r\nWhen the random numbers are limited to $ \\frac13, \\frac23, 1$\r\nthen then expected value for the maximum is uhh 22/27?\r\nUsing intense guessing skills, i figured out that when the random numbers are limited to $ \\frac {1}{n}, \\frac {2}{n}, \\frac {3}{n} ... \\frac {n \\minus{} 1}{n}, 1$\r\nthe probability is $ \\frac {(n)(n \\plus{} 1)(4n \\minus{} 1)}{6n^3}$\r\nthis can be proven using deduction.", "content_html": "In terms of n, find the expected value of the maximum element of a list of n random numbers.<br>\nIn this case, a ransom number is a randomly chosen number ranging from 0 to 1<br>\nMy guess is <img src=\"//latex.artofproblemsolving.com/6/d/c/6dcd7b9178e1481906a658a9e498140b9ab3d8c3.png\" class=\"latex\" alt=\"$ \\frac {n}{n + 1}$\" style=\"vertical-align: -14px\" width=\"44\" height=\"34\" ><br>\nFor example, if n=0, the maximum element in an empty list is cough zero.<br>\nIf n=1, the maximum element is just that random number which has expected value .5<br>\ni used pretty intense calculus to find n=2<br>\nlet's go!<br>\nSuppose each random number is limited to .5 or 1.<br>\nThe expected value for the maximum is <img src=\"//latex.artofproblemsolving.com/9/1/3/913e503b67d9bab11dff1f8c67fe0df67767e1a2.png\" class=\"latex\" alt=\"$ \\frac78$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ><br>\nWhen the random numbers are limited to <img src=\"//latex.artofproblemsolving.com/a/8/5/a85dcb96065025c3df3ad8a0b4e4d60d0899d8df.png\" class=\"latex\" alt=\"$ \\frac13, \\frac23, 1$\" style=\"vertical-align: -12px\" width=\"50\" height=\"37\" ><br>\nthen then expected value for the maximum is uhh 22/27?<br>\nUsing intense guessing skills, i figured out that when the random numbers are limited to <img src=\"//latex.artofproblemsolving.com/b/a/6/ba653432053e46747e3c1b8533d8536ead15ea7c.png\" class=\"latex\" alt=\"$ \\frac {1}{n}, \\frac {2}{n}, \\frac {3}{n} ... \\frac {n - 1}{n}, 1$\" style=\"vertical-align: -12px\" width=\"138\" height=\"37\" ><br>\nthe probability is <img src=\"//latex.artofproblemsolving.com/6/3/7/637ce75c2a93de6ad178f1c4d7c6fb1688931b28.png\" class=\"latex\" alt=\"$ \\frac {(n)(n + 1)(4n - 1)}{6n^3}$\" style=\"vertical-align: -12px\" width=\"149\" height=\"38\" ><br>\nthis can be proven using deduction.", "post_id": 4411525, "post_number": 1, "post_time_unix": 1236120554, "post_time_utc": "2009-03-03 22:49:14 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "Heh, this is an integration problem.", "content_html": "Heh, this is an integration problem.", "post_id": 4411526, "post_number": 2, "post_time_unix": 1236121997, "post_time_utc": "2009-03-03 23:13:17 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "Introduce an n+1th element. The probability that this new element is greater than max{first n elements} is equal to 1-max{first n elements}. So if we average over all sets then the average of max{first n elements}=1-P(the n+1th element is the greatest one), which is n/n+1.", "content_html": "Introduce an n+1th element. The probability that this new element is greater than max{first n elements} is equal to 1-max{first n elements}. So if we average over all sets then the average of max{first n elements}=1-P(the n+1th element is the greatest one), which is n/n+1.", "post_id": 4411527, "post_number": 3, "post_time_unix": 1236122291, "post_time_utc": "2009-03-03 23:18:11 UTC", "thanks_received": 2, "user_id": 21169, "username": "perfect628" }, { "attachments": [], "content_bbcode": "However, there are not the same density of sets with all elements < 1/2 and all elemets < 3/4.", "content_html": "However, there are not the same density of sets with all elements < 1/2 and all elemets < 3/4.", "post_id": 4411528, "post_number": 4, "post_time_unix": 1236124064, "post_time_utc": "2009-03-03 23:47:44 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "Unoriginal problem alert.\r\n\r\nhttp://www.fashionablemathematician.com/PDFs/random_expectation.pdf\r\n\r\nNo doubt that is to be expected though. :arrow:", "content_html": "Unoriginal problem alert.<br>\n<br>\n<a target=\"_blank\" href=\"http://www.fashionablemathematician.com/PDFs/random_expectation.pdf\">http://www.fashionablemathematician.com/PDFs/random_expectation.pdf</a><br>\n<br>\nNo doubt that is to be expected though. <img src=\"/assets/images/smilies/icon2.gif\" width=\"19\" height=\"19\" alt=\":arrow:\" title=\":arrow:\" class=\"bbcode_smiley\" />", "post_id": 4411529, "post_number": 5, "post_time_unix": 1236125929, "post_time_utc": "2009-03-04 00:18:49 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "When i 'make up' a problem, fractal, or observation, i can say it without doubt because the insight is mine only and is 'new' to me.\r\nBecause i had no help in making this problem, this problem is still new, at least to me.", "content_html": "When i 'make up' a problem, fractal, or observation, i can say it without doubt because the insight is mine only and is 'new' to me.<br>\nBecause i had no help in making this problem, this problem is still new, at least to me.", "post_id": 4411530, "post_number": 6, "post_time_unix": 1236137435, "post_time_utc": "2009-03-04 03:30:35 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" } ], "source": null }	Suppose a list of n independent random numbers is drawn uniformly from the interval [0,1]. In terms of n, find the expected value of the maximum element of the list.	[ "/Mathematics/ProbabilityandStatistics/Probability/IndependentStatistics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace", "/Mathematics/ProbabilityandStatistics/RandomNumbers/RandomNumber", "/Mathematics/ProbabilityandStatistics/RandomNumbers/RandomVariable", "/Mathematics/ProbabilityandStatistics/RandomNumbers/UniformDistributionTheory", "/Mathematics/ProbabilityandStatistics/RandomNumbers/UniformVariate", "/Mathematics/ProbabilityandStatistics/RankStatistics/OrderStatistic", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/BetaDistribution", "/Mathematics/ProbabilityandStatistics/StatisticalDistributions/ContinuousDistributions/UniformDistribution" ]	Use symmetry by adding an extra uniform draw so each of the n+1 observations is equally likely to be the largest.	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aops_99508	[quote="fedja"]Seems we are forgetting olympiad classics :(: remember the problem about rectangles with one integer side and $e^{2\pi i(x+y)}$? ;)[/quote] Ooops, really seems we(at least me) forgot this classic. As Fedja said, if we integrate the function $e^{2\pi i(x+y)}$ over our big rectange(say it is $R$) the value of this integral is the same as the sum of integrals over little rectangles $R_{k}$, i.e $\int_{R}e^{2\pi i(x+y)}dx dy= \sum \int_{R_{k}}e^{2\pi i(x+y)}dx dy$ (), but $\int_{a}^{b}e^{2 \pi i x}dx = \frac{1}{2 i \pi}e^{2\pi i a}(e^{2\pi i (b-a)}-1)$ and taking module in () we get that $\mid (e^{2\pi i a_{0}}-1)(e^{2\pi i b_{0}}-1) \mid \leq \sum_{k=1}^{n}\mid (e^{2\pi i a_{k}}-1)(e^{2\pi i b_{k}}-1) \mid $ and now it is only left to notice that $\mid (e^{2\pi i t}-1)\mid = \sqrt{( \cos{2\pi t }-1)^{2}+(\sin{2\pi t})^{2}}= \sqrt{ 2-2\cos{2 \pi t}}= 2 \mid \sin{\pi t}\mid $.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Funny one:\r\nA rectangle with side lengths $a_0, b_0$ is dissected into smaller rectangulars with side lengths $a_k, b_k,\\, 1\\le k\\le n$. The sides of smaller rectangulars are parallel to the corresponding sides of the big rectangular. Prove that $\|\\sin a_0\\sin b_0\|\\le \\sum_{k=1}^n\|\\sin a_k \\sin b_k\|.$", "content_html": "Funny one:<br>\nA rectangle with side lengths <img src=\"//latex.artofproblemsolving.com/7/e/c/7ec4629c50fae59e075bd8587a0dde565fc8c5ba.png\" class=\"latex\" alt=\"$a_0, b_0$\" style=\"vertical-align: -3px\" width=\"39\" height=\"16\" > is dissected into smaller rectangulars with side lengths <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/3/843595164e6188650a19cedfd3e10839c36568b8.png\" class=\"latex\" alt=\"$a_k, b_k,\\, 1\\le k\\le n$\" style=\"vertical-align: -3px\" width=\"131\" height=\"16\" >.</span> The sides of smaller rectangulars are parallel to the corresponding sides of the big rectangular. Prove that <img src=\"//latex.artofproblemsolving.com/b/a/f/baf0ed987028aa16112f661f82ec0c6de174adca.png\" class=\"latex\" alt=\"$\|\\sin a_0\\sin b_0\|\\le \\sum_{k=1}^n\|\\sin a_k \\sin b_k\|.$\" style=\"vertical-align: -20px\" width=\"256\" height=\"48\" >", "post_id": 561683, "post_number": 1, "post_time_unix": 1151603156, "post_time_utc": "2006-06-29 17:45:56 UTC", "thanks_received": 1, "user_id": 96, "username": "alekk" }, { "attachments": [], "content_bbcode": "Are you sure of the statement of the problem? Take $a_0 = b_0 = \\frac{\\pi}{2}$ and $a_k = b_k = \\frac{\\pi}{6}$ for $1 \\le k \\le 3$. It is \r\n$\|1\\cdot 1\| \\le 3\|1/2 \\cdot 1/2\|$?\r\n\r\nShould it be $\\sum_{1 \\le j, k \\le n} \|\\sin(a_j)\\sin(b_k)\|$?\r\n\r\nThen I think this is the approach: it is sufficient to prove it when we divide the original rectangle into squares. Then the statement simplifies to an easier expression; I think we can conclude by taking the (positive) square root of each side and considering Taylor series. I'll try to post full solution tonight (unless someone else does first, or if I realize I'm wrong :P)", "content_html": "Are you sure of the statement of the problem? Take <img src=\"//latex.artofproblemsolving.com/b/e/c/becda48f3ad3d16b00e0c7b402c60a5a98c1c2a0.png\" class=\"latex\" alt=\"$a_0 = b_0 = \\frac{\\pi}{2}$\" style=\"vertical-align: -12px\" width=\"94\" height=\"33\" > and <img src=\"//latex.artofproblemsolving.com/e/c/a/ecabb3c2edaaa30153851f0c71752f86c6dc348f.png\" class=\"latex\" alt=\"$a_k = b_k = \\frac{\\pi}{6}$\" style=\"vertical-align: -12px\" width=\"95\" height=\"33\" > for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/1/bc19a96f1f78c8cecaa54c8520ab5dbedc9a9afa.png\" class=\"latex\" alt=\"$1 \\le k \\le 3$\" style=\"vertical-align: -2px\" width=\"76\" height=\"15\" >.</span> It is<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/a/b/8abe8bc462e0a9a0e69d5edbaf3911832498618b.png\" class=\"latex\" alt=\"$\|1\\cdot 1\| \\le 3\|1/2 \\cdot 1/2\|$\" style=\"vertical-align: -4px\" width=\"150\" height=\"18\" >?</span><br>\n<br>\nShould it be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/2/9/4292d568e3748263f156f57107cd312f4e0699ad.png\" class=\"latex\" alt=\"$\\sum_{1 \\le j, k \\le n} \|\\sin(a_j)\\sin(b_k)\|$\" style=\"vertical-align: -23px\" width=\"175\" height=\"40\" >?</span><br>\n<br>\nThen I think this is the approach: it is sufficient to prove it when we divide the original rectangle into squares. Then the statement simplifies to an easier expression; I think we can conclude by taking the (positive) square root of each side and considering Taylor series. I'll try to post full solution tonight (unless someone else does first, or if I realize I'm wrong <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />)", "post_id": 561797, "post_number": 2, "post_time_unix": 1151607973, "post_time_utc": "2006-06-29 19:06:13 UTC", "thanks_received": 2, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "yep, you have to sum over all the small rectangles. (in your example there are $9$ smaller rectangles)", "content_html": "yep, you have to sum over all the small rectangles. (in your example there are <img src=\"//latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png\" class=\"latex\" alt=\"$9$\" width=\"8\" height=\"12\" > smaller rectangles)", "post_id": 561874, "post_number": 3, "post_time_unix": 1151613196, "post_time_utc": "2006-06-29 20:33:16 UTC", "thanks_received": 2, "user_id": 96, "username": "alekk" }, { "attachments": [], "content_bbcode": "All of them! Then I don't think my idea works, at least not as easily as I hoped... Still I am sure we should find some kind of induction from a simple case in order to win.\r\n\r\nActually, isn't it enough if we prove it for dividing a rectangle into two pieces? The inequalities just add up and are all the direction we want. \r\n\r\nThen we just need to show that if $a_0 = a_1 + a_2$ then \r\n$\|\\sin(a_0)\\sin(b_0)\| \\le \|\\sin(a_1)\\sin(b_0)\| + \|\\sin(a_2)\\sin(b_0)\|$ \r\nright? Which is of course just $\|\\sin(a_0)\| \\le \|\\sin(a_1)\| + \|\\sin(a_2)\|$. \r\nIn case of $a_1 = a_2$, we have $\|\\sin(2a_1)\| \\le 2\|\\sin(a_1)\|$ which is obvious from double angle identity for sine. But really we need to look at minimum of $\|\\sin(x)\| + \|\\sin(a-x)\|$, neh? Wait, by AM-GM an arithmetic mean is always minimized by equality, right? So we just want $\|\\sin(x)\| = \|\\sin(a-x)\|$, which means $2x-a \\equiv 0 \\pmod{\\pi}$. Then we may as well take $a_0' = a_0 \\pmod{\\pi}$ because \|sine\| is determined uniquely up to that. Then we can be sure of minimizing either when $x = a_0'/2$ or $x=0$, and in both cases the inequality is obvious.\r\n\r\nI hope there is no mistake.... :D :?", "content_html": "All of them! Then I don't think my idea works, at least not as easily as I hoped... Still I am sure we should find some kind of induction from a simple case in order to win.<br>\n<br>\nActually, isn't it enough if we prove it for dividing a rectangle into two pieces? The inequalities just add up and are all the direction we want.<br>\n<br>\nThen we just need to show that if <img src=\"//latex.artofproblemsolving.com/2/8/2/2822f13934c2e356aff205aa010a570ec4f4598e.png\" class=\"latex\" alt=\"$a_0 = a_1 + a_2$\" style=\"vertical-align: -2px\" width=\"96\" height=\"13\" > then<br>\n<img src=\"//latex.artofproblemsolving.com/9/6/8/96856f10437b1cc4b99990f0a979ddc7ab0ef3e5.png\" class=\"latex\" alt=\"$\|\\sin(a_0)\\sin(b_0)\| \\le \|\\sin(a_1)\\sin(b_0)\| + \|\\sin(a_2)\\sin(b_0)\|$\" style=\"vertical-align: -4px\" width=\"408\" height=\"18\" ><br>\nright? Which is of course just <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/b/3/1b3ff52fcbb4207bf73c9ff9ec22f89fc5f29627.png\" class=\"latex\" alt=\"$\|\\sin(a_0)\| \\le \|\\sin(a_1)\| + \|\\sin(a_2)\|$\" style=\"vertical-align: -4px\" width=\"244\" height=\"18\" >.</span><br>\nIn case of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/e/0/be01457ee96b6edbf1cff715ecd21a6c81cff64b.png\" class=\"latex\" alt=\"$a_1 = a_2$\" style=\"vertical-align: -2px\" width=\"57\" height=\"10\" >,</span> we have <img src=\"//latex.artofproblemsolving.com/a/8/2/a8203e86d8106fa4b743f2e23f323debfd886a53.png\" class=\"latex\" alt=\"$\|\\sin(2a_1)\| \\le 2\|\\sin(a_1)\|$\" style=\"vertical-align: -4px\" width=\"173\" height=\"18\" > which is obvious from double angle identity for sine. But really we need to look at minimum of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/a/4/ca4347b9d1bc990154dadd9812afb236aca81a3a.png\" class=\"latex\" alt=\"$\|\\sin(x)\| + \|\\sin(a-x)\|$\" style=\"vertical-align: -4px\" width=\"172\" height=\"18\" >,</span> neh? Wait, by AM-GM an arithmetic mean is always minimized by equality, right? So we just want <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/6/0/e604df44a617b9cebb03e60938e3232f5c48e1a6.png\" class=\"latex\" alt=\"$\|\\sin(x)\| = \|\\sin(a-x)\|$\" style=\"vertical-align: -4px\" width=\"174\" height=\"18\" >,</span> which means <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/e/1/6e1af0772a8163d796bd4d34ac4329b95c3e6136.png\" class=\"latex\" alt=\"$2x-a \\equiv 0 \\pmod{\\pi}$\" style=\"vertical-align: -4px\" width=\"157\" height=\"18\" >.</span> Then we may as well take <img src=\"//latex.artofproblemsolving.com/d/e/6/de6dc9419eb29e1e7836c1d3607bd6c4e5d23e3b.png\" class=\"latex\" alt=\"$a_0' = a_0 \\pmod{\\pi}$\" style=\"vertical-align: -4px\" width=\"131\" height=\"18\" > because \|sine\| is determined uniquely up to that. Then we can be sure of minimizing either when <img src=\"//latex.artofproblemsolving.com/1/4/4/144eea7117a9f9236f776b98c6bb4cb979ddd4dc.png\" class=\"latex\" alt=\"$x = a_0'/2$\" style=\"vertical-align: -4px\" width=\"69\" height=\"18\" > or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/4/e/34ea5b6f6de584d56d19a89cc4923a9d9a5cfa41.png\" class=\"latex\" alt=\"$x=0$\" width=\"43\" height=\"12\" >,</span> and in both cases the inequality is obvious.<br>\n<br>\nI hope there is no mistake.... <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" />", "post_id": 562728, "post_number": 4, "post_time_unix": 1151681717, "post_time_utc": "2006-06-30 15:35:17 UTC", "thanks_received": 1, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "[quote=\"Xevarion\"]Actually, isn't it enough if we prove it for dividing a rectangle into two pieces? The inequalities just add up and are all the direction we want. [/quote]\r\n\r\nWhat about this dissection?\r\n:? :?$\\begin{picture}(30, 30) \\put(0, 0){\\line(1, 0){30}}\\put(0, 10){\\line(1, 0){20}} \\put(10, 20){\\line(1, 0){20}}\\put(0, 30){\\line(1, 0){30}} \\put(0,0){\\line(0, 1){30}} \\put(20, 0){\\line(0, 1){20}} \\put(10, 10){\\line(0, 1){20}} \\put(30, 0){\\line(0, 1){30}} \\end{picture}$ :? :? :?", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Xevarion wrote:</div>\n<div class=\"bbcode_quote_body\">Actually, isn't it enough if we prove it for dividing a rectangle into two pieces? The inequalities just add up and are all the direction we want.</div>\n</div>\n<br>\nWhat about this dissection?<br>\n<img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" /> :<span style=\"white-space:nowrap;\">?<img src=\"//latex.artofproblemsolving.com/2/8/5/285eed0f4e486e806f63196a6b601d8703389081.png\" class=\"latex\" alt=\"$\\begin{picture}(30, 30) \\put(0, 0){\\line(1, 0){30}}\\put(0, 10){\\line(1, 0){20}} \\put(10, 20){\\line(1, 0){20}}\\put(0, 30){\\line(1, 0){30}} \\put(0,0){\\line(0, 1){30}} \\put(20, 0){\\line(0, 1){20}} \\put(10, 10){\\line(0, 1){20}} \\put(30, 0){\\line(0, 1){30}} \\end{picture}$\" style=\"vertical-align: 0px\" width=\"51\" height=\"51\" ></span> <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" />", "post_id": 562836, "post_number": 5, "post_time_unix": 1151687877, "post_time_utc": "2006-06-30 17:17:57 UTC", "thanks_received": 1, "user_id": 10304, "username": "rogue" }, { "attachments": [], "content_bbcode": ":o \r\n\r\nWell, that particular one can be rearranged into the kind that I like. However I am not sure how to prove that this works in general. (I am not even sure it's correct...)\r\n\r\n :blush:\r\n\r\nedit: in fact, it surely isn't. Just make the bottom right rectangle narrower, and stretch the other two to fill the space, and then I can't rearrange it any more. :(", "content_html": "<img src=\"/assets/images/smilies/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" class=\"bbcode_smiley\" /><br>\n<br>\nWell, that particular one can be rearranged into the kind that I like. However I am not sure how to prove that this works in general. (I am not even sure it's correct...)<br>\n<br>\n<img src=\"/assets/images/smilies/redface_anim.gif\" width=\"19\" height=\"19\" alt=\":blush:\" title=\":blush:\" class=\"bbcode_smiley\" /><br>\n<br>\nedit: in fact, it surely isn't. Just make the bottom right rectangle narrower, and stretch the other two to fill the space, and then I can't rearrange it any more. <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />", "post_id": 562851, "post_number": 6, "post_time_unix": 1151688964, "post_time_utc": "2006-06-30 17:36:04 UTC", "thanks_received": 2, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "Alekk, what was you idea for this problem ?", "content_html": "Alekk, what was you idea for this problem ?", "post_id": 564033, "post_number": 7, "post_time_unix": 1151791017, "post_time_utc": "2006-07-01 21:56:57 UTC", "thanks_received": 2, "user_id": 902, "username": "eugene" }, { "attachments": [], "content_bbcode": "Seems we are forgetting olympiad classics :(: remember the problem about rectangles with one integer side and $e^{2\\pi i(x+y)}$? ;)", "content_html": "Seems we are forgetting olympiad classics <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />: remember the problem about rectangles with one integer side and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/8/b/28bf2e077cb919fce534b6a1cb2e9f58690e850a.png\" class=\"latex\" alt=\"$e^{2\\pi i(x+y)}$\" width=\"62\" height=\"16\" >?</span> <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 564212, "post_number": 8, "post_time_unix": 1151806978, "post_time_utc": "2006-07-02 02:22:58 UTC", "thanks_received": 2, "user_id": 6542, "username": "fedja" }, { "attachments": [], "content_bbcode": "[quote=\"fedja\"]Seems we are forgetting olympiad classics :([/quote]\r\nToo bad my experience with olympiads is pretty much $O(f(x))$ ($x \\to \\infty$) where $f$ is in the Schwartz class...", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">fedja wrote:</div>\n<div class=\"bbcode_quote_body\">Seems we are forgetting olympiad classics <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /></div>\n</div>\nToo bad my experience with olympiads is pretty much <img src=\"//latex.artofproblemsolving.com/4/d/d/4dde9c55aadc93e5aadac4615275068107795a39.png\" class=\"latex\" alt=\"$O(f(x))$\" style=\"vertical-align: -4px\" width=\"62\" height=\"18\" > <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/7/d/0/7d0eb486837b6e2d5c5e6f6a075ea764200db0c8.png\" class=\"latex\" alt=\"$x \\to \\infty$\" style=\"vertical-align: 0px\" width=\"56\" height=\"10\" >)</span> where <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > is in the Schwartz class...", "post_id": 564245, "post_number": 9, "post_time_unix": 1151809687, "post_time_utc": "2006-07-02 03:08:07 UTC", "thanks_received": 2, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "[quote=\"fedja\"]Seems we are forgetting olympiad classics :(: remember the problem about rectangles with one integer side and $e^{2\\pi i(x+y)}$? ;)[/quote]\r\n\r\nOoops, really seems we(at least me) forgot this classic.\r\n\r\nAs Fedja said, if we integrate the function $e^{2\\pi i(x+y)}$ over our big rectange(say it is $R$) the value of this integral is the same as the sum of integrals over little rectangles $R_{k}$, i.e\r\n\r\n$\\int_{R}e^{2\\pi i(x+y)}dx dy= \\sum \\int_{R_{k}}e^{2\\pi i(x+y)}dx dy$ (), but\r\n\r\n$\\int_{a}^{b}e^{2 \\pi i x}dx = \\frac{1}{2 i \\pi}e^{2\\pi i a}(e^{2\\pi i (b-a)}-1)$ and taking module in () we get that \r\n\r\n$\\mid (e^{2\\pi i a_{0}}-1)(e^{2\\pi i b_{0}}-1) \\mid \\leq \\sum_{k=1}^{n}\\mid (e^{2\\pi i a_{k}}-1)(e^{2\\pi i b_{k}}-1) \\mid $ and now it is only left to notice that\r\n\r\n$\\mid (e^{2\\pi i t}-1)\\mid = \\sqrt{( \\cos{2\\pi t }-1)^{2}+(\\sin{2\\pi t})^{2}}= \\sqrt{ 2-2\\cos{2 \\pi t}}= 2 \\mid \\sin{\\pi t}\\mid $.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">fedja wrote:</div>\n<div class=\"bbcode_quote_body\">Seems we are forgetting olympiad classics <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />: remember the problem about rectangles with one integer side and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/8/b/28bf2e077cb919fce534b6a1cb2e9f58690e850a.png\" class=\"latex\" alt=\"$e^{2\\pi i(x+y)}$\" width=\"62\" height=\"16\" >?</span> <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\nOoops, really seems we(at least me) forgot this classic.<br>\n<br>\nAs Fedja said, if we integrate the function <img src=\"//latex.artofproblemsolving.com/2/8/b/28bf2e077cb919fce534b6a1cb2e9f58690e850a.png\" class=\"latex\" alt=\"$e^{2\\pi i(x+y)}$\" width=\"62\" height=\"16\" > over our big rectange(say it is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >)</span> the value of this integral is the same as the sum of integrals over little rectangles <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/2/3/b2360e09e8a5216e0adb6d1c10ac42e2e52320e2.png\" class=\"latex\" alt=\"$R_{k}$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" >,</span> i.e<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/6/3/263ce7c4f4784c8a3dcd3a6b6c071bdcb28ba08e.png\" class=\"latex\" alt=\"$\\int_{R}e^{2\\pi i(x+y)}dx dy= \\sum \\int_{R_{k}}e^{2\\pi i(x+y)}dx dy$\" style=\"vertical-align: -18px\" width=\"313\" height=\"42\" > (), but<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/3/3/a/33a608e2834f7ba631db70ad1696eedfef6b74f5.png\" class=\"latex\" alt=\"$\\int_{a}^{b}e^{2 \\pi i x}dx = \\frac{1}{2 i \\pi}e^{2\\pi i a}(e^{2\\pi i (b-a)}-1)$\" style=\"vertical-align: -16px\" width=\"280\" height=\"44\" > and taking module in () we get that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/6/a/b6a2dd602c9c32d4845ae6227518b9152b9a1ce7.png\" class=\"latex\" alt=\"$\\mid (e^{2\\pi i a_{0}}-1)(e^{2\\pi i b_{0}}-1) \\mid \\leq \\sum_{k=1}^{n}\\mid (e^{2\\pi i a_{k}}-1)(e^{2\\pi i b_{k}}-1) \\mid $\" style=\"vertical-align: -20px\" width=\"439\" height=\"48\" > and now it is only left to notice that<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/e/a/9ea2c12bf5e704bc2f201a5ab124c9ababc94372.png\" class=\"latex\" alt=\"$\\mid (e^{2\\pi i t}-1)\\mid = \\sqrt{( \\cos{2\\pi t }-1)^{2}+(\\sin{2\\pi t})^{2}}= \\sqrt{ 2-2\\cos{2 \\pi t}}= 2 \\mid \\sin{\\pi t}\\mid $\" style=\"vertical-align: -10px\" width=\"575\" height=\"32\" >.</span>", "post_id": 565010, "post_number": 10, "post_time_unix": 1151907855, "post_time_utc": "2006-07-03 06:24:15 UTC", "thanks_received": 3, "user_id": 902, "username": "eugene" }, { "attachments": [], "content_bbcode": "eugene, :coolspeak:", "content_html": "eugene, <img src=\"/assets/images/smilies/coolspeak.gif\" width=\"61\" height=\"20\" alt=\":coolspeak:\" title=\":coolspeak:\" class=\"bbcode_smiley\" />", "post_id": 565253, "post_number": 11, "post_time_unix": 1151940731, "post_time_utc": "2006-07-03 15:32:11 UTC", "thanks_received": 2, "user_id": 6053, "username": "zhaobin" } ], "source": null }	A rectangle with side lengths $a_0,b_0$ is dissected into smaller rectangles with side lengths $a_k,b_k$ for $1\le k\le n$. The sides of the smaller rectangles are parallel to the corresponding sides of the large rectangle. Prove that \[ \bigl\|\sin a_0\sin b_0\bigr\|\le \sum_{k=1}^n\bigl\|\sin a_k\sin b_k\bigr\|. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus/SumRule", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/DefiniteIntegrals", "/Mathematics/CalculusandAnalysis/Calculus/MultivariableCalculus/MultipleIntegral", "/Mathematics/CalculusandAnalysis/Calculus/MultivariableCalculus/MultivariateCalculus", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/AbsoluteValue", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/ComplexExponentiation", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/ComplexModulus", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/ComplexNumber", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/EulerFormula", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/ImaginaryUnit", "/Mathematics/CalculusandAnalysis/ComplexAnalysis/ComplexNumbers/i", "/Mathematics/CalculusandAnalysis/Inequalities/Inequality", "/Mathematics/Geometry/Dissection", "/Mathematics/Geometry/GeometricInequalities/Brunn-MinkowskiInequality", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Rectangles/RectangleTiling", "/Mathematics/Geometry/PlaneGeometry/Tiling/Tessellation", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sinusoid", "/Mathematics/Geometry/Trigonometry/TrigonometricInequalities", "/Mathematics/RecreationalMathematics/Dissection/Wallace-Bolyai-Gerwien Theorem/Dissection", "/Mathematics/RecreationalMathematics/Dissection/Wallace-Bolyai-Gerwien Theorem/Equidecomposable" ]	Integrate e^{2πi(x+y)} over the big rectangle and use additivity of the integral over the smaller rectangles to obtain a product of terms (e^{2πi a}-1)(e^{2πi b}-1).	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aops_99515	I should think not. We can find some $m$ (infinitely many in fact) so that $\|\pi - \frac{n}{m}\| < \frac{1}{m^2}$. Then $\frac{1}{n\sin(n)}$ is about $\frac{1}{n(\frac{1}{m^2})} \approx \frac{m}{\pi}$, which is arbitrarily large. (for rigor you should probably take two terms of the taylor series for sine so that the "about" is an inequality in the direction we want, and then bound from each side by a constant times $m$) I bet it does converge in some weaker sense, like Cesaro.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Is it true that\r\n\\begin{eqnarray} \\lim_{n \\rightarrow \\infty} \\frac{1}{n \\sin n} & = & 0 \\end{eqnarray}\r\nWhat about\r\n\\begin{eqnarray} \\underset{n \\longrightarrow \\infty}{\\lim \\sup} \\frac{1}{n \\sin n} & = & ? \\end{eqnarray}", "content_html": "Is it true that<br>\n<img src=\"//latex.artofproblemsolving.com/f/f/4/ff4653157e59a221f3f9e2165483a887ca08852d.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray} \\lim_{n \\rightarrow \\infty} \\frac{1}{n \\sin n} & = & 0 \\end{eqnarray}\" width=\"147\" height=\"37\" ><br>\nWhat about<br>\n<img src=\"//latex.artofproblemsolving.com/9/8/a/98ab56b2919c5373cc52ca100c1fe87ba2821720.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray} \\underset{n \\longrightarrow \\infty}{\\lim \\sup} \\frac{1}{n \\sin n} & = & ? \\end{eqnarray}\" width=\"164\" height=\"39\" >", "post_id": 561722, "post_number": 1, "post_time_unix": 1151605209, "post_time_utc": "2006-06-29 18:20:09 UTC", "thanks_received": 2, "user_id": 12380, "username": "{x}" }, { "attachments": [], "content_bbcode": "I should think not. We can find some $m$ (infinitely many in fact) so that $\|\\pi - \\frac{n}{m}\| < \\frac{1}{m^2}$. Then $\\frac{1}{n\\sin(n)}$ is about $\\frac{1}{n(\\frac{1}{m^2})} \\approx \\frac{m}{\\pi}$, which is arbitrarily large. \r\n\r\n(for rigor you should probably take two terms of the taylor series for sine so that the \"about\" is an inequality in the direction we want, and then bound from each side by a constant times $m$)\r\n\r\nI bet it does converge in some weaker sense, like Cesaro.", "content_html": "I should think not. We can find some <img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" > (infinitely many in fact) so that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/c/9/3c973d5c4f47a367052da6c4f40d26bbf128ce62.png\" class=\"latex\" alt=\"$\|\\pi - \\frac{n}{m}\| < \\frac{1}{m^2}$\" style=\"vertical-align: -12px\" width=\"113\" height=\"37\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/0/b/5/0b51ac4f962934da1ecd721429b61480c898123a.png\" class=\"latex\" alt=\"$\\frac{1}{n\\sin(n)}$\" style=\"vertical-align: -17px\" width=\"63\" height=\"41\" > is about <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/9/9/d9918d3b7bbcf42e1054b520575f9df60a2a278f.png\" class=\"latex\" alt=\"$\\frac{1}{n(\\frac{1}{m^2})} \\approx \\frac{m}{\\pi}$\" style=\"vertical-align: -19px\" width=\"93\" height=\"43\" >,</span> which is arbitrarily large.<br>\n<br>\n(for rigor you should probably take two terms of the taylor series for sine so that the "about" is an inequality in the direction we want, and then bound from each side by a constant times <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" >)</span><br>\n<br>\nI bet it does converge in some weaker sense, like Cesaro.", "post_id": 561733, "post_number": 2, "post_time_unix": 1151605740, "post_time_utc": "2006-06-29 18:29:00 UTC", "thanks_received": 3, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "The first fact is a corollary of Hurwitz theorem, as far as I know.\r\nOtherwise, I don't think you really need to mess up with Taylor series. Some basic inequalities should suffice.", "content_html": "The first fact is a corollary of Hurwitz theorem, as far as I know.<br>\nOtherwise, I don't think you really need to mess up with Taylor series. Some basic inequalities should suffice.", "post_id": 561753, "post_number": 3, "post_time_unix": 1151606518, "post_time_utc": "2006-06-29 18:41:58 UTC", "thanks_received": 2, "user_id": 12380, "username": "{x}" }, { "attachments": [], "content_bbcode": "For any $c$, $\\limsup \\left\|\\frac1{n\\sin cn}\\ge\\frac1{\\pi}\\right\|$. This limsup is $\\infty$ when the continued fraction for $\\frac c{\\pi}$ has arbitrarily large quotients or $\\frac c{\\pi}$ is rational.\r\n\r\nFor any integer $m$, $\|\\sin cn\|=\|\\sin(cn-m\\pi)\|< \|cn-m\\pi\|=n\\pi\\left\|\\frac{c}{\\pi}-\\frac mn\\right\|$. If $\\frac pq$ is a convergent (in lowest terms) of the continued fraction for $\\frac c{\\pi}$, $\\left\| \\frac{c}{\\pi}-\\frac pq\\right\|<\\frac1{q^2}$. There are infinitely many convergents $\\frac pq$; choosing these as our $m,n$, we have $n\\pi\\left\|\\frac{c}{\\pi}-\\frac mn\\right\|<\\frac{\\pi}n$. Taking the reciprocal and dividing by $n$, we have $\|\\frac1{n\\sin cn}\|> \\frac1{\\pi}$.\r\n\r\nXevarion's argument misses one of these factors of $n$; whether the $\\limsup$ is finite in any particular case is an interesting question. It is always nonzero.\r\n\r\nIf we drop the absolute value signs, we can show that $\\limsup \\frac1{n\\sin cn}\\ge\\frac1{4\\pi}$ and $\\liminf \\frac1{n\\sin cn}\\le-\\frac1{4\\pi}$. The convergents $\\frac pq$ alternate between being too large and too small; if we then let $n=2q$, $\\sin 2cn$ will be positive when $\\frac pq$ is low and negative when $\\frac pq$ is high. In both cases, we lose a factor of 4 by doubling.", "content_html": "For any <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png\" class=\"latex\" alt=\"$c$\" width=\"8\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/b/2/5b26b6b67c7eae47293046d6c0360e10db46fd57.png\" class=\"latex\" alt=\"$\\limsup \\left\|\\frac1{n\\sin cn}\\ge\\frac1{\\pi}\\right\|$\" style=\"vertical-align: -17px\" width=\"169\" height=\"43\" >.</span> This limsup is <img src=\"//latex.artofproblemsolving.com/b/6/7/b671f1bb7e4ee86584347d5d22f1dc8abdb5bef2.png\" class=\"latex\" alt=\"$\\infty$\" style=\"vertical-align: 0px\" width=\"17\" height=\"9\" > when the continued fraction for <img src=\"//latex.artofproblemsolving.com/a/7/3/a732d28729963a17a04831beb74bdb54555d4924.png\" class=\"latex\" alt=\"$\\frac c{\\pi}$\" style=\"vertical-align: -12px\" width=\"13\" height=\"33\" > has arbitrarily large quotients or <img src=\"//latex.artofproblemsolving.com/a/7/3/a732d28729963a17a04831beb74bdb54555d4924.png\" class=\"latex\" alt=\"$\\frac c{\\pi}$\" style=\"vertical-align: -12px\" width=\"13\" height=\"33\" > is rational.<br>\n<br>\nFor any integer <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/1/ba1359fba29c8319a263cac709a5d5a12c764521.png\" class=\"latex\" alt=\"$\|\\sin cn\|=\|\\sin(cn-m\\pi)\|< \|cn-m\\pi\|=n\\pi\\left\|\\frac{c}{\\pi}-\\frac mn\\right\|$\" style=\"vertical-align: -12px\" width=\"417\" height=\"33\" >.</span> If <img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" > is a convergent (in lowest terms) of the continued fraction for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/7/3/a732d28729963a17a04831beb74bdb54555d4924.png\" class=\"latex\" alt=\"$\\frac c{\\pi}$\" style=\"vertical-align: -12px\" width=\"13\" height=\"33\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/1/d810068472d3be2b77cee389e22bb0f5133d33c3.png\" class=\"latex\" alt=\"$\\left\| \\frac{c}{\\pi}-\\frac pq\\right\|<\\frac1{q^2}$\" style=\"vertical-align: -17px\" width=\"105\" height=\"43\" >.</span> There are infinitely many convergents <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" >;</span> choosing these as our <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/b/46ba6ed097cc88ded349bf99eea89817f135f470.png\" class=\"latex\" alt=\"$m,n$\" style=\"vertical-align: -3px\" width=\"34\" height=\"11\" >,</span> we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/c/0ec523107c647e11d549ce10c0473e3657a2340e.png\" class=\"latex\" alt=\"$n\\pi\\left\|\\frac{c}{\\pi}-\\frac mn\\right\|<\\frac{\\pi}n$\" style=\"vertical-align: -12px\" width=\"131\" height=\"33\" >.</span> Taking the reciprocal and dividing by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >,</span> we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/5/a15d2d1b9e28519d3d0b7d0c1a29bdae03232ab5.png\" class=\"latex\" alt=\"$\|\\frac1{n\\sin cn}\|> \\frac1{\\pi}$\" style=\"vertical-align: -12px\" width=\"110\" height=\"37\" >.</span><br>\n<br>\nXevarion's argument misses one of these factors of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >;</span> whether the <img src=\"//latex.artofproblemsolving.com/9/c/0/9c024d06c309d092d89ff0bc382912d809a96dea.png\" class=\"latex\" alt=\"$\\limsup$\" style=\"vertical-align: -3px\" width=\"55\" height=\"16\" > is finite in any particular case is an interesting question. It is always nonzero.<br>\n<br>\nIf we drop the absolute value signs, we can show that <img src=\"//latex.artofproblemsolving.com/8/6/b/86b62b4722d4c29ece5338b09d980d2860178f24.png\" class=\"latex\" alt=\"$\\limsup \\frac1{n\\sin cn}\\ge\\frac1{4\\pi}$\" style=\"vertical-align: -13px\" width=\"167\" height=\"37\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/d/09dcb8ee274467f3366d23c8d0bc485396ebcd68.png\" class=\"latex\" alt=\"$\\liminf \\frac1{n\\sin cn}\\le-\\frac1{4\\pi}$\" style=\"vertical-align: -13px\" width=\"176\" height=\"37\" >.</span> The convergents <img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" > alternate between being too large and too small; if we then let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/b/eeba8215078fd2f13746e8da11a731e4dbd947f5.png\" class=\"latex\" alt=\"$n=2q$\" style=\"vertical-align: -3px\" width=\"52\" height=\"15\" >,</span> <img src=\"//latex.artofproblemsolving.com/d/b/3/db311a1c61c05870de174d3cdf8917fd6d68776e.png\" class=\"latex\" alt=\"$\\sin 2cn$\" width=\"53\" height=\"12\" > will be positive when <img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" > is low and negative when <img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" > is high. In both cases, we lose a factor of 4 by doubling.", "post_id": 561774, "post_number": 4, "post_time_unix": 1151607252, "post_time_utc": "2006-06-29 18:54:12 UTC", "thanks_received": 2, "user_id": 2975, "username": "jmerry" }, { "attachments": [], "content_bbcode": ":blush: \r\nThanks for correcting my error jmerry. Can you please give some more detail on the case where the continued fraction for $\\frac{c}{\\pi}$ has arbitrarily large quotients? I'm afraid I don't see immediately why it's so ... :oops:", "content_html": "<img src=\"/assets/images/smilies/redface_anim.gif\" width=\"19\" height=\"19\" alt=\":blush:\" title=\":blush:\" class=\"bbcode_smiley\" /><br>\nThanks for correcting my error jmerry. Can you please give some more detail on the case where the continued fraction for <img src=\"//latex.artofproblemsolving.com/f/6/d/f6d5de61dbe8791c8792b439ebf1532a46c9be4e.png\" class=\"latex\" alt=\"$\\frac{c}{\\pi}$\" style=\"vertical-align: -12px\" width=\"13\" height=\"33\" > has arbitrarily large quotients? I'm afraid I don't see immediately why it's so ... <img src=\"/assets/images/smilies/blush.gif\" width=\"20\" height=\"20\" alt=\":oops:\" title=\":oops:\" class=\"bbcode_smiley\" />", "post_id": 561808, "post_number": 5, "post_time_unix": 1151608297, "post_time_utc": "2006-06-29 19:11:37 UTC", "thanks_received": 2, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "If the quotient after the convergent $\\frac pq$ is $r$, we have $\\left\|\\frac c{\\pi}-\\frac pq\\right\|\\le\\frac1{rq^2}$.\r\n\r\nI think so, anyway; it's been a while since I worked out the details. The two quantities are comparable in any case.\r\nAs a corollary, large quotients indicate unusually good rational approximations. As a famous example, here's the first few terms of the continued fraction for $\\pi$:\r\n3;7,15,1,292\r\nIf we cut off just before the 292, we get $\\frac{355}{113}$ as a convergent.", "content_html": "If the quotient after the convergent <img src=\"//latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png\" class=\"latex\" alt=\"$\\frac pq$\" style=\"vertical-align: -16px\" width=\"11\" height=\"36\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png\" class=\"latex\" alt=\"$r$\" width=\"8\" height=\"8\" >,</span> we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/7/267d2236f02904722234558e5338f6901907ffe8.png\" class=\"latex\" alt=\"$\\left\|\\frac c{\\pi}-\\frac pq\\right\|\\le\\frac1{rq^2}$\" style=\"vertical-align: -17px\" width=\"113\" height=\"43\" >.</span><br>\n<br>\nI think so, anyway; it's been a while since I worked out the details. The two quantities are comparable in any case.<br>\nAs a corollary, large quotients indicate unusually good rational approximations. As a famous example, here's the first few terms of the continued fraction for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/2/c/f2ca003a7da0de4994b4733e203b74ff52d42553.png\" class=\"latex\" alt=\"$\\pi$\" width=\"10\" height=\"8\" >:</span><br>\n3;7,15,1,292<br>\nIf we cut off just before the 292, we get <img src=\"//latex.artofproblemsolving.com/b/1/8/b18bcc289ae334e9e58f27289513f87c8628a536.png\" class=\"latex\" alt=\"$\\frac{355}{113}$\" style=\"vertical-align: -13px\" width=\"29\" height=\"38\" > as a convergent.", "post_id": 561833, "post_number": 6, "post_time_unix": 1151610300, "post_time_utc": "2006-06-29 19:45:00 UTC", "thanks_received": 2, "user_id": 2975, "username": "jmerry" }, { "attachments": [], "content_bbcode": "Oh, I get it, that's really cool! Thanks :D \r\n\r\nAre there general methods for knowing whether the continued fraction of some irrational has arbitrarily large quotients (other than actually knowing the general term, like in case of the golden ratio)?", "content_html": "Oh, I get it, that's really cool! Thanks <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\nAre there general methods for knowing whether the continued fraction of some irrational has arbitrarily large quotients (other than actually knowing the general term, like in case of the golden ratio)?", "post_id": 561838, "post_number": 7, "post_time_unix": 1151610538, "post_time_utc": "2006-06-29 19:48:58 UTC", "thanks_received": 2, "user_id": 6233, "username": "Xevarion" }, { "attachments": [], "content_bbcode": "[quote=\"Xevarion\"]... like in case of the golden ratio)?[/quote]\r\nTheorem: the quotients of the continued fraction of a number are periodic if and only if the number is a quadratic irrational - that is an irrational number that can be written in the form $\\frac{a+\\sqrt{b}}c$ or $\\frac{a-\\sqrt{b}}c$ for integers $a,b,c.$ (If the number is rational, then the continued fraction representation terminates.) Of course, the golden ratio is a quadratic irrational.\r\n\r\nSo quadratic irrationals cannot have arbitrarily large quotients, because the quotients are periodic. But we can't say anything about the converse.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Xevarion wrote:</div>\n<div class=\"bbcode_quote_body\">... like in case of the golden ratio)?</div>\n</div>\nTheorem: the quotients of the continued fraction of a number are periodic if and only if the number is a quadratic irrational - that is an irrational number that can be written in the form <img src=\"//latex.artofproblemsolving.com/0/8/4/084ba67fa86e737d3937e3c5d258c1c26e1f7134.png\" class=\"latex\" alt=\"$\\frac{a+\\sqrt{b}}c$\" style=\"vertical-align: -12px\" width=\"57\" height=\"41\" > or <img src=\"//latex.artofproblemsolving.com/6/3/3/63335515c7bd01bc8b53292e0fbd695ea347083a.png\" class=\"latex\" alt=\"$\\frac{a-\\sqrt{b}}c$\" style=\"vertical-align: -12px\" width=\"57\" height=\"41\" > for integers <img src=\"//latex.artofproblemsolving.com/3/f/0/3f054be28a57624feee47965aad91fdc1b7a6792.png\" class=\"latex\" alt=\"$a,b,c.$\" style=\"vertical-align: -3px\" width=\"45\" height=\"16\" > (If the number is rational, then the continued fraction representation terminates.) Of course, the golden ratio is a quadratic irrational.<br>\n<br>\nSo quadratic irrationals cannot have arbitrarily large quotients, because the quotients are periodic. But we can't say anything about the converse.", "post_id": 561859, "post_number": 8, "post_time_unix": 1151612570, "post_time_utc": "2006-06-29 20:22:50 UTC", "thanks_received": 2, "user_id": 2948, "username": "Kent Merryfield" } ], "source": null }	Is it true that \[ \lim_{n\to\infty}\frac{1}{n\sin n}=0? \] What is \[ \limsup_{n\to\infty}\frac{1}{n\sin n}\,? \]	[ "/Mathematics/CalculusandAnalysis/Functions/ElementaryFunction", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/OscillatingFunction", "/Mathematics/CalculusandAnalysis/Functions/Oscillation", "/Mathematics/CalculusandAnalysis/Functions/PeriodicFunction", "/Mathematics/CalculusandAnalysis/Functions/RealAnalyticFunction", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/Scalar-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/ScalarFunction", "/Mathematics/CalculusandAnalysis/Functions/Single-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis" ]	Approximate π by rationals n/m so that n is near a multiple of π, making sin n arbitrarily small and 1/(n sin n) unbounded.	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aops_995183	Using the Euclidean Algorithm, we can find the greatest common divisor of the numerator and denominator: GCD(21n+4, 14n+3)=GCD(7n+1, 14n+3)=GCD(7n+1, 7n+2)=$ 1$ #1 1959 IMO, first ever IMO problem.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "What mathematical term rhymes with texas?\r\n\r\nThe plurality of the population agrees that which 5 numbers are the most important in math?\r\n\r\nThere are only 2 known examples of which \"special number'?\r\n\r\nWhat is the mathematical term for a 3-d block \"8\"?\r\n\r\nWhat is the next word in the sequence:\r\nTwin, Cousin, _____________________\r\n?\r\n\r\nWhat did Erdos primarily use to prove his most elementary observations on prime numbers?\r\n\r\n(giveaway, hopefully)\r\nWhere is the following mathematical problem from? (competition, only)\r\n$ \\text{Prove that the fraction } \\frac {21n \\plus{} 4}{14n \\plus{} 3} \\text{ is irreducible for all n.}$", "content_html": "What mathematical term rhymes with texas?<br>\n<br>\nThe plurality of the population agrees that which 5 numbers are the most important in math?<br>\n<br>\nThere are only 2 known examples of which "special number'?<br>\n<br>\nWhat is the mathematical term for a 3-d block "8"?<br>\n<br>\nWhat is the next word in the sequence:<br>\nTwin, Cousin, _____________________<br>\n?<br>\n<br>\nWhat did Erdos primarily use to prove his most elementary observations on prime numbers?<br>\n<br>\n(giveaway, hopefully)<br>\nWhere is the following mathematical problem from? (competition, only)<br>\n<img src=\"//latex.artofproblemsolving.com/8/2/8/8284cd7cb62571c61e3583b9a12beb71d0b7a0d6.png\" class=\"latex\" alt=\"$ \\text{Prove that the fraction } \\frac {21n + 4}{14n + 3} \\text{ is irreducible for all n.}$\" style=\"vertical-align: -14px\" width=\"402\" height=\"38\" >", "post_id": 4412206, "post_number": 1, "post_time_unix": 1248919400, "post_time_utc": "2009-07-30 02:03:20 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "[hide=\"IMO\"]IMO[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">IMO</a><div class=\"cmty-hide-content\" style=\"display:none\">IMO</div>", "post_id": 4412207, "post_number": 2, "post_time_unix": 1248919441, "post_time_utc": "2009-07-30 02:04:01 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "No, IMO does not rhyme with texas\r\n[hide]jk i know what you are talking about[/hide]", "content_html": "No, IMO does not rhyme with texas<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">jk i know what you are talking about</div>", "post_id": 4412208, "post_number": 3, "post_time_unix": 1248919610, "post_time_utc": "2009-07-30 02:06:50 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "[hide=\"stuff\"]$ e, \\pi, i, 0, 1$\noctahedron?\ntwin, cousin, sexy (no, I'm serious, this is the correct answer http://mathworld.wolfram.com/SexyPrimes.html )[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">stuff</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/9/2/f/92f3301f1182814941f4f733d21774baf151f575.png\" class=\"latex\" alt=\"$ e, \\pi, i, 0, 1$\" style=\"vertical-align: -3px\" width=\"75\" height=\"16\" ><br>\noctahedron?<br>\ntwin, cousin, sexy (no, I'm serious, this is the correct answer <a target=\"_blank\" href=\"http://mathworld.wolfram.com/SexyPrimes.html\">http://mathworld.wolfram.com/SexyPrimes.html</a> )</div>", "post_id": 4412209, "post_number": 4, "post_time_unix": 1248925592, "post_time_utc": "2009-07-30 03:46:32 UTC", "thanks_received": 2, "user_id": 45289, "username": "dysfunctionalequations" }, { "attachments": [], "content_bbcode": "correct for sexy and the 5 numbers, but octahedron is probably not correct.\r\n\r\ngood job on sexy", "content_html": "correct for sexy and the 5 numbers, but octahedron is probably not correct.<br>\n<br>\ngood job on sexy", "post_id": 4412210, "post_number": 5, "post_time_unix": 1248929671, "post_time_utc": "2009-07-30 04:54:31 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "tesseract net?", "content_html": "tesseract net?", "post_id": 4412211, "post_number": 6, "post_time_unix": 1248983717, "post_time_utc": "2009-07-30 19:55:17 UTC", "thanks_received": 2, "user_id": 64866, "username": "fwolth" }, { "attachments": [], "content_bbcode": "probably not\r\n\r\nanswers tomorrow probably", "content_html": "probably not<br>\n<br>\nanswers tomorrow probably", "post_id": 4412212, "post_number": 7, "post_time_unix": 1248983816, "post_time_utc": "2009-07-30 19:56:56 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "Using the Euclidean Algorithm, we can find the greatest common divisor of the numerator and denominator:\r\n\r\nGCD(21n+4, 14n+3)=GCD(7n+1, 14n+3)=GCD(7n+1, 7n+2)=$ 1$\r\n\r\n#1 1959 IMO, first ever IMO problem.", "content_html": "Using the Euclidean Algorithm, we can find the greatest common divisor of the numerator and denominator:<br>\n<br>\nGCD(21n+4, 14n+3)=GCD(7n+1, 14n+3)=GCD(7n+1, 7n+2)<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/3/9/0/39064bdd89b3dfa0626ca59d843d926ea072830b.png\" class=\"latex\" alt=\"$ 1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" ></span><br>\n<br>\n#1 1959 IMO, first ever IMO problem.", "post_id": 4412213, "post_number": 8, "post_time_unix": 1248987821, "post_time_utc": "2009-07-30 21:03:41 UTC", "thanks_received": 2, "user_id": 61769, "username": "PowerOfPi" }, { "attachments": [], "content_bbcode": "hmmmm... that was on our first handout for Number Theory... I remember because it was the first problem that i did in the problem session! :D", "content_html": "hmmmm... that was on our first handout for Number Theory... I remember because it was the first problem that i did in the problem session! <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 4412214, "post_number": 9, "post_time_unix": 1249003304, "post_time_utc": "2009-07-31 01:21:44 UTC", "thanks_received": 2, "user_id": 58015, "username": "connaissance" }, { "attachments": [], "content_bbcode": "unfortunately for PowerofPi, Yongyi781 already answered that.\r\n\r\nI guess that it's correct too though...", "content_html": "unfortunately for PowerofPi, Yongyi781 already answered that.<br>\n<br>\nI guess that it's correct too though...", "post_id": 4412215, "post_number": 10, "post_time_unix": 1249003556, "post_time_utc": "2009-07-31 01:25:56 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "What mathematical term rhymes with texas? \r\n\r\nThe plurality of the population agrees that which 5 numbers are the most important in math? \r\n\r\nThere are only 2 known examples of which \"special number'? \r\n\r\nWhat is the mathematical term for a 3-d block \"8\"? \r\n\r\nWhat is the next word in the sequence: \r\nTwin, Cousin, _____________________ \r\n? \r\n\r\nWhat did Erdos primarily use to prove his most elementary observations on prime numbers? \r\n\r\n(giveaway, hopefully) \r\nWhere is the following mathematical problem from? (competition, only) \r\n[blah]\r\n\r\n1) nexus\r\nThis is the ring of self intersection on a klein bottle.\r\n\r\n2) the numbers in the following equation\r\n$ e^{\\pi i}\\plus{}1\\equal{}0$\r\n\r\n3) sublime number\r\nnumbers such that the following are both perfect numbers:\r\n#number of factors\r\nsum of factors\r\n\r\n4) double torus\r\n\r\n5) sexy\r\ntwin primes: separated by 2\r\ncousin primes: separated by 4\r\nsexy primes: separated by 6\r\n\r\n6) binomial coefficients\r\nHe used them to prove that they wer divisible by certain primes a certain number of times", "content_html": "What mathematical term rhymes with texas?<br>\n<br>\nThe plurality of the population agrees that which 5 numbers are the most important in math?<br>\n<br>\nThere are only 2 known examples of which "special number'?<br>\n<br>\nWhat is the mathematical term for a 3-d block "8"?<br>\n<br>\nWhat is the next word in the sequence:<br>\nTwin, Cousin, _____________________<br>\n?<br>\n<br>\nWhat did Erdos primarily use to prove his most elementary observations on prime numbers?<br>\n<br>\n(giveaway, hopefully)<br>\nWhere is the following mathematical problem from? (competition, only)<br>\n[blah]<br>\n<br>\n1) nexus<br>\nThis is the ring of self intersection on a klein bottle.<br>\n<br>\n2) the numbers in the following equation<br>\n<img src=\"//latex.artofproblemsolving.com/d/b/e/dbe29d65bac522477c7488582994dd49f76a126f.png\" class=\"latex\" alt=\"$ e^{\\pi i}+1=0$\" style=\"vertical-align: -1px\" width=\"86\" height=\"16\" ><br>\n<br>\n3) sublime number<br>\nnumbers such that the following are both perfect numbers:<br>\n#number of factors<br>\nsum of factors<br>\n<br>\n4) double torus<br>\n<br>\n5) sexy<br>\ntwin primes: separated by 2<br>\ncousin primes: separated by 4<br>\nsexy primes: separated by 6<br>\n<br>\n6) binomial coefficients<br>\nHe used them to prove that they wer divisible by certain primes a certain number of times", "post_id": 4412216, "post_number": 11, "post_time_unix": 1249083771, "post_time_utc": "2009-07-31 23:42:51 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" } ], "source": null }	1. What mathematical term rhymes with "Texas"? 2. The plurality of the population agrees that which five numbers are the most important in math? 3. There are only two known examples of which "special number"? 4. What is the mathematical term for a 3-dimensional block "8"? 5. What is the next word in the sequence: Twin, Cousin, _____________________ ? 6. What did Erdős primarily use to prove his most elementary observations on prime numbers? 7. Where is the following mathematical problem from (competition name only)? Prove that the fraction $\dfrac{21n+4}{14n+3}$ is irreducible for all integers $n$.	[ "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Use the Euclidean algorithm on the linear expressions to reduce their GCD to 1.	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aops_995242	I have run through a sketchy proof through my head as to why it is no. If there are 2 such circles, then clearly at least 2 more circles intersect at one of the 2 intersection points, PROBABLY creating more non-tangency intersections PROBABLY creating more non-tangency intersections. hooray for being rigorous.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "YAY NOW I WILL OFFICIALLY CHECK STEVENMEOW'S SPELLING\r\n\r\nHmm...\r\n\r\nHere is a Circle Problem.\r\n\r\nWe have a finite number of circles in the plane. Every intersection point of 2 circles has an even number of circles going through it. Also, at every intersection point there exist 2 circles tangent to that point. Is it possible that 2 of the circles intersect at exactly 2 points?", "content_html": "YAY NOW I WILL OFFICIALLY CHECK STEVENMEOW'S SPELLING<br>\n<br>\nHmm...<br>\n<br>\nHere is a Circle Problem.<br>\n<br>\nWe have a finite number of circles in the plane. Every intersection point of 2 circles has an even number of circles going through it. Also, at every intersection point there exist 2 circles tangent to that point. Is it possible that 2 of the circles intersect at exactly 2 points?", "post_id": 4412370, "post_number": 1, "post_time_unix": 1251946949, "post_time_utc": "2009-09-03 03:02:29 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" }, { "attachments": [], "content_bbcode": "that requires circular logic\r\n\r\nhmm very unique problem", "content_html": "that requires circular logic<br>\n<br>\nhmm very unique problem", "post_id": 4412371, "post_number": 2, "post_time_unix": 1252022331, "post_time_utc": "2009-09-03 23:58:51 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "I have run through a sketchy proof through my head as to why it is no. If there are 2 such circles, then clearly at least 2 more circles intersect at one of the 2 intersection points, PROBABLY creating more non-tangency intersections PROBABLY creating more non-tangency intersections.\r\n\r\nhooray for being rigorous.", "content_html": "I have run through a sketchy proof through my head as to why it is no. If there are 2 such circles, then clearly at least 2 more circles intersect at one of the 2 intersection points, PROBABLY creating more non-tangency intersections PROBABLY creating more non-tangency intersections.<br>\n<br>\nhooray for being rigorous.", "post_id": 4412372, "post_number": 3, "post_time_unix": 1252029117, "post_time_utc": "2009-09-04 01:51:57 UTC", "thanks_received": 2, "user_id": 37558, "username": "stevenmeow" }, { "attachments": [], "content_bbcode": "DETECTED INCORRECT SPELLING: REGOROUS\r\n\r\nAlso, that solution is COMPLETELY fake... like basically you have to prove some things about the circle's intersections, and this isn't that easy... that provides intutition why the answer is false, but...", "content_html": "DETECTED INCORRECT SPELLING: REGOROUS<br>\n<br>\nAlso, that solution is COMPLETELY fake... like basically you have to prove some things about the circle's intersections, and this isn't that easy... that provides intutition why the answer is false, but...", "post_id": 4412373, "post_number": 4, "post_time_unix": 1252034760, "post_time_utc": "2009-09-04 03:26:00 UTC", "thanks_received": 2, "user_id": 30008, "username": "pythag011" } ], "source": null }	YAY NOW I WILL OFFICIALLY CHECK STEVENMEOW'S SPELLING Here is a Circle Problem. We have a finite number of circles in the plane. Every intersection point of two circles has an even number of circles passing through it. Also, at every intersection point there exist two circles tangent at that point. Is it possible that two of the circles intersect at exactly two points?	[ "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/Curves/PlaneCurves/AlgebraicCurves", "/Mathematics/Geometry/Curves/PlaneCurves/ConicSections", "/Mathematics/Geometry/Curves/PlaneCurves/GeneralPlaneCurves", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/InversiveGeometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle-CircleIntersection" ]	Combine the even‑number condition with the required tangent pair to show any intersection must be a tangency, ruling out two distinct transversal intersections.	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aops_99528	[quote="Arvind_sn"]Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought? :huuh:[/quote] [hide="If kittens count as cats"] $666(6+1)=2167=\boxed{1512}$. :D [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought? \r\n:huuh:", "content_html": "Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought?<br>\n<img src=\"/assets/images/smilies/wtf.gif\" width=\"20\" height=\"25\" alt=\":huuh:\" title=\":huuh:\" class=\"bbcode_smiley\" />", "post_id": 561800, "post_number": 1, "post_time_unix": 1151608051, "post_time_utc": "2006-06-29 19:07:31 UTC", "thanks_received": 2, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "are kittens considered cats in your problem? You really didn't specify.", "content_html": "are kittens considered cats in your problem? You really didn't specify.", "post_id": 561811, "post_number": 2, "post_time_unix": 1151608491, "post_time_utc": "2006-06-29 19:14:51 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought? \n:huuh:[/quote]\r\n[hide=\"If kittens count as cats\"] $666(6+1)=2167=\\boxed{1512}$. :D [/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought?<br>\n<img src=\"/assets/images/smilies/wtf.gif\" width=\"20\" height=\"25\" alt=\":huuh:\" title=\":huuh:\" class=\"bbcode_smiley\" /></div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">If kittens count as cats</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/7/3/673fc54d6c8908092067bbd3d2630e58b2b27e82.png\" class=\"latex\" alt=\"$666(6+1)=2167=\\boxed{1512}$\" style=\"vertical-align: -5px\" width=\"283\" height=\"23\" >.</span> <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div>", "post_id": 561832, "post_number": 3, "post_time_unix": 1151610161, "post_time_utc": "2006-06-29 19:42:41 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "Kittens are cats. Just like puppies are dogs, and kids are humans, kittens most definitely are cats.", "content_html": "Kittens are cats. Just like puppies are dogs, and kids are humans, kittens most definitely are cats.", "post_id": 561981, "post_number": 4, "post_time_unix": 1151617511, "post_time_utc": "2006-06-29 21:45:11 UTC", "thanks_received": 1, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "INCORRECT! Kids are goats! CHILDREN are humans.\r\n\r\n[hide=\"answer\"]1512[/hide]", "content_html": "INCORRECT! Kids are goats! CHILDREN are humans.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">answer</a><div class=\"cmty-hide-content\" style=\"display:none\">1512</div>", "post_id": 561997, "post_number": 5, "post_time_unix": 1151618079, "post_time_utc": "2006-06-29 21:54:39 UTC", "thanks_received": 1, "user_id": 15223, "username": "1=2" }, { "attachments": [], "content_bbcode": "[quote]INCORRECT! Kids are goats! CHILDREN are humans. [/quote]\r\n\r\nWhatever... By the way, I have a question. You know when people hide something, it says \"read hidden text\" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say \"answer\". Please tell me!", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">INCORRECT! Kids are goats! CHILDREN are humans.</div>\n</div>\n<br>\nWhatever... By the way, I have a question. You know when people hide something, it says "read hidden text" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say "answer". Please tell me!", "post_id": 562080, "post_number": 6, "post_time_unix": 1151622783, "post_time_utc": "2006-06-29 23:13:03 UTC", "thanks_received": 2, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"][quote]INCORRECT! Kids are goats! CHILDREN are humans. [/quote]\n\nWhatever... By the way, I have a question. You know when people hide something, it says \"read hidden test\" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say \"answer\". Please tell me![/quote]\r\n\r\nInside the bracket things, it's normally \"hide\". Just type \"hide=Answer\" or whatever.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">INCORRECT! Kids are goats! CHILDREN are humans.</div>\n</div>\n<br>\nWhatever... By the way, I have a question. You know when people hide something, it says "read hidden test" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say "answer". Please tell me!</div>\n</div>\n<br>\nInside the bracket things, it's normally "hide". Just type "hide=Answer" or whatever.", "post_id": 562086, "post_number": 7, "post_time_unix": 1151622885, "post_time_utc": "2006-06-29 23:14:45 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"][quote]INCORRECT! Kids are goats! CHILDREN are humans. [/quote]\n\nWhatever... By the way, I have a question. You know when people hide something, it says \"read hidden text\" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say \"answer\". Please tell me![/quote]\r\n\r\nThis occurs from doing [hide=\"Hopefully this helps\"]\n[hide=\"Hopefully this helps\"]\n[/hide][/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">INCORRECT! Kids are goats! CHILDREN are humans.</div>\n</div>\n<br>\nWhatever... By the way, I have a question. You know when people hide something, it says "read hidden text" or something like that? How do you make it say something different? For example, nutz_for2.718281828 made it say "answer". Please tell me!</div>\n</div>\n<br>\nThis occurs from doing <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hopefully this helps</a><div class=\"cmty-hide-content\" style=\"display:none\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hopefully this helps</a><div class=\"cmty-hide-content\" style=\"display:none\"></div></div>", "post_id": 563887, "post_number": 8, "post_time_unix": 1151781008, "post_time_utc": "2006-07-01 19:10:08 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought? \n:huuh:[/quote]\r\n\r\n[hide]they [i][b]had[/b][/i] kittens. just kdding\n\n6667=2167=1512[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought?<br>\n<img src=\"/assets/images/smilies/wtf.gif\" width=\"20\" height=\"25\" alt=\":huuh:\" title=\":huuh:\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">they <i><b>had</b></i> kittens. just kdding<br>\n<br>\n6667=2167=1512</div>", "post_id": 623026, "post_number": 9, "post_time_unix": 1157503129, "post_time_utc": "2006-09-06 00:38:49 UTC", "thanks_received": 2, "user_id": 11899, "username": "moogra" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought? \n:huuh:[/quote]\r\n If the kittens are counted as cats, \r\n[hide] we have : 6667= 1512 cats.\n[/hide]\n If not, we have\n[hide] 666=216 cats[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. 6 men appear, each holding 6 bags. Those bags each have 6 cats. Each cat had 6 kittens. How many cats have the men brought?<br>\n<img src=\"/assets/images/smilies/wtf.gif\" width=\"20\" height=\"25\" alt=\":huuh:\" title=\":huuh:\" class=\"bbcode_smiley\" /></div>\n</div>\nIf the kittens are counted as cats,<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">we have : 6667= 1512 cats.</div><br>\nIf not, we have<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">666=216 cats</div>", "post_id": 623229, "post_number": 10, "post_time_unix": 1157512970, "post_time_utc": "2006-09-06 03:22:50 UTC", "thanks_received": 1, "user_id": 22123, "username": "Chocolate milk" } ], "source": null }	Billy Bob wants to have many pets. A pet company orders some men to take the specified number of cats to Billy Bob's resort. Six men appear, each holding six bags. Each bag contains six cats. Each cat has six kittens. How many cats have the men brought?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/MathematicalHumor", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle", "/Mathematics/RecreationalMathematics/Puzzles/StIvesProblem" ]	Treat each cat and its six kittens as seven cats, then multiply the numbers of men, bags, and cats per bag.	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aops_995469	I just looked at the problem and knew that the answer was $ 1,2,44,45$ (everything about it looks like a telescoping sum), but the pdf version said $ \theta_1 \plus{} \theta_2 \plus{} \theta_3 \cdot \theta_4$ so I was like :?: By the way, I have written up most of the solutions on the wiki - [[Mock AIME 1 2007-2008]] - missing #6 (diagram formatting is kind've difficult to copy) and #11 though. [hide="Solution"] By product-to-sum identities, we have $ 2\sin(x)\sin(1) \equal{} \cos(x \minus{} 1) \minus{} \cos(x \plus{} 1)$. Thus, \[ & \sum_{x \equal{} 2}^{44} [\cos(x \minus{} 1) \minus{} \cos(x \plus{} 1)][1 \plus{} \sec (x \minus{} 1) \sec (x \plus{} 1)] \\ & \equal{} \sum_{x \equal{} 2}^{44} \cos(x \minus{} 1) \minus{} \cos(x \plus{} 1) \plus{} \frac {1}{\cos(x \plus{} 1)} \minus{} \frac {1}{\cos(x \minus{} 1)} \\ & \equal{} \sum_{x \equal{} 2}^{44} \left(\frac {\sin^2(x \plus{} 1)}{\cos(x \plus{} 1)}\right) \minus{} \left(\frac {\sin^2(x \minus{} 1)}{\cos(x \minus{} 1)}\right)\\ & \equal{} \minus{} \frac {\sin(1)}{\cot(1)} \minus{} \frac {\sin(2)}{\cot(2)} \plus{} \frac {\sin(44)}{\cot(44)} \plus{} \frac {\sin(45)}{\cot(45)} \] And $ 1 \plus{} 2 \plus{} 44 \plus{} 45 \equal{} \boxed{092}$. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "15) The sum\r\n\\[ \\sum_{x \\equal{} 2}^{44}2\\sin(x)\\sin(1)[1 \\plus{} \\sec(x \\minus{} 1)\\sec(x \\plus{} 1)]\r\n\\]\r\ncan be written in the form $ \\sum_{n \\equal{} 1}^{4}( \\minus{} 1)^{n}\\frac {\\Phi(\\theta_{n})}{\\Psi(\\theta_{n})}$, where $ \\Phi, \\Psi$ are trigonometric functions and $ \\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4}$ are degrees $ \\in [0,45]$. Find $ \\theta_{1} \\plus{} \\theta_{2} \\plus{} \\theta_{3} \\plus{} \\theta_{4}$.", "content_html": "15) The sum<br>\n<img src=\"//latex.artofproblemsolving.com/0/d/0/0d0ee8b763b1e3388b2be597836f76e0aedf8893.png\" class=\"latexcenter\" alt=\"\\[ \\sum_{x = 2}^{44}2\\sin(x)\\sin(1)[1 + \\sec(x - 1)\\sec(x + 1)]\n\\]\" width=\"336\" height=\"50\" ><br>\ncan be written in the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/2/e/22e9de75e75894909014058a19ba9a3d0dfae783.png\" class=\"latex\" alt=\"$ \\sum_{n = 1}^{4}( - 1)^{n}\\frac {\\Phi(\\theta_{n})}{\\Psi(\\theta_{n})}$\" style=\"vertical-align: -20px\" width=\"120\" height=\"50\" >,</span> where <img src=\"//latex.artofproblemsolving.com/0/0/2/0021a87b6fbf9ff818d91775b25cb4d00d3a77ca.png\" class=\"latex\" alt=\"$ \\Phi, \\Psi$\" style=\"vertical-align: -3px\" width=\"34\" height=\"16\" > are trigonometric functions and <img src=\"//latex.artofproblemsolving.com/1/6/9/1699db3334374b56b5e755d6c52af37eff76f98e.png\" class=\"latex\" alt=\"$ \\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4}$\" style=\"vertical-align: -3px\" width=\"87\" height=\"16\" > are degrees <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/e/3/1e36337959f9b1a356a8fedf4299014218aa3fc6.png\" class=\"latex\" alt=\"$ \\in [0,45]$\" style=\"vertical-align: -5px\" width=\"60\" height=\"18\" >.</span> Find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/6/2/3626d6f9e3470f398c9bf056d3719a22bf627f4a.png\" class=\"latex\" alt=\"$ \\theta_{1} + \\theta_{2} + \\theta_{3} + \\theta_{4}$\" style=\"vertical-align: -2px\" width=\"129\" height=\"15\" >.</span>", "post_id": 4413011, "post_number": 1, "post_time_unix": 1207099273, "post_time_utc": "2008-04-02 01:21:13 UTC", "thanks_received": 2, "user_id": 16706, "username": "Tenoreoz" }, { "attachments": [], "content_bbcode": "We expand this sum and apply DeMoivre's. After a lot of algebra and complex number work, we see that the answer is $ \\boxed{092}$.", "content_html": "We expand this sum and apply DeMoivre's. After a lot of algebra and complex number work, we see that the answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/d/f/fdf80e5af46b15ee13875e0a732cdac5c138883a.png\" class=\"latex\" alt=\"$ \\boxed{092}$\" style=\"vertical-align: -5px\" width=\"39\" height=\"23\" >.</span>", "post_id": 4413012, "post_number": 2, "post_time_unix": 1207099463, "post_time_utc": "2008-04-02 01:24:23 UTC", "thanks_received": 2, "user_id": 9401, "username": "#H34N1" }, { "attachments": [], "content_bbcode": "I just looked at the problem and knew that the answer was $ 1,2,44,45$ (everything about it looks like a telescoping sum), but the pdf version said $ \\theta_1 \\plus{} \\theta_2 \\plus{} \\theta_3 \\cdot \\theta_4$ so I was like :?:\r\n\r\nBy the way, I have written up most of the solutions on the wiki - [[Mock AIME 1 2007-2008]] - missing #6 (diagram formatting is kind've difficult to copy) and #11 though.\r\n\r\n[hide=\"Solution\"]\nBy product-to-sum identities, we have $ 2\\sin(x)\\sin(1) \\equal{} \\cos(x \\minus{} 1) \\minus{} \\cos(x \\plus{} 1)$. Thus,\n\\[ & \\sum_{x \\equal{} 2}^{44} [\\cos(x \\minus{} 1) \\minus{} \\cos(x \\plus{} 1)][1 \\plus{} \\sec (x \\minus{} 1) \\sec (x \\plus{} 1)] \\\\\n& \\equal{} \\sum_{x \\equal{} 2}^{44} \\cos(x \\minus{} 1) \\minus{} \\cos(x \\plus{} 1) \\plus{} \\frac {1}{\\cos(x \\plus{} 1)} \\minus{} \\frac {1}{\\cos(x \\minus{} 1)} \\\\\n& \\equal{} \\sum_{x \\equal{} 2}^{44} \\left(\\frac {\\sin^2(x \\plus{} 1)}{\\cos(x \\plus{} 1)}\\right) \\minus{} \\left(\\frac {\\sin^2(x \\minus{} 1)}{\\cos(x \\minus{} 1)}\\right)\\\\ & \\equal{} \\minus{} \\frac {\\sin(1)}{\\cot(1)} \\minus{} \\frac {\\sin(2)}{\\cot(2)} \\plus{} \\frac {\\sin(44)}{\\cot(44)} \\plus{} \\frac {\\sin(45)}{\\cot(45)}\n\\]\nAnd $ 1 \\plus{} 2 \\plus{} 44 \\plus{} 45 \\equal{} \\boxed{092}$.\n[/hide]", "content_html": "I just looked at the problem and knew that the answer was <img src=\"//latex.artofproblemsolving.com/6/a/b/6abd9e50655b4e8f8b563cd0caf6f6f1b0a68ff6.png\" class=\"latex\" alt=\"$ 1,2,44,45$\" style=\"vertical-align: -3px\" width=\"78\" height=\"16\" > (everything about it looks like a telescoping sum), but the pdf version said <img src=\"//latex.artofproblemsolving.com/c/8/d/c8d28313fcf19dbbefd80a10d412dfd197ffe8aa.png\" class=\"latex\" alt=\"$ \\theta_1 + \\theta_2 + \\theta_3 \\cdot \\theta_4$\" style=\"vertical-align: -2px\" width=\"120\" height=\"15\" > so I was like <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" />:<br>\n<br>\nBy the way, I have written up most of the solutions on the wiki - <a href=\"/wiki/index.php/Mock_AIME_1_2007_2008\" class=\"bbcode_wiki\">Mock AIME 1 2007-2008</a> - missing #6 (diagram formatting is kind've difficult to copy) and #11 though.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Solution</a><div class=\"cmty-hide-content\" style=\"display:none\">By product-to-sum identities, we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/9/2/e92583da41a793e391724e2b7664dfbe385ce249.png\" class=\"latex\" alt=\"$ 2\\sin(x)\\sin(1) = \\cos(x - 1) - \\cos(x + 1)$\" style=\"vertical-align: -4px\" width=\"312\" height=\"18\" >.</span> Thus,<br>\n<pre class=\"aopscode-error aopscode-latex-error\">\\[ & \\sum_{x = 2}^{44} [\\cos(x - 1) - \\cos(x + 1)][1 + \\sec (x - 1) \\sec (x + 1)] \\\\\n& = \\sum_{x = 2}^{44} \\cos(x - 1) - \\cos(x + 1) + \\frac {1}{\\cos(x + 1)} - \\frac {1}{\\cos(x - 1)} \\\\\n& = \\sum_{x = 2}^{44} \\left(\\frac {\\sin^2(x + 1)}{\\cos(x + 1)}\\right) - \\left(\\frac {\\sin^2(x - 1)}{\\cos(x - 1)}\\right)\\\\ & = - \\frac {\\sin(1)}{\\cot(1)} - \\frac {\\sin(2)}{\\cot(2)} + \\frac {\\sin(44)}{\\cot(44)} + \\frac {\\sin(45)}{\\cot(45)}\n\\]</pre><br>\nAnd <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/1/0/210d16f7d11fb4aa8c8b4582073c7142d9133787.png\" class=\"latex\" alt=\"$ 1 + 2 + 44 + 45 = \\boxed{092}$\" style=\"vertical-align: -5px\" width=\"184\" height=\"23\" >.</span></div>", "post_id": 4413013, "post_number": 3, "post_time_unix": 1207177134, "post_time_utc": "2008-04-02 22:58:54 UTC", "thanks_received": 2, "user_id": 26584, "username": "azjps" } ], "source": null }	15) The sum \[ \sum_{x \equal{} 2}^{44}2\sin(x)\sin(1)[1 \plus{} \sec(x \minus{} 1)\sec(x \plus{} 1)] \] can be written in the form $ \sum_{n \equal{} 1}^{4}( \minus{} 1)^{n}\frac {\Phi(\theta_{n})}{\Psi(\theta_{n})}$, where $ \Phi, \Psi$ are trigonometric functions and $ \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$ are degrees $ \in [0,45]$. Find $ \theta_{1} \plus{} \theta_{2} \plus{} \theta_{3} \plus{} \theta_{4}$.	[ "/Mathematics/RecreationalMathematics" ]	Use the product‑to‑sum identity to rewrite 2sin x sin 1 as a difference of cosines, turning the whole sum into a telescoping series.	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aops_995479	Okay so write (1+i)^17 - (1-i)^17 in its polar form which would give us (sqrt 2 (cis pi/4))^17 - (sqrt 2(cis -pi/4))^17 use the power rule/whatever it is called to simplify (sqrt2)^17 (cis 17pi/4) - 17sqrt 2 (cis -17pi/4) take out 17 sqrt 2 because it is in both terms (sqrt2)^17((cis 17pi/4) - (cis -17pi/4)) add -4 pi to the first angle and 4pi to the second one (sqrt2)^17((cispi/4)-(cis -pi/4)) 512 ((cis pi/4) - (cis -pi/4) simplify from here and you get 512 i and absolute value of 512 i is 512 ....so there's the answer :rotfl:	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "5) Let $ S \\equal{} (1 \\plus{} i)^{17} \\minus{} (1 \\minus{} i)^{17}$, where $ i \\equal{} \\sqrt { \\minus{} 1}$. Find $ \|S\|$.", "content_html": "5) Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/2/c/f2cacaf80ec880030f08e0ff0cd60eeab30b9d13.png\" class=\"latex\" alt=\"$ S = (1 + i)^{17} - (1 - i)^{17}$\" style=\"vertical-align: -4px\" width=\"189\" height=\"19\" >,</span> where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/1/8/818199050466675a3bc97676f0ecbfe8c92cfb10.png\" class=\"latex\" alt=\"$ i = \\sqrt { - 1}$\" style=\"vertical-align: -2px\" width=\"69\" height=\"18\" >.</span> Find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/c/3/fc363e0ecd010c4748d3db8d5d4e498a03b09c16.png\" class=\"latex\" alt=\"$ \|S\|$\" style=\"vertical-align: -4px\" width=\"20\" height=\"18\" >.</span>", "post_id": 4413034, "post_number": 1, "post_time_unix": 1207098828, "post_time_utc": "2008-04-02 01:13:48 UTC", "thanks_received": 2, "user_id": 16706, "username": "Tenoreoz" }, { "attachments": [], "content_bbcode": "I got 256.\r\n\r\nAnybody got the same ans as me? By the way, which method do you guys use?", "content_html": "I got 256.<br>\n<br>\nAnybody got the same ans as me? By the way, which method do you guys use?", "post_id": 4413035, "post_number": 2, "post_time_unix": 1207106400, "post_time_utc": "2008-04-02 03:20:00 UTC", "thanks_received": 2, "user_id": 35111, "username": "tine" }, { "attachments": [], "content_bbcode": "Okay so write (1+i)^17 - (1-i)^17 in its polar form\r\nwhich would give us\r\n\r\n(sqrt 2 (cis pi/4))^17 - (sqrt 2(cis -pi/4))^17\r\n\r\n\r\nuse the power rule/whatever it is called to simplify\r\n\r\n(sqrt2)^17 (cis 17pi/4) - 17sqrt 2 (cis -17pi/4)\r\ntake out 17 sqrt 2 because it is in both terms\r\n\r\n(sqrt2)^17((cis 17pi/4) - (cis -17pi/4))\r\nadd -4 pi to the first angle and 4pi to the second one\r\n\r\n(sqrt2)^17((cispi/4)-(cis -pi/4))\r\n512 ((cis pi/4) - (cis -pi/4)\r\nsimplify from here and you get 512 i and absolute value of 512 i is 512 ....so there's the answer :rotfl:", "content_html": "Okay so write (1+i)^17 - (1-i)^17 in its polar form<br>\nwhich would give us<br>\n<br>\n(sqrt 2 (cis pi/4))^17 - (sqrt 2(cis -pi/4))^17<br>\n<br>\n<br>\nuse the power rule/whatever it is called to simplify<br>\n<br>\n(sqrt2)^17 (cis 17pi/4) - 17sqrt 2 (cis -17pi/4)<br>\ntake out 17 sqrt 2 because it is in both terms<br>\n<br>\n(sqrt2)^17((cis 17pi/4) - (cis -17pi/4))<br>\nadd -4 pi to the first angle and 4pi to the second one<br>\n<br>\n(sqrt2)^17((cispi/4)-(cis -pi/4))<br>\n512 ((cis pi/4) - (cis -pi/4)<br>\nsimplify from here and you get 512 i and absolute value of 512 i is 512 ....so there's the answer <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" />", "post_id": 4413036, "post_number": 3, "post_time_unix": 1208265211, "post_time_utc": "2008-04-15 13:13:31 UTC", "thanks_received": 2, "user_id": 37307, "username": "vsingh21" } ], "source": null }	Let $S=(1+i)^{17}-(1-i)^{17}$, where $i=\sqrt{-1}$. Find $\|S\|$.	[ "/Mathematics/Algebra/GeneralAlgebra/Algebra" ]	Convert (1±i) to polar form and apply De Moivre’s theorem to simplify the difference.	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aops_99570	Ah, the classic example of a function that's not Riemann integrable. Making the interval $[0,12]$ sounds like whoever posed this problem was bored with the usual - not that it makes any real difference. The key observation: any interval of positive length contains both rational and irrational numbers. Hence the supremum of $f$ over that interval is 1 and the infinum is 0. Let $P$ be any partition of $[0,12].$ Then $U(f,P)=\sum_{j}1\cdot\Delta x_{j}=12$ and $L(f,P)=\sum_{j}0\cdot\Delta x_{j}=0$ The upper integral is 12, the lower integral is zero, and the function is not Darboux integrable (hence not Riemann integrable).	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "If f(x) = (1 if x is rational) and (0 if x is irrational) on the interval [0,12]... how do you find upper and lower partitions?", "content_html": "If f(x) = (1 if x is rational) and (0 if x is irrational) on the interval [0,12]... how do you find upper and lower partitions?", "post_id": 562107, "post_number": 1, "post_time_unix": 1151623497, "post_time_utc": "2006-06-29 23:24:57 UTC", "thanks_received": 2, "user_id": 20690, "username": "Casual Mathematician" }, { "attachments": [], "content_bbcode": "Ah, the classic example of a function that's not Riemann integrable. Making the interval $[0,12]$ sounds like whoever posed this problem was bored with the usual - not that it makes any real difference.\r\n\r\nThe key observation: any interval of positive length contains both rational and irrational numbers. Hence the supremum of $f$ over that interval is 1 and the infinum is 0.\r\n\r\nLet $P$ be any partition of $[0,12].$ Then\r\n\r\n$U(f,P)=\\sum_{j}1\\cdot\\Delta x_{j}=12$ and\r\n\r\n$L(f,P)=\\sum_{j}0\\cdot\\Delta x_{j}=0$\r\n\r\nThe upper integral is 12, the lower integral is zero, and the function is not Darboux integrable (hence not Riemann integrable).", "content_html": "Ah, the classic example of a function that's not Riemann integrable. Making the interval <img src=\"//latex.artofproblemsolving.com/6/e/b/6eb3235d5b50b22e44d7dcffe8d8542843de6527.png\" class=\"latex\" alt=\"$[0,12]$\" style=\"vertical-align: -5px\" width=\"43\" height=\"18\" > sounds like whoever posed this problem was bored with the usual - not that it makes any real difference.<br>\n<br>\nThe key observation: any interval of positive length contains both rational and irrational numbers. Hence the supremum of <img src=\"//latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png\" class=\"latex\" alt=\"$f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > over that interval is 1 and the infinum is 0.<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > be any partition of <img src=\"//latex.artofproblemsolving.com/b/a/f/bafa8fefadf57a49d7719452b10f9a1558716a40.png\" class=\"latex\" alt=\"$[0,12].$\" style=\"vertical-align: -5px\" width=\"48\" height=\"18\" > Then<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/b/0/fb018d7a25e99f8f7722f04ae03e9b7f93d9dfbc.png\" class=\"latex\" alt=\"$U(f,P)=\\sum_{j}1\\cdot\\Delta x_{j}=12$\" style=\"vertical-align: -22px\" width=\"210\" height=\"40\" > and<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/4/9/b496416c5099ab5ce60dd4d2b85e279571a1e3e2.png\" class=\"latex\" alt=\"$L(f,P)=\\sum_{j}0\\cdot\\Delta x_{j}=0$\" style=\"vertical-align: -22px\" width=\"199\" height=\"40\" ><br>\n<br>\nThe upper integral is 12, the lower integral is zero, and the function is not Darboux integrable (hence not Riemann integrable).", "post_id": 563170, "post_number": 2, "post_time_unix": 1151715131, "post_time_utc": "2006-07-01 00:52:11 UTC", "thanks_received": 2, "user_id": 2948, "username": "Kent Merryfield" } ], "source": null }	Let $f:[0,12]\to\mathbb{R}$ be defined by \[ f(x)=\begin{cases} 1,& x\in\mathbb{Q},\\[4pt] 0,& x\notin\mathbb{Q}. \end{cases} \] On the interval $[0,12]$, find the upper and lower Riemann sums (upper and lower Darboux sums) for any partition.	[ "/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus", "/Mathematics/CalculusandAnalysis/Calculus/Integrals/DefiniteIntegrals", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/RealFunction", "/Mathematics/CalculusandAnalysis/Functions/Scalar-ValuedFunction", "/Mathematics/CalculusandAnalysis/Functions/ScalarFunction", "/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction" ]	Every non‑degenerate subinterval contains both rationals and irrationals, so sup f=1 and inf f=0 on each subinterval.	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aops_99582	[quote="math92"]Find the slope of this line: 6y = -1 + x[/quote] [hide]$y=\frac{1}{6}x-\frac{1}{6}$ Therefore, the slope is $\boxed{\frac{1}{6}}$[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find the slope of this line:\r\n6y = -1 + x", "content_html": "Find the slope of this line:<br>\n6y = -1 + x", "post_id": 562228, "post_number": 1, "post_time_unix": 1151631237, "post_time_utc": "2006-06-30 01:33:57 UTC", "thanks_received": 2, "user_id": 8131, "username": "math92" }, { "attachments": [], "content_bbcode": "Slope is $\\frac{1}{6}$", "content_html": "Slope is <img src=\"//latex.artofproblemsolving.com/4/3/b/43ba19be0e30b99142416ebfbe17670437c8453e.png\" class=\"latex\" alt=\"$\\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" >", "post_id": 562427, "post_number": 2, "post_time_unix": 1151648053, "post_time_utc": "2006-06-30 06:14:13 UTC", "thanks_received": 1, "user_id": 17435, "username": "ashwinrk_jain" }, { "attachments": [], "content_bbcode": "[hide]6y=x-1, divide both sides by 6 to get it in the form y=mx+b, where m is the slope. m=$\\frac{1}{6}$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">6y=x-1, divide both sides by 6 to get it in the form y=mx+b, where m is the slope. m<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/4/3/b/43ba19be0e30b99142416ebfbe17670437c8453e.png\" class=\"latex\" alt=\"$\\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ></span></div>", "post_id": 562766, "post_number": 3, "post_time_unix": 1151683078, "post_time_utc": "2006-06-30 15:57:58 UTC", "thanks_received": 2, "user_id": 8477, "username": "easyas3.14159..." }, { "attachments": [], "content_bbcode": "[quote=\"math92\"]Find the slope of this line:\n6y = -1 + x[/quote]\r\n[hide]\nLet y=mx+b. By definition m is the slope of the line. So we have to convert the equation into this form.\n\nThis is $y=\\frac{1}{6} x -1$.\n\nSo the slope is $\\frac{1}{6}$[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">math92 wrote:</div>\n<div class=\"bbcode_quote_body\">Find the slope of this line:<br>\n6y = -1 + x</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let y=mx+b. By definition m is the slope of the line. So we have to convert the equation into this form.<br>\n<br>\nThis is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/c/9/5c9a92e23c50778589878ed1a751dd4b7a3f2b36.png\" class=\"latex\" alt=\"$y=\\frac{1}{6} x -1$\" style=\"vertical-align: -12px\" width=\"87\" height=\"37\" >.</span><br>\n<br>\nSo the slope is <img src=\"//latex.artofproblemsolving.com/4/3/b/43ba19be0e30b99142416ebfbe17670437c8453e.png\" class=\"latex\" alt=\"$\\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ></div>", "post_id": 562777, "post_number": 4, "post_time_unix": 1151683389, "post_time_utc": "2006-06-30 16:03:09 UTC", "thanks_received": 1, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "[quote=\"math92\"]Find the slope of this line:\n6y = -1 + x[/quote]\r\n\r\n[hide]$y=\\frac{1}{6}x-\\frac{1}{6}$\nTherefore, the slope is $\\boxed{\\frac{1}{6}}$[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">math92 wrote:</div>\n<div class=\"bbcode_quote_body\">Find the slope of this line:<br>\n6y = -1 + x</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/6/c/a6c4494038719b2f7c5459d8e52754a9968f3e61.png\" class=\"latex\" alt=\"$y=\\frac{1}{6}x-\\frac{1}{6}$\" style=\"vertical-align: -12px\" width=\"91\" height=\"37\" ><br>\nTherefore, the slope is <img src=\"//latex.artofproblemsolving.com/c/8/5/c857173077b149d2227aee6f62e84ff6f7ead2c6.png\" class=\"latex\" alt=\"$\\boxed{\\frac{1}{6}}$\" style=\"vertical-align: -18px\" width=\"24\" height=\"48\" ></div>", "post_id": 563117, "post_number": 5, "post_time_unix": 1151708191, "post_time_utc": "2006-06-30 22:56:31 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[hide=\"slope\"]$\\frac16$[/hide]\n[hide=\"x int\"]$1$[/hide]\n[hide=\"y int\"]$\\frac{-1}{6}$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">slope</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/8/b/a8b7dd31c63b399cd72fe186a628adba5df3cad3.png\" class=\"latex\" alt=\"$\\frac16$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ></div><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">x int</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" ></div><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">y int</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/4/6/4/464195ed60cacce55d588d08e61f51836d363fa9.png\" class=\"latex\" alt=\"$\\frac{-1}{6}$\" style=\"vertical-align: -12px\" width=\"25\" height=\"37\" ></div>", "post_id": 564277, "post_number": 6, "post_time_unix": 1151813424, "post_time_utc": "2006-07-02 04:10:24 UTC", "thanks_received": 1, "user_id": 8960, "username": "bpms" } ], "source": null }	Find the slope of the line: \[ 6y = -1 + x. \]	[ "/Mathematics/Algebra/LinearAlgebra/LinearSystemsofEquations/LinearEquation" ]	Rewrite the equation in slope‑intercept form and read off the coefficient of x as the slope.	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aops_99587	[hide]At Hamden Middle you have $\frac{492}{120}=4.1$ kids per minute. At Wintergreen you have $\frac{63}{240}=.2625$ kids per minute. The difference is 3.8375 kids per minute.[/hide] hmm...I wonder how you graduate part of a kid. :)	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "If it takes 4 hours to graduate 63 kids at Wintergreen Interdistrict Magnet School and only 2 hours to graduate 492 kids at nearby Hamden Middle School, approximately how many more kids, to the nearest ten-thousandth, graduate from Hamden Middle per minute? You may not use a calculator and assume the rates are constant.", "content_html": "If it takes 4 hours to graduate 63 kids at Wintergreen Interdistrict Magnet School and only 2 hours to graduate 492 kids at nearby Hamden Middle School, approximately how many more kids, to the nearest ten-thousandth, graduate from Hamden Middle per minute? You may not use a calculator and assume the rates are constant.", "post_id": 562307, "post_number": 1, "post_time_unix": 1151634210, "post_time_utc": "2006-06-30 02:23:30 UTC", "thanks_received": 2, "user_id": 17514, "username": "Ignite168" }, { "attachments": [], "content_bbcode": "[hide]It takes 4 hours to graduate 984 kids at the middle school so the answer is $\\frac{984-63}{4\\cdot 60}=\\frac{921}{240}=\\frac{307}{80}\\approx \\boxed{4}$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">It takes 4 hours to graduate 984 kids at the middle school so the answer is <img src=\"//latex.artofproblemsolving.com/c/e/9/ce966470af1d804b2b88352d06e6a16e9871da61.png\" class=\"latex\" alt=\"$\\frac{984-63}{4\\cdot 60}=\\frac{921}{240}=\\frac{307}{80}\\approx \\boxed{4}$\" style=\"vertical-align: -13px\" width=\"227\" height=\"38\" ></div>", "post_id": 562309, "post_number": 2, "post_time_unix": 1151634556, "post_time_utc": "2006-06-30 02:29:16 UTC", "thanks_received": 2, "user_id": 11029, "username": "ch1n353ch3s54a1l" }, { "attachments": [], "content_bbcode": "Woops, problem should say to the nearest ten-thousandth, not hundredth.", "content_html": "Woops, problem should say to the nearest ten-thousandth, not hundredth.", "post_id": 562314, "post_number": 3, "post_time_unix": 1151634706, "post_time_utc": "2006-06-30 02:31:46 UTC", "thanks_received": 2, "user_id": 17514, "username": "Ignite168" }, { "attachments": [], "content_bbcode": "[hide]At Hamden Middle you have $\\frac{492}{120}=4.1$ kids per minute. At Wintergreen you have $\\frac{63}{240}=.2625$ kids per minute. The difference is 3.8375 kids per minute.[/hide]\r\nhmm...I wonder how you graduate part of a kid. :)", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">At Hamden Middle you have <img src=\"//latex.artofproblemsolving.com/b/c/4/bc4f9d6745398fce5d5d213b2d37937a9b8408b1.png\" class=\"latex\" alt=\"$\\frac{492}{120}=4.1$\" style=\"vertical-align: -13px\" width=\"77\" height=\"38\" > kids per minute. At Wintergreen you have <img src=\"//latex.artofproblemsolving.com/5/6/f/56f06113ed5946c649f62d43d1ef2a1dc86593cc.png\" class=\"latex\" alt=\"$\\frac{63}{240}=.2625$\" style=\"vertical-align: -13px\" width=\"96\" height=\"38\" > kids per minute. The difference is 3.8375 kids per minute.</div><br>\nhmm...I wonder how you graduate part of a kid. <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 562775, "post_number": 4, "post_time_unix": 1151683304, "post_time_utc": "2006-06-30 16:01:44 UTC", "thanks_received": 2, "user_id": 8477, "username": "easyas3.14159..." }, { "attachments": [], "content_bbcode": "[hide]At Hamden Middle, it will take $\\frac{492}{120}=4.1$ kids for every minute.\n\nAt Wintergree it will take $\\frac{63}{240}=0.2625$ kids for every minute. The difference hence is $3.8375$ kids per minute.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">At Hamden Middle, it will take <img src=\"//latex.artofproblemsolving.com/b/c/4/bc4f9d6745398fce5d5d213b2d37937a9b8408b1.png\" class=\"latex\" alt=\"$\\frac{492}{120}=4.1$\" style=\"vertical-align: -13px\" width=\"77\" height=\"38\" > kids for every minute.<br>\n<br>\nAt Wintergree it will take <img src=\"//latex.artofproblemsolving.com/a/d/a/ada190d30be4855a3a79eddfb997df51b6ccd8dd.png\" class=\"latex\" alt=\"$\\frac{63}{240}=0.2625$\" style=\"vertical-align: -13px\" width=\"105\" height=\"38\" > kids for every minute. The difference hence is <img src=\"//latex.artofproblemsolving.com/d/e/0/de07aa55a552a4e8491fe4788de6d2137a3a2d7b.png\" class=\"latex\" alt=\"$3.8375$\" width=\"49\" height=\"12\" > kids per minute.</div>", "post_id": 562781, "post_number": 5, "post_time_unix": 1151683622, "post_time_utc": "2006-06-30 16:07:02 UTC", "thanks_received": 1, "user_id": 11714, "username": "mathgeniuse^ln(x)" } ], "source": null }	If it takes 4 hours to graduate 63 kids at Wintergreen Interdistrict Magnet School and 2 hours to graduate 492 kids at nearby Hamden Middle School, approximately how many more kids, to the nearest ten-thousandth, graduate from Hamden Middle per minute? You may not use a calculator and assume the rates are constant.	[ "/Mathematics/Algebra/RateProblems", "/Mathematics/AppliedMathematics" ]	Convert the given time periods to minutes, compute each school's graduation rate, then subtract the rates.	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aops_99606	[b]Theorem.[/b][i] Every projective module is flat.[/i] [b]Theorem (Bass).[/b] [i]The following conditions are equivalent: (а) Every right flat $R$-module is projective. (b) $R/rad(R)$ is semisimple and for any sequence $a_{1},a_{2},\ldots$ from $rad(R)$ there exists $n\in\mathbb N$ such that $a_{n}a_{n-1}\ldots a_{1}=0$. (c) $R$ satisfies the descending chain condition for principal left ideals.[/i] The latter theorem is extremely hard to prove.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "hi can you help me ?\r\nevery r_module flat is projective?", "content_html": "hi can you help me ?<br>\nevery r_module flat is projective?", "post_id": 562364, "post_number": 1, "post_time_unix": 1151639729, "post_time_utc": "2006-06-30 03:55:29 UTC", "thanks_received": 2, "user_id": 14950, "username": "sepidetaghizade" }, { "attachments": [], "content_bbcode": "Well, what is $r$¿ The reals¿ Any ring (some doubts)¿", "content_html": "Well, what is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png\" class=\"latex\" alt=\"$r$\" width=\"8\" height=\"8\" >¿</span> The reals¿ Any ring (some doubts)¿", "post_id": 562561, "post_number": 2, "post_time_unix": 1151670639, "post_time_utc": "2006-06-30 12:30:39 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "[b]Theorem.[/b][i] Every projective module is flat.[/i]\r\n\r\n[b]Theorem (Bass).[/b] [i]The following conditions are equivalent:\n(а) Every right flat $R$-module is projective.\n(b) $R/rad(R)$ is semisimple and for any sequence $a_{1},a_{2},\\ldots$ from $rad(R)$ there exists $n\\in\\mathbb N$ such that $a_{n}a_{n-1}\\ldots a_{1}=0$.\n(c) $R$ satisfies the descending chain condition for principal left ideals.[/i]\r\n\r\nThe latter theorem is extremely hard to prove.", "content_html": "<b>Theorem.</b><i> Every projective module is flat.</i><br>\n<br>\n<b>Theorem (Bass).</b> <i>The following conditions are equivalent:<br>\n(а) Every right flat <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >-</span>module is projective.<br>\n(b) <img src=\"//latex.artofproblemsolving.com/6/7/9/67937cd408863fd200ac3e5fa29f1ab89f47df6c.png\" class=\"latex\" alt=\"$R/rad(R)$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" > is semisimple and for any sequence <img src=\"//latex.artofproblemsolving.com/6/9/2/692dafa03dc84caa8b39e7425f66381a0d6fd909.png\" class=\"latex\" alt=\"$a_{1},a_{2},\\ldots$\" style=\"vertical-align: -3px\" width=\"70\" height=\"11\" > from <img src=\"//latex.artofproblemsolving.com/0/7/d/07d59afe25080e157ee0ac86c1c4ad47043da2bb.png\" class=\"latex\" alt=\"$rad(R)$\" style=\"vertical-align: -4px\" width=\"54\" height=\"18\" > there exists <img src=\"//latex.artofproblemsolving.com/5/b/3/5b3916951a15cb10c0b14a2b8920efbeef759035.png\" class=\"latex\" alt=\"$n\\in\\mathbb N$\" style=\"vertical-align: -1px\" width=\"45\" height=\"13\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/e/9/4e9d3a44cf3c0ab9a8d6438d4f8ecbaa8c469ebd.png\" class=\"latex\" alt=\"$a_{n}a_{n-1}\\ldots a_{1}=0$\" style=\"vertical-align: -2px\" width=\"131\" height=\"15\" >.</span><br>\n(c) <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > satisfies the descending chain condition for principal left ideals.</i><br>\n<br>\nThe latter theorem is extremely hard to prove.", "post_id": 563006, "post_number": 3, "post_time_unix": 1151699560, "post_time_utc": "2006-06-30 20:32:40 UTC", "thanks_received": 2, "user_id": 19435, "username": "lofar" }, { "attachments": [], "content_bbcode": "@sepidetaghizade: No ... take $R = \\mathbb{Z}, M = \\mathbb{Q}$. Then $M$ is a flat $R$-module. But every projective $R$-module is free (since $R$ is a PID, submodules of free $R$-modules are also free), and clearly, $M$ is not.", "content_html": "@sepidetaghizade: No ... take <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/3/f/d3ff8d4f7ebc8c2b5064d3ea7533697746b304f2.png\" class=\"latex\" alt=\"$R = \\mathbb{Z}, M = \\mathbb{Q}$\" style=\"vertical-align: -3px\" width=\"115\" height=\"16\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is a flat <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >-</span>module. But every projective <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >-</span>module is free (since <img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" > is a PID, submodules of free <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/f/eff43e84f8a3bcf7b6965f0a3248bc4d3a9d0cd4.png\" class=\"latex\" alt=\"$R$\" width=\"14\" height=\"12\" >-</span>modules are also free), and clearly, <img src=\"//latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > is not.", "post_id": 563513, "post_number": 4, "post_time_unix": 1151752873, "post_time_utc": "2006-07-01 11:21:13 UTC", "thanks_received": 2, "user_id": 18458, "username": "-oo-" } ], "source": null }	hi can you help me ? every r_module flat is projective?	[ "/Mathematics/Algebra/HomologicalAlgebra/ChainCondition", "/Mathematics/Algebra/HomologicalAlgebra/DescendingChainCondition", "/Mathematics/Algebra/HomologicalAlgebra/FlatModule", "/Mathematics/Algebra/HomologicalAlgebra/Module", "/Mathematics/Algebra/HomologicalAlgebra/ProjectiveModule", "/Mathematics/Algebra/HomologicalAlgebra/R-Module", "/Mathematics/Algebra/RingTheory/JacobsonRadical", "/Mathematics/Algebra/RingTheory/LeftIdeal", "/Mathematics/Algebra/RingTheory/RightIdeal", "/Mathematics/Algebra/RingTheory/SemisimpleRing" ]	Apply Bass's theorem that characterizes rings for which every right flat module is projective.	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aops_996097	a) silly and well-known b) silly and well-known c) less silly and less well-known so hmm 200...101 has how many 2's it would be [200/2]+[200/4]+[200/8]+... - [100/2]+[100/4]+... = 100+50+25+12+6+3+1 - 50-25-12-6-3-1 then we divide by 100! which is subtracting 50+25+12+6+3+1 = 97 so 3?	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "too easy for any forum so i'll just post em here:\r\n1. Find $ \\frac{\\text{d}[f(x)]}{dx}$ if f(x) = cos x.\r\n2. Prove $ (n)(n\\plus{}1)(n\\plus{}2)$ is divisble by three. \r\n3. What is the greatest power of 2 that divides $ \\binom{200}{100}$", "content_html": "too easy for any forum so i'll just post em here:<br>\n1. Find <img src=\"//latex.artofproblemsolving.com/4/1/0/410701c9cfdddaac18a29a8cbab8de23c77da738.png\" class=\"latex\" alt=\"$ \\frac{\\text{d}[f(x)]}{dx}$\" style=\"vertical-align: -12px\" width=\"57\" height=\"38\" > if f(x) = cos x.<br>\n2. Prove <img src=\"//latex.artofproblemsolving.com/1/0/f/10f86f5520ac84cec7d75e3978fd21d1312da106.png\" class=\"latex\" alt=\"$ (n)(n+1)(n+2)$\" style=\"vertical-align: -4px\" width=\"136\" height=\"18\" > is divisble by three.<br>\n3. What is the greatest power of 2 that divides <img src=\"//latex.artofproblemsolving.com/d/c/4/dc464bd821bba491b2ae26444508140dac87ba01.png\" class=\"latex\" alt=\"$ \\binom{200}{100}$\" style=\"vertical-align: -22px\" width=\"51\" height=\"53\" >", "post_id": 4414107, "post_number": 1, "post_time_unix": 1220209215, "post_time_utc": "2008-08-31 19:00:15 UTC", "thanks_received": 2, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "a) silly and well-known\r\nb) silly and well-known\r\nc) less silly and less well-known so\r\n\r\nhmm 200...101 has how many 2's\r\n\r\nit would be [200/2]+[200/4]+[200/8]+... - [100/2]+[100/4]+...\r\n= 100+50+25+12+6+3+1 - 50-25-12-6-3-1\r\n\r\nthen we divide by 100! which is subtracting 50+25+12+6+3+1 = 97\r\n\r\nso 3?", "content_html": "a) silly and well-known<br>\nb) silly and well-known<br>\nc) less silly and less well-known so<br>\n<br>\nhmm 200...101 has how many 2's<br>\n<br>\nit would be [200/2]+[200/4]+[200/8]+... - [100/2]+[100/4]+...<br>\n= 100+50+25+12+6+3+1 - 50-25-12-6-3-1<br>\n<br>\nthen we divide by 100! which is subtracting 50+25+12+6+3+1 = 97<br>\n<br>\nso 3?", "post_id": 4414108, "post_number": 2, "post_time_unix": 1220228282, "post_time_utc": "2008-09-01 00:18:02 UTC", "thanks_received": 2, "user_id": 18909, "username": "not_trig" }, { "attachments": [], "content_bbcode": "no that was a very very easy AIME problem made slightly harder someone who made MOP should know :P", "content_html": "no that was a very very easy AIME problem made slightly harder someone who made MOP should know <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />", "post_id": 4414109, "post_number": 3, "post_time_unix": 1220236613, "post_time_utc": "2008-09-01 02:36:53 UTC", "thanks_received": 2, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "97 how did you come up with 3?!!!! you wrote 97!\r\ncan you do the second one please oh please pretty please?", "content_html": "97 how did you come up with 3?!!!! you wrote 97!<br>\ncan you do the second one please oh please pretty please?", "post_id": 4414110, "post_number": 4, "post_time_unix": 1220293673, "post_time_utc": "2008-09-01 18:27:53 UTC", "thanks_received": 2, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "Hint for the second one (for Poincare): think of the algebraic definition of a factorial.", "content_html": "Hint for the second one (for Poincare): think of the algebraic definition of a factorial.", "post_id": 4414111, "post_number": 5, "post_time_unix": 1220391262, "post_time_utc": "2008-09-02 21:34:22 UTC", "thanks_received": 2, "user_id": 23588, "username": "n0vad3m0n" }, { "attachments": [], "content_bbcode": "for #2 is n an integer? if so then do pigeon hole principle", "content_html": "for #2 is n an integer? if so then do pigeon hole principle", "post_id": 4414112, "post_number": 6, "post_time_unix": 1221252400, "post_time_utc": "2008-09-12 20:46:40 UTC", "thanks_received": 2, "user_id": 47992, "username": "Sephiroth" }, { "attachments": [], "content_bbcode": "actually, you kinda reduced the AIME problem to one that could very easily be on a mathcounts test", "content_html": "actually, you kinda reduced the AIME problem to one that could very easily be on a mathcounts test", "post_id": 4414113, "post_number": 7, "post_time_unix": 1229601377, "post_time_utc": "2008-12-18 11:56:17 UTC", "thanks_received": 2, "user_id": 42575, "username": "5849206328x" } ], "source": null }	1. Find $\dfrac{d}{dx}f(x)$ if $f(x)=\cos x$. 2. Prove that $n(n+1)(n+2)$ is divisible by 3 for all integers $n$. 3. What is the greatest power of 2 that divides $\dbinom{200}{100}$?	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/Divisible", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/GreatestDividingExponent", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Parity" ]	Count the exponent of 2 in the binomial coefficient using Legendre's formula (sum of floor divisions) and subtract the contributions from the denominator.	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aops_99615	Let $P_{1}P_{2}...P_{n}$ a regular polygon with center $O$, then $[P_{1}OP_{2}]=\frac{1}{2}r^{2}sen(\frac{2\pi}{n})$ so $[P_{1}P_{2}...P_{n}]=\frac{n}{2}r^{2}sen(\frac{2\pi}{n})$ and as $n \to \infty$: $[P_{1}P_{2}...P_{n}]=\frac{n}{2}r^{2}\frac{2\pi}{n}$ so $[P_{1}P_{2}...P_{n}]=\pi r^{2}$.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove: \r\n\\[ \\pi = \\displaystyle\\lim_{n\\to\\infty}n\\sin(\\frac{\\pi}{n}) \\]\r\n\r\nI don't really use the L' Hospital Rule to solve it, because there's a geometric way to do it..........", "content_html": "Prove:<br>\n<img src=\"//latex.artofproblemsolving.com/0/7/8/078e43b4cd375abab0055076bdee77f50d65dafa.png\" class=\"latexcenter\" alt=\"\\[ \\pi = \\displaystyle\\lim_{n\\to\\infty}n\\sin(\\frac{\\pi}{n}) \\]\" width=\"137\" height=\"33\" ><br>\n<br>\nI don't really use the L' Hospital Rule to solve it, because there's a geometric way to do it..........", "post_id": 562450, "post_number": 1, "post_time_unix": 1151653077, "post_time_utc": "2006-06-30 07:37:57 UTC", "thanks_received": 4, "user_id": 13170, "username": "brianchung11" }, { "attachments": [], "content_bbcode": "Hi !\r\n\r\nWe have $\\lim_{x \\to 0}{\\frac{sinx}{x}= 1}$ and with $x=\\frac{\\pi}{n}$ it's O.K. ! ;)", "content_html": "Hi !<br>\n<br>\nWe have <img src=\"//latex.artofproblemsolving.com/2/4/6/246ae631e29ff59140dcd94c0036a6b669de63de.png\" class=\"latex\" alt=\"$\\lim_{x \\to 0}{\\frac{sinx}{x}= 1}$\" style=\"vertical-align: -12px\" width=\"102\" height=\"37\" > and with <img src=\"//latex.artofproblemsolving.com/a/6/3/a6339f8f87e6553a9003c9255a6b97dabc7883ad.png\" class=\"latex\" alt=\"$x=\\frac{\\pi}{n}$\" style=\"vertical-align: -12px\" width=\"48\" height=\"33\" > it's O.K. ! <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 562457, "post_number": 2, "post_time_unix": 1151655017, "post_time_utc": "2006-06-30 08:10:17 UTC", "thanks_received": 2, "user_id": 13040, "username": "mstoenescu" }, { "attachments": [], "content_bbcode": "I've forgotten about this, and i've constructed a regular n-gon with its circumradius is 1. When n tends to infinity, it'll be a circle.......", "content_html": "I've forgotten about this, and i've constructed a regular n-gon with its circumradius is 1. When n tends to infinity, it'll be a circle.......", "post_id": 562560, "post_number": 3, "post_time_unix": 1151670618, "post_time_utc": "2006-06-30 12:30:18 UTC", "thanks_received": 2, "user_id": 13170, "username": "brianchung11" }, { "attachments": [], "content_bbcode": "Let $P_{1}P_{2}...P_{n}$ a regular polygon with center $O$, then $[P_{1}OP_{2}]=\\frac{1}{2}r^{2}sen(\\frac{2\\pi}{n})$ so $[P_{1}P_{2}...P_{n}]=\\frac{n}{2}r^{2}sen(\\frac{2\\pi}{n})$ and as $n \\to \\infty$: $[P_{1}P_{2}...P_{n}]=\\frac{n}{2}r^{2}\\frac{2\\pi}{n}$ so $[P_{1}P_{2}...P_{n}]=\\pi r^{2}$.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/f/f/b/ffbe189caf0f43373b399604c12d42636a7adc08.png\" class=\"latex\" alt=\"$P_{1}P_{2}...P_{n}$\" style=\"vertical-align: -2px\" width=\"72\" height=\"15\" > a regular polygon with center <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/1/d/51da37d984564162c87710ca27bea422f657fb73.png\" class=\"latex\" alt=\"$O$\" width=\"13\" height=\"12\" >,</span> then <img src=\"//latex.artofproblemsolving.com/1/b/f/1bfd29e479a08b30c85af9eda33305b003760661.png\" class=\"latex\" alt=\"$[P_{1}OP_{2}]=\\frac{1}{2}r^{2}sen(\\frac{2\\pi}{n})$\" style=\"vertical-align: -12px\" width=\"181\" height=\"37\" > so <img src=\"//latex.artofproblemsolving.com/5/1/d/51d4fdf1369e6d2eb47d131b4c1c07555682cf3a.png\" class=\"latex\" alt=\"$[P_{1}P_{2}...P_{n}]=\\frac{n}{2}r^{2}sen(\\frac{2\\pi}{n})$\" style=\"vertical-align: -12px\" width=\"204\" height=\"37\" > and as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/3/4/63405adfa323cd12386536930115984575a36cf9.png\" class=\"latex\" alt=\"$n \\to \\infty$\" style=\"vertical-align: 0px\" width=\"56\" height=\"10\" >:</span> <img src=\"//latex.artofproblemsolving.com/3/d/b/3db3343b172eb1d0a596de96d4269e69329095fe.png\" class=\"latex\" alt=\"$[P_{1}P_{2}...P_{n}]=\\frac{n}{2}r^{2}\\frac{2\\pi}{n}$\" style=\"vertical-align: -12px\" width=\"162\" height=\"37\" > so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/f/7/7f7fc69b882ccfe320b7c930f8dbebb4016537e6.png\" class=\"latex\" alt=\"$[P_{1}P_{2}...P_{n}]=\\pi r^{2}$\" style=\"vertical-align: -5px\" width=\"134\" height=\"20\" >.</span>", "post_id": 565960, "post_number": 4, "post_time_unix": 1152020801, "post_time_utc": "2006-07-04 13:46:41 UTC", "thanks_received": 2, "user_id": 11966, "username": "hucht" } ], "source": null }	Prove: \[ \pi=\lim_{n\to\infty}n\sin\!\left(\frac{\pi}{n}\right). \]	[ "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis", "/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/UnitCircle", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/Area", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/PolygonArea", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularPolygon", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine" ]	Compare the area of an inscribed regular n‑gon with the circle’s area to obtain the limit n·sin(π/n) = π.	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aops_996154	considering your mathematical ability, I'd expect an overkill solution to come with that, but whatever [hide] We know that $ 2(xy\plus{}xz\plus{}yz)\equal{}1337$, so $ xy\plus{}xz\plus{}yz\equal{}\frac{1337}{2}$. Applying AM-GM to the 3 terms, we find that \[ \frac{xy\plus{}xz\plus{}yz}{3}\geq \sqrt[3]{x^2y^2z^2}\]Obviously, we're looking for the maximum of $ xyz$, and it should be obvious from here what it is [/hide] I'm pretty sure there's another way that basically kills the problem	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "What are yours?\r\n\r\nmine:\r\n\r\n1) Being in existence\r\n2) Trying to do math\r\n\r\nIn the meantime, here's an easy problem\r\n\r\nBob has a rectangular prism, and 1337 square inches of wrapping paper, which covers the prism completely without extra wrapping paper. What is the maximum possible volume of the prism?", "content_html": "What are yours?<br>\n<br>\nmine:<br>\n<br>\n1) Being in existence<br>\n2) Trying to do math<br>\n<br>\nIn the meantime, here's an easy problem<br>\n<br>\nBob has a rectangular prism, and 1337 square inches of wrapping paper, which covers the prism completely without extra wrapping paper. What is the maximum possible volume of the prism?", "post_id": 4414296, "post_number": 1, "post_time_unix": 1208112771, "post_time_utc": "2008-04-13 18:52:51 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "$ \\left(\\sqrt {\\dfrac{1337}{6}}\\right)^3$\r\n\r\n$ \\approx 3326.37 \\text{ in}^3$", "content_html": "<img src=\"//latex.artofproblemsolving.com/3/4/a/34aa98fa4c64afd9fbd6d710850df1fe316939a7.png\" class=\"latex\" alt=\"$ \\left(\\sqrt {\\dfrac{1337}{6}}\\right)^3$\" style=\"vertical-align: -22px\" width=\"92\" height=\"56\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/0/e/f0ef48aceb496378bc47a5f761ce07e950ffe32e.png\" class=\"latex\" alt=\"$ \\approx 3326.37 \\text{ in}^3$\" width=\"104\" height=\"15\" >", "post_id": 4414297, "post_number": 2, "post_time_unix": 1208120499, "post_time_utc": "2008-04-13 21:01:39 UTC", "thanks_received": 1, "user_id": 38100, "username": "distracted523" }, { "attachments": [], "content_bbcode": "considering your mathematical ability, I'd expect an overkill solution to come with that, but whatever\r\n\r\n[hide]\nWe know that $ 2(xy\\plus{}xz\\plus{}yz)\\equal{}1337$, so $ xy\\plus{}xz\\plus{}yz\\equal{}\\frac{1337}{2}$. Applying AM-GM to the 3 terms, we find that \\[ \\frac{xy\\plus{}xz\\plus{}yz}{3}\\geq \\sqrt[3]{x^2y^2z^2}\\]Obviously, we're looking for the maximum of $ xyz$, and it should be obvious from here what it is\n[/hide]\r\n\r\nI'm pretty sure there's another way that basically kills the problem", "content_html": "considering your mathematical ability, I'd expect an overkill solution to come with that, but whatever<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">We know that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/0/c/d0ce7128c2fca0799108a2c17c647c60776cb030.png\" class=\"latex\" alt=\"$ 2(xy+xz+yz)=1337$\" style=\"vertical-align: -4px\" width=\"186\" height=\"18\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/b/6/2b696fee88c3a4008a324c6aaba1f0fd4a472049.png\" class=\"latex\" alt=\"$ xy+xz+yz=\\frac{1337}{2}$\" style=\"vertical-align: -12px\" width=\"165\" height=\"37\" >.</span> Applying AM-GM to the 3 terms, we find that <img src=\"//latex.artofproblemsolving.com/8/3/b/83b4f2da81cbab1a6a8d9bf2f006b5b642b99aaf.png\" class=\"latexcenter\" alt=\"\\[ \\frac{xy+xz+yz}{3}\\geq \\sqrt[3]{x^2y^2z^2}\\]\" width=\"199\" height=\"35\" >Obviously, we're looking for the maximum of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/9/e0952299feef19841a0155f5a659b01ea2844225.png\" class=\"latex\" alt=\"$ xyz$\" style=\"vertical-align: -3px\" width=\"28\" height=\"11\" >,</span> and it should be obvious from here what it is</div><br>\n<br>\nI'm pretty sure there's another way that basically kills the problem", "post_id": 4414298, "post_number": 3, "post_time_unix": 1208124620, "post_time_utc": "2008-04-13 22:10:20 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "/me wonders why alan didn't use the volume/SA formulas for a cube\r\n\r\n/me wonders whether that was an insult or a snide half-praise\r\n\r\n/me wonders if alan=ClarkeBot", "content_html": "/me wonders why alan didn't use the volume/SA formulas for a cube<br>\n<br>\n/me wonders whether that was an insult or a snide half-praise<br>\n<br>\n/me wonders if alan=ClarkeBot", "post_id": 4414299, "post_number": 4, "post_time_unix": 1208221695, "post_time_utc": "2008-04-15 01:08:15 UTC", "thanks_received": 1, "user_id": 38100, "username": "distracted523" }, { "attachments": [], "content_bbcode": "meh, I didn't feel like leaving the question of why the maximum is a cube", "content_html": "meh, I didn't feel like leaving the question of why the maximum is a cube", "post_id": 4414300, "post_number": 5, "post_time_unix": 1208377603, "post_time_utc": "2008-04-16 20:26:43 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" } ], "source": null }	Bob has a rectangular prism, and 1337 square inches of wrapping paper, which covers the prism completely without extra wrapping paper. What is the maximum possible volume of the prism?	[ "/Mathematics/Geometry/GeometricInequalities", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Height", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Length", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Solid", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/SolidGeometry", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Width", "/Mathematics/Geometry/SolidGeometry/Volume" ]	Apply AM‑GM to the three face‑area products to bound the volume under a fixed surface area	[ -0.0001208186149597168, -0.054840087890625, 0.020599365234375, -0.00927734375, -0.0010194778442382812, 0.01617431640625, 0.021820068359375, 0.003391265869140625, 0.01149749755859375, -0.0269317626953125, -0.023590087890625, -0.0379638671875, -0.0006537437438964844, -0.004985809326171875, -0.0043182373046875, -0.003887176513671875, 0.01049041748046875, -0.016204833984375, 0.0296173095703125, -0.00030803680419921875, -0.0103302001953125, -0.0252532958984375, 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aops_996158	Meh, first thing that comes to my mind is to take each side (mod 4). For p that are 3 (mod 4) there are no solutions, but I'm not going to find out if all the ones that are 1 (mod 4) do have solutions.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "find the sum of all prime numbers $ p<100$ such that there do not exist $ x,y\\in \\mathbb{Z}$ such that $ x^2\\plus{}y^2\\equal{}p$.", "content_html": "find the sum of all prime numbers <img src=\"//latex.artofproblemsolving.com/0/2/b/02b76d39a8b5c48bf1c4493b8f21a2790e36c43b.png\" class=\"latex\" alt=\"$ p<100$\" style=\"vertical-align: -3px\" width=\"61\" height=\"16\" > such that there do not exist <img src=\"//latex.artofproblemsolving.com/0/8/3/083f78bcc3ded5bf2af500703ac3f48814166eba.png\" class=\"latex\" alt=\"$ x,y\\in \\mathbb{Z}$\" style=\"vertical-align: -3px\" width=\"62\" height=\"15\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/d/1/5d143ed0d5f1eddeb0f271f07dc673b149eb3f29.png\" class=\"latex\" alt=\"$ x^2+y^2=p$\" style=\"vertical-align: -3px\" width=\"90\" height=\"18\" >.</span>", "post_id": 4414326, "post_number": 1, "post_time_unix": 1220221656, "post_time_utc": "2008-08-31 22:27:36 UTC", "thanks_received": 1, "user_id": 42575, "username": "5849206328x" }, { "attachments": [], "content_bbcode": "Meh, first thing that comes to my mind is to take each side (mod 4). For p that are 3 (mod 4) there are no solutions, but I'm not going to find out if all the ones that are 1 (mod 4) do have solutions.", "content_html": "Meh, first thing that comes to my mind is to take each side (mod 4). For p that are 3 (mod 4) there are no solutions, but I'm not going to find out if all the ones that are 1 (mod 4) do have solutions.", "post_id": 4414327, "post_number": 2, "post_time_unix": 1220221852, "post_time_utc": "2008-08-31 22:30:52 UTC", "thanks_received": 1, "user_id": 28420, "username": "xpmath" }, { "attachments": [], "content_bbcode": "Any prime that is 1 mod 4 can indeed be expressed as the sum of two squares. This is well known.", "content_html": "Any prime that is 1 mod 4 can indeed be expressed as the sum of two squares. This is well known.", "post_id": 4414328, "post_number": 3, "post_time_unix": 1220228871, "post_time_utc": "2008-09-01 00:27:51 UTC", "thanks_received": 1, "user_id": 26057, "username": "tjhance" }, { "attachments": [], "content_bbcode": "Yay, you posted a math problem. Now I have got to post more....lets see.", "content_html": "Yay, you posted a math problem. Now I have got to post more....lets see.", "post_id": 4414329, "post_number": 4, "post_time_unix": 1220231234, "post_time_utc": "2008-09-01 01:07:14 UTC", "thanks_received": 1, "user_id": 35887, "username": "shentang" }, { "attachments": [], "content_bbcode": "Ah. I was being stupid, as usual.", "content_html": "Ah. I was being stupid, as usual.", "post_id": 4414330, "post_number": 5, "post_time_unix": 1220234603, "post_time_utc": "2008-09-01 02:03:23 UTC", "thanks_received": 1, "user_id": 28420, "username": "xpmath" } ], "source": null }	Find the sum of all prime numbers $p<100$ such that there do not exist integers $x,y$ with $x^2+y^2=p$.	[ "/Mathematics/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/PrimeNumbers/PrimeRepresentations" ]	Apply Fermat's theorem that a prime is a sum of two squares exactly when it is 2 or congruent to 1 modulo 4.	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aops_99648	Yes, I think such a path does exist. Old trick: Among all cycles with straight edges and having our $n$ points as vertices, choose the one with minimal perimeter. If the segments $xy,zt$ were to intersect, replacing them with one of the pairs $(xz,yt),\ (xt,yz)$ (just one of them works; the proper one has to be chosen; this can be shown by reasoning on an $n$-cycle drawn as a convex polygon, with $xy,zt$ as two of its sides) would keep the graph connected, but would make the perimeter strictly smaller. This contradicts the minimality assumption. I hope it's Ok.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "given $n$ points in the plane; prove or disprove, that we can always find a path from a starting point such that we pass all other points and come back to the same, starting point; with the condition that there are no crosses in our path (in other words: the path should not cross itself) .", "content_html": "given <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > points in the plane; prove or disprove, that we can always find a path from a starting point such that we pass all other points and come back to the same, starting point; with the condition that there are no crosses in our path (in other words: the path should not cross itself) .", "post_id": 562585, "post_number": 1, "post_time_unix": 1151672217, "post_time_utc": "2006-06-30 12:56:57 UTC", "thanks_received": 2, "user_id": 18450, "username": "e^pi" }, { "attachments": [], "content_bbcode": "Yes, I think such a path does exist. Old trick:\r\n\r\nAmong all cycles with straight edges and having our $n$ points as vertices, choose the one with minimal perimeter. If the segments $xy,zt$ were to intersect, replacing them with one of the pairs $(xz,yt),\\ (xt,yz)$ (just one of them works; the proper one has to be chosen; this can be shown by reasoning on an $n$-cycle drawn as a convex polygon, with $xy,zt$ as two of its sides) would keep the graph connected, but would make the perimeter strictly smaller. This contradicts the minimality assumption.\r\n\r\nI hope it's Ok.", "content_html": "Yes, I think such a path does exist. Old trick:<br>\n<br>\nAmong all cycles with straight edges and having our <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > points as vertices, choose the one with minimal perimeter. If the segments <img src=\"//latex.artofproblemsolving.com/4/7/3/473f114bb000c708d50cb9b7dce6f3b4618f07f9.png\" class=\"latex\" alt=\"$xy,zt$\" style=\"vertical-align: -3px\" width=\"43\" height=\"15\" > were to intersect, replacing them with one of the pairs <img src=\"//latex.artofproblemsolving.com/e/0/2/e026f6a117808602cff371914a98501fb6da3e83.png\" class=\"latex\" alt=\"$(xz,yt),\\ (xt,yz)$\" style=\"vertical-align: -4px\" width=\"128\" height=\"18\" > (just one of them works; the proper one has to be chosen; this can be shown by reasoning on an <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >-</span>cycle drawn as a convex polygon, with <img src=\"//latex.artofproblemsolving.com/4/7/3/473f114bb000c708d50cb9b7dce6f3b4618f07f9.png\" class=\"latex\" alt=\"$xy,zt$\" style=\"vertical-align: -3px\" width=\"43\" height=\"15\" > as two of its sides) would keep the graph connected, but would make the perimeter strictly smaller. This contradicts the minimality assumption.<br>\n<br>\nI hope it's Ok.", "post_id": 562661, "post_number": 2, "post_time_unix": 1151678204, "post_time_utc": "2006-06-30 14:36:44 UTC", "thanks_received": 2, "user_id": 26, "username": "grobber" }, { "attachments": [], "content_bbcode": "yes triangle ineq", "content_html": "yes triangle ineq", "post_id": 562664, "post_number": 3, "post_time_unix": 1151678395, "post_time_utc": "2006-06-30 14:39:55 UTC", "thanks_received": 2, "user_id": 3431, "username": "manuel" }, { "attachments": [], "content_bbcode": "Changing $(xy,zt)$ to either one of those two pairs decreases the length (yes, by the triangle inequality); however, one of these moves keeps the graph connected, while one does not. This is what I meant when I said that we have to choose properly between the two pairs of edges we want to change the initial pair into.", "content_html": "Changing <img src=\"//latex.artofproblemsolving.com/4/8/8/48806aa76304f8dcd04fa3981065351fa31a8c8c.png\" class=\"latex\" alt=\"$(xy,zt)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" > to either one of those two pairs decreases the length (yes, by the triangle inequality); however, one of these moves keeps the graph connected, while one does not. This is what I meant when I said that we have to choose properly between the two pairs of edges we want to change the initial pair into.", "post_id": 562687, "post_number": 4, "post_time_unix": 1151679962, "post_time_utc": "2006-06-30 15:06:02 UTC", "thanks_received": 2, "user_id": 26, "username": "grobber" }, { "attachments": [], "content_bbcode": "my initial reaction upon reading this question was to take the biggest convex hull of points of the set and so on.\r\nthus there always exist such path without any crossings.\r\n\r\nEDIT: as pointed out to me my \"perfect_radio\" i dont think this idea works since we are dealing with cycles here.", "content_html": "my initial reaction upon reading this question was to take the biggest convex hull of points of the set and so on.<br>\nthus there always exist such path without any crossings.<br>\n<br>\nEDIT: as pointed out to me my "perfect_radio" i dont think this idea works since we are dealing with cycles here.", "post_id": 564147, "post_number": 5, "post_time_unix": 1151799957, "post_time_utc": "2006-07-02 00:25:57 UTC", "thanks_received": 2, "user_id": 8278, "username": "amirhtlusa" }, { "attachments": [], "content_bbcode": "I disagree: It is not always possible to construct such a path. \r\n\r\nE. g. consider the points $(0,0)$, $(1,0)$ and $(2,0)$, which cannot be \r\nchoined by a path in the given way. (I suppose that if a bond is contained \r\nin another bond then this counts as a crossing, or doesn't it?) \r\n\r\nBut I suppose everything is OK if not all points are collinear.", "content_html": "I disagree: It is not always possible to construct such a path.<br>\n<br>\nE. g. consider the points <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/9/6/f9603ca3089464e548fc6f1366bc474e7efef8d9.png\" class=\"latex\" alt=\"$(0,0)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" >,</span> <img src=\"//latex.artofproblemsolving.com/2/5/a/25a306093bbd9e0699c65dba033c9483cd379dc4.png\" class=\"latex\" alt=\"$(1,0)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/0/c/c0c67f1e6c098104b1cb8f1cdd6bb16f689ac3e6.png\" class=\"latex\" alt=\"$(2,0)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" >,</span> which cannot be<br>\nchoined by a path in the given way. (I suppose that if a bond is contained<br>\nin another bond then this counts as a crossing, or doesn't it?)<br>\n<br>\nBut I suppose everything is OK if not all points are collinear.", "post_id": 565110, "post_number": 6, "post_time_unix": 1151926228, "post_time_utc": "2006-07-03 11:30:28 UTC", "thanks_received": 2, "user_id": 4420, "username": "solyaris" }, { "attachments": [], "content_bbcode": "Of course. Since in such problems collinear points give rise to all sorts of pathological examples, I, for one, assumed even more: no three points are collinear. These are the interesting cases anyway :).", "content_html": "Of course. Since in such problems collinear points give rise to all sorts of pathological examples, I, for one, assumed even more: no three points are collinear. These are the interesting cases anyway <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />.", "post_id": 565129, "post_number": 7, "post_time_unix": 1151930203, "post_time_utc": "2006-07-03 12:36:43 UTC", "thanks_received": 2, "user_id": 26, "username": "grobber" } ], "source": null }	Given $n$ points in the plane, prove or disprove that we can always find a path starting at one of the points, visiting all the other points, and returning to the starting point, with the path having no self-intersections (i.e., the path is a simple closed polygonal chain through all $n$ points).	[ "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/PlanarPolygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/SimplePolygon", "/Mathematics/Geometry/Points/Point" ]	Choose a Hamiltonian cycle of minimal total length; any crossing can be uncrossed to shorten the perimeter, contradicting minimality.	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aops_99652	[hide]There are 43 ways to chose the first person on the handshake and 42 ways to chose the second person on the handshake. This means that there are 4342 ways of shaking people's hands. But since order doesn't matter it is 4321=903. $903$[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Billy Bob went to a reunion where he met some friends from college. Totally, 43 people came. Each person shook hands with each other person exactly once. How many handshakes were there?", "content_html": "Billy Bob went to a reunion where he met some friends from college. Totally, 43 people came. Each person shook hands with each other person exactly once. How many handshakes were there?", "post_id": 562600, "post_number": 1, "post_time_unix": 1151673501, "post_time_utc": "2006-06-30 13:18:21 UTC", "thanks_received": 2, "user_id": 17283, "username": "Arvind_sn" }, { "attachments": [], "content_bbcode": "there were 43 people so the first person shook hands with 42 people and the second with 41 and so on.\r\n\r\n43+42+41+40+...+1=\r\n\r\n(4342)/2 = 903\r\n\r\n\r\nthere were 903 hand shakes", "content_html": "there were 43 people so the first person shook hands with 42 people and the second with 41 and so on.<br>\n<br>\n43+42+41+40+...+1=<br>\n<br>\n(4342)/2 = 903<br>\n<br>\n<br>\nthere were 903 hand shakes", "post_id": 562631, "post_number": 2, "post_time_unix": 1151676797, "post_time_utc": "2006-06-30 14:13:17 UTC", "thanks_received": 2, "user_id": 6572, "username": "SuperSut" }, { "attachments": [], "content_bbcode": "[hide=\"solution\"]There are 43+42+41+.....+2+1 handshakes=$\\frac{4344}{2}=946$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">solution</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43+42+41+.....+2+1 handshakes<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/d/5/9/d59c681b0e0f3926dc259e26e8170089c60cd09d.png\" class=\"latex\" alt=\"$\\frac{4344}{2}=946$\" style=\"vertical-align: -12px\" width=\"108\" height=\"37\" ></span></div>", "post_id": 562700, "post_number": 3, "post_time_unix": 1151680661, "post_time_utc": "2006-06-30 15:17:41 UTC", "thanks_received": 1, "user_id": 15223, "username": "1=2" }, { "attachments": [], "content_bbcode": "[quote=\"SuperSut\"](4342)/2 = 903[/quote]\r\nThe formula for 1+2+3+...+n is $\\frac{n(n+1)}{2}$", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">SuperSut wrote:</div>\n<div class=\"bbcode_quote_body\">(4342)/2 = 903</div>\n</div>\nThe formula for 1+2+3+...+n is <img src=\"//latex.artofproblemsolving.com/6/4/f/64f8070867b03b0dc224db7e741135e07f4502c6.png\" class=\"latex\" alt=\"$\\frac{n(n+1)}{2}$\" style=\"vertical-align: -12px\" width=\"69\" height=\"38\" >", "post_id": 562718, "post_number": 4, "post_time_unix": 1151681247, "post_time_utc": "2006-06-30 15:27:27 UTC", "thanks_received": 2, "user_id": 17780, "username": "redcomet46" }, { "attachments": [], "content_bbcode": "[hide]There are 43 ways to chose the first person on the handshake and 42 ways to chose the second person on the handshake. This means that there are 4342 ways of shaking people's hands. But since order doesn't matter it is 4321=903.\n\n$903$[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43 ways to chose the first person on the handshake and 42 ways to chose the second person on the handshake. This means that there are 4342 ways of shaking people's hands. But since order doesn't matter it is 4321=903.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/3/e/c/3ecb3193a0638957575fb912b9b3af3372c2e0c9.png\" class=\"latex\" alt=\"$903$\" width=\"26\" height=\"12\" ></div>", "post_id": 562772, "post_number": 5, "post_time_unix": 1151683238, "post_time_utc": "2006-06-30 16:00:38 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "[quote=\"nutz_for2.718281828\"][hide=\"solution\"]There are 43+42+41+.....+2+1 handshakes=$\\frac{4344}{2}=946$[/hide][/quote]\r\nIt's $42+41+...+1$ not $43+42+...+1$.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nutz_for2.718281828 wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">solution</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43+42+41+.....+2+1 handshakes<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/d/5/9/d59c681b0e0f3926dc259e26e8170089c60cd09d.png\" class=\"latex\" alt=\"$\\frac{4344}{2}=946$\" style=\"vertical-align: -12px\" width=\"108\" height=\"37\" ></span></div></div>\n</div>\nIt's <img src=\"//latex.artofproblemsolving.com/e/4/5/e4537e79477eafb7ad7b93abde42aaaafa9a315b.png\" class=\"latex\" alt=\"$42+41+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"13\" > not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/0/4a0a76a878b74ab9af9bdd849d0a128592a391e9.png\" class=\"latex\" alt=\"$43+42+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"14\" >.</span>", "post_id": 563074, "post_number": 6, "post_time_unix": 1151704961, "post_time_utc": "2006-06-30 22:02:41 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "[quote=\"lotrgreengrapes7926\"][quote=\"nutz_for2.718281828\"][hide=\"solution\"]There are 43+42+41+.....+2+1 handshakes=$\\frac{4344}{2}=946$[/hide][/quote]\nIt's $42+41+...+1$ not $43+42+...+1$.[/quote]\r\n\r\nBilly Bob is the 44th person.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">lotrgreengrapes7926 wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nutz_for2.718281828 wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">solution</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43+42+41+.....+2+1 handshakes<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/d/5/9/d59c681b0e0f3926dc259e26e8170089c60cd09d.png\" class=\"latex\" alt=\"$\\frac{4344}{2}=946$\" style=\"vertical-align: -12px\" width=\"108\" height=\"37\" ></span></div></div>\n</div>\nIt's <img src=\"//latex.artofproblemsolving.com/e/4/5/e4537e79477eafb7ad7b93abde42aaaafa9a315b.png\" class=\"latex\" alt=\"$42+41+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"13\" > not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/0/4a0a76a878b74ab9af9bdd849d0a128592a391e9.png\" class=\"latex\" alt=\"$43+42+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"14\" >.</span></div>\n</div>\n<br>\nBilly Bob is the 44th person.", "post_id": 563119, "post_number": 7, "post_time_unix": 1151708361, "post_time_utc": "2006-06-30 22:59:21 UTC", "thanks_received": 2, "user_id": 18270, "username": "SplashD" }, { "attachments": [], "content_bbcode": "[quote=\"lotrgreengrapes7926\"][quote=\"nutz_for2.718281828\"][hide=\"solution\"]There are 43+42+41+.....+2+1 handshakes=$\\frac{4344}{2}=946$[/hide][/quote]\nIt's $42+41+...+1$ not $43+42+...+1$.[/quote]\r\nHe's right which is why the answer is $903$.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">lotrgreengrapes7926 wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">nutz_for2.718281828 wrote:</div>\n<div class=\"bbcode_quote_body\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">solution</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43+42+41+.....+2+1 handshakes<span style=\"white-space:nowrap;\">=<img src=\"//latex.artofproblemsolving.com/d/5/9/d59c681b0e0f3926dc259e26e8170089c60cd09d.png\" class=\"latex\" alt=\"$\\frac{4344}{2}=946$\" style=\"vertical-align: -12px\" width=\"108\" height=\"37\" ></span></div></div>\n</div>\nIt's <img src=\"//latex.artofproblemsolving.com/e/4/5/e4537e79477eafb7ad7b93abde42aaaafa9a315b.png\" class=\"latex\" alt=\"$42+41+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"13\" > not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/0/4a0a76a878b74ab9af9bdd849d0a128592a391e9.png\" class=\"latex\" alt=\"$43+42+...+1$\" style=\"vertical-align: -1px\" width=\"126\" height=\"14\" >.</span></div>\n</div>\nHe's right which is why the answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/e/c/3ecb3193a0638957575fb912b9b3af3372c2e0c9.png\" class=\"latex\" alt=\"$903$\" width=\"26\" height=\"12\" >.</span>", "post_id": 564273, "post_number": 8, "post_time_unix": 1151813311, "post_time_utc": "2006-07-02 04:08:31 UTC", "thanks_received": 2, "user_id": 8960, "username": "bpms" }, { "attachments": [], "content_bbcode": "The answer is 946\r\nFirst you add the biggest and smallest numbers, to get 44, then you multiply by the number of numbers there are, to get 4443=1892, then divide by two to get 946\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\nMy brother taught me that a couple of years ago :lol:", "content_html": "The answer is 946<br>\nFirst you add the biggest and smallest numbers, to get 44, then you multiply by the number of numbers there are, to get 4443=1892, then divide by two to get 946<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\nMy brother taught me that a couple of years ago <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />", "post_id": 576593, "post_number": 9, "post_time_unix": 1153168972, "post_time_utc": "2006-07-17 20:42:52 UTC", "thanks_received": 2, "user_id": 19932, "username": "kishiman" }, { "attachments": [], "content_bbcode": "[hide=\"hi\"]There are 43 people at the party. So, the first person shakes hands with 42 people second with 41, third with 40 etc. \n\n42+41+40......+3+2+1 = 4321 aka n(n+1)/2 . \n\nSo, the answer is [b]903[/b][/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">hi</a><div class=\"cmty-hide-content\" style=\"display:none\">There are 43 people at the party. So, the first person shakes hands with 42 people second with 41, third with 40 etc.<br>\n<br>\n42+41+40......+3+2+1 = 4321 aka n(n+1)/2 .<br>\n<br>\nSo, the answer is <b>903</b></div>", "post_id": 576619, "post_number": 10, "post_time_unix": 1153171646, "post_time_utc": "2006-07-17 21:27:26 UTC", "thanks_received": 2, "user_id": 10770, "username": "nonie" }, { "attachments": [], "content_bbcode": "[quote=\"kishiman\"]The answer is 946\nFirst you add the biggest and smallest numbers, to get 44, then you multiply by the number of numbers there are, to get 4443=1892, then divide by two to get 946\n\n\n\n\n\n\n\n\n\nMy brother taught me that a couple of years ago :lol:[/quote]\r\n\r\nNo, their right. If one person shakes 43 times, either he shakes a hand twice of he shakes his own hand.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">kishiman wrote:</div>\n<div class=\"bbcode_quote_body\">The answer is 946<br>\nFirst you add the biggest and smallest numbers, to get 44, then you multiply by the number of numbers there are, to get 4443=1892, then divide by two to get 946<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\n<br>\nMy brother taught me that a couple of years ago <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\nNo, their right. If one person shakes 43 times, either he shakes a hand twice of he shakes his own hand.", "post_id": 577146, "post_number": 11, "post_time_unix": 1153231899, "post_time_utc": "2006-07-18 14:11:39 UTC", "thanks_received": 2, "user_id": 15223, "username": "1=2" }, { "attachments": [], "content_bbcode": "903 is correct, by the triangular number formula already mentioned earlier", "content_html": "903 is correct, by the triangular number formula already mentioned earlier", "post_id": 590927, "post_number": 12, "post_time_unix": 1154543634, "post_time_utc": "2006-08-02 18:33:54 UTC", "thanks_received": 1, "user_id": 17840, "username": "daermon" }, { "attachments": [], "content_bbcode": "Well, if Billy Bob was one of the 43 people, the answer is 903. If there are actually 44 people at the party, the answer's 946.", "content_html": "Well, if Billy Bob was one of the 43 people, the answer is 903. If there are actually 44 people at the party, the answer's 946.", "post_id": 591046, "post_number": 13, "post_time_unix": 1154548161, "post_time_utc": "2006-08-02 19:49:21 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "I actually got 903. I think the problem needs to be clearer.[/hide]", "content_html": "I actually got 903. I think the problem needs to be clearer.[/hide]", "post_id": 592359, "post_number": 14, "post_time_unix": 1154634565, "post_time_utc": "2006-08-03 19:49:25 UTC", "thanks_received": 2, "user_id": 8131, "username": "math92" }, { "attachments": [], "content_bbcode": "[quote=\"Arvind_sn\"]Billy Bob went to a reunion where he met some friends from college. Totally, 43 people came. Each person shook hands with each other person exactly once. How many handshakes were there?[/quote]\r\n\r\n[hide]4342/2=4321=43+860=903[/hide]", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Arvind_sn wrote:</div>\n<div class=\"bbcode_quote_body\">Billy Bob went to a reunion where he met some friends from college. Totally, 43 people came. Each person shook hands with each other person exactly once. How many handshakes were there?</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">4342/2=4321=43+860=903</div>", "post_id": 615259, "post_number": 15, "post_time_unix": 1156822897, "post_time_utc": "2006-08-29 03:41:37 UTC", "thanks_received": 2, "user_id": 11899, "username": "moogra" } ], "source": null }	Billy Bob went to a reunion where he met some friends from college. Totally, $43$ people came. Each person shook hands with each other person exactly once. How many handshakes were there?	[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Count unordered pairs of people using the combination formula n choose 2.	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aops_99655	maybe... [hide] $a=\sqrt{2}$ $b=\log_{\sqrt{2}}3$ $a^b=\sqrt{2}^{\log_{\sqrt{2}}3}=3$[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Give an instance of $2$ different irrational numbers $a$ and $b$ such that $a^b$ is definitely a rational number.\r\n\r\nIs such a phenomenon possible?\r\n\r\n??", "content_html": "Give an instance of <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > different irrational numbers <img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" width=\"8\" height=\"12\" > such that <img src=\"//latex.artofproblemsolving.com/3/a/8/3a8be4a03400384a2719b865cf815f46d4f857be.png\" class=\"latex\" alt=\"$a^b$\" width=\"15\" height=\"15\" > is definitely a rational number.<br>\n<br>\nIs such a phenomenon possible?<br>\n<br>\n??", "post_id": 562613, "post_number": 1, "post_time_unix": 1151675449, "post_time_utc": "2006-06-30 13:50:49 UTC", "thanks_received": 2, "user_id": 16952, "username": "xxxyyyy" }, { "attachments": [], "content_bbcode": "maybe...\r\n[hide]\n\n$a=\\sqrt{2}$\n$b=\\log_{\\sqrt{2}}3$\n$a^b=\\sqrt{2}^{\\log_{\\sqrt{2}}3}=3$[/hide]", "content_html": "maybe...<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/a/6/aa673781398d74ddd0622734d333392eaf4c3754.png\" class=\"latex\" alt=\"$a=\\sqrt{2}$\" style=\"vertical-align: -1px\" width=\"58\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/4/c/3/4c3becf24236eb76eae23c67ce70a4aadd2f476c.png\" class=\"latex\" alt=\"$b=\\log_{\\sqrt{2}}3$\" style=\"vertical-align: -6px\" width=\"85\" height=\"19\" ><br>\n<img src=\"//latex.artofproblemsolving.com/7/2/a/72a6ea2252843c907ead7ca4546666357067c1aa.png\" class=\"latex\" alt=\"$a^b=\\sqrt{2}^{\\log_{\\sqrt{2}}3}=3$\" style=\"vertical-align: -1px\" width=\"140\" height=\"23\" ></div>", "post_id": 562621, "post_number": 2, "post_time_unix": 1151676239, "post_time_utc": "2006-06-30 14:03:59 UTC", "thanks_received": 2, "user_id": 17962, "username": "hydro" }, { "attachments": [], "content_bbcode": "[hide]\ntake $\\sqrt[x]{x}$ for any natural number $x$ that makes this irrational. now consider the number $(\\sqrt[x]{x})^{\\sqrt[x]{x}}$. If this is rational then we're done. If not, raise it to the $\\sqrt[x]{x}$ power again. If this is rational then we're done (because the number before is irrational). if not, continue this prcoess until the $x$th time, then we have:\n\n$(\\sqrt[x]{x})^{\\sqrt[x]{x}^x} = (\\sqrt[x]{x})^x = x$\nwhich is rational. Therefore, there's infinite numbers that fits your criteria.\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">take <img src=\"//latex.artofproblemsolving.com/a/c/4/ac46a865e1bf3e3a52706a44d022e361232ee8d5.png\" class=\"latex\" alt=\"$\\sqrt[x]{x}$\" style=\"vertical-align: -3px\" width=\"27\" height=\"18\" > for any natural number <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > that makes this irrational. now consider the number <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/b/c/ebc7840a0f313fd333f1ee5bf31b1e76996a0ae2.png\" class=\"latex\" alt=\"$(\\sqrt[x]{x})^{\\sqrt[x]{x}}$\" style=\"vertical-align: -4px\" width=\"63\" height=\"21\" >.</span> If this is rational then we're done. If not, raise it to the <img src=\"//latex.artofproblemsolving.com/a/c/4/ac46a865e1bf3e3a52706a44d022e361232ee8d5.png\" class=\"latex\" alt=\"$\\sqrt[x]{x}$\" style=\"vertical-align: -3px\" width=\"27\" height=\"18\" > power again. If this is rational then we're done (because the number before is irrational). if not, continue this prcoess until the <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" >t</span>h time, then we have:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/7/6/e76356f0f230f93296936ae596a7766c2ed47feb.png\" class=\"latex\" alt=\"$(\\sqrt[x]{x})^{\\sqrt[x]{x}^x} = (\\sqrt[x]{x})^x = x$\" style=\"vertical-align: -4px\" width=\"178\" height=\"21\" ><br>\nwhich is rational. Therefore, there's infinite numbers that fits your criteria.</div>", "post_id": 562749, "post_number": 3, "post_time_unix": 1151682518, "post_time_utc": "2006-06-30 15:48:38 UTC", "thanks_received": 1, "user_id": 10705, "username": "pkerichang" } ], "source": null }	Give an instance of two different irrational numbers $a$ and $b$ such that $a^b$ is a rational number. Is such a phenomenon possible?	[ "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/IrrationalNumber", "/Mathematics/Algebra/NumberTheory/IrrationalNumbers/SquareRootof2", "/Mathematics/Algebra/NumberTheory/RationalNumbers/QuadraticIrrationalNumber", "/Mathematics/Algebra/NumberTheory/RationalNumbers/QuadraticSurd", "/Mathematics/Algebra/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/NumberTheory/IrrationalNumbers/IrrationalNumber", "/Mathematics/NumberTheory/IrrationalNumbers/SquareRootof2", "/Mathematics/NumberTheory/IrrationalNumbers/Surd", "/Mathematics/NumberTheory/RationalNumbers/FieldofRationals", "/Mathematics/NumberTheory/RationalNumbers/QuadraticIrrationalNumber", "/Mathematics/NumberTheory/RationalNumbers/QuadraticSurd", "/Mathematics/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/NumberTheory/RealNumbers/R", "/Mathematics/NumberTheory/RealNumbers/RealNumber", "/Mathematics/NumberTheory/RealNumbers/Reals", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Pick an irrational base a and define b = log_a (rational), so a^b equals that rational number.	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aops_996604	aaw come one no one wants to solve it :( [hide="Hint"]$ \displaystyle\lim_{\Delta x\to\0}}\frac {\sin(x \plus{} \Delta x \minus{} \sin(x))}{\Delta x}$ [/hide] EDIT: in the limit the change in x approaches zero	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "So my brother said Halliday and Resnick is too complicated for me so i will just use his university physics textbook. \r\n\r\nHere is a pretty simple calculus problem:\r\nProve $ \\frac {\\text{d}[\\sin(x)]}{\\text{d}x} \\equal{} cos (x)$", "content_html": "So my brother said Halliday and Resnick is too complicated for me so i will just use his university physics textbook.<br>\n<br>\nHere is a pretty simple calculus problem:<br>\nProve <img src=\"//latex.artofproblemsolving.com/7/d/2/7d27d24c6a95bcd3cc3f1f0f8299cd5308ccbe77.png\" class=\"latex\" alt=\"$ \\frac {\\text{d}[\\sin(x)]}{\\text{d}x} = cos (x)$\" style=\"vertical-align: -12px\" width=\"143\" height=\"38\" >", "post_id": 4415138, "post_number": 1, "post_time_unix": 1218637223, "post_time_utc": "2008-08-13 14:20:23 UTC", "thanks_received": 1, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "aaw come one no one wants to solve it :( \r\n\r\n[hide=\"Hint\"]$ \\displaystyle\\lim_{\\Delta x\\to\\0}}\\frac {\\sin(x \\plus{} \\Delta x \\minus{} \\sin(x))}{\\Delta x}$\n\n[/hide]\r\nEDIT: in the limit the change in x approaches zero", "content_html": "aaw come one no one wants to solve it <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hint</a><div class=\"cmty-hide-content\" style=\"display:none\"><span class=\"aopscode-error aopscode-latex-error\">$ \\displaystyle\\lim_{\\Delta x\\to\\0}}\\frac {\\sin(x + \\Delta x - \\sin(x))}{\\Delta x}$</span></div><br>\nEDIT: in the limit the change in x approaches zero", "post_id": 4415139, "post_number": 2, "post_time_unix": 1218643631, "post_time_utc": "2008-08-13 16:07:11 UTC", "thanks_received": 1, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "huh, proving??\r\ni just memorized that...", "content_html": "huh, proving??<br>\ni just memorized that...", "post_id": 4415140, "post_number": 3, "post_time_unix": 1218675375, "post_time_utc": "2008-08-14 00:56:15 UTC", "thanks_received": 1, "user_id": 41147, "username": "FantasyLover" }, { "attachments": [], "content_bbcode": "there are LOT of problems that not only need a full understanding of this limit but also the proof", "content_html": "there are LOT of problems that not only need a full understanding of this limit but also the proof", "post_id": 4415141, "post_number": 4, "post_time_unix": 1218682469, "post_time_utc": "2008-08-14 02:54:29 UTC", "thanks_received": 1, "user_id": 36435, "username": "Poincare" } ], "source": null }	Prove \[ \frac{d}{dx}\big[\sin x\big]=\cos x. \]	[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differential", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Epsilon-DeltaDefinition", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Epsilon-DeltaProof", "/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit", "/Mathematics/CalculusandAnalysis/Functions/ElementaryFunction", "/Mathematics/CalculusandAnalysis/Functions/Function", "/Mathematics/CalculusandAnalysis/Functions/RealFunction" ]	Apply the sine addition formula to rewrite the difference quotient and use the limit sin h/h → 1.	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aops_99679	A somewhat trivial solution is $4^3=1^3+5(1^2)+58=64$ [quote="sunchips"] $(y+1)^3<y^3+5y^2+58<(y+2)^3$ for general, then go through small cases not covered.[/quote] The inequality doesn't work for $y=1$ but I guess that's what you meant by "general".	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find $\\ x,y\\in\\mathbb{N}$ such that $\\ x^3-y^3=5y^2+58$.", "content_html": "Find <img src=\"//latex.artofproblemsolving.com/c/f/b/cfb575110ed0deca25e5723082b6a2fa64075bd9.png\" class=\"latex\" alt=\"$\\ x,y\\in\\mathbb{N}$\" style=\"vertical-align: -3px\" width=\"68\" height=\"15\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/6/d/d6d2da98d5d2bbebcfcaf6c6da869457bf286496.png\" class=\"latex\" alt=\"$\\ x^3-y^3=5y^2+58$\" style=\"vertical-align: -3px\" width=\"152\" height=\"18\" >.</span>", "post_id": 562684, "post_number": 1, "post_time_unix": 1151679852, "post_time_utc": "2006-06-30 15:04:12 UTC", "thanks_received": 2, "user_id": 17823, "username": "stancioiu sorin" }, { "attachments": [], "content_bbcode": "$(y+1)^3<y^3+5y^2+58<(y+2)^3$ for general, then go through small cases not covered.", "content_html": "<img src=\"//latex.artofproblemsolving.com/8/8/9/8898ef0308a4c04d37937200140a347ef03974a4.png\" class=\"latex\" alt=\"$(y+1)^3<y^3+5y^2+58<(y+2)^3$\" style=\"vertical-align: -4px\" width=\"277\" height=\"19\" > for general, then go through small cases not covered.", "post_id": 562848, "post_number": 2, "post_time_unix": 1151688784, "post_time_utc": "2006-06-30 17:33:04 UTC", "thanks_received": 1, "user_id": 20477, "username": "sunchips" }, { "attachments": [], "content_bbcode": "A somewhat trivial solution is $4^3=1^3+5(1^2)+58=64$\r\n[quote=\"sunchips\"] $(y+1)^3<y^3+5y^2+58<(y+2)^3$ for general, then go through small cases not covered.[/quote]\r\nThe inequality doesn't work for $y=1$ but I guess that's what you meant by \"general\".", "content_html": "A somewhat trivial solution is <img src=\"//latex.artofproblemsolving.com/0/a/5/0a57c155b29f28ec694266a8903376aa1a006a21.png\" class=\"latex\" alt=\"$4^3=1^3+5(1^2)+58=64$\" style=\"vertical-align: -4px\" width=\"202\" height=\"19\" >\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">sunchips wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/8/8/9/8898ef0308a4c04d37937200140a347ef03974a4.png\" class=\"latex\" alt=\"$(y+1)^3<y^3+5y^2+58<(y+2)^3$\" style=\"vertical-align: -4px\" width=\"277\" height=\"19\" > for general, then go through small cases not covered.</div>\n</div>\nThe inequality doesn't work for <img src=\"//latex.artofproblemsolving.com/6/c/3/6c3b4834614e5e7e189cb7d621cfb4fb0bd36dd7.png\" class=\"latex\" alt=\"$y=1$\" style=\"vertical-align: -3px\" width=\"42\" height=\"15\" > but I guess that's what you meant by "general".", "post_id": 562859, "post_number": 3, "post_time_unix": 1151689309, "post_time_utc": "2006-06-30 17:41:49 UTC", "thanks_received": 2, "user_id": 16287, "username": "quantum leap" } ], "source": null }	Find $x,y\in\mathbb{N}$ such that \[ x^3-y^3=5y^2+58. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/CubicEquation", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/DiophantineEquation3rdPowers", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/N", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/WholeNumber", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/Polynomials/CubicEquation", "/Mathematics/Algebra/Polynomials/CubicPolynomial", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation3rdPowers", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus" ]	Bound the right‑hand side between (y+1)^3 and (y+2)^3 to force x to be y+1 or y+2.	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aops_99692	We consider the field $\mathbb{F}_{p}$ (that one $\mod p$) and it's extension $\mathbb{F}_{p^{3}}$ given by $\mathbb{F}_{p}[\sqrt[3]2]$. We want that $a^{3}+2b^{3}+4c^{3}-6abc = 0$ and it suffices to show that this only happens for $a=b=c=0$ in $\mathbb{F}_{p}$. Let $\zeta \neq , \zeta^{3}=1$, so let the roots of $x^{3}-2$ be $\sqrt[3]2, \zeta \sqrt[3]2 , \zeta^{2}\sqrt[3]2$ (clearly $\zeta \in \mathbb{F}_{p}$). Then we have $0=a^{3}+2b^{3}+4c^{3}-6abc =$ $= (a+\sqrt[3]2 b+\sqrt[3]4 c)(a+\zeta \sqrt[3]2 b+\zeta^{2}\sqrt[3]4 c)(a+\zeta^{2}\sqrt[3]2 b+\zeta \sqrt[3]4 c)$, so one factor is zero (we are in a field!). WLOG let be $a+\sqrt[3]2 b+\sqrt[3]4 c=0$. Since $a,b,c \in \mathbb{F_{p}}$, when they would not be all [b][i]0[/i][/b] we would have a polynomial of second degree over $\mathbb{F}_{p}$ with root $\sqrt[3]2$, namely $a+bx+cx^{2}$. But the latter is impossible. Thus $a=b=c=0$ and we are done. And we have shown even more: only numbers can be of that form for that all prime divisors for that $2$ is not a perfect cube occure a mutltiple of three times in the factorisation; The natural question occurs: is this criterion sufficient¿ Answer: yes (but I don't have an elementary proof or even have some dumb mistake...). So I propose: Let $P$ be the set of prime numbers such that $x^{3}\equiv 2 \mod p$ has no solution. Then $n$ is of type $n=a^{3}+2b^{3}+4c^{3}-6abc$ iff every prime factor from $P$ occurs $0,3,6,9,...$ times in $n$. (note that this is very similar to the condition on $n$ being the sum of two squares and this is no random event)	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $p$ be prime. When there is no integer $a$ such that $a^3 \\equiv 2 (mod p)$, prove that there are no integers $a,b,c$ such that\r\n$a^3 +2b^3 +4c^3 - 6abc = p$.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png\" class=\"latex\" alt=\"$p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > be prime. When there is no integer <img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/2/c/32c1d889810055bb43799e56a962deabe4b08ae4.png\" class=\"latex\" alt=\"$a^3 \\equiv 2 (mod p)$\" style=\"vertical-align: -4px\" width=\"106\" height=\"19\" >,</span> prove that there are no integers <img src=\"//latex.artofproblemsolving.com/a/5/b/a5b29b358e825defa3e11b7d903d43ec31e5909a.png\" class=\"latex\" alt=\"$a,b,c$\" style=\"vertical-align: -3px\" width=\"41\" height=\"16\" > such that<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/4/6/7466960a27faf406bc62719fb35b47813099414f.png\" class=\"latex\" alt=\"$a^3 +2b^3 +4c^3 - 6abc = p$\" style=\"vertical-align: -3px\" width=\"200\" height=\"18\" >.</span>", "post_id": 562810, "post_number": 1, "post_time_unix": 1151685775, "post_time_utc": "2006-06-30 16:42:55 UTC", "thanks_received": 2, "user_id": 1098, "username": "Parkdoosung" }, { "attachments": [], "content_bbcode": "We consider the field $\\mathbb{F}_{p}$ (that one $\\mod p$) and it's extension $\\mathbb{F}_{p^{3}}$ given by $\\mathbb{F}_{p}[\\sqrt[3]2]$.\r\nWe want that $a^{3}+2b^{3}+4c^{3}-6abc = 0$ and it suffices to show that this only happens for $a=b=c=0$ in $\\mathbb{F}_{p}$.\r\nLet $\\zeta \\neq , \\zeta^{3}=1$,\r\nso let the roots of $x^{3}-2$ be $\\sqrt[3]2, \\zeta \\sqrt[3]2 , \\zeta^{2}\\sqrt[3]2$ (clearly $\\zeta \\in \\mathbb{F}_{p}$). Then we have\r\n$0=a^{3}+2b^{3}+4c^{3}-6abc =$\r\n$= (a+\\sqrt[3]2 b+\\sqrt[3]4 c)(a+\\zeta \\sqrt[3]2 b+\\zeta^{2}\\sqrt[3]4 c)(a+\\zeta^{2}\\sqrt[3]2 b+\\zeta \\sqrt[3]4 c)$,\r\nso one factor is zero (we are in a field!).\r\nWLOG let be $a+\\sqrt[3]2 b+\\sqrt[3]4 c=0$. Since $a,b,c \\in \\mathbb{F_{p}}$, when they would not be all [b][i]0[/i][/b] we would have a polynomial of second degree over $\\mathbb{F}_{p}$ with root $\\sqrt[3]2$, namely $a+bx+cx^{2}$. But the latter is impossible.\r\nThus $a=b=c=0$ and we are done.\r\nAnd we have shown even more: only numbers can be of that form for that all prime divisors for that $2$ is not a perfect cube occure a mutltiple of three times in the factorisation;\r\nThe natural question occurs: is this criterion sufficient¿\r\nAnswer: yes (but I don't have an elementary proof or even have some dumb mistake...).\r\n\r\nSo I propose:\r\n\r\nLet $P$ be the set of prime numbers such that $x^{3}\\equiv 2 \\mod p$ has no solution. Then $n$ is of type $n=a^{3}+2b^{3}+4c^{3}-6abc$ iff every prime factor from $P$ occurs $0,3,6,9,...$ times in $n$.\r\n(note that this is very similar to the condition on $n$ being the sum of two squares and this is no random event)", "content_html": "We consider the field <img src=\"//latex.artofproblemsolving.com/f/a/8/fa8b62027d0275ff2f082d38baf3e86aee45c779.png\" class=\"latex\" alt=\"$\\mathbb{F}_{p}$\" style=\"vertical-align: -4px\" width=\"17\" height=\"16\" > (that one <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/f/9/0f965eb9407014fe0e4fc671eed0acbfd84eddb2.png\" class=\"latex\" alt=\"$\\mod p$\" style=\"vertical-align: -3px\" width=\"62\" height=\"16\" >)</span> and it's extension <img src=\"//latex.artofproblemsolving.com/e/b/b/ebb23add54854a585dbddb677e679d50561a20a9.png\" class=\"latex\" alt=\"$\\mathbb{F}_{p^{3}}$\" style=\"vertical-align: -4px\" width=\"23\" height=\"16\" > given by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/1/e01f2f4641bcf0a3f05f9bf41e5bfd45f98916cf.png\" class=\"latex\" alt=\"$\\mathbb{F}_{p}[\\sqrt[3]2]$\" style=\"vertical-align: -5px\" width=\"51\" height=\"22\" >.</span><br>\nWe want that <img src=\"//latex.artofproblemsolving.com/1/c/f/1cf43858ae57b2e642f0e680dd1c1947ad5b3a85.png\" class=\"latex\" alt=\"$a^{3}+2b^{3}+4c^{3}-6abc = 0$\" style=\"vertical-align: -1px\" width=\"199\" height=\"16\" > and it suffices to show that this only happens for <img src=\"//latex.artofproblemsolving.com/f/3/5/f357e4a021e8d3ae6e9b48511227593f74b6be25.png\" class=\"latex\" alt=\"$a=b=c=0$\" width=\"106\" height=\"12\" > in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/8/fa8b62027d0275ff2f082d38baf3e86aee45c779.png\" class=\"latex\" alt=\"$\\mathbb{F}_{p}$\" style=\"vertical-align: -4px\" width=\"17\" height=\"16\" >.</span><br>\nLet <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/a/3/0a3df8e0f488c8c1ce006756a06f3e57646b12b9.png\" class=\"latex\" alt=\"$\\zeta \\neq , \\zeta^{3}=1$\" style=\"vertical-align: -4px\" width=\"85\" height=\"19\" >,</span><br>\nso let the roots of <img src=\"//latex.artofproblemsolving.com/5/1/8/518696c9f6072db3fa24bb49a824b10f7e1cb757.png\" class=\"latex\" alt=\"$x^{3}-2$\" width=\"48\" height=\"15\" > be <img src=\"//latex.artofproblemsolving.com/6/8/e/68e8b7876e5f01771bfea8c900c6edcb8c070c7d.png\" class=\"latex\" alt=\"$\\sqrt[3]2, \\zeta \\sqrt[3]2 , \\zeta^{2}\\sqrt[3]2$\" style=\"vertical-align: -3px\" width=\"117\" height=\"20\" > (clearly <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/9/26915b7e30020ea9f4f2daeb93293215e6c111af.png\" class=\"latex\" alt=\"$\\zeta \\in \\mathbb{F}_{p}$\" style=\"vertical-align: -4px\" width=\"49\" height=\"17\" >)</span>. Then we have<br>\n<img src=\"//latex.artofproblemsolving.com/4/1/4/4141f0ebf41030799f25f8b205742e98bd4550ce.png\" class=\"latex\" alt=\"$0=a^{3}+2b^{3}+4c^{3}-6abc =$\" style=\"vertical-align: -1px\" width=\"218\" height=\"16\" ><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/3/2/d32c5bf3dcacb300cbb8183c50398aa3f1f9fc16.png\" class=\"latex\" alt=\"$= (a+\\sqrt[3]2 b+\\sqrt[3]4 c)(a+\\zeta \\sqrt[3]2 b+\\zeta^{2}\\sqrt[3]4 c)(a+\\zeta^{2}\\sqrt[3]2 b+\\zeta \\sqrt[3]4 c)$\" style=\"vertical-align: -4px\" width=\"470\" height=\"21\" >,</span><br>\nso one factor is zero (we are in a field!).<br>\nWLOG let be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/6/0/f60de9437a280f6031fb7341f7a228bd7d4b83d2.png\" class=\"latex\" alt=\"$a+\\sqrt[3]2 b+\\sqrt[3]4 c=0$\" style=\"vertical-align: -1px\" width=\"152\" height=\"18\" >.</span> Since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/b/c6b031096511bacbc6eb8b7e6db01b785fc485a1.png\" class=\"latex\" alt=\"$a,b,c \\in \\mathbb{F_{p}}$\" style=\"vertical-align: -3px\" width=\"77\" height=\"16\" >,</span> when they would not be all <b><i>0</i></b> we would have a polynomial of second degree over <img src=\"//latex.artofproblemsolving.com/f/a/8/fa8b62027d0275ff2f082d38baf3e86aee45c779.png\" class=\"latex\" alt=\"$\\mathbb{F}_{p}$\" style=\"vertical-align: -4px\" width=\"17\" height=\"16\" > with root <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/d/2/fd2db1af029a3aaaad5a3ee0ea826875cec0022e.png\" class=\"latex\" alt=\"$\\sqrt[3]2$\" style=\"vertical-align: -1px\" width=\"25\" height=\"18\" >,</span> namely <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/c/4ac6d0b3a11bf0c258bed6ff91a12eef18128980.png\" class=\"latex\" alt=\"$a+bx+cx^{2}$\" style=\"vertical-align: -1px\" width=\"96\" height=\"16\" >.</span> But the latter is impossible.<br>\nThus <img src=\"//latex.artofproblemsolving.com/f/3/5/f357e4a021e8d3ae6e9b48511227593f74b6be25.png\" class=\"latex\" alt=\"$a=b=c=0$\" width=\"106\" height=\"12\" > and we are done.<br>\nAnd we have shown even more: only numbers can be of that form for that all prime divisors for that <img src=\"//latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png\" class=\"latex\" alt=\"$2$\" width=\"8\" height=\"12\" > is not a perfect cube occure a mutltiple of three times in the factorisation;<br>\nThe natural question occurs: is this criterion sufficient¿<br>\nAnswer: yes (but I don't have an elementary proof or even have some dumb mistake...).<br>\n<br>\nSo I propose:<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > be the set of prime numbers such that <img src=\"//latex.artofproblemsolving.com/5/8/2/582ffd1b284f087bbd149360e9bc1ccc167d4eca.png\" class=\"latex\" alt=\"$x^{3}\\equiv 2 \\mod p$\" style=\"vertical-align: -3px\" width=\"113\" height=\"18\" > has no solution. Then <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > is of type <img src=\"//latex.artofproblemsolving.com/d/1/2/d120dd3b17c8729ce2a0bb2d494b3470810fbc58.png\" class=\"latex\" alt=\"$n=a^{3}+2b^{3}+4c^{3}-6abc$\" style=\"vertical-align: -1px\" width=\"202\" height=\"16\" > iff every prime factor from <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > occurs <img src=\"//latex.artofproblemsolving.com/c/d/9/cd916a156d67a57526e030a8123bbe9938c4dbec.png\" class=\"latex\" alt=\"$0,3,6,9,...$\" style=\"vertical-align: -3px\" width=\"82\" height=\"16\" > times in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >.</span><br>\n(note that this is very similar to the condition on <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > being the sum of two squares and this is no random event)", "post_id": 563192, "post_number": 2, "post_time_unix": 1151716041, "post_time_utc": "2006-07-01 01:07:21 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" }, { "attachments": [], "content_bbcode": "Nice solution :lol: \r\nI also proved your proposed problem, but I used that $Z[\\sqrt[3]2]$ is UFD, which is not an elementary fact. :(", "content_html": "Nice solution <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /><br>\nI also proved your proposed problem, but I used that <img src=\"//latex.artofproblemsolving.com/7/c/e/7ce740697644c2d263048b3adfad0822193fcefe.png\" class=\"latex\" alt=\"$Z[\\sqrt[3]2]$\" style=\"vertical-align: -5px\" width=\"46\" height=\"22\" > is UFD, which is not an elementary fact. <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />", "post_id": 563570, "post_number": 3, "post_time_unix": 1151758724, "post_time_utc": "2006-07-01 12:58:44 UTC", "thanks_received": 2, "user_id": 1098, "username": "Parkdoosung" }, { "attachments": [], "content_bbcode": "I used that it has class number $1$, but thats equivalent.", "content_html": "I used that it has class number <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png\" class=\"latex\" alt=\"$1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> but thats equivalent.", "post_id": 563645, "post_number": 4, "post_time_unix": 1151764160, "post_time_utc": "2006-07-01 14:29:20 UTC", "thanks_received": 2, "user_id": 5787, "username": "ZetaX" } ], "source": null }	Let $p$ be a prime. When there is no integer $a$ such that $a^3\equiv 2\pmod p$, prove that there are no integers $a,b,c$ such that \[ a^3+2b^3+4c^3-6abc=p. \]	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity", "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/NumberTheory/AlgebraicNumberTheory", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/Algebra/NumberTheory/Congruences/Mod", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/DiophantineEquation3rdPowers", "/Mathematics/Algebra/Polynomials/CubicEquation", "/Mathematics/Algebra/Polynomials/CubicPolynomial", "/Mathematics/Algebra/Polynomials/EisensteinsIrreducibilityCriterion", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/IrreduciblePolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialFactorTheorem", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/PolynomialIdentity", "/Mathematics/Algebra/Polynomials/PolynomialRoots", "/Mathematics/Algebra/Polynomials/PrimeDivisor", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/NumberTheory/AlgebraicNumberTheory", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Factor the cubic using the three cube‑roots of 2 and a primitive cube root of unity, forcing a linear relation that would give a cube root of 2 in 𝔽_p, a contradiction.	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aops_996931	\[ \begin{align}a^3\plus{}b^3\plus{}c^3\minus{}3abc&\equal{}(a\plus{}b\plus{}c)(a^2\plus{}b^2\plus{}c^2)\minus{}\sum_{\text{cyc}}(a^2b\plus{}ab^2)\minus{}3abc\\&\equal{}(a\plus{}b\plus{}c)(a^2\plus{}b^2\plus{}c^2)\minus{}\sum_{\text{cyc}}(a^2b\plus{}ab^2\plus{}abc)\\&\equal{}(a\plus{}b\plus{}c)(a^2\plus{}b^2\plus{}c^2\minus{}\sum_{\text{cyc}}(ab(a\plus{}b\plus{}c))\\&\equal{}(a\plus{}b\plus{}c)\left(a^2\plus{}b^2\plus{}c^2\minus{}\sum_{\text{cyc}}ab\right)\\&\equal{}(a\plus{}b\plus{}c)(a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac)\end{align}.\]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Factor $ a^3\\plus{}b^3\\plus{}c^3\\minus{}3abc$\r\n\r\nSource: Mathematical Olympiad Treasures \r\n\r\n(extra points if you prove an identity out of this)", "content_html": "Factor <img src=\"//latex.artofproblemsolving.com/7/3/c/73c8233fb6fc0b3c579f13dcb2d78dccf49a2590.png\" class=\"latex\" alt=\"$ a^3+b^3+c^3-3abc$\" style=\"vertical-align: -1px\" width=\"149\" height=\"16\" ><br>\n<br>\nSource: Mathematical Olympiad Treasures<br>\n<br>\n(extra points if you prove an identity out of this)", "post_id": 4415895, "post_number": 1, "post_time_unix": 1249241988, "post_time_utc": "2009-08-02 19:39:48 UTC", "thanks_received": 1, "user_id": 36435, "username": "Poincare" }, { "attachments": [], "content_bbcode": "$ (a\\plus{}b\\plus{}c)(a^2\\plus{}b^2\\plus{}c^2\\minus{}ab\\minus{}bc\\minus{}ca)$. sorry, memmed.", "content_html": "<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/5/8/858935c7d6b5c1b7aa6cfe4627c2e7f398e83d52.png\" class=\"latex\" alt=\"$ (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$\" style=\"vertical-align: -4px\" width=\"306\" height=\"19\" >.</span> sorry, memmed.", "post_id": 4415896, "post_number": 2, "post_time_unix": 1249243461, "post_time_utc": "2009-08-02 20:04:21 UTC", "thanks_received": 1, "user_id": 32890, "username": "gauss1181" }, { "attachments": [], "content_bbcode": "\\[ \\begin{align}a^3\\plus{}b^3\\plus{}c^3\\minus{}3abc&\\equal{}(a\\plus{}b\\plus{}c)(a^2\\plus{}b^2\\plus{}c^2)\\minus{}\\sum_{\\text{cyc}}(a^2b\\plus{}ab^2)\\minus{}3abc\\\\&\\equal{}(a\\plus{}b\\plus{}c)(a^2\\plus{}b^2\\plus{}c^2)\\minus{}\\sum_{\\text{cyc}}(a^2b\\plus{}ab^2\\plus{}abc)\\\\&\\equal{}(a\\plus{}b\\plus{}c)(a^2\\plus{}b^2\\plus{}c^2\\minus{}\\sum_{\\text{cyc}}(ab(a\\plus{}b\\plus{}c))\\\\&\\equal{}(a\\plus{}b\\plus{}c)\\left(a^2\\plus{}b^2\\plus{}c^2\\minus{}\\sum_{\\text{cyc}}ab\\right)\\\\&\\equal{}(a\\plus{}b\\plus{}c)(a^2\\plus{}b^2\\plus{}c^2\\minus{}ab\\minus{}bc\\minus{}ac)\\end{align}.\\]", "content_html": "<pre class=\"aopscode-error aopscode-latex-error\">\\[ \\begin{align}a^3+b^3+c^3-3abc&=(a+b+c)(a^2+b^2+c^2)-\\sum_{\\text{cyc}}(a^2b+ab^2)-3abc\\\\&=(a+b+c)(a^2+b^2+c^2)-\\sum_{\\text{cyc}}(a^2b+ab^2+abc)\\\\&=(a+b+c)(a^2+b^2+c^2-\\sum_{\\text{cyc}}(ab(a+b+c))\\\\&=(a+b+c)\\left(a^2+b^2+c^2-\\sum_{\\text{cyc}}ab\\right)\\\\&=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)\\end{align}.\\]</pre>", "post_id": 4415897, "post_number": 3, "post_time_unix": 1249313376, "post_time_utc": "2009-08-03 15:29:36 UTC", "thanks_received": 1, "user_id": 60109, "username": "RoFlLoLcOpT" } ], "source": null }	Factor $a^3 + b^3 + c^3 - 3abc$. Source: Mathematical Olympiad Treasures (Extra points if you prove an identity out of this.)	[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity", "/Mathematics/Algebra/AlgebraicIdentities/TrinomialIdentity", "/Mathematics/Algebra/AlgebraicOperations/PolynomialTransformations", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/PolynomialIdentity", "/Mathematics/Algebra/Polynomials/SymmetricPolynomial", "/Mathematics/Algebra/Products/Product", "/Mathematics/Algebra/Sums/Sum" ]	Recognize that a³+b³+c³−3abc factors as (a+b+c)·(a²+b²+c²−ab−bc−ca).	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aops_99695	Let $n=2m-1$ and $s=\mid S\mid $. Then \[a_{n}= \sum_{s=1}^{n}2 \sum_{t=s}^{m-1}{m-1-t\choose s-1}= 2 \sum_{s=1}^{n}{m-s\choose s}= 2(F_{m+1}-1)\] where $2t$ or $2t+1$ stand for the smallest element of $S$. Similarly, if $n=2m$ and $s=\mid S\mid $ then \[a_{n}= \sum_{s=1}^{n}\sum_{t=s}^{m}{m-t\choose s-1}+\sum_{t=s}^{m-1}{m-1-t\choose s-1}= (F_{m+2}-1)+(F_{m+1}-1) = F_{m+3}-2.\]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Let $a_n$ be the number of nonempty subsets $S$ such that\r\n(i) $S \\subseteq {1,2,\\ldots,n}$\r\n(ii) all elements of $S$ have the same parity\r\n(iii) each element $k \\in S$ satisfies $k \\ge 2\|S\|$, where $\|S\|$ is the number of elements in $\|S\|$\r\nProve that \\[ a_{2m-1} = 2(F_{m+1} -1) \\] and \\[ a_{2m} = F_{m+3}-2 \\] for all $m \\ge 1$, where $F_n$ is the $n^{th}$ Fibonacci number.", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/6/f/1/6f1a2b6b7c193f9ba2b2134a59f1da5addcfbc98.png\" class=\"latex\" alt=\"$a_n$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > be the number of nonempty subsets <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > such that<br>\n(i) <img src=\"//latex.artofproblemsolving.com/a/d/4/ad4ae4575f6bb9915d5244452a262664635fbd77.png\" class=\"latex\" alt=\"$S \\subseteq {1,2,\\ldots,n}$\" style=\"vertical-align: -3px\" width=\"113\" height=\"16\" ><br>\n(ii) all elements of <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > have the same parity<br>\n(iii) each element <img src=\"//latex.artofproblemsolving.com/0/d/4/0d457c6c10a552e339986f47f93e7c8c45694126.png\" class=\"latex\" alt=\"$k \\in S$\" style=\"vertical-align: -1px\" width=\"44\" height=\"13\" > satisfies <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/8/c684fa5fbec2391ace41eac3623d7889f160b6bb.png\" class=\"latex\" alt=\"$k \\ge 2\|S\|$\" style=\"vertical-align: -4px\" width=\"63\" height=\"18\" >,</span> where <img src=\"//latex.artofproblemsolving.com/1/1/c/11ceba9d7455309f60178fe82bb6e81e6c01fa83.png\" class=\"latex\" alt=\"$\|S\|$\" style=\"vertical-align: -4px\" width=\"20\" height=\"18\" > is the number of elements in <img src=\"//latex.artofproblemsolving.com/1/1/c/11ceba9d7455309f60178fe82bb6e81e6c01fa83.png\" class=\"latex\" alt=\"$\|S\|$\" style=\"vertical-align: -4px\" width=\"20\" height=\"18\" ><br>\nProve that <img src=\"//latex.artofproblemsolving.com/7/3/d/73d4edccd71e88f894f42b3c4b7bf47a6e222e95.png\" class=\"latexcenter\" alt=\"\\[ a_{2m-1} = 2(F_{m+1} -1) \\]\" width=\"163\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/e/a/a/eaac8c646ffa7ee94d96f6681ef166925f729dbb.png\" class=\"latexcenter\" alt=\"\\[ a_{2m} = F_{m+3}-2 \\]\" width=\"124\" height=\"17\" > for all <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/d/4/1d42c8080f1f22bae152b5dc35922de7178496be.png\" class=\"latex\" alt=\"$m \\ge 1$\" style=\"vertical-align: -2px\" width=\"48\" height=\"14\" >,</span> where <img src=\"//latex.artofproblemsolving.com/b/8/4/b842b4552478b69732a9aea702309d8c51e43056.png\" class=\"latex\" alt=\"$F_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > is the <img src=\"//latex.artofproblemsolving.com/2/f/a/2fa62a365c27d4f2c298262b3e8c7c224550c20a.png\" class=\"latex\" alt=\"$n^{th}$\" width=\"23\" height=\"15\" > Fibonacci number.", "post_id": 562820, "post_number": 1, "post_time_unix": 1151686546, "post_time_utc": "2006-06-30 16:55:46 UTC", "thanks_received": 2, "user_id": 2556, "username": "Optimosis" }, { "attachments": [], "content_bbcode": "Let $n=2m-1$ and $s=\\mid S\\mid $. Then \\[a_{n}= \\sum_{s=1}^{n}2 \\sum_{t=s}^{m-1}{m-1-t\\choose s-1}= 2 \\sum_{s=1}^{n}{m-s\\choose s}= 2(F_{m+1}-1)\\] where $2t$ or $2t+1$ stand for the smallest element of $S$. \r\n\r\nSimilarly, if $n=2m$ and $s=\\mid S\\mid $ then \\[a_{n}= \\sum_{s=1}^{n}\\sum_{t=s}^{m}{m-t\\choose s-1}+\\sum_{t=s}^{m-1}{m-1-t\\choose s-1}= (F_{m+2}-1)+(F_{m+1}-1) = F_{m+3}-2.\\]", "content_html": "Let <img src=\"//latex.artofproblemsolving.com/7/2/c/72c5e5eca42e4a07a4fe92d17bd1a4dcb6b4582c.png\" class=\"latex\" alt=\"$n=2m-1$\" style=\"vertical-align: 0px\" width=\"90\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/3/8/b3882d35190bb32559d3af97d894b8aeced27366.png\" class=\"latex\" alt=\"$s=\\mid S\\mid $\" style=\"vertical-align: -4px\" width=\"58\" height=\"18\" >.</span> Then <img src=\"//latex.artofproblemsolving.com/0/2/3/02351279f33265222122b48c206fcb11a5a6abd7.png\" class=\"latexcenter\" alt=\"\\[a_{n}= \\sum_{s=1}^{n}2 \\sum_{t=s}^{m-1}{m-1-t\\choose s-1}= 2 \\sum_{s=1}^{n}{m-s\\choose s}= 2(F_{m+1}-1)\\]\" width=\"477\" height=\"53\" > where <img src=\"//latex.artofproblemsolving.com/f/6/5/f65450756c44b33df63c304f031fdfc2751a0d56.png\" class=\"latex\" alt=\"$2t$\" width=\"15\" height=\"12\" > or <img src=\"//latex.artofproblemsolving.com/e/4/8/e486b00859288870f521055a85f726e52f3e56e9.png\" class=\"latex\" alt=\"$2t+1$\" style=\"vertical-align: -1px\" width=\"46\" height=\"13\" > stand for the smallest element of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" >.</span><br>\n<br>\nSimilarly, if <img src=\"//latex.artofproblemsolving.com/0/f/2/0f28f7175430049f8e55d0a8fd07d6b710bf2b03.png\" class=\"latex\" alt=\"$n=2m$\" width=\"59\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/b/3/8/b3882d35190bb32559d3af97d894b8aeced27366.png\" class=\"latex\" alt=\"$s=\\mid S\\mid $\" style=\"vertical-align: -4px\" width=\"58\" height=\"18\" > then <img src=\"//latex.artofproblemsolving.com/4/4/9/449db45595408266b8a94c5d1dcd6f1e5ecd6e6f.png\" class=\"latexcenter\" alt=\"\\[a_{n}= \\sum_{s=1}^{n}\\sum_{t=s}^{m}{m-t\\choose s-1}+\\sum_{t=s}^{m-1}{m-1-t\\choose s-1}= (F_{m+2}-1)+(F_{m+1}-1) = F_{m+3}-2.\\]\" width=\"610\" height=\"53\" >", "post_id": 565070, "post_number": 2, "post_time_unix": 1151920217, "post_time_utc": "2006-07-03 09:50:17 UTC", "thanks_received": 2, "user_id": 8090, "username": "maxal" } ], "source": null }	Let $a_n$ be the number of nonempty subsets $S$ such that (i) $S \subseteq \{1,2,\ldots,n\}$, (ii) all elements of $S$ have the same parity, (iii) each element $k\in S$ satisfies $k \ge 2\|S\|$, where $\|S\|$ denotes the number of elements of $S$. Prove that for all $m\ge 1$, \[ a_{2m-1}=2\bigl(F_{m+1}-1\bigr) \quad\text{and}\quad a_{2m}=F_{m+3}-2, \] where $F_n$ is the $n$th Fibonacci number.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration" ]	Reduce the subset count to a sum of binomial coefficients C(m‑s, s), which equals a Fibonacci number.	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aops_99720	[hide="1"]$x^{2}+y^{2}= kxy$ with $k \in \mathbb{N}$ $x^{2}-kxy+y^{2}= 0 \Rightarrow x_{1/2}= \frac{ky \pm \sqrt{k^{2}y^{2}-4y^{2}}}{2}$. Hence $k^{2}-4 = m^{2}\Rightarrow (k-m)(k+m) = 4$, that gives $k = 2$. So $(a, a)$ is solution, with $a \in \mathbb{N}$.[/hide] [hide="Hint for 2"]Pythagorean triples[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Find all positive $x$ and $y$ such that\r\n\r\n$\\frac{x^{2}+y^{2}}{xy}$ is an integer.\r\n\r\nalso find all positive $x$ and $y$ such that\r\n$x+y+\\sqrt{x^{2}+y^{2}}=\\frac{xy}{2}$", "content_html": "Find all positive <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > such that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/6/e/26e23cfe52eb1cf47112e757c22920dadd2e56a3.png\" class=\"latex\" alt=\"$\\frac{x^{2}+y^{2}}{xy}$\" style=\"vertical-align: -16px\" width=\"59\" height=\"42\" > is an integer.<br>\n<br>\nalso find all positive <img src=\"//latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png\" class=\"latex\" alt=\"$x$\" width=\"10\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png\" class=\"latex\" alt=\"$y$\" style=\"vertical-align: -3px\" width=\"9\" height=\"11\" > such that<br>\n<img src=\"//latex.artofproblemsolving.com/8/0/b/80b6f7adc94409e0122a55938e6cffccc21205ee.png\" class=\"latex\" alt=\"$x+y+\\sqrt{x^{2}+y^{2}}=\\frac{xy}{2}$\" style=\"vertical-align: -12px\" width=\"185\" height=\"33\" >", "post_id": 563073, "post_number": 1, "post_time_unix": 1151704638, "post_time_utc": "2006-06-30 21:57:18 UTC", "thanks_received": 2, "user_id": 10301, "username": "maokid7" }, { "attachments": [], "content_bbcode": "Do you mean positive integers?", "content_html": "Do you mean positive integers?", "post_id": 563121, "post_number": 2, "post_time_unix": 1151708589, "post_time_utc": "2006-06-30 23:03:09 UTC", "thanks_received": 2, "user_id": 17438, "username": "Jiang" }, { "attachments": [], "content_bbcode": "o yes...sorry i forgot to mention that. :D", "content_html": "o yes...sorry i forgot to mention that. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 563125, "post_number": 3, "post_time_unix": 1151709387, "post_time_utc": "2006-06-30 23:16:27 UTC", "thanks_received": 2, "user_id": 10301, "username": "maokid7" }, { "attachments": [], "content_bbcode": "[hide=\"1\"]$x^{2}+y^{2}= kxy$ with $k \\in \\mathbb{N}$\n$x^{2}-kxy+y^{2}= 0 \\Rightarrow x_{1/2}= \\frac{ky \\pm \\sqrt{k^{2}y^{2}-4y^{2}}}{2}$.\nHence $k^{2}-4 = m^{2}\\Rightarrow (k-m)(k+m) = 4$, that gives $k = 2$.\nSo $(a, a)$ is solution, with $a \\in \\mathbb{N}$.[/hide]\n\n[hide=\"Hint for 2\"]Pythagorean triples[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">1</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/3/0/a/30ac0749259b712af05515e38e6d63920ade8add.png\" class=\"latex\" alt=\"$x^{2}+y^{2}= kxy$\" style=\"vertical-align: -3px\" width=\"111\" height=\"18\" > with <img src=\"//latex.artofproblemsolving.com/7/6/7/767a53f43be056f9333f71e3e39f59aa0f8bca35.png\" class=\"latex\" alt=\"$k \\in \\mathbb{N}$\" style=\"vertical-align: -1px\" width=\"45\" height=\"13\" ><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/9/2/a9287666beaff625b51bdf3bec4af6628d0b48d6.png\" class=\"latex\" alt=\"$x^{2}-kxy+y^{2}= 0 \\Rightarrow x_{1/2}= \\frac{ky \\pm \\sqrt{k^{2}y^{2}-4y^{2}}}{2}$\" style=\"vertical-align: -12px\" width=\"368\" height=\"39\" >.</span><br>\nHence <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/b/f/dbf65a782bed236688ae4f315379f4759c553eb6.png\" class=\"latex\" alt=\"$k^{2}-4 = m^{2}\\Rightarrow (k-m)(k+m) = 4$\" style=\"vertical-align: -4px\" width=\"282\" height=\"19\" >,</span> that gives <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/4/8e44ca2cae6be42148fe1b1fd8b5d55bd6b599bd.png\" class=\"latex\" alt=\"$k = 2$\" width=\"43\" height=\"12\" >.</span><br>\nSo <img src=\"//latex.artofproblemsolving.com/a/9/7/a972055a2c2274518c4eb811d7344cd89ac76419.png\" class=\"latex\" alt=\"$(a, a)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > is solution, with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/a/d2a74338dcca72fb1317929a3f768d8ebf682feb.png\" class=\"latex\" alt=\"$a \\in \\mathbb{N}$\" style=\"vertical-align: -1px\" width=\"44\" height=\"13\" >.</span></div><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hint for 2</a><div class=\"cmty-hide-content\" style=\"display:none\">Pythagorean triples</div>", "post_id": 563429, "post_number": 4, "post_time_unix": 1151740881, "post_time_utc": "2006-07-01 08:01:21 UTC", "thanks_received": 2, "user_id": 5839, "username": "Andreas" }, { "attachments": [], "content_bbcode": "can we do this reasoning for 1?\r\n\r\n[hide]\n\n$xy\\mid x^{2}+y^{2}\\implies x^{2}+y^{2}\\equiv 0 \\mod x$\n\nThus, $y=kx$ for some $k$ integer\n\nSo the expression becomes $\\frac{x^{2}+k^{2}x^{2}}{kx^{2}}=\\frac{k^{2}+1}{k}$\n\nbut we have that $\\gcd (k,k^{2}+1)=1$, so necessary $k=1$. Therefore, the only solutions are $x=y$ $\\forall x \\in \\mathbb{N}$[/hide]", "content_html": "can we do this reasoning for 1?<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/d/8/c/d8c78787a4d996b18541e75173080ebeb5f30bdb.png\" class=\"latex\" alt=\"$xy\\mid x^{2}+y^{2}\\implies x^{2}+y^{2}\\equiv 0 \\mod x$\" style=\"vertical-align: -4px\" width=\"295\" height=\"19\" ><br>\n<br>\nThus, <img src=\"//latex.artofproblemsolving.com/c/d/1/cd14499bebbf7855623388869c0910c4a004bf77.png\" class=\"latex\" alt=\"$y=kx$\" style=\"vertical-align: -3px\" width=\"54\" height=\"16\" > for some <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > integer<br>\n<br>\nSo the expression becomes <img src=\"//latex.artofproblemsolving.com/f/4/0/f400bd52a25f063b4ee1c49c34b4acce978efd3e.png\" class=\"latex\" alt=\"$\\frac{x^{2}+k^{2}x^{2}}{kx^{2}}=\\frac{k^{2}+1}{k}$\" style=\"vertical-align: -12px\" width=\"154\" height=\"39\" ><br>\n<br>\nbut we have that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/2/c62455bdf05287e8c81df8d3ed26bbbfe8e63621.png\" class=\"latex\" alt=\"$\\gcd (k,k^{2}+1)=1$\" style=\"vertical-align: -4px\" width=\"140\" height=\"19\" >,</span> so necessary <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/e/d/5ed8470bc717a49aafe4556c7e2c2e11ddc7b687.png\" class=\"latex\" alt=\"$k=1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"13\" >.</span> Therefore, the only solutions are <img src=\"//latex.artofproblemsolving.com/0/1/1/0110f1bb04c51b2a4d431bf14489fc1b48b37421.png\" class=\"latex\" alt=\"$x=y$\" style=\"vertical-align: -3px\" width=\"43\" height=\"11\" > <img src=\"//latex.artofproblemsolving.com/3/b/1/3b17d467c756a6c8fa5cc172f986a2e5a6698f31.png\" class=\"latex\" alt=\"$\\forall x \\in \\mathbb{N}$\" style=\"vertical-align: -1px\" width=\"55\" height=\"14\" ></div>", "post_id": 563464, "post_number": 5, "post_time_unix": 1151745863, "post_time_utc": "2006-07-01 09:24:23 UTC", "thanks_received": 2, "user_id": 17962, "username": "hydro" }, { "attachments": [], "content_bbcode": "[quote=\"hydro\"]\n\n$x^{2}+y^{2}\\equiv 0 \\mod x$\n\nThus, $y=kx$ for some $k$ integer\n\n[/quote]\r\nTry $x=9$ and $y=6.$ ;)", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">hydro wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/5/6/1/5614639ce2a795f88855bea2d8121a32ea0d32c0.png\" class=\"latex\" alt=\"$x^{2}+y^{2}\\equiv 0 \\mod x$\" style=\"vertical-align: -3px\" width=\"153\" height=\"18\" ><br>\n<br>\nThus, <img src=\"//latex.artofproblemsolving.com/c/d/1/cd14499bebbf7855623388869c0910c4a004bf77.png\" class=\"latex\" alt=\"$y=kx$\" style=\"vertical-align: -3px\" width=\"54\" height=\"16\" > for some <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > integer</div>\n</div>\nTry <img src=\"//latex.artofproblemsolving.com/5/b/1/5b1d8cdec48abe8293a4169a8afce4fc8a8348e7.png\" class=\"latex\" alt=\"$x=9$\" width=\"43\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/d/c/0/dc0a5c05c0be0d0bba84e827a5c965209d356275.png\" class=\"latex\" alt=\"$y=6.$\" style=\"vertical-align: -3px\" width=\"46\" height=\"16\" > <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 563472, "post_number": 6, "post_time_unix": 1151746518, "post_time_utc": "2006-07-01 09:35:18 UTC", "thanks_received": 2, "user_id": 12908, "username": "arqady" } ], "source": null }	Find all positive $x$ and $y$ such that \[ \frac{x^{2}+y^{2}}{xy}\in\mathbb{Z}. \] Find all positive $x$ and $y$ such that \[ x+y+\sqrt{x^{2}+y^{2}}=\frac{xy}{2}. \]	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/Algebra/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/N", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/RationalInteger", "/Mathematics/Algebra/NumberTheory/Integers/Z", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/NumberTheory/Arithmetic/Fractions", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/PythagoreanTriad", "/Mathematics/NumberTheory/DiophantineEquations/PythagoreanTriangle", "/Mathematics/NumberTheory/DiophantineEquations/PythagoreanTriple", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Set (x²+y²)/(xy)=k∈ℕ, then require the quadratic discriminant to be a perfect square, forcing k=2.	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aops_997246	Let us divide the problem into three cases: If $ x < 0$, we also have $ x > 0$, which leads to a contradiction, so $ x\not < 0$. If $ x \equal{} 0$, we also have $ x > 0$, which leads to a contradiction, so $ x\ne0$. So combining the 2, we have $ x\not\le0$... I'm not sure how to proceed from here though...	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove that $ x > 0$ for all $ x > 0$.\r\n\r\n[color=red]Warning: May cause hysterical laughter.[/color]", "content_html": "Prove that <img src=\"//latex.artofproblemsolving.com/8/3/3/8331222cf49f7a74fb425545e8b7e0e2fb3611dd.png\" class=\"latex\" alt=\"$ x > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" > for all <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/3/8331222cf49f7a74fb425545e8b7e0e2fb3611dd.png\" class=\"latex\" alt=\"$ x > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >.</span><br>\n<br>\n<span style=\"color:red\">Warning: May cause hysterical laughter.</span>", "post_id": 4416651, "post_number": 1, "post_time_unix": 1219789972, "post_time_utc": "2008-08-26 22:32:52 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "Let us divide the problem into three cases:\r\nIf $ x < 0$, we also have $ x > 0$, which leads to a contradiction, so $ x\\not < 0$.\r\nIf $ x \\equal{} 0$, we also have $ x > 0$, which leads to a contradiction, so $ x\\ne0$.\r\nSo combining the 2, we have $ x\\not\\le0$...\r\nI'm not sure how to proceed from here though...", "content_html": "Let us divide the problem into three cases:<br>\nIf <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/2/f/12f777405b504e05e83faec1e624f7447ea6d610.png\" class=\"latex\" alt=\"$ x < 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >,</span> we also have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/3/8331222cf49f7a74fb425545e8b7e0e2fb3611dd.png\" class=\"latex\" alt=\"$ x > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >,</span> which leads to a contradiction, so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/4/5/045453ce0c9e4d88925d0e8bd7a372f85003462f.png\" class=\"latex\" alt=\"$ x\\not < 0$\" style=\"vertical-align: -4px\" width=\"43\" height=\"17\" >.</span><br>\nIf <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/6/e/66ed918a9623c43b07e30672568c80408e47e4ac.png\" class=\"latex\" alt=\"$ x = 0$\" width=\"43\" height=\"12\" >,</span> we also have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/3/8331222cf49f7a74fb425545e8b7e0e2fb3611dd.png\" class=\"latex\" alt=\"$ x > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >,</span> which leads to a contradiction, so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/0/a/50a82d8e6462fc5fe754b10151a646a78a7ded53.png\" class=\"latex\" alt=\"$ x\\ne0$\" style=\"vertical-align: -4px\" width=\"43\" height=\"17\" >.</span><br>\nSo combining the 2, we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/1/4/714c7651b871da88edaf044cbea6dd69241e0128.png\" class=\"latex\" alt=\"$ x\\not\\le0$\" style=\"vertical-align: -4px\" width=\"43\" height=\"17\" >.</span>..<br>\nI'm not sure how to proceed from here though...", "post_id": 4416652, "post_number": 2, "post_time_unix": 1219877857, "post_time_utc": "2008-08-27 22:57:37 UTC", "thanks_received": 2, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "Lolz... that was the best proof I've ever seen :rotfl:", "content_html": "Lolz... that was the best proof I've ever seen <img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" />", "post_id": 4416653, "post_number": 3, "post_time_unix": 1219950277, "post_time_utc": "2008-08-28 19:04:37 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "The set of all x less than or equal to 0 is the complement of all x greater that or equal to 0, so the complement of x less that or equal to 0 is the complement of the complement of all x greater than 0, which is equal to x>0. Q.E.D.\r\n\r\nP.S. Sorry, I have no Latex skills whatsoever.", "content_html": "The set of all x less than or equal to 0 is the complement of all x greater that or equal to 0, so the complement of x less that or equal to 0 is the complement of the complement of all x greater than 0, which is equal to x>0. Q.E.D.<br>\n<br>\nP.S. Sorry, I have no Latex skills whatsoever.", "post_id": 4416654, "post_number": 4, "post_time_unix": 1224797107, "post_time_utc": "2008-10-23 21:25:07 UTC", "thanks_received": 2, "user_id": 44832, "username": "ThinkFlow" } ], "source": null }	Prove that $x>0$ for all $x>0$.	[ "/Mathematics/FoundationsofMathematics/MathematicalProblems/SolvedProblems" ]	Notice the premise x>0 is exactly the claim, so the implication holds trivially.	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aops_997252	[hide="Suppose they can intersect..."]Uh playing with this for 3 minutes yields the following result: $ A_8 \equal{} \frac 1{0!} \plus{} \frac 1{1!}\binom 82 \plus{} \frac 1{2!}\binom 82 \binom 62 \plus{} \frac 1{3!} \binom 82 \binom 62 \binom 42 \plus{} \frac 1{4!} \binom 82 \binom 62 \binom 42 \binom 22$ I don't know how to express this in summation notation... ... but the numerical answer is $ \boxed{764}$. ... Obviously, from my verbose use of factorials and binom's, you can tell how I derived it. I'd like to nominate this problem for the AIME. :D[/hide] But they can't intersect, so... [hide="Hopefully correct solution"] This is the same as Catalan numbers, except that we can choose either 2, 4, 6, or 8 not to participate. If we remove any number of points, the remaining set is still cyclic, so we can "re-calibrate" them, so to say. So this is $ \binom 88 C_8 \plus{} \binom 86C_6 \plus{} \binom 84 C_4 \plus{} \binom 82 C_2 \plus{} \binom 80 C_0$. Since $ C_0 \equal{} 1,C_2 \equal{} 1,C_4 \equal{} 2,C_6 \equal{} 5,C_8 \equal{} 14$: $ \begin{align*}\binom 88 C_8 & \plus{} \binom 86C_6 \plus{} \binom 84 C_4 \plus{} \binom 82 C_2 \plus{} \binom 80 C_0 \\ & \equal{} 14 \plus{} 140 \plus{} 140 \plus{} 28 \plus{} 1 \\ & \equal{} \boxed{323}$. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "8 points $ P_1, P_2, ... P_8$ are equally spaced around a circle in that order. How many ways can you draw line segments between these points such that: (a) no $ P_i$ is the endpoint of more than 1 segment, and (b) none of the lines intersect?\r\n\r\nHint: try to make a recurrence similar to the one for Catalan numbers.", "content_html": "8 points <img src=\"//latex.artofproblemsolving.com/f/4/5/f45de189b0961b41640a4015ab01b1dda9caf9a2.png\" class=\"latex\" alt=\"$ P_1, P_2, ... P_8$\" style=\"vertical-align: -3px\" width=\"87\" height=\"16\" > are equally spaced around a circle in that order. How many ways can you draw line segments between these points such that: (a) no <img src=\"//latex.artofproblemsolving.com/b/a/5/ba586b6f359b9cc4fb98f160fa2eaac9cd32ca7a.png\" class=\"latex\" alt=\"$ P_i$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > is the endpoint of more than 1 segment, and (b) none of the lines intersect?<br>\n<br>\nHint: try to make a recurrence similar to the one for Catalan numbers.", "post_id": 4416681, "post_number": 1, "post_time_unix": 1231359401, "post_time_utc": "2009-01-07 20:16:41 UTC", "thanks_received": 1, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "Hm this is very similar to a problem we did at a Nationals practice... I will guess the answer is 14 :D\r\n\r\nObviously we can't have an odd number of people \"landlocked\" to the left or right of a line. Therefore, $ P_1$ only has 4 choices of people to shake hands with. (\"Shake hands? What?\" Alkjash will understand :lol:). If he chooses a person immediately to the next of him, this becomes a problem involving 6 people. If on the other hand he chooses a person not immediately to the next of him, this becomes a problem involving two people on one side and 4 on the other.\r\n\r\nSo $ C_8 \\equal{} 2C_6 \\plus{} 2(C_2C_4)$. Obviously, $ C_2 \\equal{} 1$, so:\r\n\r\n$ C_8 \\equal{} 2C_6 \\plus{} 2C_4 \\equal{} 2(C_6 \\plus{} C_4)$.\r\n\r\nAnd $ C_6$ can be calculated similarly. $ C_6 \\equal{} 2C_4 \\plus{} C_2^2 \\equal{} 2C_4 \\plus{} 1$.\r\n\r\nAnd $ C_4 \\equal{} 2$.\r\n\r\nSo $ C_6 \\equal{} 5$, and $ C_8 \\equal{} 2(5 \\plus{} 2) \\equal{} \\boxed{14}$.\r\n\r\nThat's if I didn't memorize it in the first place :)", "content_html": "Hm this is very similar to a problem we did at a Nationals practice... I will guess the answer is 14 <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\nObviously we can't have an odd number of people "landlocked" to the left or right of a line. Therefore, <img src=\"//latex.artofproblemsolving.com/0/5/b/05bbb85d8240ec9731a369e296d5171e4a886f26.png\" class=\"latex\" alt=\"$ P_1$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > only has 4 choices of people to shake hands with. ("Shake hands? What?" Alkjash will understand <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />). If he chooses a person immediately to the next of him, this becomes a problem involving 6 people. If on the other hand he chooses a person not immediately to the next of him, this becomes a problem involving two people on one side and 4 on the other.<br>\n<br>\nSo <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/b/0/cb099edbb652b2436e50ace5d83258b407e02090.png\" class=\"latex\" alt=\"$ C_8 = 2C_6 + 2(C_2C_4)$\" style=\"vertical-align: -4px\" width=\"159\" height=\"18\" >.</span> Obviously, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/8/1/f811a3e399dcc99a8502b8e1a0134eef3c49fac0.png\" class=\"latex\" alt=\"$ C_2 = 1$\" style=\"vertical-align: -2px\" width=\"52\" height=\"15\" >,</span> so:<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/1/e/81e8d3d16e8ac294c7ca9d1377661bd0d38d11f8.png\" class=\"latex\" alt=\"$ C_8 = 2C_6 + 2C_4 = 2(C_6 + C_4)$\" style=\"vertical-align: -4px\" width=\"235\" height=\"18\" >.</span><br>\n<br>\nAnd <img src=\"//latex.artofproblemsolving.com/f/c/e/fce993d8ff6db71f32242a49d54feb0cd2986408.png\" class=\"latex\" alt=\"$ C_6$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > can be calculated similarly. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/8/f/e8f708cde833ec2b7cdcc11fb4e7188e6ff6d975.png\" class=\"latex\" alt=\"$ C_6 = 2C_4 + C_2^2 = 2C_4 + 1$\" style=\"vertical-align: -4px\" width=\"202\" height=\"19\" >.</span><br>\n<br>\nAnd <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/8/6/586fac290132ac912146f55bdec7f7c1fa3312a7.png\" class=\"latex\" alt=\"$ C_4 = 2$\" style=\"vertical-align: -2px\" width=\"53\" height=\"15\" >.</span><br>\n<br>\nSo <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/0/7/f073dd33c2de9b7b8db9b6db466b7f62dd8e7e9a.png\" class=\"latex\" alt=\"$ C_6 = 5$\" style=\"vertical-align: -2px\" width=\"53\" height=\"15\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/d/1/7d1196d2490f5b53771ae73cc1b10096c73eb72e.png\" class=\"latex\" alt=\"$ C_8 = 2(5 + 2) = \\boxed{14}$\" style=\"vertical-align: -5px\" width=\"162\" height=\"23\" >.</span><br>\n<br>\nThat's if I didn't memorize it in the first place <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />", "post_id": 4416682, "post_number": 2, "post_time_unix": 1231538991, "post_time_utc": "2009-01-09 22:09:51 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "Hmmm...\r\nnote that I didn't say each person must shake hands - only that they can shake hands with at most one other person.\r\nI thought I gave that hint with the status...", "content_html": "Hmmm...<br>\nnote that I didn't say each person must shake hands - only that they can shake hands with at most one other person.<br>\nI thought I gave that hint with the status...", "post_id": 4416683, "post_number": 3, "post_time_unix": 1231892950, "post_time_utc": "2009-01-14 00:29:10 UTC", "thanks_received": 1, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "Hm I have to find some way to edit your post :D", "content_html": "Hm I have to find some way to edit your post <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />", "post_id": 4416684, "post_number": 4, "post_time_unix": 1232239328, "post_time_utc": "2009-01-18 00:42:08 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "[hide=\"Suppose they can intersect...\"]Uh playing with this for 3 minutes yields the following result:\n\n$ A_8 \\equal{} \\frac 1{0!} \\plus{} \\frac 1{1!}\\binom 82 \\plus{} \\frac 1{2!}\\binom 82 \\binom 62 \\plus{} \\frac 1{3!} \\binom 82 \\binom 62 \\binom 42 \\plus{} \\frac 1{4!} \\binom 82 \\binom 62 \\binom 42 \\binom 22$\n\nI don't know how to express this in summation notation...\n\n... but the numerical answer is $ \\boxed{764}$.\n\n... Obviously, from my verbose use of factorials and binom's, you can tell how I derived it.\n\nI'd like to nominate this problem for the AIME. :D[/hide]\n\nBut they can't intersect, so...\n\n[hide=\"Hopefully correct solution\"]\nThis is the same as Catalan numbers, except that we can choose either 2, 4, 6, or 8 not to participate. If we remove any number of points, the remaining set is still cyclic, so we can \"re-calibrate\" them, so to say.\n\nSo this is $ \\binom 88 C_8 \\plus{} \\binom 86C_6 \\plus{} \\binom 84 C_4 \\plus{} \\binom 82 C_2 \\plus{} \\binom 80 C_0$.\nSince $ C_0 \\equal{} 1,C_2 \\equal{} 1,C_4 \\equal{} 2,C_6 \\equal{} 5,C_8 \\equal{} 14$:\n\n$ \\begin{align}\\binom 88 C_8 & \\plus{} \\binom 86C_6 \\plus{} \\binom 84 C_4 \\plus{} \\binom 82 C_2 \\plus{} \\binom 80 C_0 \\\\\n& \\equal{} 14 \\plus{} 140 \\plus{} 140 \\plus{} 28 \\plus{} 1 \\\\\n& \\equal{} \\boxed{323}$.\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Suppose they can intersect...</a><div class=\"cmty-hide-content\" style=\"display:none\">Uh playing with this for 3 minutes yields the following result:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/5/c/d5ce4139cb8fda64aac863e4429d496f60ce845c.png\" class=\"latex\" alt=\"$ A_8 = \\frac 1{0!} + \\frac 1{1!}\\binom 82 + \\frac 1{2!}\\binom 82 \\binom 62 + \\frac 1{3!} \\binom 82 \\binom 62 \\binom 42 + \\frac 1{4!} \\binom 82 \\binom 62 \\binom 42 \\binom 22$\" style=\"vertical-align: -22px\" width=\"593\" height=\"53\" ><br>\n<br>\nI don't know how to express this in summation notation...<br>\n<br>\n... but the numerical answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/8/1/881624834151432fbaa2e5863af298d57f9fca86.png\" class=\"latex\" alt=\"$ \\boxed{764}$\" style=\"vertical-align: -5px\" width=\"39\" height=\"23\" >.</span><br>\n<br>\n... Obviously, from my verbose use of factorials and binom's, you can tell how I derived it.<br>\n<br>\nI'd like to nominate this problem for the AIME. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div><br>\n<br>\nBut they can't intersect, so...<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hopefully correct solution</a><div class=\"cmty-hide-content\" style=\"display:none\">This is the same as Catalan numbers, except that we can choose either 2, 4, 6, or 8 not to participate. If we remove any number of points, the remaining set is still cyclic, so we can "re-calibrate" them, so to say.<br>\n<br>\nSo this is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/4/01496b49b43353d8a201beedf585427c40892787.png\" class=\"latex\" alt=\"$ \\binom 88 C_8 + \\binom 86C_6 + \\binom 84 C_4 + \\binom 82 C_2 + \\binom 80 C_0$\" style=\"vertical-align: -22px\" width=\"376\" height=\"53\" >.</span><br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/7/a/d7af29d5926b3c7215a31fc8ebb08acf1e5284fe.png\" class=\"latex\" alt=\"$ C_0 = 1,C_2 = 1,C_4 = 2,C_6 = 5,C_8 = 14$\" style=\"vertical-align: -3px\" width=\"310\" height=\"16\" >:</span><br>\n<br>\n<span style=\"white-space:nowrap;\"><span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align}\\binom 88 C_8 & + \\binom 86C_6 + \\binom 84 C_4 + \\binom 82 C_2 + \\binom 80 C_0 \\\\\n& = 14 + 140 + 140 + 28 + 1 \\\\\n& = \\boxed{323}$</span>.</span></div>", "post_id": 4416685, "post_number": 5, "post_time_unix": 1233882210, "post_time_utc": "2009-02-06 01:03:30 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "I do believe the answer is right.\r\n\r\nTry to find a direct recursion and solve that way.", "content_html": "I do believe the answer is right.<br>\n<br>\nTry to find a direct recursion and solve that way.", "post_id": 4416686, "post_number": 6, "post_time_unix": 1233957761, "post_time_utc": "2009-02-06 22:02:41 UTC", "thanks_received": 1, "user_id": 27067, "username": "alkjash" } ], "source": null }	8 points $P_1,P_2,\dots,P_8$ are equally spaced around a circle in that order. How many ways can you draw line segments between these points such that (a) no $P_i$ is the endpoint of more than one segment, and (b) none of the line segments intersect?	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/BracedPolygon" ]	Select an even subset of points to pair and count its non‑crossing matchings via Catalan numbers, then sum over all even subset sizes.	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aops_99726	[hide]Um, I am pretty sure that the base has a length of 1, then the area of the base is $\frac{\sqrt{3}}{4}$. Then the height is $h=\sqrt{(1)^{2}-(\frac{\sqrt{3}}{3})^{2}}$, which is equal to $\frac{\sqrt{6}}{3}$. So the volume is $\frac{\sqrt{6}}{3}\frac{\sqrt{3}}{4}\frac{1}{3}$. This equals $\frac{\sqrt{2}}{12}$. $\frac{\sqrt{2}}{12}$. JClarke could you show your work for how you did the question?[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "I know that the volume of a tetrahedron is equal to the area of the base x 1/3 of its height. I'm not sure if i'm getting the height right though.\r\n\r\nIs the area of a regular tetrahedron with length 1 [hide]3/16[/hide]? I got $\\sqrt2/2$ for the height.", "content_html": "I know that the volume of a tetrahedron is equal to the area of the base x 1/3 of its height. I'm not sure if i'm getting the height right though.<br>\n<br>\nIs the area of a regular tetrahedron with length 1 <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">3/16</div>? I got <img src=\"//latex.artofproblemsolving.com/0/5/9/0592b45f3300b9d9813d75ec9bbec817ef8cdfdd.png\" class=\"latex\" alt=\"$\\sqrt2/2$\" style=\"vertical-align: -4px\" width=\"41\" height=\"21\" > for the height.", "post_id": 563168, "post_number": 1, "post_time_unix": 1151714886, "post_time_utc": "2006-07-01 00:48:06 UTC", "thanks_received": 2, "user_id": 18442, "username": "13375P34K43V312" }, { "attachments": [], "content_bbcode": "[hide] I got $\\frac{\\sqrt6}{3}$ for the height. :? [/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">I got <img src=\"//latex.artofproblemsolving.com/8/c/4/8c44c6eddcf5f7f30639daffaf6b3d2e3eda1ff9.png\" class=\"latex\" alt=\"$\\frac{\\sqrt6}{3}$\" style=\"vertical-align: -12px\" width=\"26\" height=\"41\" > for the height. <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" /></div>", "post_id": 563191, "post_number": 2, "post_time_unix": 1151715996, "post_time_utc": "2006-07-01 01:06:36 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "[hide]Um, I am pretty sure that the base has a length of 1, then the area of the base is $\\frac{\\sqrt{3}}{4}$. Then the height is \n\n$h=\\sqrt{(1)^{2}-(\\frac{\\sqrt{3}}{3})^{2}}$, which is equal to $\\frac{\\sqrt{6}}{3}$.\n\nSo the volume is $\\frac{\\sqrt{6}}{3}\\frac{\\sqrt{3}}{4}\\frac{1}{3}$. This equals $\\frac{\\sqrt{2}}{12}$.\n\n$\\frac{\\sqrt{2}}{12}$.\n\nJClarke could you show your work for how you did the question?[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Um, I am pretty sure that the base has a length of 1, then the area of the base is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/9/b/b9b30d7166f38d8f18e66dfd6e9473880efaca2e.png\" class=\"latex\" alt=\"$\\frac{\\sqrt{3}}{4}$\" style=\"vertical-align: -13px\" width=\"26\" height=\"41\" >.</span> Then the height is<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/1/ea1d31bde8a1dc06c1a799cca8c288b1de18362c.png\" class=\"latex\" alt=\"$h=\\sqrt{(1)^{2}-(\\frac{\\sqrt{3}}{3})^{2}}$\" style=\"vertical-align: -16px\" width=\"155\" height=\"53\" >,</span> which is equal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/1/0/410b52010405ed20bc351b711ef399794b07cefc.png\" class=\"latex\" alt=\"$\\frac{\\sqrt{6}}{3}$\" style=\"vertical-align: -12px\" width=\"26\" height=\"41\" >.</span><br>\n<br>\nSo the volume is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/6/6/b6678d4bb43c322f590fda453af701b2955a764b.png\" class=\"latex\" alt=\"$\\frac{\\sqrt{6}}{3}\\frac{\\sqrt{3}}{4}\\frac{1}{3}$\" style=\"vertical-align: -13px\" width=\"102\" height=\"41\" >.</span> This equals <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/c/d/2cd716f5a1975497cef619196ccbe9e4ec46a2ee.png\" class=\"latex\" alt=\"$\\frac{\\sqrt{2}}{12}$\" style=\"vertical-align: -13px\" width=\"26\" height=\"41\" >.</span><br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/c/d/2cd716f5a1975497cef619196ccbe9e4ec46a2ee.png\" class=\"latex\" alt=\"$\\frac{\\sqrt{2}}{12}$\" style=\"vertical-align: -13px\" width=\"26\" height=\"41\" >.</span><br>\n<br>\nJClarke could you show your work for how you did the question?</div>", "post_id": 563194, "post_number": 3, "post_time_unix": 1151716090, "post_time_utc": "2006-07-01 01:08:10 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "If $\\frac{\\sqrt2}{2}$ was the height, the volume would be $\\frac{1}{3}\\frac{\\sqrt2}{2}\\frac{\\sqrt3}{4}=\\frac{\\sqrt6}{24}$. :huh:", "content_html": "If <img src=\"//latex.artofproblemsolving.com/1/e/6/1e61d03caa371f28c51f555bf685f1296ff8de8f.png\" class=\"latex\" alt=\"$\\frac{\\sqrt2}{2}$\" style=\"vertical-align: -12px\" width=\"26\" height=\"41\" > was the height, the volume would be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/0/e/a0e1f29b53d1c5e87a108abb29a301fb4784bf6c.png\" class=\"latex\" alt=\"$\\frac{1}{3}\\frac{\\sqrt2}{2}\\frac{\\sqrt3}{4}=\\frac{\\sqrt6}{24}$\" style=\"vertical-align: -13px\" width=\"155\" height=\"41\" >.</span> <img src=\"/assets/images/smilies/huh.gif\" width=\"20\" height=\"20\" alt=\":huh:\" title=\":huh:\" class=\"bbcode_smiley\" />", "post_id": 563198, "post_number": 4, "post_time_unix": 1151716250, "post_time_utc": "2006-07-01 01:10:50 UTC", "thanks_received": 2, "user_id": 18001, "username": "lotrgreengrapes7926" }, { "attachments": [], "content_bbcode": "Yes, so could you show how you did it j clarke and we will help you.", "content_html": "Yes, so could you show how you did it j clarke and we will help you.", "post_id": 563207, "post_number": 5, "post_time_unix": 1151716485, "post_time_utc": "2006-07-01 01:14:45 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "I used a formula for my sample problem, the volume of a tetrahedron is $\\frac{s^{3}\\sqrt{2}}{12}$.", "content_html": "I used a formula for my sample problem, the volume of a tetrahedron is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/9/b/69b876f1f8097d04355f746c615debc248e50988.png\" class=\"latex\" alt=\"$\\frac{s^{3}\\sqrt{2}}{12}$\" style=\"vertical-align: -13px\" width=\"42\" height=\"41\" >.</span>", "post_id": 563264, "post_number": 6, "post_time_unix": 1151718510, "post_time_utc": "2006-07-01 01:48:30 UTC", "thanks_received": 2, "user_id": 20692, "username": "mad_skillz_aops" }, { "attachments": [], "content_bbcode": "[quote=\"mad_skillz_aops\"]I used a formula for my sample problem, the volume of a tetrahedron is $\\frac{s^{3}\\sqrt{2}}{12}$.[/quote]\r\n\r\nActually it is $\\frac{s^{3}\\sqrt{3}}{12}$. But, we showed theway if you didn't know the formula.\r\n\r\nEDIT:Never mind, I am wrong.", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">mad_skillz_aops wrote:</div>\n<div class=\"bbcode_quote_body\">I used a formula for my sample problem, the volume of a tetrahedron is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/9/b/69b876f1f8097d04355f746c615debc248e50988.png\" class=\"latex\" alt=\"$\\frac{s^{3}\\sqrt{2}}{12}$\" style=\"vertical-align: -13px\" width=\"42\" height=\"41\" >.</span></div>\n</div>\n<br>\nActually it is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/a/d2ad7e4f12c961156da72edf66edc77e3f0d3249.png\" class=\"latex\" alt=\"$\\frac{s^{3}\\sqrt{3}}{12}$\" style=\"vertical-align: -13px\" width=\"42\" height=\"41\" >.</span> But, we showed theway if you didn't know the formula.<br>\n<br>\nEDIT:Never mind, I am wrong.", "post_id": 563275, "post_number": 7, "post_time_unix": 1151719005, "post_time_utc": "2006-07-01 01:56:45 UTC", "thanks_received": 2, "user_id": 11714, "username": "mathgeniuse^ln(x)" }, { "attachments": [], "content_bbcode": "Nope, you have the height right, but you did the multiplication wrong.", "content_html": "Nope, you have the height right, but you did the multiplication wrong.", "post_id": 563297, "post_number": 8, "post_time_unix": 1151719824, "post_time_utc": "2006-07-01 02:10:24 UTC", "thanks_received": 2, "user_id": 20692, "username": "mad_skillz_aops" }, { "attachments": [], "content_bbcode": "Yep, you are right. I didn't know what I was thinking of.\r\n\r\nYOu are absolutely correct.\r\n\r\nI edited my previous posts.", "content_html": "Yep, you are right. I didn't know what I was thinking of.<br>\n<br>\nYOu are absolutely correct.<br>\n<br>\nI edited my previous posts.", "post_id": 563744, "post_number": 9, "post_time_unix": 1151768740, "post_time_utc": "2006-07-01 15:45:40 UTC", "thanks_received": 1, "user_id": 11714, "username": "mathgeniuse^ln(x)" } ], "source": null }	I know that the volume of a tetrahedron is equal to the area of the base times $\tfrac{1}{3}$ of its height. Is the area of a regular tetrahedron with edge length $1$ equal to $\tfrac{3}{16}$? I got $\tfrac{\sqrt{2}}{2}$ for the height.	[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes/Hypertetrahedron", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes/Simplex", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Height", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Length", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/Solid", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/SolidGeometry", "/Mathematics/Geometry/SolidGeometry/Polyhedra/PlatonicSolids", "/Mathematics/Geometry/SolidGeometry/Polyhedra/Tetrahedra", "/Mathematics/Geometry/SolidGeometry/Volume/VolumeTheorem" ]	Use the Pythagorean theorem on the triangle from a vertex to the centroid of the equilateral base to find the tetrahedron's height.	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aops_997268	[hide="(1)"] Haha, I will be mocked for actually typing up a solution to this... First, rearrange the inequality: $ 2xyz \leq xy\plus{}xz\plus{}yz \leq \frac 7{27}\plus{}2xyz$. AM-GM: $ \frac{x\plus{}y\plus{}z}3 \equal{} \frac 13 \geq \sqrt[3]{xyz} \implies xyz \leq \frac 1{27} \implies 2xyz \leq \frac 2{27}$. AM-HM: $ \frac {x\plus{}y\plus{}z}3 \equal{} \frac 13 \geq \frac 3{1/x\plus{}1/y\plus{}1/z} \implies 9 \leq \frac 1x \plus{} \frac 1y \plus{} \frac 1z$ So $ \frac 1x \plus{} \frac 1y \plus{} \frac 1z \geq 9$. We'll prove this first: $ xy\plus{}xz\plus{}yz \geq 2xyz$. If any of $ x,y,z\equal{}0$, then obviously this is true, so suppose none of them equal 0. Rewrite this as $ xyz\left(\frac 1x\plus{}\frac 1y\plus{}\frac 1z\right) \geq 2xyz$. We can divide by $ xyz$, and the result follows trivially by AM-HM. Next, we have $ xy\plus{}xz\plus{}yz \leq \frac 7{27}\plus{}2xyz$, blah I have to have dinner now.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Trivialty at its best:\r\n\r\nProve\r\n$ 0 \\leq yz \\plus{} zx \\plus{} xy \\minus{} 2xyz \\leq 7/27$\r\nfor non-negative x, y, z with $ x\\plus{}y\\plus{}z\\equal{}1$.\r\n\r\nWeird but easy:\r\n\r\nProve that the zeros of\r\n$ x^5 \\plus{} ax^4 \\plus{} bx^3 \\plus{}cx^2 \\plus{} dx \\plus{} e \\equal{} 0$\r\ncannot all be real if $ 2a^2 < 5b$.", "content_html": "Trivialty at its best:<br>\n<br>\nProve<br>\n<img src=\"//latex.artofproblemsolving.com/0/c/5/0c5274ea2b5e864bcc91e31e44c4e9445ab8d75d.png\" class=\"latex\" alt=\"$ 0 \\leq yz + zx + xy - 2xyz \\leq 7/27$\" style=\"vertical-align: -4px\" width=\"257\" height=\"18\" ><br>\nfor non-negative x, y, z with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/f/7/6f771c112f540d922c9eaf5c760276ccadbba9e6.png\" class=\"latex\" alt=\"$ x+y+z=1$\" style=\"vertical-align: -3px\" width=\"106\" height=\"15\" >.</span><br>\n<br>\nWeird but easy:<br>\n<br>\nProve that the zeros of<br>\n<img src=\"//latex.artofproblemsolving.com/5/c/3/5c360040b6e55e8c10b0b26cc6edaecc616475e7.png\" class=\"latex\" alt=\"$ x^5 + ax^4 + bx^3 +cx^2 + dx + e = 0$\" style=\"vertical-align: -1px\" width=\"269\" height=\"16\" ><br>\ncannot all be real if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/3/a/93a378f06c56fc756b331addc2e611a35f25b342.png\" class=\"latex\" alt=\"$ 2a^2 < 5b$\" style=\"vertical-align: 0px\" width=\"67\" height=\"15\" >.</span>", "post_id": 4416727, "post_number": 1, "post_time_unix": 1240347764, "post_time_utc": "2009-04-21 21:02:44 UTC", "thanks_received": 1, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "[hide=\"(1)\"]\nHaha, I will be mocked for actually typing up a solution to this...\n\nFirst, rearrange the inequality: $ 2xyz \\leq xy\\plus{}xz\\plus{}yz \\leq \\frac 7{27}\\plus{}2xyz$.\n\nAM-GM: $ \\frac{x\\plus{}y\\plus{}z}3 \\equal{} \\frac 13 \\geq \\sqrt[3]{xyz} \\implies xyz \\leq \\frac 1{27} \\implies 2xyz \\leq \\frac 2{27}$.\n\nAM-HM: $ \\frac {x\\plus{}y\\plus{}z}3 \\equal{} \\frac 13 \\geq \\frac 3{1/x\\plus{}1/y\\plus{}1/z} \\implies 9 \\leq \\frac 1x \\plus{} \\frac 1y \\plus{} \\frac 1z$\n\nSo $ \\frac 1x \\plus{} \\frac 1y \\plus{} \\frac 1z \\geq 9$.\n\nWe'll prove this first: $ xy\\plus{}xz\\plus{}yz \\geq 2xyz$. If any of $ x,y,z\\equal{}0$, then obviously this is true, so suppose none of them equal 0. Rewrite this as $ xyz\\left(\\frac 1x\\plus{}\\frac 1y\\plus{}\\frac 1z\\right) \\geq 2xyz$. We can divide by $ xyz$, and the result follows trivially by AM-HM.\n\nNext, we have $ xy\\plus{}xz\\plus{}yz \\leq \\frac 7{27}\\plus{}2xyz$, blah I have to have dinner now.[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">(1)</a><div class=\"cmty-hide-content\" style=\"display:none\">Haha, I will be mocked for actually typing up a solution to this...<br>\n<br>\nFirst, rearrange the inequality: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/6/da65ebdcd9a396ec9ac66dfc86e2c7321d26970a.png\" class=\"latex\" alt=\"$ 2xyz \\leq xy+xz+yz \\leq \\frac 7{27}+2xyz$\" style=\"vertical-align: -12px\" width=\"271\" height=\"37\" >.</span><br>\n<br>\nAM-GM: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/8/5/6853c25cd6c100db03d3ea12d5cc4506ffeb1b9d.png\" class=\"latex\" alt=\"$ \\frac{x+y+z}3 = \\frac 13 \\geq \\sqrt[3]{xyz} \\implies xyz \\leq \\frac 1{27} \\implies 2xyz \\leq \\frac 2{27}$\" style=\"vertical-align: -12px\" width=\"442\" height=\"37\" >.</span><br>\n<br>\nAM-HM: <img src=\"//latex.artofproblemsolving.com/9/3/9/939b2003d674a8c6baa714e7116eddcff807a8b0.png\" class=\"latex\" alt=\"$ \\frac {x+y+z}3 = \\frac 13 \\geq \\frac 3{1/x+1/y+1/z} \\implies 9 \\leq \\frac 1x + \\frac 1y + \\frac 1z$\" style=\"vertical-align: -17px\" width=\"438\" height=\"42\" ><br>\n<br>\nSo <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/4/3/a43296d6fee0ccd349286943f5a13be6e7f86049.png\" class=\"latex\" alt=\"$ \\frac 1x + \\frac 1y + \\frac 1z \\geq 9$\" style=\"vertical-align: -16px\" width=\"118\" height=\"40\" >.</span><br>\n<br>\nWe'll prove this first: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/4/c/a4c78a43ea3f5a0dd308b8a9b85297627561585d.png\" class=\"latex\" alt=\"$ xy+xz+yz \\geq 2xyz$\" style=\"vertical-align: -3px\" width=\"164\" height=\"15\" >.</span> If any of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/d/8ed51bcab12f42ce43e173d779eeeb5e37319c31.png\" class=\"latex\" alt=\"$ x,y,z=0$\" style=\"vertical-align: -3px\" width=\"78\" height=\"16\" >,</span> then obviously this is true, so suppose none of them equal 0. Rewrite this as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/5/ef5c1acca99e6f3b82e92bc7495a9ca84b2c4979.png\" class=\"latex\" alt=\"$ xyz\\left(\\frac 1x+\\frac 1y+\\frac 1z\\right) \\geq 2xyz$\" style=\"vertical-align: -17px\" width=\"206\" height=\"43\" >.</span> We can divide by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/0/9/e0952299feef19841a0155f5a659b01ea2844225.png\" class=\"latex\" alt=\"$ xyz$\" style=\"vertical-align: -3px\" width=\"28\" height=\"11\" >,</span> and the result follows trivially by AM-HM.<br>\n<br>\nNext, we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/e/46e92235359366c70c32a53c475abcc8257ec914.png\" class=\"latex\" alt=\"$ xy+xz+yz \\leq \\frac 7{27}+2xyz$\" style=\"vertical-align: -12px\" width=\"209\" height=\"37\" >,</span> blah I have to have dinner now.</div>", "post_id": 4416728, "post_number": 2, "post_time_unix": 1240353258, "post_time_utc": "2009-04-21 22:34:18 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "just let x=1/a, y=1/b, z=1/c, then we need ab+bc+ca=abc and we need to show that 27(a+b+c-2) le 7(ab+bc+ac) which I think is well known.", "content_html": "just let x=1/a, y=1/b, z=1/c, then we need ab+bc+ca=abc and we need to show that 27(a+b+c-2) le 7(ab+bc+ac) which I think is well known.", "post_id": 4416729, "post_number": 3, "post_time_unix": 1240356026, "post_time_utc": "2009-04-21 23:20:26 UTC", "thanks_received": 1, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "dude what that is a pretty weird inequality that is probably not well known (nor known at all?)\r\n\r\nanyhow 1 is homogenize and schur, 2 is expand and it becomes sym a^2 > sym ab which is true by muirhead if the roots are positive then if the roots some are negative the RHS goes down while the LHS stays the same.", "content_html": "dude what that is a pretty weird inequality that is probably not well known (nor known at all?)<br>\n<br>\nanyhow 1 is homogenize and schur, 2 is expand and it becomes sym a^2 > sym ab which is true by muirhead if the roots are positive then if the roots some are negative the RHS goes down while the LHS stays the same.", "post_id": 4416730, "post_number": 4, "post_time_unix": 1240360228, "post_time_utc": "2009-04-22 00:30:28 UTC", "thanks_received": 1, "user_id": 34136, "username": "CatalystOfNostalgia" }, { "attachments": [], "content_bbcode": "huh not well known. i've seen it cited as something like \"and we are done\" in preoly at least once, but okay that was mayhaps handwaving on both my and that person's part. it also turns out not to be true, it seems, if a=2, b=2, c=4.\r\n\r\nalso posting a homogenization-schur or muirhead solution for something like that is rather strange in my eyes, but for something as \"trivial\" as this I cannot find anything else, at least with only cases of power mean and random manipulations and substitutions.", "content_html": "huh not well known. i've seen it cited as something like "and we are done" in preoly at least once, but okay that was mayhaps handwaving on both my and that person's part. it also turns out not to be true, it seems, if a=2, b=2, c=4.<br>\n<br>\nalso posting a homogenization-schur or muirhead solution for something like that is rather strange in my eyes, but for something as "trivial" as this I cannot find anything else, at least with only cases of power mean and random manipulations and substitutions.", "post_id": 4416731, "post_number": 5, "post_time_unix": 1240365684, "post_time_utc": "2009-04-22 02:01:24 UTC", "thanks_received": 1, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "Um, $ \\frac 12 \\plus{} \\frac 12 \\plus{} \\frac 14 \\neq 1$ ;)", "content_html": "Um, <img src=\"//latex.artofproblemsolving.com/2/4/d/24d95b03f466082eab5d3ebe6dff523c76e2dc1c.png\" class=\"latex\" alt=\"$ \\frac 12 + \\frac 12 + \\frac 14 \\neq 1$\" style=\"vertical-align: -13px\" width=\"116\" height=\"37\" > <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" />", "post_id": 4416732, "post_number": 6, "post_time_unix": 1240408873, "post_time_utc": "2009-04-22 14:01:13 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "2, 4, and 4 is pretty clearly what I meant.", "content_html": "2, 4, and 4 is pretty clearly what I meant.", "post_id": 4416733, "post_number": 7, "post_time_unix": 1240424870, "post_time_utc": "2009-04-22 18:27:50 UTC", "thanks_received": 1, "user_id": 28419, "username": "Temperal" }, { "attachments": [], "content_bbcode": "But the inequality is true with 2,4,4 :) ($ \\frac 14 < \\frac 7{27}$)", "content_html": "But the inequality is true with 2,4,4 <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /> <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/9/f/2/9f2f0ad1080a677709c7bd5560844125b6bf93f1.png\" class=\"latex\" alt=\"$ \\frac 14 < \\frac 7{27}$\" style=\"vertical-align: -13px\" width=\"58\" height=\"37\" >)</span>", "post_id": 4416734, "post_number": 8, "post_time_unix": 1240427926, "post_time_utc": "2009-04-22 19:18:46 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "of course; it's equivalent to the original inequality. I meant in general in my post, which was why I was incorrect.", "content_html": "of course; it's equivalent to the original inequality. I meant in general in my post, which was why I was incorrect.", "post_id": 4416735, "post_number": 9, "post_time_unix": 1240432777, "post_time_utc": "2009-04-22 20:39:37 UTC", "thanks_received": 1, "user_id": 28419, "username": "Temperal" } ], "source": null }	Triviality at its best: Prove \[ 0 \le yz + zx + xy - 2xyz \le \frac{7}{27} \] for nonnegative $x,y,z$ with $x+y+z=1$. Weird but easy: Prove that the zeros of \[ x^5 + a x^4 + b x^3 + c x^2 + d x + e = 0 \] cannot all be real if $2a^2 < 5b$.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuinticEquation", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialRoots", "/Mathematics/Algebra/Polynomials/RealPolynomial" ]	Apply AM‑HM to obtain 1/x+1/y+1/z ≥ 9, then multiply by xyz to get xy+yz+zx ≥ 2xyz; use AM‑GM to bound xyz ≤ 1/27 for the upper side.	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aops_997284	Good job posting problems from Wikipedia: http://en.wikipedia.org/wiki/Fermat_number#Factorization_of_Fermat_numbers Unfortunately, it doesn't show the proof :( Edit: Fine I'll steal a proof then :D [hide] Let $ b\equal{}2^{2^{n\minus{}2}}(2^{2^{n\minus{}1}}\minus{}1)$. Since \[ 2^{2^n}\plus{}1\equiv 0\pmod p,\]we have\[ \begin{align*}b^2&\equal{}2^{2^{n\minus{}1}}(2^{2^n}\minus{}2\cdot 2^{2^{n\minus{}1}}\plus{}1)\equiv \minus{}2\cdot 2^{2^n}\\ &\equiv \minus{}2\cdot 2^{2^n}\plus{}2(2^{2^n}\plus{}1)\equiv 2\pmod p.\]Also it follows that\[ b^{2^{n\plus{}1}}\equiv 2^{2^{n}}\equiv \minus{}1\pmod p,\]and thus,\[ b^{2^{n\plus{}2}}\equiv 1\pmod p.\]Consequently, according to the lemma, $ \text{ord}_p b\equal{}2^j$ for some $ j\leq n\plus{}2$. However, if $ j<n\plus{}2$ and $ e\equal{}\text{ord}_p b$, then by the same lemma, \[ b^{e2^{m\plus{}1\minus{}j}}\minus{}1\equal{}b^{2^{m\plus{}1}}\minus{}1\equiv 2^{2^m}\minus{}1\equiv 0\pmod p,\] which contradicts the given. Hence,\[ \text{ord}_p b\equal{}2^{m\plus{}2}.\] The numbers $ p$ and $ b$ are coprime. Therefore, applying Fermat's Little Theorem, and using the lemma, we obtain\[ p\minus{}1\equal{}k\,\text{ord}_p b\equiv k2^{m\plus{}2}.\;\;\blacksquare\][/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "For $ n \\geq 2$ show that any prime divisor $ p$ of $ 2^{2^n} \\plus{} 1$ satisfies:\r\n$ p \\equiv 1 \\pmod{2^{n \\plus{} 2}}$.\r\n\r\nFind all positive integers $ n$ that are squares (quadratic residue or 0) mod $ p$ for every prime $ p$.\r\n\r\nShow that the odd prime factors of any squarefree number representable as the sum of two positive perfect squares are of the form $ 4n \\plus{} 1$.", "content_html": "For <img src=\"//latex.artofproblemsolving.com/1/4/a/14a5fd3da3f4c1da4f9747b18896a66b7e5ac54c.png\" class=\"latex\" alt=\"$ n \\geq 2$\" style=\"vertical-align: -2px\" width=\"43\" height=\"14\" > show that any prime divisor <img src=\"//latex.artofproblemsolving.com/6/5/f/65f279776d6c37b2790a28c7a1616a695c9e1170.png\" class=\"latex\" alt=\"$ p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > of <img src=\"//latex.artofproblemsolving.com/b/e/8/be81d402e5463865ce607dda2b81b48d655bfb6d.png\" class=\"latex\" alt=\"$ 2^{2^n} + 1$\" style=\"vertical-align: -1px\" width=\"55\" height=\"17\" > satisfies:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/8/2/682614f78fafd4f0bbdb923336802f448ea1cb15.png\" class=\"latex\" alt=\"$ p \\equiv 1 \\pmod{2^{n + 2}}$\" style=\"vertical-align: -4px\" width=\"140\" height=\"19\" >.</span><br>\n<br>\nFind all positive integers <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > that are squares (quadratic residue or 0) mod <img src=\"//latex.artofproblemsolving.com/6/5/f/65f279776d6c37b2790a28c7a1616a695c9e1170.png\" class=\"latex\" alt=\"$ p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > for every prime <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/5/f/65f279776d6c37b2790a28c7a1616a695c9e1170.png\" class=\"latex\" alt=\"$ p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" >.</span><br>\n<br>\nShow that the odd prime factors of any squarefree number representable as the sum of two positive perfect squares are of the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/7/077eeb97e3ff7906440c03682e9201bedcb17767.png\" class=\"latex\" alt=\"$ 4n + 1$\" style=\"vertical-align: -1px\" width=\"50\" height=\"13\" >.</span>", "post_id": 4416796, "post_number": 1, "post_time_unix": 1247603176, "post_time_utc": "2009-07-14 20:26:16 UTC", "thanks_received": 2, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "$ 17 \\neq \\minus{}1\\pmod{16}$@@@@@@@@@@@@@@@", "content_html": "<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/1/3/e13d4e540016a81dba075559683a9dc8e9f2e107.png\" class=\"latex\" alt=\"$ 17 \\neq -1\\pmod{16}$\" style=\"vertical-align: -4px\" width=\"145\" height=\"18\" >@</span>@@@@@@@@@@@@@@", "post_id": 4416797, "post_number": 2, "post_time_unix": 1247613917, "post_time_utc": "2009-07-14 23:25:17 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "$ 257 \\neq\\minus{}1\\pmod{32}$@@@@@@@@@@@@@@@@@@@@@@@@\r\n\r\nHehe.", "content_html": "<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/0/b/d0b462361a50520aebc2234e82d21516f1146d1c.png\" class=\"latex\" alt=\"$ 257 \\neq-1\\pmod{32}$\" style=\"vertical-align: -4px\" width=\"154\" height=\"18\" >@</span>@@@@@@@@@@@@@@@@@@@@@@@<br>\n<br>\nHehe.", "post_id": 4416798, "post_number": 3, "post_time_unix": 1247665946, "post_time_utc": "2009-07-15 13:52:26 UTC", "thanks_received": 2, "user_id": 29190, "username": "Math Geek" }, { "attachments": [], "content_bbcode": "Darn.\r\nI need to get better at math.", "content_html": "Darn.<br>\nI need to get better at math.", "post_id": 4416799, "post_number": 4, "post_time_unix": 1247670182, "post_time_utc": "2009-07-15 15:03:02 UTC", "thanks_received": 2, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "Good job posting problems from Wikipedia:\r\nhttp://en.wikipedia.org/wiki/Fermat_number#Factorization_of_Fermat_numbers\r\n\r\nUnfortunately, it doesn't show the proof :(\r\n\r\nEdit: Fine I'll steal a proof then :D\r\n\r\n[hide]\nLet $ b\\equal{}2^{2^{n\\minus{}2}}(2^{2^{n\\minus{}1}}\\minus{}1)$. Since\n\\[ 2^{2^n}\\plus{}1\\equiv 0\\pmod p,\\]we have\\[ \\begin{align}b^2&\\equal{}2^{2^{n\\minus{}1}}(2^{2^n}\\minus{}2\\cdot 2^{2^{n\\minus{}1}}\\plus{}1)\\equiv \\minus{}2\\cdot 2^{2^n}\\\\\n&\\equiv \\minus{}2\\cdot 2^{2^n}\\plus{}2(2^{2^n}\\plus{}1)\\equiv 2\\pmod p.\\]Also it follows that\\[ b^{2^{n\\plus{}1}}\\equiv 2^{2^{n}}\\equiv \\minus{}1\\pmod p,\\]and thus,\\[ b^{2^{n\\plus{}2}}\\equiv 1\\pmod p.\\]Consequently, according to the lemma, $ \\text{ord}_p b\\equal{}2^j$ for some $ j\\leq n\\plus{}2$. However, if $ j<n\\plus{}2$ and $ e\\equal{}\\text{ord}_p b$, then by the same lemma, \\[ b^{e2^{m\\plus{}1\\minus{}j}}\\minus{}1\\equal{}b^{2^{m\\plus{}1}}\\minus{}1\\equiv 2^{2^m}\\minus{}1\\equiv 0\\pmod p,\\] which contradicts the given. Hence,\\[ \\text{ord}_p b\\equal{}2^{m\\plus{}2}.\\] The numbers $ p$ and $ b$ are coprime. Therefore, applying Fermat's Little Theorem, and using the lemma, we obtain\\[ p\\minus{}1\\equal{}k\\,\\text{ord}_p b\\equiv k2^{m\\plus{}2}.\\;\\;\\blacksquare\\][/hide]", "content_html": "Good job posting problems from Wikipedia:<br>\n<a target=\"_blank\" href=\"http://en.wikipedia.org/wiki/Fermat_number#Factorization_of_Fermat_numbers\">http://en.wikipedia.org/wiki/Fermat_number#Factorization_of_Fermat_numbers</a><br>\n<br>\nUnfortunately, it doesn't show the proof <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /><br>\n<br>\nEdit: Fine I'll steal a proof then <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/1/83163c56171616160ef3c83e092916dc96c660d6.png\" class=\"latex\" alt=\"$ b=2^{2^{n-2}}(2^{2^{n-1}}-1)$\" style=\"vertical-align: -4px\" width=\"154\" height=\"23\" >.</span> Since<br>\n<img src=\"//latex.artofproblemsolving.com/2/c/e/2ce69a989725f3e2e3d9d29814d6614be7367107.png\" class=\"latexcenter\" alt=\"\\[ 2^{2^n}+1\\equiv 0\\pmod p,\\]\" width=\"175\" height=\"20\" >we have<pre class=\"aopscode-error aopscode-latex-error\">\\[ \\begin{align}b^2&=2^{2^{n-1}}(2^{2^n}-2\\cdot 2^{2^{n-1}}+1)\\equiv -2\\cdot 2^{2^n}\\\\\n&\\equiv -2\\cdot 2^{2^n}+2(2^{2^n}+1)\\equiv 2\\pmod p.\\]</pre>Also it follows that<img src=\"//latex.artofproblemsolving.com/2/9/3/293056d528ea97acecf4074fd683021a74339e53.png\" class=\"latexcenter\" alt=\"\\[ b^{2^{n+1}}\\equiv 2^{2^{n}}\\equiv -1\\pmod p,\\]\" width=\"220\" height=\"23\" >and thus,<img src=\"//latex.artofproblemsolving.com/8/e/8/8e81703af0bf7966ed71decd1c300a07767e3bee.png\" class=\"latexcenter\" alt=\"\\[ b^{2^{n+2}}\\equiv 1\\pmod p.\\]\" width=\"156\" height=\"23\" >Consequently, according to the lemma, <img src=\"//latex.artofproblemsolving.com/4/b/d/4bdd0a21a5bc56865519eb5647892bb1ff6c1dad.png\" class=\"latex\" alt=\"$ \\text{ord}_p b=2^j$\" style=\"vertical-align: -4px\" width=\"78\" height=\"19\" > for some <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/8/a/28a8f7dded32bc668236ca39d96fb742bdaa9ec9.png\" class=\"latex\" alt=\"$ j\\leq n+2$\" style=\"vertical-align: -3px\" width=\"75\" height=\"16\" >.</span> However, if <img src=\"//latex.artofproblemsolving.com/4/4/b/44bb7cc61600c3703da374dcd8c2528334d46fc3.png\" class=\"latex\" alt=\"$ j<n+2$\" style=\"vertical-align: -3px\" width=\"75\" height=\"16\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/e/3/fe3ceff37cc91c1c77957c144688b9ba11f8378d.png\" class=\"latex\" alt=\"$ e=\\text{ord}_p b$\" style=\"vertical-align: -4px\" width=\"72\" height=\"17\" >,</span> then by the same lemma, <img src=\"//latex.artofproblemsolving.com/8/a/f/8af9e04eff2dc663a34d4f62147c8c489375b135.png\" class=\"latexcenter\" alt=\"\\[ b^{e2^{m+1-j}}-1=b^{2^{m+1}}-1\\equiv 2^{2^m}-1\\equiv 0\\pmod p,\\]\" width=\"390\" height=\"23\" > which contradicts the given. Hence,<img src=\"//latex.artofproblemsolving.com/1/5/4/15417c4f198cd8d3965bb2802f463bcba89c17e3.png\" class=\"latexcenter\" alt=\"\\[ \\text{ord}_p b=2^{m+2}.\\]\" width=\"105\" height=\"20\" > The numbers <img src=\"//latex.artofproblemsolving.com/6/5/f/65f279776d6c37b2790a28c7a1616a695c9e1170.png\" class=\"latex\" alt=\"$ p$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > and <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > are coprime. Therefore, applying Fermat's Little Theorem, and using the lemma, we obtain<img src=\"//latex.artofproblemsolving.com/8/1/8/81893117539fc0f916a1d3a021e900fce8b778e7.png\" class=\"latexcenter\" alt=\"\\[ p-1=k\\,\\text{ord}_p b\\equiv k2^{m+2}.\\;\\;\\blacksquare\\]\" width=\"219\" height=\"20\" ></div>", "post_id": 4416800, "post_number": 5, "post_time_unix": 1247862901, "post_time_utc": "2009-07-17 20:35:01 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" } ], "source": null }	1. For $n\ge 2$ show that any prime divisor $p$ of $2^{2^n}+1$ satisfies \[ p\equiv 1\pmod{2^{n+2}}. \] 2. Find all positive integers $n$ that are quadratic residues (including $0$) modulo every prime $p$. 3. Show that the odd prime factors of any squarefree integer representable as the sum of two positive perfect squares are of the form $4k+1$.	[ "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/FermatsLittleTheorem", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModuloOrder", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties" ]	Show that a specially constructed element has order 2^{n+2} modulo p, forcing p‑1 to be divisible by 2^{n+2}.	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aops_9973	JBL wrote:6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last 3 digits. Spoiler: [hide]Let the digits be a :le: b :le: c :le: d :le: e :le: f. We permute these digits to the number f b a d c e. then (f-e) + (b - d) + (a - e) :le: 9, since f-e :le: 9 and the others are at most 0.[/hide] JBL wrote:2. What regular polygon has the same number of diagonals as sides? Twice as many? 3 times as many? Spoiler:[hide]The formula for the number of diagonals of an n-gon is n(n+3)/2. so n = n(n-3)/2 => 2=n-3 => n=5. generalizing, if a n-gon has a times as many diagonals as sides, 2a = n-3, n=2a+3.[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "1. Prove that nC(n - 1, k - 1) = kC(n, k) using a combinatorial argument. (That means you cannot use the factorial formula for C(n, r))\r\n\r\n2. What regular polygon has the same number of diagonals as sides? Twice as many? 3 times as many?\r\n\r\n3. How many ways can you select 3 distinct digits from {0, 1, ..., 9} so that no two consecutive digits have been selected? What if we consider 0 and 9 to be consecutive?\r\n\r\n4. In how many ways can n students be seperated into two teams such that each team has to have at least 1 member? What about 3 teams?\r\n\r\n5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?\r\n\r\n6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last 3 digits.", "content_html": "1. Prove that nC(n - 1, k - 1) = kC(n, k) using a combinatorial argument. (That means you cannot use the factorial formula for C(n, r))<br>\n<br>\n2. What regular polygon has the same number of diagonals as sides? Twice as many? 3 times as many?<br>\n<br>\n3. How many ways can you select 3 distinct digits from {0, 1, ..., 9} so that no two consecutive digits have been selected? What if we consider 0 and 9 to be consecutive?<br>\n<br>\n4. In how many ways can n students be seperated into two teams such that each team has to have at least 1 member? What about 3 teams?<br>\n<br>\n5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?<br>\n<br>\n6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last 3 digits.", "post_id": 62517, "post_number": 1, "post_time_unix": 1074021733, "post_time_utc": "2004-01-13 19:22:13 UTC", "thanks_received": 2, "user_id": 1430, "username": "JBL" }, { "attachments": [], "content_bbcode": "JBL wrote:6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last 3 digits.\n\nSpoiler: [hide]Let the digits be a :le: b :le: c :le: d :le: e :le: f. We permute these digits to the number f b a d c e. then (f-e) + (b - d) + (a - e) :le: 9, since f-e :le: 9 and the others are at most 0.[/hide]\n\nJBL wrote:2. What regular polygon has the same number of diagonals as sides? Twice as many? 3 times as many? \n\n\n\nSpoiler:[hide]The formula for the number of diagonals of an n-gon is n(n+3)/2. so n = n(n-3)/2 => 2=n-3 => n=5. generalizing, if a n-gon has a times as many diagonals as sides, 2a = n-3, n=2a+3.[/hide]", "content_html": "JBL wrote:6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last 3 digits.<br>\n<br>\nSpoiler: <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let the digits be a :le: b :le: c :le: d :le: e :le: f. We permute these digits to the number f b a d c e. then (f-e) + (b - d) + (a - e) :le: 9, since f-e :le: 9 and the others are at most 0.</div><br>\n<br>\nJBL wrote:2. What regular polygon has the same number of diagonals as sides? Twice as many? 3 times as many?<br>\n<br>\n<br>\n<br>\nSpoiler:<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">The formula for the number of diagonals of an n-gon is n(n+3)/2. so n = n(n-3)/2 => 2=n-3 => n=5. generalizing, if a n-gon has a times as many diagonals as sides, 2a = n-3, n=2a+3.</div>", "post_id": 62552, "post_number": 2, "post_time_unix": 1074030773, "post_time_utc": "2004-01-13 21:52:53 UTC", "thanks_received": 2, "user_id": 1374, "username": "tetrahedr0n" }, { "attachments": [], "content_bbcode": "Tetrahedron -- for number 6, you need to look at the positive difference. The scheme you described might give you a difference less than -9.", "content_html": "Tetrahedron -- for number 6, you need to look at the positive difference. The scheme you described might give you a difference less than -9.", "post_id": 62553, "post_number": 3, "post_time_unix": 1074031006, "post_time_utc": "2004-01-13 21:56:46 UTC", "thanks_received": 2, "user_id": 1430, "username": "JBL" }, { "attachments": [], "content_bbcode": "4. In how many ways can n students be seperated into two teams such that each team has to have at least 1 member? What about 3 teams?\r\n\r\nI'm assuming that the two teams are distinguishable. So in the case where there are two groups, we can think each student will either be in or out of our group (and if he's out of our group, he's in the other one). So that's 2^n, but this doesn't account for the times when there are n students in one group and 0 in the other. This happens exactly twice (all in team A, or all in team B), so 2^n-2. With similar logic, we have 3^n, but we want the cases where one group has 0, 2 groups have 0, 3 groups have 0. One group has 0 in 3(nC2-2) cases... we subtract 2 cause in 2 of those cases, all will be in one group, and none in the other. 2 Groups have 0 in 3 cases since it means they're all in one group, 3 groups have 0 in no cases. Is it 3^n -3C(n,2) +3 I think that's probablly too high... enumerating the cases for n=4, im pretty sure this is wrong.", "content_html": "4. In how many ways can n students be seperated into two teams such that each team has to have at least 1 member? What about 3 teams?<br>\n<br>\nI'm assuming that the two teams are distinguishable. So in the case where there are two groups, we can think each student will either be in or out of our group (and if he's out of our group, he's in the other one). So that's 2^n, but this doesn't account for the times when there are n students in one group and 0 in the other. This happens exactly twice (all in team A, or all in team B), so 2^n-2. With similar logic, we have 3^n, but we want the cases where one group has 0, 2 groups have 0, 3 groups have 0. One group has 0 in 3(nC2-2) cases... we subtract 2 cause in 2 of those cases, all will be in one group, and none in the other. 2 Groups have 0 in 3 cases since it means they're all in one group, 3 groups have 0 in no cases. Is it 3^n -3C(n,2) +3 I think that's probablly too high... enumerating the cases for n=4, im pretty sure this is wrong.", "post_id": 62578, "post_number": 4, "post_time_unix": 1074038816, "post_time_utc": "2004-01-14 00:06:56 UTC", "thanks_received": 2, "user_id": 1515, "username": "cats..." }, { "attachments": [], "content_bbcode": "Are the teams distinguishable for #4? I presume the students are, but it's still an interesting problem if they aren't, at least for more than 2 teams.", "content_html": "Are the teams distinguishable for #4? I presume the students are, but it's still an interesting problem if they aren't, at least for more than 2 teams.", "post_id": 62585, "post_number": 5, "post_time_unix": 1074039888, "post_time_utc": "2004-01-14 00:24:48 UTC", "thanks_received": 2, "user_id": 1233, "username": "ComplexZeta" }, { "attachments": [], "content_bbcode": "1): [hide]suppose we have n people, and we want to to choose a k person team, in which one person is designated captain. On the one hand, the number of such teams is nC(n-1,k-1) because we can first choose a captain from the n people, and then out of the remaining n-1 people, we choose k-1 to be the rest of the team. On the other hand, this number is C(n,k)k because we can choose a k person team from the n people first, and then each of the k people could be designated captain.[/hide]", "content_html": "1): <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">suppose we have n people, and we want to to choose a k person team, in which one person is designated captain. On the one hand, the number of such teams is nC(n-1,k-1) because we can first choose a captain from the n people, and then out of the remaining n-1 people, we choose k-1 to be the rest of the team. On the other hand, this number is C(n,k)k because we can choose a k person team from the n people first, and then each of the k people could be designated captain.</div>", "post_id": 62588, "post_number": 6, "post_time_unix": 1074041078, "post_time_utc": "2004-01-14 00:44:38 UTC", "thanks_received": 2, "user_id": 1362, "username": "zscool" }, { "attachments": [], "content_bbcode": "[quote=\"JBL\"]3. How many ways can you select 3 distinct digits from {0, 1, ..., 9} so that no two consecutive digits have been selected? [/quote]\n\nI think it is 56 = C(10,3) - 98 + 8\n\nWhy ?\n. C(10,3) = choice of 3 among 10\n. 98 = 9 ways to choose 2 consecutive integers, 8 ways to choose the remaining integer\n. 8 = in the previous count we count twice the case where the three integers are consecutive.\n\n[quote=\"JBL\"]What if we consider 0 and 9 to be consecutive?\n[/quote]\r\n\r\n56-6=50 ? \r\nBecause we just have to remove the case 0 b 9 where 2<=b<=7 (case b=1 or 8 removed in the previous count).", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">JBL wrote:</div>\n<div class=\"bbcode_quote_body\">3. How many ways can you select 3 distinct digits from {0, 1, ..., 9} so that no two consecutive digits have been selected?</div>\n</div>\n<br>\nI think it is 56 = C(10,3) - 98 + 8<br>\n<br>\nWhy ?<br>\n. C(10,3) = choice of 3 among 10<br>\n. 98 = 9 ways to choose 2 consecutive integers, 8 ways to choose the remaining integer<br>\n. 8 = in the previous count we count twice the case where the three integers are consecutive.<br>\n\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">JBL wrote:</div>\n<div class=\"bbcode_quote_body\">What if we consider 0 and 9 to be consecutive?</div>\n</div>\n<br>\n56-6=50 ?<br>\nBecause we just have to remove the case 0 b 9 where 2<=b<=7 (case b=1 or 8 removed in the previous count).", "post_id": 62855, "post_number": 7, "post_time_unix": 1074164107, "post_time_utc": "2004-01-15 10:55:07 UTC", "thanks_received": 2, "user_id": 1633, "username": "belenos" }, { "attachments": [], "content_bbcode": "[quote=\"JBL\"]5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?\n[/quote]\r\n\r\n\r\nIgnoring for now integer beginning with '0'. If we choose six distinct integer we can build one and only one 6 increasing digits integer : C(10,6). \r\nNow we must remove the integer beginning with 0 : C(9,5)\r\n\r\nTotal C(10,6)- C(9,5) = 84.\r\n\r\ncorrect ??", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">JBL wrote:</div>\n<div class=\"bbcode_quote_body\">5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?</div>\n</div>\n<br>\n<br>\nIgnoring for now integer beginning with '0'. If we choose six distinct integer we can build one and only one 6 increasing digits integer : C(10,6).<br>\nNow we must remove the integer beginning with 0 : C(9,5)<br>\n<br>\nTotal C(10,6)- C(9,5) = 84.<br>\n<br>\ncorrect ??", "post_id": 62857, "post_number": 8, "post_time_unix": 1074166150, "post_time_utc": "2004-01-15 11:29:10 UTC", "thanks_received": 2, "user_id": 1633, "username": "belenos" }, { "attachments": [], "content_bbcode": "[quote=\"JBL\"]5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?[/quote]\r\n\r\n\r\nI think I have an other (and to my opinion smarter) proof.\r\n\r\nI claim that a 6-digit number whose digits are distinct and increase from left to right can be written :\r\na * 111111 + b * 11111 + c * 1111 + d * 111 + e * 11 + f * 1\r\nwith () a+b+c+d+e+f<=9 and a>0,b>0,..,f >0.\r\nAnd conversely such number is a good one.\r\n\r\nWith a'=a+1, ...,f'=f+1, () is equivalent to :\r\na'+b'+c'+d'+e'+f'<=3 or () a'+b'+c'+d'+e'+f' + g =3 with g >= 0.\r\n\r\nThe number of solutions of () is well known by 'De moivre' theorem, it is C(7+3-1,7-1)=C(9,6)=84 :mrgreen:", "content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">JBL wrote:</div>\n<div class=\"bbcode_quote_body\">5. How many 6-digit numbers are there whose digits are distinct and increase from left to right?</div>\n</div>\n<br>\n<br>\nI think I have an other (and to my opinion smarter) proof.<br>\n<br>\nI claim that a 6-digit number whose digits are distinct and increase from left to right can be written :<br>\na * 111111 + b * 11111 + c * 1111 + d * 111 + e * 11 + f * 1<br>\nwith () a+b+c+d+e+f<=9 and a>0,b>0,..,f >0.<br>\nAnd conversely such number is a good one.<br>\n<br>\nWith a'=a+1, ...,f'=f+1, () is equivalent to :<br>\na'+b'+c'+d'+e'+f'<=3 or () a'+b'+c'+d'+e'+f' + g =3 with g >= 0.<br>\n<br>\nThe number of solutions of () is well known by 'De moivre' theorem, it is C(7+3-1,7-1)=C(9,6)=84 :mrgreen:", "post_id": 62869, "post_number": 9, "post_time_unix": 1074174118, "post_time_utc": "2004-01-15 13:41:58 UTC", "thanks_received": 2, "user_id": 1633, "username": "belenos" }, { "attachments": [], "content_bbcode": "just a note for 3 and 5\r\n\r\n3: in general, if we want to take a subset of k elements from a set of n elements s.t. there are not two consecutive elements, then the number of such subsets is C(n-k+1,k) because in any subset of k elements from n-k+1 elements, we add 1 spaces between each element (k-1 spaces in total), and end up with a subset of k elements from a set of n elements (n-k+1 + k-1 = n), and because we added the spaces, no two are consecutive.\r\n\r\n5: if the digits are in increasing order from left to right, then 0 wont be an issue, since if the first number is not 0, none of the numbers are 0 so it is just C(9,6); for each subset of 6 numbers from {1,2,...,9} only one arrangement has them in increasing order.", "content_html": "just a note for 3 and 5<br>\n<br>\n3: in general, if we want to take a subset of k elements from a set of n elements s.t. there are not two consecutive elements, then the number of such subsets is C(n-k+1,k) because in any subset of k elements from n-k+1 elements, we add 1 spaces between each element (k-1 spaces in total), and end up with a subset of k elements from a set of n elements (n-k+1 + k-1 = n), and because we added the spaces, no two are consecutive.<br>\n<br>\n5: if the digits are in increasing order from left to right, then 0 wont be an issue, since if the first number is not 0, none of the numbers are 0 so it is just C(9,6); for each subset of 6 numbers from {1,2,...,9} only one arrangement has them in increasing order.", "post_id": 62876, "post_number": 10, "post_time_unix": 1074184123, "post_time_utc": "2004-01-15 16:28:43 UTC", "thanks_received": 2, "user_id": 1362, "username": "zscool" } ], "source": null }	1. Prove that $n\binom{n-1}{k-1}=k\binom{n}{k}$ using a combinatorial argument. 2. What regular polygon has the same number of diagonals as sides? Twice as many? Three times as many? 3. How many ways can you select 3 distinct digits from $\{0,1,\dots,9\}$ so that no two consecutive digits have been selected? What if 0 and 9 are considered consecutive? 4. In how many ways can $n$ students be separated into two teams such that each team has at least one member? What about three teams? 5. How many 6-digit numbers are there whose digits are distinct and increase from left to right? 6. Prove that the digits of any six-digit number can be permuted so that the sum of the first three digits differs by at most 9 from the sum of the last three digits.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]	Sort the digits and arrange them so that large and small digits are paired across the two halves, guaranteeing the three‑digit sums differ by at most 9.	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aops_997320	since $ a'_{n} \approx a_{n \plus{} 1} \minus{} a_{n} \equal{} \ln n$ then $ a_{n} \approx n\ln n$ thus serie $ \sum^{N} 1/a_{n} \approx \sum^{N} \frac {1}{n\ln n}\approx \ln \ln N \to \infty$ as $ N \to \infty$	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "A problem from Mathematical Reflections.\r\n\r\nLet $ a_1 \\equal{} 1$ and $ a_n \\equal{} a_{n \\minus{} 1} \\plus{} \\ln (n)$. Prove that the sequence $ \\sum_{i \\equal{} 1}^{n}\\frac {1}{a_i}$ is divergent.\r\n\r\nI'll give later a solution.", "content_html": "A problem from Mathematical Reflections.<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/4/b/e/4be795893930539766acffe8b31f75726e96b136.png\" class=\"latex\" alt=\"$ a_1 = 1$\" style=\"vertical-align: -2px\" width=\"49\" height=\"14\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/c/b/4cb7d0755935aaf2aa2000152a57d76aff433b79.png\" class=\"latex\" alt=\"$ a_n = a_{n - 1} + \\ln (n)$\" style=\"vertical-align: -4px\" width=\"139\" height=\"18\" >.</span> Prove that the sequence <img src=\"//latex.artofproblemsolving.com/5/e/d/5ed99767a1fe85e55b5d21f77c167a2764a44682.png\" class=\"latex\" alt=\"$ \\sum_{i = 1}^{n}\\frac {1}{a_i}$\" style=\"vertical-align: -20px\" width=\"46\" height=\"48\" > is divergent.<br>\n<br>\nI'll give later a solution.", "post_id": 4416925, "post_number": 1, "post_time_unix": 1212102155, "post_time_utc": "2008-05-29 23:02:35 UTC", "thanks_received": 2, "user_id": 43011, "username": "Amazigh" }, { "attachments": [], "content_bbcode": "since $ a'_{n} \\approx a_{n \\plus{} 1} \\minus{} a_{n} \\equal{} \\ln n$ \r\nthen $ a_{n} \\approx n\\ln n$ thus serie $ \\sum^{N} 1/a_{n} \\approx \\sum^{N} \\frac {1}{n\\ln n}\\approx \\ln \\ln N \\to \\infty$ as $ N \\to \\infty$", "content_html": "since <img src=\"//latex.artofproblemsolving.com/5/6/3/56329f017ed353c2baac4f7f40471f64c3208c09.png\" class=\"latex\" alt=\"$ a'_{n} \\approx a_{n + 1} - a_{n} = \\ln n$\" style=\"vertical-align: -4px\" width=\"172\" height=\"18\" ><br>\nthen <img src=\"//latex.artofproblemsolving.com/d/5/b/d5b45f6c0dfd29a60c570e2767684ccfe94cc54e.png\" class=\"latex\" alt=\"$ a_{n} \\approx n\\ln n$\" style=\"vertical-align: -2px\" width=\"85\" height=\"15\" > thus serie <img src=\"//latex.artofproblemsolving.com/9/b/0/9b03860a30f42dae5208287a176bdef53732f2fc.png\" class=\"latex\" alt=\"$ \\sum^{N} 1/a_{n} \\approx \\sum^{N} \\frac {1}{n\\ln n}\\approx \\ln \\ln N \\to \\infty$\" style=\"vertical-align: -12px\" width=\"288\" height=\"42\" > as <img src=\"//latex.artofproblemsolving.com/d/0/f/d0f64bf0e828b9853425de581a3f9357640baa9c.png\" class=\"latex\" alt=\"$ N \\to \\infty$\" style=\"vertical-align: 0px\" width=\"62\" height=\"13\" >", "post_id": 4416926, "post_number": 2, "post_time_unix": 1231932857, "post_time_utc": "2009-01-14 11:34:17 UTC", "thanks_received": 2, "user_id": 18200, "username": "Extremal" } ], "source": null }	Let $a_1=1$ and $a_n=a_{n-1}+\ln n$ for $n\ge2$. Prove that the sequence $\sum_{i=1}^{n}\frac{1}{a_i}$ is divergent.	[ "/Mathematics/CalculusandAnalysis/Calculus", "/Mathematics/CalculusandAnalysis/Series/Convergence" ]	Estimate a_n by the cumulative sum of ln k, yielding a_n ≈ n ln n, then compare 1/a_n to 1/(n ln n) which diverges.	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aops_997330	[hide] We will consider four cases. Case 1: $ a$ is divisible by 10. So trivial, I'll let you guys do that Case 2: $ a$ is divisible by 2, but not 5. Let $ a\equal{}2b$. We wish to prove that $ 2^{22}b^{22}\equiv 4b^2$ (mod 100). We can simplify this to the congruence $ 2^{20}b^{22}\equiv b^2$ (mod 25). Since $ a$ is not divisible by $ 5$, $ b$ is not divisible by $ 5$, so we can divide by $ b^2$ to obtain $ (2b)^{20}\equiv 1$ (mod 25), which is true by the generalization of FLT. Just make this stuff go in the forwards direction. Case 3: $ a$ is divisible by 5, but not 2. Just let $ a\equal{}5b$ and do the exact same stuff, except you change to mod 4 instead of 25. Case 4: $ a$ is relatively prime to 100. We can quickly find that $ a^{40}\equiv 1$ (mod 100). If we find all the square roots of 1 in mod 100, we find that $ a^{20}$ could be either 1, 49, 51, or 99. If it was 49 or 99, then the square root of $ a^{20}$ would have to end in 3 or 7, but the square root of $ a^{20}$ is $ a^{10}$, which is a perfect square, thus it can't end in 3 or 7. If we had $ a^{20}\equiv 51$ (mod 100), then $ a^{20}\equal{}100k\plus{}51$. If we take mod 4, we have $ a^{20}\equiv 3$ (mod 4), which is impossible since $ a^{20}$ is a perfect square. Thus, $ a^{20}\equiv 1$ (mod 100) and the result follows. [/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "5. Find the remainder when the coefficient of $ x^{64}$ in the expansion of $ (x \\plus{} 1)(x \\plus{} 2)(x \\plus{} 3)(x \\plus{} 4)(x \\plus{} 5) \\cdots (x \\plus{} 100)$ is divided by $ 1000$. (If you don't feel like wasting your time, don't do it :lol: )\r\n\r\nThe answer is [hide]540[/hide].\r\n\r\n6. Prove that for every integer $ n$, $ n^k \\equiv n^{k \\plus{} 20} \\pmod {100}$ for all integers $ k \\ge 2$.", "content_html": "5. Find the remainder when the coefficient of <img src=\"//latex.artofproblemsolving.com/f/a/4/fa4e86089aee00a8aeacd9125d553aa7c6f119c3.png\" class=\"latex\" alt=\"$ x^{64}$\" width=\"23\" height=\"15\" > in the expansion of <img src=\"//latex.artofproblemsolving.com/4/2/c/42c4c365365bce570164b6143c82b08b8d6bb3a6.png\" class=\"latex\" alt=\"$ (x + 1)(x + 2)(x + 3)(x + 4)(x + 5) \\cdots (x + 100)$\" style=\"vertical-align: -4px\" width=\"378\" height=\"18\" > is divided by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/a/d8aaf57e4fd28a845e3bec1125606da30f0a67fb.png\" class=\"latex\" alt=\"$ 1000$\" style=\"vertical-align: 0px\" width=\"35\" height=\"13\" >.</span> (If you don't feel like wasting your time, don't do it <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" /> )<br>\n<br>\nThe answer is <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">540</div>.<br>\n<br>\n6. Prove that for every integer <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" >,</span> <img src=\"//latex.artofproblemsolving.com/a/4/b/a4b94e32e90e0d132854d2dcdf02cac0803db9ed.png\" class=\"latex\" alt=\"$ n^k \\equiv n^{k + 20} \\pmod {100}$\" style=\"vertical-align: -4px\" width=\"174\" height=\"20\" > for all integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/9/f/99fdbe79b6f809fad073a9bd05d5463f450ad0ee.png\" class=\"latex\" alt=\"$ k \\ge 2$\" style=\"vertical-align: -2px\" width=\"43\" height=\"15\" >.</span>", "post_id": 4416984, "post_number": 1, "post_time_unix": 1209864341, "post_time_utc": "2008-05-04 01:25:41 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "$ n$ is not necessarily congruent to $ n^{21}$ in mod 100 :wink:", "content_html": "<img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > is not necessarily congruent to <img src=\"//latex.artofproblemsolving.com/5/1/6/516f67d82de44fbec25a25837b792d07f2ceb974.png\" class=\"latex\" alt=\"$ n^{21}$\" width=\"23\" height=\"15\" > in mod 100 <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />", "post_id": 4416985, "post_number": 2, "post_time_unix": 1209865558, "post_time_utc": "2008-05-04 01:45:58 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "Sigh\r\n\r\nIt may be me miswriting the question, or it may be alanchou's misreading the question :D \r\n\r\nThis is why I introduced the value $ M$. Informally, it basically means that after some power, the last two digits repeat itself every 20 powers.\r\n\r\nActually, it can be proven that $ M$ is at most $ 2$ in all cases. I'll go edit the problem now.", "content_html": "Sigh<br>\n<br>\nIt may be me miswriting the question, or it may be alanchou's misreading the question <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\nThis is why I introduced the value <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/3/2/23293d929381045b3b629c3bdfc5d8d61362dcda.png\" class=\"latex\" alt=\"$ M$\" width=\"19\" height=\"12\" >.</span> Informally, it basically means that after some power, the last two digits repeat itself every 20 powers.<br>\n<br>\nActually, it can be proven that <img src=\"//latex.artofproblemsolving.com/2/3/2/23293d929381045b3b629c3bdfc5d8d61362dcda.png\" class=\"latex\" alt=\"$ M$\" width=\"19\" height=\"12\" > is at most <img src=\"//latex.artofproblemsolving.com/c/d/6/cd6096c4067edc08d300656b4bb46837ac0b8834.png\" class=\"latex\" alt=\"$ 2$\" width=\"8\" height=\"12\" > in all cases. I'll go edit the problem now.", "post_id": 4416986, "post_number": 3, "post_time_unix": 1209865893, "post_time_utc": "2008-05-04 01:51:33 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "good job, now first off\r\n\r\n5) do you know the answer, and do you know how to do it in a mathematical, non-calculator way?\r\n\r\n6) the induction is way too trivial. Just prove that for all integers $ a$, $ a^{22}\\equiv a^2\\text{ (mod 100)}$. That might take a while though...", "content_html": "good job, now first off<br>\n<br>\n5) do you know the answer, and do you know how to do it in a mathematical, non-calculator way?<br>\n<br>\n6) the induction is way too trivial. Just prove that for all integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/5/6/65610a523ad2553d5ac39be1591fa25984dbb292.png\" class=\"latex\" alt=\"$ a^{22}\\equiv a^2\\text{ (mod 100)}$\" style=\"vertical-align: -4px\" width=\"144\" height=\"19\" >.</span> That might take a while though...", "post_id": 4416987, "post_number": 4, "post_time_unix": 1209866346, "post_time_utc": "2008-05-04 01:59:06 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "5. [hide=\"Hint\"]Choose 64 of those binomials to be $ x$'s. Then there's 36 constants to choose from. Just find sum of the products of all possible combinations of 36 constants. The fact that they are 1, 2, 3, ..., 100 should help you. Then, you only have to find the last three digits of that![/hide]\r\n\r\n6. Yes, so it is impossible to prove that for [b]all[/b] integers $ a$. That would require an infinite number of cases! First, see if you can prove that a subset of integers works, then that all other integers follow from that subset.", "content_html": "5. <a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hint</a><div class=\"cmty-hide-content\" style=\"display:none\">Choose 64 of those binomials to be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" >'</span>s. Then there's 36 constants to choose from. Just find sum of the products of all possible combinations of 36 constants. The fact that they are 1, 2, 3, ..., 100 should help you. Then, you only have to find the last three digits of that!</div><br>\n<br>\n6. Yes, so it is impossible to prove that for <b>all</b> integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" >.</span> That would require an infinite number of cases! First, see if you can prove that a subset of integers works, then that all other integers follow from that subset.", "post_id": 4416988, "post_number": 5, "post_time_unix": 1209866597, "post_time_utc": "2008-05-04 02:03:17 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "I proved the result for all integers $ a$, it's not really that hard.\r\n\r\nand for #5, that's the part I got, but I'm not interested in multiplying out or finding the last 3 digits even, and that doesn't account for adding all them up.", "content_html": "I proved the result for all integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" >,</span> it's not really that hard.<br>\n<br>\nand for #5, that's the part I got, but I'm not interested in multiplying out or finding the last 3 digits even, and that doesn't account for adding all them up.", "post_id": 4416989, "post_number": 6, "post_time_unix": 1209866742, "post_time_utc": "2008-05-04 02:05:42 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "#5 was a problem I stole directly from a future AIME (don't ask which one...)\r\n\r\n#6... alanchou, would you mind posting a proof for us? I think I posted those problems so that people could solve them! Be sure to enclose them in (hide)(/hide) tags.", "content_html": "#5 was a problem I stole directly from a future AIME (don't ask which one...)<br>\n<br>\n#6... alanchou, would you mind posting a proof for us? I think I posted those problems so that people could solve them! Be sure to enclose them in (hide)(/hide) tags.", "post_id": 4416990, "post_number": 7, "post_time_unix": 1209867309, "post_time_utc": "2008-05-04 02:15:09 UTC", "thanks_received": 1, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "[hide]\nWe will consider four cases.\n\nCase 1: $ a$ is divisible by 10.\n\nSo trivial, I'll let you guys do that\n\nCase 2: $ a$ is divisible by 2, but not 5.\n\nLet $ a\\equal{}2b$. We wish to prove that $ 2^{22}b^{22}\\equiv 4b^2$ (mod 100). We can simplify this to the congruence $ 2^{20}b^{22}\\equiv b^2$ (mod 25). Since $ a$ is not divisible by $ 5$, $ b$ is not divisible by $ 5$, so we can divide by $ b^2$ to obtain $ (2b)^{20}\\equiv 1$ (mod 25), which is true by the generalization of FLT. Just make this stuff go in the forwards direction.\n\nCase 3: $ a$ is divisible by 5, but not 2.\n\nJust let $ a\\equal{}5b$ and do the exact same stuff, except you change to mod 4 instead of 25.\n\nCase 4: $ a$ is relatively prime to 100.\n\nWe can quickly find that $ a^{40}\\equiv 1$ (mod 100). If we find all the square roots of 1 in mod 100, we find that $ a^{20}$ could be either 1, 49, 51, or 99. If it was 49 or 99, then the square root of $ a^{20}$ would have to end in 3 or 7, but the square root of $ a^{20}$ is $ a^{10}$, which is a perfect square, thus it can't end in 3 or 7. If we had $ a^{20}\\equiv 51$ (mod 100), then $ a^{20}\\equal{}100k\\plus{}51$. If we take mod 4, we have $ a^{20}\\equiv 3$ (mod 4), which is impossible since $ a^{20}$ is a perfect square. Thus, $ a^{20}\\equiv 1$ (mod 100) and the result follows.\n[/hide]", "content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">We will consider four cases.<br>\n<br>\nCase 1: <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > is divisible by 10.<br>\n<br>\nSo trivial, I'll let you guys do that<br>\n<br>\nCase 2: <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > is divisible by 2, but not 5.<br>\n<br>\nLet <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/c/c/8cc3ee07e92eb48d4a50169c5fdbea5af81ffdc4.png\" class=\"latex\" alt=\"$ a=2b$\" width=\"51\" height=\"12\" >.</span> We wish to prove that <img src=\"//latex.artofproblemsolving.com/d/4/5/d45be8320f1123233d9fb4989de26a06a6d57959.png\" class=\"latex\" alt=\"$ 2^{22}b^{22}\\equiv 4b^2$\" style=\"vertical-align: 0px\" width=\"92\" height=\"15\" > (mod 100). We can simplify this to the congruence <img src=\"//latex.artofproblemsolving.com/c/c/b/ccbf8ac3af22e904e3c823074cb473f3e47dcf9e.png\" class=\"latex\" alt=\"$ 2^{20}b^{22}\\equiv b^2$\" width=\"83\" height=\"15\" > (mod 25). Since <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > is not divisible by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/7/e/e7e9e6247026cfd2c28a759660e83db3e20bd2c4.png\" class=\"latex\" alt=\"$ 5$\" width=\"8\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > is not divisible by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/7/e/e7e9e6247026cfd2c28a759660e83db3e20bd2c4.png\" class=\"latex\" alt=\"$ 5$\" width=\"8\" height=\"12\" >,</span> so we can divide by <img src=\"//latex.artofproblemsolving.com/0/8/5/08547519206cffe2981fd7e72c602c747482e7d9.png\" class=\"latex\" alt=\"$ b^2$\" width=\"14\" height=\"15\" > to obtain <img src=\"//latex.artofproblemsolving.com/f/1/2/f12a2ddae798c483405ec7b50ebe39c127fd2652.png\" class=\"latex\" alt=\"$ (2b)^{20}\\equiv 1$\" style=\"vertical-align: -4px\" width=\"77\" height=\"19\" > (mod 25), which is true by the generalization of FLT. Just make this stuff go in the forwards direction.<br>\n<br>\nCase 3: <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > is divisible by 5, but not 2.<br>\n<br>\nJust let <img src=\"//latex.artofproblemsolving.com/e/d/8/ed8ccae47dd29ff0d6dfbe2251d9755dd77eb8d7.png\" class=\"latex\" alt=\"$ a=5b$\" width=\"51\" height=\"12\" > and do the exact same stuff, except you change to mod 4 instead of 25.<br>\n<br>\nCase 4: <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > is relatively prime to 100.<br>\n<br>\nWe can quickly find that <img src=\"//latex.artofproblemsolving.com/2/9/0/2904cd5dd890c9a0852eb9e25620633771664adc.png\" class=\"latex\" alt=\"$ a^{40}\\equiv 1$\" style=\"vertical-align: 0px\" width=\"56\" height=\"15\" > (mod 100). If we find all the square roots of 1 in mod 100, we find that <img src=\"//latex.artofproblemsolving.com/5/d/d/5dd700ce2531c4d3a8bc5d1667cf4a9b65e0d087.png\" class=\"latex\" alt=\"$ a^{20}$\" width=\"22\" height=\"15\" > could be either 1, 49, 51, or 99. If it was 49 or 99, then the square root of <img src=\"//latex.artofproblemsolving.com/5/d/d/5dd700ce2531c4d3a8bc5d1667cf4a9b65e0d087.png\" class=\"latex\" alt=\"$ a^{20}$\" width=\"22\" height=\"15\" > would have to end in 3 or 7, but the square root of <img src=\"//latex.artofproblemsolving.com/5/d/d/5dd700ce2531c4d3a8bc5d1667cf4a9b65e0d087.png\" class=\"latex\" alt=\"$ a^{20}$\" width=\"22\" height=\"15\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/0/8/708026084ee2d6e118442f623a695dd3220ff77c.png\" class=\"latex\" alt=\"$ a^{10}$\" width=\"22\" height=\"15\" >,</span> which is a perfect square, thus it can't end in 3 or 7. If we had <img src=\"//latex.artofproblemsolving.com/4/4/f/44f600356a4da65d70d30ceb720bc9d0631bc577.png\" class=\"latex\" alt=\"$ a^{20}\\equiv 51$\" style=\"vertical-align: 0px\" width=\"65\" height=\"15\" > (mod 100), then <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/d/9/9d9cca294685d248d331b9b9d11504a08783dcec.png\" class=\"latex\" alt=\"$ a^{20}=100k+51$\" style=\"vertical-align: -1px\" width=\"124\" height=\"16\" >.</span> If we take mod 4, we have <img src=\"//latex.artofproblemsolving.com/6/c/e/6ce077c5653c40fc13b6e39f15260501823e83f8.png\" class=\"latex\" alt=\"$ a^{20}\\equiv 3$\" width=\"56\" height=\"15\" > (mod 4), which is impossible since <img src=\"//latex.artofproblemsolving.com/5/d/d/5dd700ce2531c4d3a8bc5d1667cf4a9b65e0d087.png\" class=\"latex\" alt=\"$ a^{20}$\" width=\"22\" height=\"15\" > is a perfect square. Thus, <img src=\"//latex.artofproblemsolving.com/b/4/a/b4af6e6a8d8f60ccc4610c1ce8d855fb055c2b87.png\" class=\"latex\" alt=\"$ a^{20}\\equiv 1$\" style=\"vertical-align: 0px\" width=\"56\" height=\"15\" > (mod 100) and the result follows.</div>", "post_id": 4416991, "post_number": 8, "post_time_unix": 1209867931, "post_time_utc": "2008-05-04 02:25:31 UTC", "thanks_received": 1, "user_id": 26904, "username": "alanchou" } ], "source": null }	5. Find the remainder when the coefficient of $x^{64}$ in the expansion of \[ (x+1)(x+2)(x+3)\cdots(x+100) \] is divided by $1000$. 6. Prove that for every integer $n$, \[ n^k \equiv n^{k+20} \pmod{100} \] for all integers $k\ge 2$.	[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/EulersTotientTheorem", "/Mathematics/NumberTheory/Congruences/FermatsLittleTheorem", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]	Apply Euler/Fermat theorem (via CRT) to show a^{20} ≡ 1 (mod 100) for all a, handling the cases where a shares factors with 100 separately.	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aops_997333	Let the side lengths be $ a$ and $ b$. $ ab\equal{}2a\plus{}2b$ $ ab\minus{}2a\equal{}2b$ $ a(b\minus{}2)\equal{}2b$ $ a\equal{}\frac{2b}{b\minus{}2}$ So any positive $ b>2$ generates a rectangle that satisfies the required relation.	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "Prove that there are an infinite number of rectangles with the property that its area is numerically equal to its perimeter. Show all work.", "content_html": "Prove that there are an infinite number of rectangles with the property that its area is numerically equal to its perimeter. Show all work.", "post_id": 4417002, "post_number": 1, "post_time_unix": 1209693878, "post_time_utc": "2008-05-02 02:04:38 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" }, { "attachments": [], "content_bbcode": "Let the side lengths be $ a$ and $ b$.\r\n$ ab\\equal{}2a\\plus{}2b$\r\n$ ab\\minus{}2a\\equal{}2b$\r\n$ a(b\\minus{}2)\\equal{}2b$\r\n$ a\\equal{}\\frac{2b}{b\\minus{}2}$\r\n\r\nSo any positive $ b>2$ generates a rectangle that satisfies the required relation.", "content_html": "Let the side lengths be <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" >.</span><br>\n<img src=\"//latex.artofproblemsolving.com/9/9/5/9952ec24ebe368695b80f948ee93978268a9d878.png\" class=\"latex\" alt=\"$ ab=2a+2b$\" style=\"vertical-align: -1px\" width=\"99\" height=\"14\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/9/3/39354a46e6a8c0ca03fc7352f8f86a556c518aaa.png\" class=\"latex\" alt=\"$ ab-2a=2b$\" width=\"99\" height=\"12\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/e/d/cedf931e2c53f286c770f4ae6f3c4464100a7814.png\" class=\"latex\" alt=\"$ a(b-2)=2b$\" style=\"vertical-align: -4px\" width=\"104\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/0/c/2/0c268b2fa53e699147a609570a30ff1e2b5118a8.png\" class=\"latex\" alt=\"$ a=\\frac{2b}{b-2}$\" style=\"vertical-align: -12px\" width=\"75\" height=\"37\" ><br>\n<br>\nSo any positive <img src=\"//latex.artofproblemsolving.com/9/1/6/916c5b3b819af44030429bba6553b3e61ef484bf.png\" class=\"latex\" alt=\"$ b>2$\" style=\"vertical-align: 0px\" width=\"40\" height=\"13\" > generates a rectangle that satisfies the required relation.", "post_id": 4417003, "post_number": 2, "post_time_unix": 1209731987, "post_time_utc": "2008-05-02 12:39:47 UTC", "thanks_received": 2, "user_id": 27067, "username": "alkjash" }, { "attachments": [], "content_bbcode": "BASIC ALGEBRA IS SOOOOOOO TOUGH", "content_html": "BASIC ALGEBRA IS SOOOOOOO TOUGH", "post_id": 4417004, "post_number": 3, "post_time_unix": 1209778638, "post_time_utc": "2008-05-03 01:37:18 UTC", "thanks_received": 2, "user_id": 21169, "username": "perfect628" } ], "source": null }	Prove that there are an infinite number of rectangles with the property that its area is numerically equal to its perimeter. Show all work.	[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/AlgebraicExpression", "/Mathematics/Algebra/AlgebraicOperations", "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/GeometricInequalities/Chapple-EulerInequality", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Quadrilateral", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle" ]	Set the equation area = perimeter (ab = 2a+2b) and solve for one side to obtain a parametrization producing infinitely many rectangles.	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aops_997346	Ans: [hide]x=7548 y=5683 I'm too lazy to say how I got it :) but it took only a few minutes :D[/hide] If anyone's too lazy to do alanchou's one... I'll give an easier problem... [hide]x^2+y^2=23457457568678674 Find x and y where x>y. (there are two answers... so it should be easier!!!! (ok... not really))[/hide]	null	{ "competition": null, "dataset": "AOPS", "posts": [ { "attachments": [], "content_bbcode": "$ x^2\\plus{}y^2\\equal{}89268793$\r\n\r\nyou may not just try each possible value of x and hope y is an integer.\r\n\r\n(x and y are possible integers such that x>y)", "content_html": "<img src=\"//latex.artofproblemsolving.com/6/6/d/66da5ba0f849c3005e188f89c7c15c9e8d78a889.png\" class=\"latex\" alt=\"$ x^2+y^2=89268793$\" style=\"vertical-align: -3px\" width=\"153\" height=\"18\" ><br>\n<br>\nyou may not just try each possible value of x and hope y is an integer.<br>\n<br>\n(x and y are possible integers such that x>y)", "post_id": 4417060, "post_number": 1, "post_time_unix": 1211159265, "post_time_utc": "2008-05-19 01:07:45 UTC", "thanks_received": 2, "user_id": 26904, "username": "alanchou" }, { "attachments": [], "content_bbcode": "mods might help, but the number is kinda big.", "content_html": "mods might help, but the number is kinda big.", "post_id": 4417061, "post_number": 2, "post_time_unix": 1211159613, "post_time_utc": "2008-05-19 01:13:33 UTC", "thanks_received": 2, "user_id": 28965, "username": "mihail911" }, { "attachments": [], "content_bbcode": "Ans:\r\n[hide]x=7548 \ny=5683 I'm too lazy to say how I got it :) but it took only a few minutes :D[/hide]\n\nIf anyone's too lazy to do alanchou's one... I'll give an easier problem...\n\n[hide]x^2+y^2=23457457568678674\nFind x and y where x>y. (there are two answers... so it should be easier!!!! (ok... not really))[/hide]", "content_html": "Ans:<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">x=7548<br>\ny=5683 I'm too lazy to say how I got it <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /> but it took only a few minutes <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div><br>\n<br>\nIf anyone's too lazy to do alanchou's one... I'll give an easier problem...<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">x^2+y^2=23457457568678674<br>\nFind x and y where x>y. (there are two answers... so it should be easier!!!! (ok... not really))</div>", "post_id": 4417062, "post_number": 3, "post_time_unix": 1211242034, "post_time_utc": "2008-05-20 00:07:14 UTC", "thanks_received": 2, "user_id": 38433, "username": "ac-king" }, { "attachments": [], "content_bbcode": "$ x \\equal{} 1029743865$ and $ y \\equal{} 1446982907$. I'm too lazy to find the other one.", "content_html": "<img src=\"//latex.artofproblemsolving.com/0/e/1/0e1ddc87a265eb30fde34883647ac981697a0d69.png\" class=\"latex\" alt=\"$ x = 1029743865$\" style=\"vertical-align: 0px\" width=\"124\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/a/46a3725d0b0ac58d0108949d2dd7ca7a1366b7f4.png\" class=\"latex\" alt=\"$ y = 1446982907$\" style=\"vertical-align: -3px\" width=\"124\" height=\"16\" >.</span> I'm too lazy to find the other one.", "post_id": 4417063, "post_number": 4, "post_time_unix": 1211243482, "post_time_utc": "2008-05-20 00:31:22 UTC", "thanks_received": 2, "user_id": 40253, "username": "Yongyi781" } ], "source": null }	Given integers x and y with x > y, solve \[ x^2 + y^2 = 89\,268\,793. \]	[ "/Mathematics/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation2ndPowers", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Numbers", "/Mathematics/RecreationalMathematics/MathematicalHumor/Proof", "/Mathematics/RecreationalMathematics/Puzzles" ]	Factor the integer and apply the sum‑of‑two‑squares theorem to construct x and y.	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