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aops_997351
|
er, better as in better chance of picking the holocaust museum?
technically, the chances if you picked first or last of getting that are the same, if information isn't provided beforehand what others picked/what was left. either way, the "intriguing puzzle" seems pretty silly to me.
oh, the joys of nitpicking
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "In my social studies class, there are 20 kids including me. We were each assigned a random topic to write about for the Washington D.C. trip. The way the random drawing works is that my teacher puts 20 places in a hat and the students take turns one-by-one drawing a slip from the hat. I was the last person to draw... and I got the Holocaust Museum!\r\n\r\nWould my chances have been better or worse if I had picked first?\r\n\r\nTemperal is as of now strictly prohibited to nitpick my posts from now on. :D",
"content_html": "In my social studies class, there are 20 kids including me. We were each assigned a random topic to write about for the Washington D.C. trip. The way the random drawing works is that my teacher puts 20 places in a hat and the students take turns one-by-one drawing a slip from the hat. I was the last person to draw... and I got the Holocaust Museum!<br>\n<br>\nWould my chances have been better or worse if I had picked first?<br>\n<br>\nTemperal is as of now strictly prohibited to nitpick my posts from now on. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4417087,
"post_number": 1,
"post_time_unix": 1211420882,
"post_time_utc": "2008-05-22 01:48:02 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "er, better as in better chance of picking the holocaust museum?\r\n\r\ntechnically, the chances if you picked first or last of getting that are the same, if information isn't provided beforehand what others picked/what was left. either way, the \"intriguing puzzle\" seems pretty silly to me.\r\n\r\noh, the joys of nitpicking",
"content_html": "er, better as in better chance of picking the holocaust museum?<br>\n<br>\ntechnically, the chances if you picked first or last of getting that are the same, if information isn't provided beforehand what others picked/what was left. either way, the "intriguing puzzle" seems pretty silly to me.<br>\n<br>\noh, the joys of nitpicking",
"post_id": 4417088,
"post_number": 2,
"post_time_unix": 1211421521,
"post_time_utc": "2008-05-22 01:58:41 UTC",
"thanks_received": 2,
"user_id": 28419,
"username": "Temperal"
},
{
"attachments": [],
"content_bbcode": "Congratulations, Temperal, you have earned your spot on my banned nitpickers list. :)",
"content_html": "Congratulations, Temperal, you have earned your spot on my banned nitpickers list. <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 4417089,
"post_number": 3,
"post_time_unix": 1211422211,
"post_time_utc": "2008-05-22 02:10:11 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
There are 20 students including me. Each student is assigned a random topic by drawing one slip from a hat containing 20 places; students draw one by one without replacement. I was the last person to draw and I got the Holocaust Museum. Would my chances have been better or worse if I had picked first?
|
[
"/Mathematics/ProbabilityandStatistics/Probability/ProbabilityAxioms",
"/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace",
"/Mathematics/ProbabilityandStatistics/Probability/SampleSpace"
] |
All positions are equally likely to receive any given slip, so picking first or last gives the same chance.
|
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] |
aops_997417
|
We can build generating functions and then [i]convolute[/i] (i.e. multiply) them together.
Apples $ \equal{} A(x) \equal{} 1 \plus{} x^2 \plus{} x^4 \plus{} x^6 \plus{} \dots \equal{} \sum_{n \equal{} 0}^\infty x^{2n} \equal{} \frac 1{1 \minus{} x^2}$
Bananas $ \equal{} B(x) \equal{} 1 \plus{} x^5 \plus{} x^{10} \plus{} x^{15} \plus{} \dots \equal{} \sum_{n \equal{} 0}^\infty x^{5n} \equal{} \frac 1{1 \minus{} x^5}$
Oranges $ \equal{} O(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} x^3 \plus{} x^4 \equal{} \frac {1 \minus{} x^5}{1 \minus{} x}$
Pears $ \equal{} P(x) \equal{} 1 \plus{} x \equal{} \frac {1 \minus{} x^2}{1 \minus{} x}$
Multiplying:
$ G(x) \equal{} A(x)B(x)O(x)P(x) \equal{} \frac 1{1 \minus{} x^2} \cdot \frac 1{1 \minus{} x^5} \cdot \frac {1 \minus{} x^5}{1 \minus{} x} \cdot \frac {1 \minus{} x^2}{1 \minus{} x}$
Notice that almost everything cancels out! We are left with
$ G(x) \equal{} \frac 1{(1 \minus{} x)^2}$
We can find a generating function for this by noting that this is simply the derivative of $ \frac 1{1 \minus{} x}$, whose generating function is simply $ \sum_{n \equal{} 0}^\infty x^n$. Therefore, the generating function of $ \frac 1{(1 \minus{} x)^2} \equal{} \sum_{n \equal{} 1}^\infty nx^{n \minus{} 1} \equal{} \sum_{n \equal{} 0}^\infty (n \plus{} 1)x^n$. Therefore, the coefficient of $ x^n$ in $ G(x)$ is $ n \plus{} 1$. :lol:
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "In how many ways can we fill a bag with $ n$ fruits subject to the following constraints?\r\n\r\n- The number of apples must be even.\r\n- The number of bananas must be a multiple of $ 5$.\r\n- There can be at most four oranges.\r\n- There can be at most one pear.\r\n- There can only be apples, bananas, oranges, and pears in this bag, no other fruits.\r\n\r\nOnce you've found it, [i]prove[/i] that it's true.",
"content_html": "In how many ways can we fill a bag with <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > fruits subject to the following constraints?<br>\n<br>\n- The number of apples must be even.<br>\n- The number of bananas must be a multiple of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/7/e/e7e9e6247026cfd2c28a759660e83db3e20bd2c4.png\" class=\"latex\" alt=\"$ 5$\" width=\"8\" height=\"12\" >.</span><br>\n- There can be at most four oranges.<br>\n- There can be at most one pear.<br>\n- There can only be apples, bananas, oranges, and pears in this bag, no other fruits.<br>\n<br>\nOnce you've found it, <i>prove</i> that it's true.",
"post_id": 4417391,
"post_number": 1,
"post_time_unix": 1214576835,
"post_time_utc": "2008-06-27 14:27:15 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "if n=0, it is 1. \r\n\r\nif n=1, it is 2\r\n\r\nif n=2, it is 3\r\n\r\nyay! I've made a lot of progress! I will guess $ n \\plus{} 1$. :rotfl: :rotfl: :rotfl:",
"content_html": "if n=0, it is 1.<br>\n<br>\nif n=1, it is 2<br>\n<br>\nif n=2, it is 3<br>\n<br>\nyay! I've made a lot of progress! I will guess <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/a/4aa5a556284375bb64e46a7d366eb15e5980d8ee.png\" class=\"latex\" alt=\"$ n + 1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >.</span> <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": 4417392,
"post_number": 2,
"post_time_unix": 1214580627,
"post_time_utc": "2008-06-27 15:30:27 UTC",
"thanks_received": 1,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "Can you PROVE ( :D ) it?",
"content_html": "Can you PROVE ( <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> ) it?",
"post_id": 4417393,
"post_number": 3,
"post_time_unix": 1214584089,
"post_time_utc": "2008-06-27 16:28:09 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "was I even right?",
"content_html": "was I even right?",
"post_id": 4417394,
"post_number": 4,
"post_time_unix": 1214585511,
"post_time_utc": "2008-06-27 16:51:51 UTC",
"thanks_received": 1,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "It's your job to find out. BY PROVING IT (or disproving it). :)",
"content_html": "It's your job to find out. BY PROVING IT (or disproving it). <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 4417395,
"post_number": 5,
"post_time_unix": 1214586762,
"post_time_utc": "2008-06-27 17:12:42 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Because tinytim is an idiot, he is obviously incorrect. Therefore the answer is not $ n\\plus{}1$.\r\n\r\nQ.E.D.",
"content_html": "Because tinytim is an idiot, he is obviously incorrect. Therefore the answer is not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/4/f/74f5cfe5ca7406962695c43bcf016f023ff50349.png\" class=\"latex\" alt=\"$ n+1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >.</span><br>\n<br>\nQ.E.D.",
"post_id": 4417396,
"post_number": 6,
"post_time_unix": 1214589171,
"post_time_utc": "2008-06-27 17:52:51 UTC",
"thanks_received": 1,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "...... I'll have you know that that proof is not very rigorous, and as such, you receive a 0 out of 7 for that proof.",
"content_html": "...... I'll have you know that that proof is not very rigorous, and as such, you receive a 0 out of 7 for that proof.",
"post_id": 4417397,
"post_number": 7,
"post_time_unix": 1214589755,
"post_time_utc": "2008-06-27 18:02:35 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "By the [b]dictionary.com Theorem[/b], the definition of idiot is \"an utterly foolish or senseless person\". Now I will prove that the [i]Tinytim Idiocy Theorem[/i]:\r\n\r\nAssume that tinytim isn't an idiot, then that would contradict the name of the [i]Tinytim Idiocy Theorem[/i], thus tinytim is an idiot. \r\n\r\nBy the Transitive Property, tinytim is an utterly foolish and or senseless person. Because tinytim is an utterly foolish and or senseless person, he never gets any math problems correct. Thus, the answer is not $ n\\plus{}1$.\r\n\r\nQ.E.D.",
"content_html": "By the <b><a target=\"_blank\" href=\"http://dictionary.com/\">dictionary.com</a> Theorem</b>, the definition of idiot is "an utterly foolish or senseless person". Now I will prove that the <i>Tinytim Idiocy Theorem</i>:<br>\n<br>\nAssume that tinytim isn't an idiot, then that would contradict the name of the <i>Tinytim Idiocy Theorem</i>, thus tinytim is an idiot.<br>\n<br>\nBy the Transitive Property, tinytim is an utterly foolish and or senseless person. Because tinytim is an utterly foolish and or senseless person, he never gets any math problems correct. Thus, the answer is not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/4/f/74f5cfe5ca7406962695c43bcf016f023ff50349.png\" class=\"latex\" alt=\"$ n+1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >.</span><br>\n<br>\nQ.E.D.",
"post_id": 4417398,
"post_number": 8,
"post_time_unix": 1214594186,
"post_time_utc": "2008-06-27 19:16:26 UTC",
"thanks_received": 1,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "*sigh*\r\n\r\nThe message is too small. Please make the message longer before submitting.",
"content_html": "*sigh*<br>\n<br>\nThe message is too small. Please make the message longer before submitting.",
"post_id": 4417399,
"post_number": 9,
"post_time_unix": 1214598352,
"post_time_utc": "2008-06-27 20:25:52 UTC",
"thanks_received": 1,
"user_id": 40880,
"username": "leoxnlin"
},
{
"attachments": [],
"content_bbcode": "Well you see... um... that statement is incorrect, because it IS in fact $ n \\plus{} 1$. Therefore, your proof is wrong and the [i]Tinytim Idiocy Theorem[/i] is satisfied.",
"content_html": "Well you see... um... that statement is incorrect, because it IS in fact <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/a/4aa5a556284375bb64e46a7d366eb15e5980d8ee.png\" class=\"latex\" alt=\"$ n + 1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >.</span> Therefore, your proof is wrong and the <i>Tinytim Idiocy Theorem</i> is satisfied.",
"post_id": 4417400,
"post_number": 10,
"post_time_unix": 1214598507,
"post_time_utc": "2008-06-27 20:28:27 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I think we can now use the Author Said So Theorem.",
"content_html": "I think we can now use the Author Said So Theorem.",
"post_id": 4417401,
"post_number": 11,
"post_time_unix": 1214599534,
"post_time_utc": "2008-06-27 20:45:34 UTC",
"thanks_received": 2,
"user_id": 37564,
"username": "zephyredx"
},
{
"attachments": [],
"content_bbcode": "Incorrect. I ban the use of that theorem.",
"content_html": "Incorrect. I ban the use of that theorem.",
"post_id": 4417402,
"post_number": 12,
"post_time_unix": 1214603621,
"post_time_utc": "2008-06-27 21:53:41 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "OMG!! I guessed right!!! Wish that was an USAMO problem. :rotfl: :rotfl:",
"content_html": "OMG!! I guessed right!!! Wish that was an USAMO problem. <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": 4417403,
"post_number": 13,
"post_time_unix": 1214605162,
"post_time_utc": "2008-06-27 22:19:22 UTC",
"thanks_received": 2,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "Really? You'd get a 0. :P",
"content_html": "Really? You'd get a 0. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4417404,
"post_number": 14,
"post_time_unix": 1214606178,
"post_time_utc": "2008-06-27 22:36:18 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "You didn't have to ruin it Yongyi. Gosh! :o :mad: :( \r\n\r\nnow I will never talk to you again!",
"content_html": "You didn't have to ruin it Yongyi. Gosh! <img src=\"/assets/images/smilies/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/mad.gif\" width=\"20\" height=\"20\" alt=\":mad:\" title=\":mad:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /><br>\n<br>\nnow I will never talk to you again!",
"post_id": 4417405,
"post_number": 15,
"post_time_unix": 1214613315,
"post_time_utc": "2008-06-28 00:35:15 UTC",
"thanks_received": 2,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "I meant you'd get a zero for the proving part, since the USAMO graders don't care about the answer. :P \r\n\r\nBut... but... you're my friend... why won't you talk to me anymore? :(",
"content_html": "I meant you'd get a zero for the proving part, since the USAMO graders don't care about the answer. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /><br>\n<br>\nBut... but... you're my friend... why won't you talk to me anymore? <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 4417406,
"post_number": 16,
"post_time_unix": 1214613451,
"post_time_utc": "2008-06-28 00:37:31 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Can you give a proof Yongyi? :maybe: Me wants to know...",
"content_html": "Can you give a proof Yongyi? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" /> Me wants to know...",
"post_id": 4417407,
"post_number": 17,
"post_time_unix": 1214622353,
"post_time_utc": "2008-06-28 03:05:53 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
},
{
"attachments": [],
"content_bbcode": "We can build generating functions and then [i]convolute[/i] (i.e. multiply) them together.\r\n\r\nApples $ \\equal{} A(x) \\equal{} 1 \\plus{} x^2 \\plus{} x^4 \\plus{} x^6 \\plus{} \\dots \\equal{} \\sum_{n \\equal{} 0}^\\infty x^{2n} \\equal{} \\frac 1{1 \\minus{} x^2}$\r\n\r\nBananas $ \\equal{} B(x) \\equal{} 1 \\plus{} x^5 \\plus{} x^{10} \\plus{} x^{15} \\plus{} \\dots \\equal{} \\sum_{n \\equal{} 0}^\\infty x^{5n} \\equal{} \\frac 1{1 \\minus{} x^5}$\r\n\r\nOranges $ \\equal{} O(x) \\equal{} 1 \\plus{} x \\plus{} x^2 \\plus{} x^3 \\plus{} x^4 \\equal{} \\frac {1 \\minus{} x^5}{1 \\minus{} x}$\r\n\r\nPears $ \\equal{} P(x) \\equal{} 1 \\plus{} x \\equal{} \\frac {1 \\minus{} x^2}{1 \\minus{} x}$\r\n\r\nMultiplying:\r\n\r\n$ G(x) \\equal{} A(x)B(x)O(x)P(x) \\equal{} \\frac 1{1 \\minus{} x^2} \\cdot \\frac 1{1 \\minus{} x^5} \\cdot \\frac {1 \\minus{} x^5}{1 \\minus{} x} \\cdot \\frac {1 \\minus{} x^2}{1 \\minus{} x}$\r\n\r\nNotice that almost everything cancels out! We are left with\r\n$ G(x) \\equal{} \\frac 1{(1 \\minus{} x)^2}$\r\n\r\nWe can find a generating function for this by noting that this is simply the derivative of $ \\frac 1{1 \\minus{} x}$, whose generating function is simply $ \\sum_{n \\equal{} 0}^\\infty x^n$. Therefore, the generating function of $ \\frac 1{(1 \\minus{} x)^2} \\equal{} \\sum_{n \\equal{} 1}^\\infty nx^{n \\minus{} 1} \\equal{} \\sum_{n \\equal{} 0}^\\infty (n \\plus{} 1)x^n$. Therefore, the coefficient of $ x^n$ in $ G(x)$ is $ n \\plus{} 1$. :lol:",
"content_html": "We can build generating functions and then <i>convolute</i> (i.e. multiply) them together.<br>\n<br>\nApples <img src=\"//latex.artofproblemsolving.com/c/e/e/cee57fa71758e4a3bd1f599b3b7b1f9dd4769521.png\" class=\"latex\" alt=\"$ = A(x) = 1 + x^2 + x^4 + x^6 + \\dots = \\sum_{n = 0}^\\infty x^{2n} = \\frac 1{1 - x^2}$\" style=\"vertical-align: -20px\" width=\"409\" height=\"48\" ><br>\n<br>\nBananas <img src=\"//latex.artofproblemsolving.com/d/1/f/d1fd6fa2c16393cee6f7ca29b99d7757b198d817.png\" class=\"latex\" alt=\"$ = B(x) = 1 + x^5 + x^{10} + x^{15} + \\dots = \\sum_{n = 0}^\\infty x^{5n} = \\frac 1{1 - x^5}$\" style=\"vertical-align: -20px\" width=\"423\" height=\"48\" ><br>\n<br>\nOranges <img src=\"//latex.artofproblemsolving.com/6/6/7/6671014c01ab4f8cd12a2baf0d4edcbcd50928f4.png\" class=\"latex\" alt=\"$ = O(x) = 1 + x + x^2 + x^3 + x^4 = \\frac {1 - x^5}{1 - x}$\" style=\"vertical-align: -13px\" width=\"320\" height=\"39\" ><br>\n<br>\nPears <img src=\"//latex.artofproblemsolving.com/2/7/b/27bf7faac1b1663d26f10b6dcfd025a96841e1bf.png\" class=\"latex\" alt=\"$ = P(x) = 1 + x = \\frac {1 - x^2}{1 - x}$\" style=\"vertical-align: -13px\" width=\"199\" height=\"39\" ><br>\n<br>\nMultiplying:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/4/8/f/48fd603f6b784c4aa414c16bf70a90016d1858b8.png\" class=\"latex\" alt=\"$ G(x) = A(x)B(x)O(x)P(x) = \\frac 1{1 - x^2} \\cdot \\frac 1{1 - x^5} \\cdot \\frac {1 - x^5}{1 - x} \\cdot \\frac {1 - x^2}{1 - x}$\" style=\"vertical-align: -13px\" width=\"492\" height=\"39\" ><br>\n<br>\nNotice that almost everything cancels out! We are left with<br>\n<img src=\"//latex.artofproblemsolving.com/5/d/e/5dee1f41c42a69fd35e3074eea8d14f55b587f67.png\" class=\"latex\" alt=\"$ G(x) = \\frac 1{(1 - x)^2}$\" style=\"vertical-align: -17px\" width=\"128\" height=\"41\" ><br>\n<br>\nWe can find a generating function for this by noting that this is simply the derivative of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/6/c/56c96392357527b7f0319d699778b91fb9baae9c.png\" class=\"latex\" alt=\"$ \\frac 1{1 - x}$\" style=\"vertical-align: -13px\" width=\"44\" height=\"37\" >,</span> whose generating function is simply <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/6/9/f6945a5d5e88aebce56b5fe6b480328dbd2f1c1f.png\" class=\"latex\" alt=\"$ \\sum_{n = 0}^\\infty x^n$\" style=\"vertical-align: -20px\" width=\"47\" height=\"48\" >.</span> Therefore, the generating function of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/e/2/4e247be2d79a22bc59e064398ea2ab51f02e03ef.png\" class=\"latex\" alt=\"$ \\frac 1{(1 - x)^2} = \\sum_{n = 1}^\\infty nx^{n - 1} = \\sum_{n = 0}^\\infty (n + 1)x^n$\" style=\"vertical-align: -20px\" width=\"292\" height=\"48\" >.</span> Therefore, the coefficient of <img src=\"//latex.artofproblemsolving.com/b/8/7/b87a321d2a049c259884c4a0aec367d80e8ffca9.png\" class=\"latex\" alt=\"$ x^n$\" width=\"18\" height=\"12\" > in <img src=\"//latex.artofproblemsolving.com/5/2/8/5284c8693dbcfabcb8fb0a0cb4bf3888cbec5c69.png\" class=\"latex\" alt=\"$ G(x)$\" style=\"vertical-align: -4px\" width=\"37\" height=\"18\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/a/4aa5a556284375bb64e46a7d366eb15e5980d8ee.png\" class=\"latex\" alt=\"$ n + 1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >.</span> <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />",
"post_id": 4417408,
"post_number": 18,
"post_time_unix": 1214690243,
"post_time_utc": "2008-06-28 21:57:23 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
In how many ways can we fill a bag with \(n\) fruits subject to the following constraints?
- The number of apples must be even.
- The number of bananas must be a multiple of \(5\).
- There can be at most four oranges.
- There can be at most one pear.
- The bag contains only apples, bananas, oranges, and pears.
(Provide a proof of your answer.)
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/Enumeration",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics"
] |
Encode each fruit’s restriction with a generating function and multiply them, using cancellations to get a simple closed form.
|
[
0.00005125999450683594,
-0.03857421875,
-0.00991058349609375,
0.01580810546875,
0.00028252601623535156,
-0.013671875,
0.031463623046875,
-0.050872802734375,
0.00033736228942871094,
-0.027099609375,
0.0172271728515625,
-0.0108642578125,
-0.0012340545654296875,
-0.0273895263671875,
-0.032196044921875,
0.00025391578674316406,
0.004001617431640625,
0.01776123046875,
0.023193359375,
-0.00323486328125,
-0.00576019287109375,
-0.029022216796875,
0.00146484375,
0.0182037353515625,
0.00232696533203125,
-0.0103607177734375,
-0.01146697998046875,
0.033050537109375,
0.0020732879638671875,
-0.003902435302734375,
0.01436614990234375,
0.0130767822265625,
0.011016845703125,
-0.03326416015625,
-0.0014467239379882812,
0.000537872314453125,
0.01181793212890625,
-0.0016231536865234375,
0.004039764404296875,
-0.0487060546875,
0.006458282470703125,
-0.02374267578125,
-0.00344085693359375,
0.01337432861328125,
0.011871337890625,
-0.00630950927734375,
-0.002429962158203125,
-0.004329681396484375,
-0.041900634765625,
0.0037784576416015625,
-0.006069183349609375,
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aops_997432
|
For number 2.
We move b^2 to the other side to get (a+b)(a-b)=2(2k+1)
Since a+b is equivalent to a-b mod 2. (a+b)(a-b) is either odd or divisible by 4.
We see 2(2k+1)=2 mod 4 so there are no integers a and b that satisfy the equation.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "1. Let $ (a_1, b_1), (a_2, b_2), \\dots, (a_n, b_n)$ be all possible ordered pairs of positive integers $ (a, b)$ such that $ a^2 \\equal{} 1337 \\plus{} b^2$. Find $ \\left(\\sum_{i \\equal{} 1}^n a_i, \\sum_{i \\equal{} 1}^n b_i\\right)$.\r\n\r\n2. Prove (yes, [b]prove[/b]) that there exists no ordered pairs of positive integers $ (a, b)$ such that $ a^2 \\equal{} 2(2k \\plus{} 1) \\plus{} b^2$ where $ k$ denotes a nonnegative integer.",
"content_html": "1. Let <img src=\"//latex.artofproblemsolving.com/9/a/4/9a457de62e80b998eaa879332ce0b88c5566396e.png\" class=\"latex\" alt=\"$ (a_1, b_1), (a_2, b_2), \\dots, (a_n, b_n)$\" style=\"vertical-align: -4px\" width=\"213\" height=\"18\" > be all possible ordered pairs of positive integers <img src=\"//latex.artofproblemsolving.com/0/4/2/04225db55e416d3e2e940edecc247f62aed3b536.png\" class=\"latex\" alt=\"$ (a, b)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/a/d8a8b859896a74b44e050f37c4ed769003873b1d.png\" class=\"latex\" alt=\"$ a^2 = 1337 + b^2$\" style=\"vertical-align: -1px\" width=\"114\" height=\"16\" >.</span> Find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/3/f/33fa71dbdc808287e12b4a664ea9a0d3a784cb77.png\" class=\"latex\" alt=\"$ \\left(\\sum_{i = 1}^n a_i, \\sum_{i = 1}^n b_i\\right)$\" style=\"vertical-align: -22px\" width=\"118\" height=\"53\" >.</span><br>\n<br>\n2. Prove (yes, <b>prove</b>) that there exists no ordered pairs of positive integers <img src=\"//latex.artofproblemsolving.com/0/4/2/04225db55e416d3e2e940edecc247f62aed3b536.png\" class=\"latex\" alt=\"$ (a, b)$\" style=\"vertical-align: -4px\" width=\"38\" height=\"18\" > such that <img src=\"//latex.artofproblemsolving.com/2/4/7/247f06c3edf10e0efe316d4c931f260b30110b1a.png\" class=\"latex\" alt=\"$ a^2 = 2(2k + 1) + b^2$\" style=\"vertical-align: -4px\" width=\"151\" height=\"19\" > where <img src=\"//latex.artofproblemsolving.com/d/1/0/d10af8b3fc779f307fe5c87020e433b00a41801d.png\" class=\"latex\" alt=\"$ k$\" width=\"9\" height=\"12\" > denotes a nonnegative integer.",
"post_id": 4417458,
"post_number": 1,
"post_time_unix": 1215726460,
"post_time_utc": "2008-07-10 21:47:40 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "For number 2. \r\n\r\nWe move b^2 to the other side to get (a+b)(a-b)=2(2k+1)\r\nSince a+b is equivalent to a-b mod 2. (a+b)(a-b) is either odd or divisible by 4.\r\nWe see 2(2k+1)=2 mod 4 so there are no integers a and b that satisfy the equation.",
"content_html": "For number 2.<br>\n<br>\nWe move b^2 to the other side to get (a+b)(a-b)=2(2k+1)<br>\nSince a+b is equivalent to a-b mod 2. (a+b)(a-b) is either odd or divisible by 4.<br>\nWe see 2(2k+1)=2 mod 4 so there are no integers a and b that satisfy the equation.",
"post_id": 4417459,
"post_number": 2,
"post_time_unix": 1215732502,
"post_time_utc": "2008-07-10 23:28:22 UTC",
"thanks_received": 2,
"user_id": 20380,
"username": "budi713"
},
{
"attachments": [],
"content_bbcode": "Yay! You get half a smiley! Solve #1 so I can give you a choice of these smileys:\r\n\r\n(smileys not working yet: cannot function when only half its head is working)\r\n\r\nEdit: wait, you did, on stevenmeow's blog! Yay!\r\n\r\nChoose a smiley for me to give you!",
"content_html": "Yay! You get half a smiley! Solve #1 so I can give you a choice of these smileys:<br>\n<br>\n(smileys not working yet: cannot function when only half its head is working)<br>\n<br>\nEdit: wait, you did, on stevenmeow's blog! Yay!<br>\n<br>\nChoose a smiley for me to give you!",
"post_id": 4417460,
"post_number": 3,
"post_time_unix": 1215740325,
"post_time_utc": "2008-07-11 01:38:45 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": ":football: :diablo: :love: \r\n\r\nTogether, especially the last one, these represent Chuck Norris.",
"content_html": "<img src=\"/assets/images/smilies/football.gif\" width=\"23\" height=\"28\" alt=\":football:\" title=\":football:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/diablo.gif\" width=\"37\" height=\"31\" alt=\":diablo:\" title=\":diablo:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/wub.gif\" width=\"22\" height=\"29\" alt=\":love:\" title=\":love:\" class=\"bbcode_smiley\" /><br>\n<br>\nTogether, especially the last one, these represent Chuck Norris.",
"post_id": 4417461,
"post_number": 4,
"post_time_unix": 1215741696,
"post_time_utc": "2008-07-11 02:01:36 UTC",
"thanks_received": 2,
"user_id": 37259,
"username": "math154"
},
{
"attachments": [],
"content_bbcode": "Cool I want :o",
"content_html": "Cool I want <img src=\"/assets/images/smilies/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" class=\"bbcode_smiley\" />",
"post_id": 4417462,
"post_number": 5,
"post_time_unix": 1215742349,
"post_time_utc": "2008-07-11 02:12:29 UTC",
"thanks_received": 2,
"user_id": 20380,
"username": "budi713"
},
{
"attachments": [],
"content_bbcode": "I gave it to you! :o :D :o",
"content_html": "I gave it to you! <img src=\"/assets/images/smilies/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" 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/ohmy.gif\" width=\"20\" height=\"20\" alt=\":o\" title=\":o\" class=\"bbcode_smiley\" />",
"post_id": 4417463,
"post_number": 6,
"post_time_unix": 1215808844,
"post_time_utc": "2008-07-11 20:40:44 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
1. Let \((a_1,b_1),(a_2,b_2),\dots,(a_n,b_n)\) be all ordered pairs of positive integers \((a,b)\) such that \(a^2=1337+b^2\). Find \(\left(\sum_{i=1}^n a_i,\ \sum_{i=1}^n b_i\right)\).
2. Prove that there exists no ordered pair of positive integers \((a,b)\) such that \(a^2=2(2k+1)+b^2\), where \(k\) is a nonnegative integer.
|
[
"/Mathematics/NumberTheory/DiophantineEquations/Diophantine",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation2ndPowers",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/RationalInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus"
] |
Use (a+b)(a-b)=2(2k+1) and parity to show the product must be odd or ≡0 mod 4, contradicting RHS ≡2 mod 4.
|
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aops_997440
|
It IS a five!!!
$ f(t) \equal{} \sin(2t)$
$ f'(t) \equal{} 2\cos(2t)$
$ f''(t) \equal{} \minus{} 4\sin(2t)$
$ f^{(3)}(t) \equal{} \minus{} 8\cos(2t)$
$ f^{(4)}(t) \equal{} 16\sin(2t)$
$ f^{(5)}(t) \equal{} 32\cos(2t)$
I will hopefully be taking AP Calculus BC next year when I am in 9th grade...
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "I got the AP Exam results back today! It seems I got $ x$, where\r\n\r\n$ \\frac {d^x}{dt^x}\\sin(2t) \\equal{} 32\\cos(2t)$\r\n\r\nIf you can solve that, then you will know what grade I got!! :D",
"content_html": "I got the AP Exam results back today! It seems I got <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" >,</span> where<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/5/9/3/593d2c2ddb6f730831c748704014e1b5c410685d.png\" class=\"latex\" alt=\"$ \\frac {d^x}{dt^x}\\sin(2t) = 32\\cos(2t)$\" style=\"vertical-align: -12px\" width=\"181\" height=\"37\" ><br>\n<br>\nIf you can solve that, then you will know what grade I got!! <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4417488,
"post_number": 1,
"post_time_unix": 1216324564,
"post_time_utc": "2008-07-17 19:56:04 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Wow aren't you in like 7th grade? Did you even take the AP course?",
"content_html": "Wow aren't you in like 7th grade? Did you even take the AP course?",
"post_id": 4417489,
"post_number": 2,
"post_time_unix": 1216324931,
"post_time_utc": "2008-07-17 20:02:11 UTC",
"thanks_received": 1,
"user_id": 23588,
"username": "n0vad3m0n"
},
{
"attachments": [],
"content_bbcode": "I'm in 8th grade, and yeah I took AP Calculus at my high school.",
"content_html": "I'm in 8th grade, and yeah I took AP Calculus at my high school.",
"post_id": 4417490,
"post_number": 3,
"post_time_unix": 1216325091,
"post_time_utc": "2008-07-17 20:04:51 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Arghh...Child genii... How many grades ahead would that be? Like, over 9000?\r\n\r\nIs the solution to that a five? I'm not smart, so don't make fun of me. I'm just guessing based on my knowledge about you and your math beastliness (and the fact that I saw a 32 and a 2 in the equation)",
"content_html": "Arghh...Child genii... How many grades ahead would that be? Like, over 9000?<br>\n<br>\nIs the solution to that a five? I'm not smart, so don't make fun of me. I'm just guessing based on my knowledge about you and your math beastliness (and the fact that I saw a 32 and a 2 in the equation)",
"post_id": 4417491,
"post_number": 4,
"post_time_unix": 1216336913,
"post_time_utc": "2008-07-17 23:21:53 UTC",
"thanks_received": 1,
"user_id": 29190,
"username": "Math Geek"
},
{
"attachments": [],
"content_bbcode": "Don't worry. You're much more beast than me. I'm not even as good as alkjash, stevenmeow, Bobby Shen, tinytim, you know...",
"content_html": "Don't worry. You're much more beast than me. I'm not even as good as alkjash, stevenmeow, Bobby Shen, tinytim, you know...",
"post_id": 4417492,
"post_number": 5,
"post_time_unix": 1216339789,
"post_time_utc": "2008-07-18 00:09:49 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "hmm... I know tinytim. he's a complete idiot. How can he possibly be smarter than you???",
"content_html": "hmm... I know tinytim. he's a complete idiot. How can he possibly be smarter than you???",
"post_id": 4417493,
"post_number": 6,
"post_time_unix": 1216343270,
"post_time_utc": "2008-07-18 01:07:50 UTC",
"thanks_received": 1,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "hmm i'm too lazy to do anything beyond the derivative mentally",
"content_html": "hmm i'm too lazy to do anything beyond the derivative mentally",
"post_id": 4417494,
"post_number": 7,
"post_time_unix": 1216345605,
"post_time_utc": "2008-07-18 01:46:45 UTC",
"thanks_received": 1,
"user_id": 9401,
"username": "#H34N1"
},
{
"attachments": [],
"content_bbcode": "[quote]Don't worry. You're much more beast than me. [/quote]\r\n\r\nCome on, Yongyi, lying isn't good for you. :P",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\">Don't worry. You're much more beast than me.</div>\n</div>\n<br>\nCome on, Yongyi, lying isn't good for you. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4417495,
"post_number": 8,
"post_time_unix": 1216387765,
"post_time_utc": "2008-07-18 13:29:25 UTC",
"thanks_received": 1,
"user_id": 29190,
"username": "Math Geek"
},
{
"attachments": [],
"content_bbcode": "It IS a five!!! \r\n\r\n$ f(t) \\equal{} \\sin(2t)$\r\n$ f'(t) \\equal{} 2\\cos(2t)$\r\n$ f''(t) \\equal{} \\minus{} 4\\sin(2t)$\r\n$ f^{(3)}(t) \\equal{} \\minus{} 8\\cos(2t)$\r\n$ f^{(4)}(t) \\equal{} 16\\sin(2t)$\r\n$ f^{(5)}(t) \\equal{} 32\\cos(2t)$\r\n\r\nI will hopefully be taking AP Calculus BC next year when I am in 9th grade...",
"content_html": "It IS a five!!!<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/f/d/afd44131d3fb96cd0e0ae3e8924c78a762562188.png\" class=\"latex\" alt=\"$ f(t) = \\sin(2t)$\" style=\"vertical-align: -4px\" width=\"106\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/1/8/c185949e4c40528fbec7f9f7ce42c5ed8bf52e18.png\" class=\"latex\" alt=\"$ f'(t) = 2\\cos(2t)$\" style=\"vertical-align: -4px\" width=\"125\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/e/b/5eb3bfcd377248b80d2a1e027d9d3d09befc3c21.png\" class=\"latex\" alt=\"$ f''(t) = - 4\\sin(2t)$\" style=\"vertical-align: -4px\" width=\"141\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/a/0/ca0b8547ed84ac22e363e1e537589a329c74a115.png\" class=\"latex\" alt=\"$ f^{(3)}(t) = - 8\\cos(2t)$\" style=\"vertical-align: -4px\" width=\"152\" height=\"20\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/7/f/a7f9087f96a3198e0ed688f4e7b9ff11383af612.png\" class=\"latex\" alt=\"$ f^{(4)}(t) = 16\\sin(2t)$\" style=\"vertical-align: -4px\" width=\"145\" height=\"20\" ><br>\n<img src=\"//latex.artofproblemsolving.com/0/9/5/095b8ed03641c23d059bad745b8251f53a529614.png\" class=\"latex\" alt=\"$ f^{(5)}(t) = 32\\cos(2t)$\" style=\"vertical-align: -4px\" width=\"147\" height=\"20\" ><br>\n<br>\nI will hopefully be taking AP Calculus BC next year when I am in 9th grade...",
"post_id": 4417496,
"post_number": 9,
"post_time_unix": 1216824014,
"post_time_utc": "2008-07-23 14:40:14 UTC",
"thanks_received": 1,
"user_id": 45289,
"username": "dysfunctionalequations"
}
],
"source": null
}
|
I got the AP Exam results back today! It seems I got \(x\), where
\[
\frac{d^x}{dt^x}\sin(2t)=32\cos(2t).
\]
|
[
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/ChainRule",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation",
"/Mathematics/CalculusandAnalysis/Functions/ElementaryFunction",
"/Mathematics/CalculusandAnalysis/Functions/Function",
"/Mathematics/CalculusandAnalysis/Functions/PeriodicFunction",
"/Mathematics/CalculusandAnalysis/Functions/RealAnalyticFunction",
"/Mathematics/CalculusandAnalysis/Functions/RealFunction"
] |
Track the cyclic derivative pattern of sin(2t), noting each differentiation multiplies by 2, to see that the 5th derivative yields 32 cos(2t).
|
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aops_997452
|
Ooh! Me, pick me!
Blah, this is easy:
[hide][hide][hide][hide][hide]Let $ a$, $ b$, and $ c$ be expressed as $ \frac {p_1} {q_1}$, $ \frac {p_2} {q_2}$, and $ \frac {p_3} {q_3}$ for integral $ p_k$ and $ q_k$. Setting the equation up as such, we multiply by $ (q_1\cdot q_2 \cdot q_3)^n$ on both sides to get $ (p_1q_2q_3)^n\plus{}(p_2q_1q_3)^n\equal{}(p_3q_1q_2)^n$. Because these new terms in the equation are all integers, there is no solution to the original equation by FLT.$ \blacksquare$[/hide][/hide][/hide][/hide][/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Prove that there are no rational solutions to\r\n\\[ a^n \\plus{} b^n \\equal{} c^n\r\n\\]\r\nfor all integers $ n > 2$. (Hint: Fermat's Last Theorem is now properly a theorem; use it to your advantage!)",
"content_html": "Prove that there are no rational solutions to<br>\n<img src=\"//latex.artofproblemsolving.com/8/a/4/8a43ee59d139500e2d7bea62617cab173fae8c04.png\" class=\"latexcenter\" alt=\"\\[ a^n + b^n = c^n\n\\]\" width=\"97\" height=\"14\" ><br>\nfor all integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/1/ce1ef9a7fb9bef9bdf2acab958bf035cb23ad213.png\" class=\"latex\" alt=\"$ n > 2$\" style=\"vertical-align: 0px\" width=\"43\" height=\"12\" >.</span> (Hint: Fermat's Last Theorem is now properly a theorem; use it to your advantage!)",
"post_id": 4417546,
"post_number": 1,
"post_time_unix": 1217537129,
"post_time_utc": "2008-07-31 20:45:29 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "That [b]is [/b] fermat's last theorem :P :P",
"content_html": "That <b>is </b> fermat's last theorem <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4417547,
"post_number": 2,
"post_time_unix": 1217544973,
"post_time_utc": "2008-07-31 22:56:13 UTC",
"thanks_received": 2,
"user_id": 36435,
"username": "Poincare"
},
{
"attachments": [],
"content_bbcode": "No, Fermat's Last Theorem says there are no INTEGRAL solutions, however proving this is just one extra step :D",
"content_html": "No, Fermat's Last Theorem says there are no INTEGRAL solutions, however proving this is just one extra step <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4417548,
"post_number": 3,
"post_time_unix": 1217545088,
"post_time_utc": "2008-07-31 22:58:08 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Ooh! Me, pick me!\r\n\r\nBlah, this is easy:\r\n\r\n[hide][hide][hide][hide][hide]Let $ a$, $ b$, and $ c$ be expressed as $ \\frac {p_1} {q_1}$, $ \\frac {p_2} {q_2}$, and $ \\frac {p_3} {q_3}$ for integral $ p_k$ and $ q_k$. Setting the equation up as such, we multiply by $ (q_1\\cdot q_2 \\cdot q_3)^n$ on both sides to get $ (p_1q_2q_3)^n\\plus{}(p_2q_1q_3)^n\\equal{}(p_3q_1q_2)^n$. Because these new terms in the equation are all integers, there is no solution to the original equation by FLT.$ \\blacksquare$[/hide][/hide][/hide][/hide][/hide]",
"content_html": "Ooh! Me, pick me!<br>\n<br>\nBlah, this is easy:<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" >,</span> and <img src=\"//latex.artofproblemsolving.com/b/1/4/b144c3decf04b3f7a907c07e4f369f1e02bb9adc.png\" class=\"latex\" alt=\"$ c$\" width=\"8\" height=\"8\" > be expressed as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/1/f/71fe2a4963b5092a0b687f7e6eb48d445d1e538d.png\" class=\"latex\" alt=\"$ \\frac {p_1} {q_1}$\" style=\"vertical-align: -16px\" width=\"19\" height=\"36\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/a/f/4af81297777417b579e285c4bc91231e1c7e420e.png\" class=\"latex\" alt=\"$ \\frac {p_2} {q_2}$\" style=\"vertical-align: -16px\" width=\"19\" height=\"36\" >,</span> and <img src=\"//latex.artofproblemsolving.com/e/c/c/eccae4bfe67df5152152c6b7d6f42c40b23676f8.png\" class=\"latex\" alt=\"$ \\frac {p_3} {q_3}$\" style=\"vertical-align: -16px\" width=\"19\" height=\"36\" > for integral <img src=\"//latex.artofproblemsolving.com/9/5/0/950eb9c2339a4f3c11404f4c2f6c1ce8e764115f.png\" class=\"latex\" alt=\"$ p_k$\" style=\"vertical-align: -3px\" width=\"17\" height=\"11\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/6/46629f96344e67b40c32bc0981b60bac0a91525e.png\" class=\"latex\" alt=\"$ q_k$\" style=\"vertical-align: -3px\" width=\"15\" height=\"11\" >.</span> Setting the equation up as such, we multiply by <img src=\"//latex.artofproblemsolving.com/5/f/6/5f64cd4d09e89e4cc9a7d6bc536974c63b82a648.png\" class=\"latex\" alt=\"$ (q_1\\cdot q_2 \\cdot q_3)^n$\" style=\"vertical-align: -4px\" width=\"95\" height=\"18\" > on both sides to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/e/5/5e5c59e96d96b05d3956ea780fc33eda20f7b127.png\" class=\"latex\" alt=\"$ (p_1q_2q_3)^n+(p_2q_1q_3)^n=(p_3q_1q_2)^n$\" style=\"vertical-align: -4px\" width=\"258\" height=\"18\" >.</span> Because these new terms in the equation are all integers, there is no solution to the original equation by FLT<span style=\"white-space:nowrap;\">.<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ></span></div></div></div></div></div>",
"post_id": 4417549,
"post_number": 4,
"post_time_unix": 1217774830,
"post_time_utc": "2008-08-03 14:47:10 UTC",
"thanks_received": 2,
"user_id": 29190,
"username": "Math Geek"
},
{
"attachments": [],
"content_bbcode": "Yay!!!! :D",
"content_html": "Yay!!!! <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4417550,
"post_number": 5,
"post_time_unix": 1217980362,
"post_time_utc": "2008-08-05 23:52:42 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
Prove that there are no rational solutions to
\[
a^n + b^n = c^n
\]
for all integers \(n>2\).
|
[
"/Mathematics/NumberTheory/DiophantineEquations/Diophantine",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation",
"/Mathematics/NumberTheory/DiophantineEquations/FermatEquation",
"/Mathematics/NumberTheory/DiophantineEquations/FermatsLastTheorem",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/RationalNumbers/FieldofRationals",
"/Mathematics/NumberTheory/RationalNumbers/Q",
"/Mathematics/NumberTheory/RationalNumbers/RationalNumber"
] |
Clear denominators to turn the rational equation into an integer equation of the same form and apply Fermat's Last Theorem.
|
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aops_997471
|
Something like this:
[asy]draw(shift(left) * unitcircle);
draw(shift(right) * unitcircle);
draw((-2, -2)--(-2, 2));
draw((2, -2)--(2, 2));
dot("A", (-2, 0), W);
dot("B", (0, 0));
dot("C", (2, 0));[/asy]
Or
[asy]draw(shift(left) * unitcircle);
draw(shift(right) * unitcircle);
draw((-18/5, 7/10)--(2/5, -23/10));
draw((18/5, -7/10)--(-2/5, 23/10));
dot("A", (-8/5, -4/5), SW);
dot("B", (0, 0));
dot("C", (8/5, 4/5), NE);[/asy]
Etc... and we have to prove that $ A, B, C$ are collinear.
Keep in mind that the two circles can be different sizes as well - but I don't have the Asymptote skillz to do that :(
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Construct two tangent circles and two parallel lines tangent to each of these circles. Let the two circles be $ \\phi$ and $ \\lambda$ and let the lines be $ l_1$ and $ l_2$. Prove that the three tangential points are collinear",
"content_html": "Construct two tangent circles and two parallel lines tangent to each of these circles. Let the two circles be <img src=\"//latex.artofproblemsolving.com/b/6/3/b638ff89a2669a672e5f7fc55401b4bd2559af28.png\" class=\"latex\" alt=\"$ \\phi$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > and <img src=\"//latex.artofproblemsolving.com/8/5/5/8552bce7a3a733289714687da1a5c0eae52220ef.png\" class=\"latex\" alt=\"$ \\lambda$\" width=\"10\" height=\"13\" > and let the lines be <img src=\"//latex.artofproblemsolving.com/2/d/2/2d252618fdcfd8acf491e27f5391e6b33622e916.png\" class=\"latex\" alt=\"$ l_1$\" style=\"vertical-align: -2px\" width=\"11\" height=\"15\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/f/6/df648b35da98285b56206dda5bfee4f79d41317e.png\" class=\"latex\" alt=\"$ l_2$\" style=\"vertical-align: -2px\" width=\"11\" height=\"15\" >.</span> Prove that the three tangential points are collinear",
"post_id": 4417629,
"post_number": 1,
"post_time_unix": 1219689443,
"post_time_utc": "2008-08-25 18:37:23 UTC",
"thanks_received": 2,
"user_id": 9401,
"username": "#H34N1"
},
{
"attachments": [],
"content_bbcode": "asymtote skills not 1337 enough to solve problem :(",
"content_html": "asymtote skills not 1337 enough to solve problem <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 4417630,
"post_number": 2,
"post_time_unix": 1219691500,
"post_time_utc": "2008-08-25 19:11:40 UTC",
"thanks_received": 2,
"user_id": 36435,
"username": "Poincare"
},
{
"attachments": [],
"content_bbcode": "I dont understand.....So CONFUSED!!!\r\n\r\nI cannot picture this in my head? are you sure there are 3 tangental points?\r\n\r\nI think im having my retarded moment.",
"content_html": "I dont understand.....So CONFUSED!!!<br>\n<br>\nI cannot picture this in my head? are you sure there are 3 tangental points?<br>\n<br>\nI think im having my retarded moment.",
"post_id": 4417631,
"post_number": 3,
"post_time_unix": 1219717096,
"post_time_utc": "2008-08-26 02:18:16 UTC",
"thanks_received": 2,
"user_id": 31435,
"username": "abacadaea"
},
{
"attachments": [],
"content_bbcode": "Something like this:\r\n[asy]draw(shift(left) * unitcircle);\ndraw(shift(right) * unitcircle);\ndraw((-2, -2)--(-2, 2));\ndraw((2, -2)--(2, 2));\ndot(\"A\", (-2, 0), W);\ndot(\"B\", (0, 0));\ndot(\"C\", (2, 0));[/asy]\r\nOr\r\n[asy]draw(shift(left) * unitcircle);\ndraw(shift(right) * unitcircle);\ndraw((-18/5, 7/10)--(2/5, -23/10));\ndraw((18/5, -7/10)--(-2/5, 23/10));\ndot(\"A\", (-8/5, -4/5), SW);\ndot(\"B\", (0, 0));\ndot(\"C\", (8/5, 4/5), NE);[/asy]\r\n\r\nEtc... and we have to prove that $ A, B, C$ are collinear.\r\nKeep in mind that the two circles can be different sizes as well - but I don't have the Asymptote skillz to do that :(",
"content_html": "Something like this:<br>\n<img src=\"//latex.artofproblemsolving.com/d/0/f/d0f2110c8afc7f4e48b142e4f8083c074b16fcb6.png\" class=\"asy-image\" width=\"252\" height=\"208\" alt=\"[asy]draw(shift(left) * unitcircle);\ndraw(shift(right) * unitcircle);\ndraw((-2, -2)--(-2, 2));\ndraw((2, -2)--(2, 2));\ndot("A", (-2, 0), W);\ndot("B", (0, 0));\ndot("C", (2, 0));[/asy]\"><br>\nOr<br>\n<img src=\"//latex.artofproblemsolving.com/a/5/0/a50052ed99d28d1e565ab4162a2d465be440b093.png\" class=\"asy-image\" width=\"252\" height=\"162\" alt=\"[asy]draw(shift(left) * unitcircle);\ndraw(shift(right) * unitcircle);\ndraw((-18/5, 7/10)--(2/5, -23/10));\ndraw((18/5, -7/10)--(-2/5, 23/10));\ndot("A", (-8/5, -4/5), SW);\ndot("B", (0, 0));\ndot("C", (8/5, 4/5), NE);[/asy]\"><br>\n<br>\nEtc... and we have to prove that <img src=\"//latex.artofproblemsolving.com/f/a/8/fa84f3c223b5c99f41024167f1a1429c4ee86263.png\" class=\"latex\" alt=\"$ A, B, C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > are collinear.<br>\nKeep in mind that the two circles can be different sizes as well - but I don't have the Asymptote skillz to do that <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 4417632,
"post_number": 4,
"post_time_unix": 1219757405,
"post_time_utc": "2008-08-26 13:30:05 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "It's a badly worded problem. It should be, draw two distinct parallel lines such that each is tangent to exactly one of the two circles.",
"content_html": "It's a badly worded problem. It should be, draw two distinct parallel lines such that each is tangent to exactly one of the two circles.",
"post_id": 4417633,
"post_number": 5,
"post_time_unix": 1220128111,
"post_time_utc": "2008-08-30 20:28:31 UTC",
"thanks_received": 2,
"user_id": 29190,
"username": "Math Geek"
}
],
"source": null
}
|
Construct two tangent circles and two parallel lines tangent to each of these circles. Let the two circles be \(\phi\) and \(\lambda\) and let the lines be \(l_1\) and \(l_2\). Prove that the three tangential points are collinear.
|
[
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry",
"/Mathematics/Geometry/GeometricConstruction/GeometricProblemsofAntiquity",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle-CircleTangents",
"/Mathematics/Geometry/PlaneGeometry/Circles/CircleTangentLine",
"/Mathematics/Geometry/PlaneGeometry/Circles/TangencyTheorem",
"/Mathematics/Geometry/PlaneGeometry/Circles/TangentCircles"
] |
Apply the homothety sending one circle onto the other; its center (the circles’ contact point) aligns the tangent points with the parallel lines.
|
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aops_99748
|
[quote="10000th User"]It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.
1) How long does it take for all three together to mow the lawn?
2) How long does it take for each one of them individually to mow the lawn?[/quote]
[hide]
$\frac{1}{p}+\frac{1}{s}=\frac{1}{24}$
$\frac{1}{s}+\frac{1}{j}=\frac{1}{30}$
$\frac{1}{j}+\frac{1}{p}=\frac{1}{20}$
[/hide]
[hide="1"]$2(\frac{1}{p}+\frac{1}{s}+\frac{1}{j})=\frac{1}{8}$
$\boxed{16}$ minutes[/hide]
[hide="2"]$\frac{1}{p}+\frac{1}{s}+\frac{1}{j}=\frac{1}{16}$
Subtract each equation from above:
1/j=1/48
John: 48 minutes
1/p=7/240
Peter:240/7 minutes
1/s=1/80
Sara: 80 minutes[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
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"content_bbcode": "It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.\r\n\r\n1) How long does it take for all three together to mow the lawn?\r\n\r\n2) How long does it take for each one of them individually to mow the lawn?",
"content_html": "It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.<br>\n<br>\n1) How long does it take for all three together to mow the lawn?<br>\n<br>\n2) How long does it take for each one of them individually to mow the lawn?",
"post_id": 563260,
"post_number": 1,
"post_time_unix": 1151718307,
"post_time_utc": "2006-07-01 01:45:07 UTC",
"thanks_received": 2,
"user_id": 10153,
"username": "10000th User"
},
{
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"content_bbcode": "[quote=\"10000th User\"]It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.\n\n1) How long does it take for all three together to mow the lawn?\n\n2) How long does it take for each one of them individually to mow the lawn?[/quote]\r\n[hide]\n$\\frac{1}{p}+\\frac{1}{s}=\\frac{1}{24}$\n\n$\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{30}$\n\n$\\frac{1}{j}+\\frac{1}{p}=\\frac{1}{20}$\n[/hide]\n[hide=\"1\"]$2(\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j})=\\frac{1}{8}$\n\n$\\boxed{16}$ minutes[/hide]\n\n[hide=\"2\"]$\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{16}$\n\nSubtract each equation from above:\n\n1/j=1/48\nJohn: 48 minutes\n\n1/p=7/240\n\nPeter:240/7 minutes\n\n1/s=1/80\n\nSara: 80 minutes[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">10000th User wrote:</div>\n<div class=\"bbcode_quote_body\">It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.<br>\n<br>\n1) How long does it take for all three together to mow the lawn?<br>\n<br>\n2) How long does it take for each one of them individually to mow the lawn?</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\"><img src=\"//latex.artofproblemsolving.com/1/1/4/1141eb96cfa601790c9541fbdbf55d3f75735897.png\" class=\"latex\" alt=\"$\\frac{1}{p}+\\frac{1}{s}=\\frac{1}{24}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/1/5/d1531da224516e8ef0e438180ba2b07494d750a5.png\" class=\"latex\" alt=\"$\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{30}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/c/d/5/cd5bd7c2a7804740068f60bf27cf0533b1457d62.png\" class=\"latex\" alt=\"$\\frac{1}{j}+\\frac{1}{p}=\\frac{1}{20}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" ></div><br>\n<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/8/e/c/8ec68d4f9ba0090e74b6eef89a24c24fc2743301.png\" class=\"latex\" alt=\"$2(\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j})=\\frac{1}{8}$\" style=\"vertical-align: -16px\" width=\"142\" height=\"40\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/1/2/2/122be32b3f050309c88e59fbbfdda0fd5e32a498.png\" class=\"latex\" alt=\"$\\boxed{16}$\" style=\"vertical-align: -5px\" width=\"30\" height=\"23\" > minutes</div><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">2</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/5/e/d/5edb7a2759e79809fd09f65909e9cdf96bca0c8a.png\" class=\"latex\" alt=\"$\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{16}$\" style=\"vertical-align: -16px\" width=\"128\" height=\"40\" ><br>\n<br>\nSubtract each equation from above:<br>\n<br>\n1/j=1/48<br>\nJohn: 48 minutes<br>\n<br>\n1/p=7/240<br>\n<br>\nPeter:240/7 minutes<br>\n<br>\n1/s=1/80<br>\n<br>\nSara: 80 minutes</div>",
"post_id": 563285,
"post_number": 2,
"post_time_unix": 1151719362,
"post_time_utc": "2006-07-01 02:02:42 UTC",
"thanks_received": 2,
"user_id": 11029,
"username": "ch1n353ch3s54a1l"
},
{
"attachments": [],
"content_bbcode": "[hide]$\\frac{1}{p}+\\frac{1}{s}=\\frac{1}{24}$, \n$\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{30}$, \n$\\frac{1}{j}+\\frac{1}{p}=\\frac{1}{20}$.\n\nAdding the three equations and dividing by 2, $\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{16}$, so it takes them 16 minutes.[/hide]\n[hide]Subtracting each of the first three equations from the last, we get\n\n$\\frac{1}{j}=\\frac{1}{48}$, so 48 minutes.\n$\\frac{1}{p}=\\frac{7}{240}$, so 240/7 minutes.\n$\\frac{1}{s}=\\frac{1}{80}$, so 80 minutes.[/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/1/1/4/1141eb96cfa601790c9541fbdbf55d3f75735897.png\" class=\"latex\" alt=\"$\\frac{1}{p}+\\frac{1}{s}=\\frac{1}{24}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" >,</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/5/d1531da224516e8ef0e438180ba2b07494d750a5.png\" class=\"latex\" alt=\"$\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{30}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" >,</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/d/5/cd5bd7c2a7804740068f60bf27cf0533b1457d62.png\" class=\"latex\" alt=\"$\\frac{1}{j}+\\frac{1}{p}=\\frac{1}{20}$\" style=\"vertical-align: -16px\" width=\"93\" height=\"40\" >.</span><br>\n<br>\nAdding the three equations and dividing by 2, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/e/d/5edb7a2759e79809fd09f65909e9cdf96bca0c8a.png\" class=\"latex\" alt=\"$\\frac{1}{p}+\\frac{1}{s}+\\frac{1}{j}=\\frac{1}{16}$\" style=\"vertical-align: -16px\" width=\"128\" height=\"40\" >,</span> so it takes them 16 minutes.</div><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\">Subtracting each of the first three equations from the last, we get<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/d/dadb48345de2b6d5c92598ed0bb3b591f9791464.png\" class=\"latex\" alt=\"$\\frac{1}{j}=\\frac{1}{48}$\" style=\"vertical-align: -16px\" width=\"58\" height=\"40\" >,</span> so 48 minutes.<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/3/f53ae53d71c21323216722ccaa3d7f8122157e6d.png\" class=\"latex\" alt=\"$\\frac{1}{p}=\\frac{7}{240}$\" style=\"vertical-align: -16px\" width=\"67\" height=\"40\" >,</span> so 240/7 minutes.<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/1/d/11d82a7086971120390cd6304f35dce5e5b889d2.png\" class=\"latex\" alt=\"$\\frac{1}{s}=\\frac{1}{80}$\" style=\"vertical-align: -12px\" width=\"58\" height=\"37\" >,</span> so 80 minutes.</div>",
"post_id": 563300,
"post_number": 3,
"post_time_unix": 1151719990,
"post_time_utc": "2006-07-01 02:13:10 UTC",
"thanks_received": 2,
"user_id": 11295,
"username": "pianoforte"
},
{
"attachments": [],
"content_bbcode": "[hide]\nJust use simulataneous equations. :P [/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\">Just use simulataneous equations. <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /></div>",
"post_id": 563333,
"post_number": 4,
"post_time_unix": 1151723210,
"post_time_utc": "2006-07-01 03:06:50 UTC",
"thanks_received": 2,
"user_id": 20633,
"username": "D.P.L"
}
],
"source": null
}
|
It takes Peter and Sara together 24 minutes to mow the lawn. It takes Sara and John together 30 minutes to mow the lawn. It takes John and Peter together 20 minutes to mow the lawn.
1) How long does it take for all three together to mow the lawn?
2) How long does it take for each one of them individually to mow the lawn?
|
[
"/Mathematics/Algebra/RateProblems"
] |
Express each pair's mowing time as a sum of individual rates, then add the three equations to obtain the total combined rate.
|
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aops_997491
|
Someone mentioned dot product...
$ \begin{align*}\bold u \cdot \bold v &\equal{} \|{\bold u}\| \|\bold v\| \cos \theta \\
(1, 1, 1) \cdot \bold i &\equal{} \|(1, 1, 1)\| \|(1, 0, 0)\| \cos \theta \\
1 &\equal{} \sqrt 3 \cos \theta \\
\cos \theta &\equal{} \frac 1{\sqrt 3} \\
\theta &\equal{} \cos^{\minus{}1} \frac 1{\sqrt 3} \approx 54.74^\circ\end{align*}$
Hm, who votes I should put a harder problem?
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let line $ AB$ be represented by the function $ z \\equal{} x \\equal{} y$ in 3-space. What is the angle line $ AB$ makes with any one of the coordinate axes? Express your answer in degrees to the nearest hundredth.\r\n\r\n[hide=\"Easier problem\"]\nLet line $ AB$ be represented by the function $ y \\equal{} x$ in 2-space. What is the angle line $ AB$ makes with any one of the coordinate axes? Express your answer in degrees to the nearest... uh... whole number?\n[/hide]",
"content_html": "Let line <img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" > be represented by the function <img src=\"//latex.artofproblemsolving.com/b/6/b/b6bb1cc1f43c7aca83a2df79ab79946def9627cc.png\" class=\"latex\" alt=\"$ z = x = y$\" style=\"vertical-align: -3px\" width=\"77\" height=\"11\" > in 3-space. What is the angle line <img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" > makes with any one of the coordinate axes? Express your answer in degrees to the nearest hundredth.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Easier problem</a><div class=\"cmty-hide-content\" style=\"display:none\">Let line <img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" > be represented by the function <img src=\"//latex.artofproblemsolving.com/6/7/9/6797d933bb7d43ea7757b68d02a2ca08632bf1e4.png\" class=\"latex\" alt=\"$ y = x$\" style=\"vertical-align: -3px\" width=\"44\" height=\"11\" > in 2-space. What is the angle line <img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" > makes with any one of the coordinate axes? Express your answer in degrees to the nearest... uh... whole number?</div>",
"post_id": 4417715,
"post_number": 1,
"post_time_unix": 1221697977,
"post_time_utc": "2008-09-18 00:32:57 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "It's $ 54.74$ degrees right?\r\n\r\nSince with this angle, a point $ (0.57735,0.57735,0.57735)$ is exactly $ 1$ unit away from the origin.",
"content_html": "It's <img src=\"//latex.artofproblemsolving.com/7/b/f/7bf035eddc5f4b0f29ce28a22ea7766c7cd1d2bb.png\" class=\"latex\" alt=\"$ 54.74$\" style=\"vertical-align: 0px\" width=\"41\" height=\"13\" > degrees right?<br>\n<br>\nSince with this angle, a point <img src=\"//latex.artofproblemsolving.com/9/7/5/9757578fecc69a30948e815e0b157a3b3d04765e.png\" class=\"latex\" alt=\"$ (0.57735,0.57735,0.57735)$\" style=\"vertical-align: -4px\" width=\"207\" height=\"18\" > is exactly <img src=\"//latex.artofproblemsolving.com/3/9/0/39064bdd89b3dfa0626ca59d843d926ea072830b.png\" class=\"latex\" alt=\"$ 1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" > unit away from the origin.",
"post_id": 4417716,
"post_number": 2,
"post_time_unix": 1221698634,
"post_time_utc": "2008-09-18 00:43:54 UTC",
"thanks_received": 2,
"user_id": 22481,
"username": "ProtestanT"
},
{
"attachments": [],
"content_bbcode": "NOOOO I CAN'T SOLVE IT *SOB* Oh wait... I can... WHY AM I SO STUPID???\r\n\r\nDraw triangle: $ \\tan^{\\minus{}1} \\sqrt{2}$ DUH",
"content_html": "NOOOO I CAN'T SOLVE IT *SOB* Oh wait... I can... WHY AM I SO STUPID???<br>\n<br>\nDraw triangle: <img src=\"//latex.artofproblemsolving.com/8/c/b/8cb937664cb16666e007a02b8ebc030c0487318e.png\" class=\"latex\" alt=\"$ \\tan^{-1} \\sqrt{2}$\" style=\"vertical-align: -1px\" width=\"72\" height=\"18\" > DUH",
"post_id": 4417717,
"post_number": 3,
"post_time_unix": 1221759988,
"post_time_utc": "2008-09-18 17:46:28 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
},
{
"attachments": [],
"content_bbcode": "just dot product it",
"content_html": "just dot product it",
"post_id": 4417718,
"post_number": 4,
"post_time_unix": 1221765639,
"post_time_utc": "2008-09-18 19:20:39 UTC",
"thanks_received": 2,
"user_id": 34136,
"username": "CatalystOfNostalgia"
},
{
"attachments": [],
"content_bbcode": "What do you get when you cross a cat and a light bulb?\r\n\r\n[hide]$ \\text{Cat Light bulb} \\sin \\theta$[/hide]\r\n\r\nharharhar",
"content_html": "What do you get when you cross a cat and a light bulb?<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/5/d/b/5db92e97945c18c47460858f33e5e634d15f89bb.png\" class=\"latex\" alt=\"$ \\text{Cat Light bulb} \\sin \\theta$\" style=\"vertical-align: -3px\" width=\"145\" height=\"16\" ></div><br>\n<br>\nharharhar",
"post_id": 4417719,
"post_number": 5,
"post_time_unix": 1221766631,
"post_time_utc": "2008-09-18 19:37:11 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
},
{
"attachments": [],
"content_bbcode": "Someone mentioned dot product...\r\n\r\n$ \\begin{align*}\\bold u \\cdot \\bold v &\\equal{} \\|{\\bold u}\\| \\|\\bold v\\| \\cos \\theta \\\\\r\n(1, 1, 1) \\cdot \\bold i &\\equal{} \\|(1, 1, 1)\\| \\|(1, 0, 0)\\| \\cos \\theta \\\\\r\n1 &\\equal{} \\sqrt 3 \\cos \\theta \\\\\r\n\\cos \\theta &\\equal{} \\frac 1{\\sqrt 3} \\\\\r\n\\theta &\\equal{} \\cos^{\\minus{}1} \\frac 1{\\sqrt 3} \\approx 54.74^\\circ\\end{align*}$\r\n\r\nHm, who votes I should put a harder problem?",
"content_html": "Someone mentioned dot product...<br>\n<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align*}\\bold u \\cdot \\bold v &= \\|{\\bold u}\\| \\|\\bold v\\| \\cos \\theta \\\\\n(1, 1, 1) \\cdot \\bold i &= \\|(1, 1, 1)\\| \\|(1, 0, 0)\\| \\cos \\theta \\\\\n1 &= \\sqrt 3 \\cos \\theta \\\\\n\\cos \\theta &= \\frac 1{\\sqrt 3} \\\\\n\\theta &= \\cos^{-1} \\frac 1{\\sqrt 3} \\approx 54.74^\\circ\\end{align*}$</span><br>\n<br>\nHm, who votes I should put a harder problem?",
"post_id": 4417720,
"post_number": 6,
"post_time_unix": 1221769085,
"post_time_utc": "2008-09-18 20:18:05 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I can't imagine why you would use dot products when there is a much simpler way...",
"content_html": "I can't imagine why you would use dot products when there is a much simpler way...",
"post_id": 4417721,
"post_number": 7,
"post_time_unix": 1221787748,
"post_time_utc": "2008-09-19 01:29:08 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
}
],
"source": null
}
|
Let line \(AB\) be represented by the function \(z = x = y\) in \(\mathbb{R}^3\). What is the angle that line \(AB\) makes with any one of the coordinate axes? Express your answer in degrees to the nearest hundredth.
|
[
"/Mathematics/Geometry/CoordinateGeometry/Axis",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/x-Axis",
"/Mathematics/Geometry/CoordinateGeometry/y-Axis",
"/Mathematics/Geometry/CoordinateGeometry/z-Axis",
"/Mathematics/Geometry/LineGeometry/Lines/Line",
"/Mathematics/Geometry/LineGeometry/Lines/Line-LineAngle",
"/Mathematics/Geometry/LineGeometry/Lines/StraightLine",
"/Mathematics/Geometry/Trigonometry/Angles/AcuteAngle",
"/Mathematics/Geometry/Trigonometry/Angles/Angle",
"/Mathematics/Geometry/Trigonometry/Angles/Degree",
"/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry"
] |
Use the dot‑product formula to find the angle between the line’s direction vector (1,1,1) and a coordinate axis unit vector.
|
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1 going from (-,+,+,+) to (+,-,-,-)
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Sorry, this is slightly easier (but not if you hate geometry-combinatorics :D)\r\n\r\nHow many lines $ l$ in 4-space exist such that the angle between $ l$ and the x, y, z, and w axes are all equal?\r\n\r\n[hide=\"The easy version\"]\nHow many lines $ l$ in 3-space exist such that the angle between $ l$ and the x, y, and z axes are all equal?\n[/hide]\n[hide=\"The riduculously easy version\"]\nHow many lines $ l$ in 2-space exist such that the angle between $ l$ and the x and y axes are equal?[/hide]\n[hide=\"Let us not even go there...\"]\nHow many lines $ l$ in 1-space exist such that the angle between $ l$ and the x axis is... equal?[/hide]",
"content_html": "Sorry, this is slightly easier (but not if you hate geometry-combinatorics <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />)<br>\n<br>\nHow many lines <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > in 4-space exist such that the angle between <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > and the x, y, z, and w axes are all equal?<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">The easy version</a><div class=\"cmty-hide-content\" style=\"display:none\">How many lines <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > in 3-space exist such that the angle between <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > and the x, y, and z axes are all equal?</div><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">The riduculously easy version</a><div class=\"cmty-hide-content\" style=\"display:none\">How many lines <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > in 2-space exist such that the angle between <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > and the x and y axes are equal?</div><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Let us not even go there...</a><div class=\"cmty-hide-content\" style=\"display:none\">How many lines <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > in 1-space exist such that the angle between <img src=\"//latex.artofproblemsolving.com/e/5/4/e54bfd68a71029d50fa2beb366f31db782fb0c47.png\" class=\"latex\" alt=\"$ l$\" width=\"5\" height=\"12\" > and the x axis is... equal?</div>",
"post_id": 4417722,
"post_number": 1,
"post_time_unix": 1221769476,
"post_time_utc": "2008-09-18 20:24:36 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "8 lines:\r\n1 going from (-,-,-,-) to (+,+,+,+)\r\n1 going from (-,-,-,+) to (+,+,+,-)\r\n1 going from (-,-,+,-) to (+,+,-,+)\r\n1 going from (-,-,+,+) to (+,+,-,-)\r\n1 going from (-,+,-,-) to (+,-,+,+)\r\n1 going from (-,+,-,+) to (+,-,+,-)\r\n1 going from (-,+,+,-) to (+,-,-,+)\r\n1 going from (-,+,+,+) to (+,-,-,-)",
"content_html": "8 lines:<br>\n1 going from (-,-,-,-) to (+,+,+,+)<br>\n1 going from (-,-,-,+) to (+,+,+,-)<br>\n1 going from (-,-,+,-) to (+,+,-,+)<br>\n1 going from (-,-,+,+) to (+,+,-,-)<br>\n1 going from (-,+,-,-) to (+,-,+,+)<br>\n1 going from (-,+,-,+) to (+,-,+,-)<br>\n1 going from (-,+,+,-) to (+,-,-,+)<br>\n1 going from (-,+,+,+) to (+,-,-,-)",
"post_id": 4417723,
"post_number": 2,
"post_time_unix": 1221787646,
"post_time_utc": "2008-09-19 01:27:26 UTC",
"thanks_received": 2,
"user_id": 22481,
"username": "ProtestanT"
},
{
"attachments": [],
"content_bbcode": "I think it's \"officially\" called (at least this is what it was called in the [i]Hard Problems[/i] (not that I bought it, I just saw an excerpt (nested parentheses! (read GEB by Douglas Hofstadter)))) \"combinatorial geometry\". :)",
"content_html": "I think it's "officially" called (at least this is what it was called in the <i>Hard Problems</i> (not that I bought it, I just saw an excerpt (nested parentheses! (read GEB by Douglas Hofstadter)))) "combinatorial geometry". <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 4417724,
"post_number": 3,
"post_time_unix": 1221787889,
"post_time_utc": "2008-09-19 01:31:29 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
},
{
"attachments": [],
"content_bbcode": "blah not really, combinatorial geometry usually deals with like \"how many configurations\"\r\n\r\nthis falls more into pure combo because there is not all that much geo involved",
"content_html": "blah not really, combinatorial geometry usually deals with like "how many configurations"<br>\n<br>\nthis falls more into pure combo because there is not all that much geo involved",
"post_id": 4417725,
"post_number": 4,
"post_time_unix": 1221790878,
"post_time_utc": "2008-09-19 02:21:18 UTC",
"thanks_received": 2,
"user_id": 18909,
"username": "not_trig"
},
{
"attachments": [],
"content_bbcode": "i agree with not_trig.\r\nGeo-combo is where like you have a problem like \"How many solutions are there for A if the the area of ABCD has to be the same?\" and i just think of it as just simple combo cause sometimes i get overwhelmed by big names :(",
"content_html": "i agree with not_trig.<br>\nGeo-combo is where like you have a problem like "How many solutions are there for A if the the area of ABCD has to be the same?" and i just think of it as just simple combo cause sometimes i get overwhelmed by big names <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 4417726,
"post_number": 5,
"post_time_unix": 1221951666,
"post_time_utc": "2008-09-20 23:01:06 UTC",
"thanks_received": 1,
"user_id": 36435,
"username": "Poincare"
},
{
"attachments": [],
"content_bbcode": "I think we can induct on the number of dimensions.",
"content_html": "I think we can induct on the number of dimensions.",
"post_id": 4417727,
"post_number": 6,
"post_time_unix": 1222058495,
"post_time_utc": "2008-09-22 04:41:35 UTC",
"thanks_received": 1,
"user_id": 37564,
"username": "zephyredx"
},
{
"attachments": [],
"content_bbcode": "Induct? Why would you do that? Observe:\r\nIf we have the vector $ \\bold v$ with components $ (v_1, v_2, v_3, \\dots, v_n)$, orthogonal unit vectors $ \\bold e_1, \\bold e_2, \\dots, \\bold e_n$, and angles $ \\theta_1, \\theta_2, \\dots, \\theta_n$, then our statement is equivalent to\r\n\\[ |\\theta_1| \\equal{} |\\theta_2| \\equal{} \\dots \\equal{} |\\theta_n|\r\n\\]\r\nIn addition, since for any $ \\bold e_i$ and $ \\bold e_j$, $ \\|\\bold e_i\\| \\equal{} \\|\\bold e_j\\|$, we can multiply the above equation with $ \\|\\bold v\\| \\|\\bold e_i\\|$ from $ 1 \\le i \\le n$, we get\r\n\\[ |\\|\\bold v\\| \\|\\bold e_1\\| \\cos \\theta_1| \\equal{} |\\|\\bold v\\| \\|\\bold e_2\\| \\cos \\theta_2| \\equal{} \\dots \\equal{} |\\|\\bold v\\| \\|\\bold e_n\\| \\cos \\theta_n|,\r\n\\]\r\nwhich, by the definition of dot product, is equivalent to\r\n\\[ |\\bold v \\cdot \\bold e_1| \\equal{} |\\bold v \\cdot \\bold e_2| \\equal{} \\dots \\equal{} |\\bold v \\cdot \\bold e_n|\r\n\\]\r\nHowever, by taking the dot product with the orthogonal unit vectors $ \\bold e_1 \\equal{} (1, 0, 0, \\dots)$, $ \\bold e_2 \\equal{} (0, 1, 0, \\dots), \\dots$, we obtain the individual components of $ \\bold v$ one-by-one. In other words, the above equation is equivalent to:\r\n\\[ |v_1| \\equal{} |v_2| \\equal{} |v_3| \\equal{} \\dots \\equal{} |v_n|\r\n\\]\r\nso $ v_k \\equal{} c$ or $ v_k \\equal{} \\minus{} c$ for fixed $ c$ and $ 1 \\le k \\le n$.\r\n\r\nNow, since each component of the vector has two possible values, $ c$ or $ \\minus{} c$, it follows that there are $ 2^n$ possible values of $ \\bold v$ that fit the property.\r\n\r\nHowever, the problem asks for the number of [b]lines[/b] that fit the property. Since for any $ \\bold v$, $ \\minus{} \\bold v$ lies on the same line as $ \\bold v$, this essentially halves the possibilities, so the answer is in fact $ \\frac {2^n}2$, or\r\n\\[ 2^{n \\minus{} 1},\r\n\\]\r\nas desired.",
"content_html": "Induct? Why would you do that? Observe:<br>\nIf we have the vector <img src=\"//latex.artofproblemsolving.com/f/c/3/fc3bfe65cd827d821c140f0bc8fe5a05b6caa188.png\" class=\"latex\" alt=\"$ \\bold v$\" width=\"11\" height=\"9\" > with components <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/e/a/aea38ec695ae77d598be5bc3b8fa8a664cb6461a.png\" class=\"latex\" alt=\"$ (v_1, v_2, v_3, \\dots, v_n)$\" style=\"vertical-align: -4px\" width=\"136\" height=\"18\" >,</span> orthogonal unit vectors <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/b/e/1beaf63d829dfa5f6d245aa47303f39029ea03bd.png\" class=\"latex\" alt=\"$ \\bold e_1, \\bold e_2, \\dots, \\bold e_n$\" style=\"vertical-align: -3px\" width=\"100\" height=\"12\" >,</span> and angles <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/b/26bd9ca3baa614d35ce8c5d0904ed9f29a4e5d4c.png\" class=\"latex\" alt=\"$ \\theta_1, \\theta_2, \\dots, \\theta_n$\" style=\"vertical-align: -3px\" width=\"97\" height=\"16\" >,</span> then our statement is equivalent to<br>\n<img src=\"//latex.artofproblemsolving.com/8/b/6/8b69a3b70d92812179c81b586ffd16bc6a3b7568.png\" class=\"latexcenter\" alt=\"\\[ |\\theta_1| = |\\theta_2| = \\dots = |\\theta_n|\n\\]\" width=\"170\" height=\"18\" ><br>\nIn addition, since for any <img src=\"//latex.artofproblemsolving.com/c/4/f/c4f2d2acfa93e0c1e89cb48a29fe6ab93c5421ad.png\" class=\"latex\" alt=\"$ \\bold e_i$\" style=\"vertical-align: -2px\" width=\"14\" height=\"11\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/2/3/c230e6b720e5a115aa46fde18edf9962c6ff8736.png\" class=\"latex\" alt=\"$ \\bold e_j$\" style=\"vertical-align: -4px\" width=\"15\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/4/a/14ab6269f3cc1f2ccb4aa9692f989e425571a112.png\" class=\"latex\" alt=\"$ \\|\\bold e_i\\| = \\|\\bold e_j\\|$\" style=\"vertical-align: -5px\" width=\"90\" height=\"19\" >,</span> we can multiply the above equation with <img src=\"//latex.artofproblemsolving.com/1/a/5/1a5e303badc96d64bd5cbb5cae6362dd4256bae7.png\" class=\"latex\" alt=\"$ \\|\\bold v\\| \\|\\bold e_i\\|$\" style=\"vertical-align: -5px\" width=\"60\" height=\"19\" > from <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/0/0701c3595af80ef98659c75d9a70b881ab5f96b7.png\" class=\"latex\" alt=\"$ 1 \\le i \\le n$\" style=\"vertical-align: -2px\" width=\"74\" height=\"15\" >,</span> we get<br>\n<img src=\"//latex.artofproblemsolving.com/3/9/b/39be173f5bf16db6b065ea7d345d137dcb7865ac.png\" class=\"latexcenter\" alt=\"\\[ |\\|\\bold v\\| \\|\\bold e_1\\| \\cos \\theta_1| = |\\|\\bold v\\| \\|\\bold e_2\\| \\cos \\theta_2| = \\dots = |\\|\\bold v\\| \\|\\bold e_n\\| \\cos \\theta_n|,\n\\]\" width=\"462\" height=\"19\" ><br>\nwhich, by the definition of dot product, is equivalent to<br>\n<img src=\"//latex.artofproblemsolving.com/f/0/f/f0fbc967017785edcddba56334befbe10f087e18.png\" class=\"latexcenter\" alt=\"\\[ |\\bold v \\cdot \\bold e_1| = |\\bold v \\cdot \\bold e_2| = \\dots = |\\bold v \\cdot \\bold e_n|\n\\]\" width=\"246\" height=\"18\" ><br>\nHowever, by taking the dot product with the orthogonal unit vectors <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/6/7/56781fc5e9a2bb3bb193595b264c11b189b63979.png\" class=\"latex\" alt=\"$ \\bold e_1 = (1, 0, 0, \\dots)$\" style=\"vertical-align: -4px\" width=\"129\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/e/f/bef1e5108491f8c61ffa7291dd7b6601c25c84fe.png\" class=\"latex\" alt=\"$ \\bold e_2 = (0, 1, 0, \\dots), \\dots$\" style=\"vertical-align: -4px\" width=\"159\" height=\"18\" >,</span> we obtain the individual components of <img src=\"//latex.artofproblemsolving.com/f/c/3/fc3bfe65cd827d821c140f0bc8fe5a05b6caa188.png\" class=\"latex\" alt=\"$ \\bold v$\" width=\"11\" height=\"9\" > one-by-one. In other words, the above equation is equivalent to:<br>\n<img src=\"//latex.artofproblemsolving.com/f/1/7/f172bc2069bf51069f0cc92836d04f8b06018472.png\" class=\"latexcenter\" alt=\"\\[ |v_1| = |v_2| = |v_3| = \\dots = |v_n|\n\\]\" width=\"221\" height=\"18\" ><br>\nso <img src=\"//latex.artofproblemsolving.com/f/5/c/f5c30a468d66bd425fd802927a46b906a689892b.png\" class=\"latex\" alt=\"$ v_k = c$\" style=\"vertical-align: -2px\" width=\"49\" height=\"10\" > or <img src=\"//latex.artofproblemsolving.com/5/6/a/56a9bed6794ff0f077968049a2f44bc7f8d37e21.png\" class=\"latex\" alt=\"$ v_k = - c$\" style=\"vertical-align: -2px\" width=\"63\" height=\"10\" > for fixed <img src=\"//latex.artofproblemsolving.com/b/1/4/b144c3decf04b3f7a907c07e4f369f1e02bb9adc.png\" class=\"latex\" alt=\"$ c$\" width=\"8\" height=\"8\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/6/5/16542f45cd5541960b722b8a4e5386b350bd3d7a.png\" class=\"latex\" alt=\"$ 1 \\le k \\le n$\" style=\"vertical-align: -2px\" width=\"78\" height=\"15\" >.</span><br>\n<br>\nNow, since each component of the vector has two possible values, <img src=\"//latex.artofproblemsolving.com/b/1/4/b144c3decf04b3f7a907c07e4f369f1e02bb9adc.png\" class=\"latex\" alt=\"$ c$\" width=\"8\" height=\"8\" > or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/e/e/bee97eb9fa1f72c92d069c8ebcb82022e00d984a.png\" class=\"latex\" alt=\"$ - c$\" width=\"22\" height=\"8\" >,</span> it follows that there are <img src=\"//latex.artofproblemsolving.com/7/e/c/7ecb30543eb242134de2b2e378960f5399ed99bf.png\" class=\"latex\" alt=\"$ 2^n$\" width=\"17\" height=\"12\" > possible values of <img src=\"//latex.artofproblemsolving.com/f/c/3/fc3bfe65cd827d821c140f0bc8fe5a05b6caa188.png\" class=\"latex\" alt=\"$ \\bold v$\" width=\"11\" height=\"9\" > that fit the property.<br>\n<br>\nHowever, the problem asks for the number of <b>lines</b> that fit the property. Since for any <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/c/3/fc3bfe65cd827d821c140f0bc8fe5a05b6caa188.png\" class=\"latex\" alt=\"$ \\bold v$\" width=\"11\" height=\"9\" >,</span> <img src=\"//latex.artofproblemsolving.com/5/b/8/5b8efcb28a56b30415c24ef20998634a9f53025f.png\" class=\"latex\" alt=\"$ - \\bold v$\" width=\"25\" height=\"9\" > lies on the same line as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/c/3/fc3bfe65cd827d821c140f0bc8fe5a05b6caa188.png\" class=\"latex\" alt=\"$ \\bold v$\" width=\"11\" height=\"9\" >,</span> this essentially halves the possibilities, so the answer is in fact <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/e/eee530e2758d849fe4939f66425d5028cf67b4d7.png\" class=\"latex\" alt=\"$ \\frac {2^n}2$\" style=\"vertical-align: -12px\" width=\"20\" height=\"37\" >,</span> or<br>\n<img src=\"//latex.artofproblemsolving.com/5/1/f/51f61a1b30c5f9765bc1d0c135764820bfcbe8ad.png\" class=\"latexcenter\" alt=\"\\[ 2^{n - 1},\n\\]\" width=\"38\" height=\"18\" ><br>\nas desired.",
"post_id": 4417728,
"post_number": 7,
"post_time_unix": 1222125138,
"post_time_utc": "2008-09-22 23:12:18 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
How many lines \(l\) in 4-space exist such that the angles between \(l\) and the \(x\)-, \(y\)-, \(z\)-, and \(w\)-axes are all equal?
Easy version: How many lines \(l\) in 3-space exist such that the angles between \(l\) and the \(x\)-, \(y\)-, and \(z\)-axes are all equal?
Ridiculously easy version: How many lines \(l\) in 2-space exist such that the angles between \(l\) and the \(x\)- and \(y\)-axes are equal?
One-dimensional version: How many lines \(l\) in 1-space exist such that the angle between \(l\) and the \(x\)-axis is equal?
|
[
"/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry",
"/Mathematics/Geometry/CoordinateGeometry/Axis",
"/Mathematics/Geometry/CoordinateGeometry/Cartesian",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/CoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/Coordinates",
"/Mathematics/Geometry/LineGeometry/Lines/Line",
"/Mathematics/Geometry/LineGeometry/Lines/Line-LineAngle",
"/Mathematics/Geometry/MultidimensionalGeometry/Four-DimensionalGeometry/4-DGeometry",
"/Mathematics/Geometry/MultidimensionalGeometry/Four-DimensionalGeometry/4-DimensionalGeometry",
"/Mathematics/Geometry/MultidimensionalGeometry/Four-DimensionalGeometry/8-Cell",
"/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry/Hypercube",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Equal angles imply a direction vector whose components have equal magnitude, so count sign choices of ±1 up to opposite direction.
|
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aops_997529
|
So am I supposed to solve this?
[hide="A very cynical solution"]
The shortest path is the path defined by $ P_1P_2P_3\dots P_n$. Assume for the sake of contradiction that that is not the shortest path. Then there must be some point $ A$ which is adjoined to a non-adjacent point $ C$.
[b]Lemma 1.[/b] Any path must have $ n$ segments.
[b]Proof.[/b] Each segment has 1 destination point. There are $ n$ destination points. Therefore, there must be $ n \cdot 1 \equal{} n$ segments.
[b]Lemma 2.[/b] There must exist a segment that intersects with the segment $ AC$.
[b]Proof.[/b] The line $ AC$ divides the polygon into two distinct polygons. There must be a line connecting one point from each of these polygons, otherwise a path would not be able to be constructed, and hence that line intersects with $ AC$.
Call that intersecting line $ BD$.
[b]Lemma 3.[/b] A shorter path can be made by made by interchanging the two intersecting segments $ AC$ and $ BD$ with two nonintersecting segments $ AB$ and $ CD$.
[b]Proof.[/b] Call the point of intersection $ P$. Then by the Triangle inequality:
\[ AB < AP \plus{} BP \\
CD < PC \plus{} PD
\]
But since $ AP \plus{} PC \equal{} AC$ and $ BP \plus{} PD \equal{} BD$, we have:
\[ AB < AC \\
CD < BD
\]
And adding up, $ AB \plus{} CD < AC \plus{} BD$.
So we have shown that any two segments which intersect can be replaced by two shorter segments which don't. If we repeat this method with all such pairs of segments, we will eventually end up with a graph with no intersections. This will be the graph with the shortest length.
But this is just the graph that traces the perimeter of the polygon by Lemma 0, or $ P_1P_2P_3\dots P_n. \blacksquare$
[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Unfortunately, I realized I had posted it before, so I can't use it...\r\n\r\nWe have $ P_1P_2\\ldots P_n$ is a convex polygon. An ant starts at $ P_1.$ He must go to each of the other vertices of the polygon and then return to $ P_1.$ Find, with proof, the shortest path. Write paths like $ P_1 \\minus{} P_5 \\minus{} P_8 \\minus{} P_2 \\minus{} P_3 \\minus{} P_1.$",
"content_html": "Unfortunately, I realized I had posted it before, so I can't use it...<br>\n<br>\nWe have <img src=\"//latex.artofproblemsolving.com/6/1/d/61df1ab1a3d3ee9db1912501e30860cb2b3621e3.png\" class=\"latex\" alt=\"$ P_1P_2\\ldots P_n$\" style=\"vertical-align: -2px\" width=\"85\" height=\"15\" > is a convex polygon. An ant starts at <img src=\"//latex.artofproblemsolving.com/6/f/8/6f8c7224736f6c5cc0fd3f1007a821b92921ae54.png\" class=\"latex\" alt=\"$ P_1.$\" style=\"vertical-align: -2px\" width=\"22\" height=\"15\" > He must go to each of the other vertices of the polygon and then return to <img src=\"//latex.artofproblemsolving.com/6/f/8/6f8c7224736f6c5cc0fd3f1007a821b92921ae54.png\" class=\"latex\" alt=\"$ P_1.$\" style=\"vertical-align: -2px\" width=\"22\" height=\"15\" > Find, with proof, the shortest path. Write paths like <img src=\"//latex.artofproblemsolving.com/d/0/f/d0f0f22e53b77ef6ceef35e89080aa698723de41.png\" class=\"latex\" alt=\"$ P_1 - P_5 - P_8 - P_2 - P_3 - P_1.$\" style=\"vertical-align: -2px\" width=\"229\" height=\"15\" >",
"post_id": 4417869,
"post_number": 1,
"post_time_unix": 1225318934,
"post_time_utc": "2008-10-29 22:22:14 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "So am I supposed to solve this?\r\n\r\n[hide=\"A very cynical solution\"]\nThe shortest path is the path defined by $ P_1P_2P_3\\dots P_n$. Assume for the sake of contradiction that that is not the shortest path. Then there must be some point $ A$ which is adjoined to a non-adjacent point $ C$.\n\n[b]Lemma 1.[/b] Any path must have $ n$ segments.\n[b]Proof.[/b] Each segment has 1 destination point. There are $ n$ destination points. Therefore, there must be $ n \\cdot 1 \\equal{} n$ segments.\n\n[b]Lemma 2.[/b] There must exist a segment that intersects with the segment $ AC$.\n[b]Proof.[/b] The line $ AC$ divides the polygon into two distinct polygons. There must be a line connecting one point from each of these polygons, otherwise a path would not be able to be constructed, and hence that line intersects with $ AC$.\n\nCall that intersecting line $ BD$.\n\n[b]Lemma 3.[/b] A shorter path can be made by made by interchanging the two intersecting segments $ AC$ and $ BD$ with two nonintersecting segments $ AB$ and $ CD$.\n\n[b]Proof.[/b] Call the point of intersection $ P$. Then by the Triangle inequality:\n\\[ AB < AP \\plus{} BP \\\\\nCD < PC \\plus{} PD\n\\]\nBut since $ AP \\plus{} PC \\equal{} AC$ and $ BP \\plus{} PD \\equal{} BD$, we have:\n\\[ AB < AC \\\\\nCD < BD\n\\]\nAnd adding up, $ AB \\plus{} CD < AC \\plus{} BD$.\n\nSo we have shown that any two segments which intersect can be replaced by two shorter segments which don't. If we repeat this method with all such pairs of segments, we will eventually end up with a graph with no intersections. This will be the graph with the shortest length.\n\nBut this is just the graph that traces the perimeter of the polygon by Lemma 0, or $ P_1P_2P_3\\dots P_n. \\blacksquare$\n[/hide]",
"content_html": "So am I supposed to solve this?<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">A very cynical solution</a><div class=\"cmty-hide-content\" style=\"display:none\">The shortest path is the path defined by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/7/837dab36d52af83073c426d1f77bf3ffc2933c60.png\" class=\"latex\" alt=\"$ P_1P_2P_3\\dots P_n$\" style=\"vertical-align: -2px\" width=\"104\" height=\"15\" >.</span> Assume for the sake of contradiction that that is not the shortest path. Then there must be some point <img src=\"//latex.artofproblemsolving.com/a/f/a/afa1e039a54a9d6ce6141d013c997827f98f4add.png\" class=\"latex\" alt=\"$ A$\" width=\"13\" height=\"13\" > which is adjoined to a non-adjacent point <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" >.</span><br>\n<br>\n<b>Lemma 1.</b> Any path must have <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > segments.<br>\n<b>Proof.</b> Each segment has 1 destination point. There are <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > destination points. Therefore, there must be <img src=\"//latex.artofproblemsolving.com/d/f/1/df143b2bfc8f67591b499ffe71cc2d135069f420.png\" class=\"latex\" alt=\"$ n \\cdot 1 = n$\" style=\"vertical-align: 0px\" width=\"68\" height=\"12\" > segments.<br>\n<br>\n<b>Lemma 2.</b> There must exist a segment that intersects with the segment <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/5/0756a0f5f7effcf224e6e829c46c2db34927e86f.png\" class=\"latex\" alt=\"$ AC$\" width=\"27\" height=\"13\" >.</span><br>\n<b>Proof.</b> The line <img src=\"//latex.artofproblemsolving.com/0/7/5/0756a0f5f7effcf224e6e829c46c2db34927e86f.png\" class=\"latex\" alt=\"$ AC$\" width=\"27\" height=\"13\" > divides the polygon into two distinct polygons. There must be a line connecting one point from each of these polygons, otherwise a path would not be able to be constructed, and hence that line intersects with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/5/0756a0f5f7effcf224e6e829c46c2db34927e86f.png\" class=\"latex\" alt=\"$ AC$\" width=\"27\" height=\"13\" >.</span><br>\n<br>\nCall that intersecting line <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/f/0/8f060a3b3720e24643ce978a77df88b8d425f0e4.png\" class=\"latex\" alt=\"$ BD$\" width=\"29\" height=\"12\" >.</span><br>\n<br>\n<b>Lemma 3.</b> A shorter path can be made by made by interchanging the two intersecting segments <img src=\"//latex.artofproblemsolving.com/0/7/5/0756a0f5f7effcf224e6e829c46c2db34927e86f.png\" class=\"latex\" alt=\"$ AC$\" width=\"27\" height=\"13\" > and <img src=\"//latex.artofproblemsolving.com/8/f/0/8f060a3b3720e24643ce978a77df88b8d425f0e4.png\" class=\"latex\" alt=\"$ BD$\" width=\"29\" height=\"12\" > with two nonintersecting segments <img src=\"//latex.artofproblemsolving.com/c/2/c/c2c1015b840bca4492903e3615afa5eee7cef242.png\" class=\"latex\" alt=\"$ AB$\" width=\"27\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/4/ba442f9231826f849adfd6b270c42a50da61f82d.png\" class=\"latex\" alt=\"$ CD$\" width=\"29\" height=\"12\" >.</span><br>\n<br>\n<b>Proof.</b> Call the point of intersection <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/b/1/fb1b7d554ea7d9094511c084a2682639363022f5.png\" class=\"latex\" alt=\"$ P$\" width=\"14\" height=\"12\" >.</span> Then by the Triangle inequality:<br>\n<img src=\"//latex.artofproblemsolving.com/d/b/4/db43cf99fb06944621466c60fcd94767da7c1713.png\" class=\"latexcenter\" alt=\"\\[ AB < AP + BP \\\\\nCD < PC + PD\n\\]\" width=\"265\" height=\"14\" ><br>\nBut since <img src=\"//latex.artofproblemsolving.com/0/6/2/062b696f3f1aaa7bd930afd17090062a162db5c7.png\" class=\"latex\" alt=\"$ AP + PC = AC$\" style=\"vertical-align: -1px\" width=\"130\" height=\"14\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/3/4/f34ff7626ce7c0dce1bd7232e7218b5c0b79b8e3.png\" class=\"latex\" alt=\"$ BP + PD = BD$\" style=\"vertical-align: -1px\" width=\"135\" height=\"14\" >,</span> we have:<br>\n<img src=\"//latex.artofproblemsolving.com/8/b/f/8bfaac67306927201afb45589acef84b5eaa4d55.png\" class=\"latexcenter\" alt=\"\\[ AB < AC \\\\\nCD < BD\n\\]\" width=\"164\" height=\"13\" ><br>\nAnd adding up, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/2/b/c2bdf05ea3c0b9710c2d006db72b3b5a7c58e8ab.png\" class=\"latex\" alt=\"$ AB + CD < AC + BD$\" style=\"vertical-align: -1px\" width=\"184\" height=\"14\" >.</span><br>\n<br>\nSo we have shown that any two segments which intersect can be replaced by two shorter segments which don't. If we repeat this method with all such pairs of segments, we will eventually end up with a graph with no intersections. This will be the graph with the shortest length.<br>\n<br>\nBut this is just the graph that traces the perimeter of the polygon by Lemma 0, or <img src=\"//latex.artofproblemsolving.com/d/f/9/df98caf84043f4a7d30710e70cdb6be5d7cfbc25.png\" class=\"latex\" alt=\"$ P_1P_2P_3\\dots P_n. \\blacksquare$\" style=\"vertical-align: -2px\" width=\"123\" height=\"15\" ></div>",
"post_id": 4417870,
"post_number": 2,
"post_time_unix": 1225322456,
"post_time_utc": "2008-10-29 23:20:56 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
Let \(P_1P_2\ldots P_n\) be a convex polygon. An ant starts at \(P_1\). He must visit each of the other vertices of the polygon and then return to \(P_1\). Find, with proof, the shortest possible path. Write paths in the form \(P_1-P_5-P_8-P_2-P_3-P_1\), etc.
|
[
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/GraphTheory/Paths",
"/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional",
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry",
"/Mathematics/Geometry/PlaneGeometry/Polygons/ConvexPolygon",
"/Mathematics/Geometry/PlaneGeometry/Polygons/SimplePolygon",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Eliminate crossing edges by swapping intersecting segments for the two non‑crossing sides, which shortens the tour.
|
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aops_997543
|
Yes it seems to be..
The sum equals $ (\frac{1}{2} \plus{} \frac{1}{4} \plus{} \frac{1}{8} \ldots) \plus{} (\frac{1}{4} \plus{} \frac{1}{8} \plus{} \frac {1}{16} \ldots) \plus{} \ldots \equal{} 1 \plus{} \frac {1}{2} \plus{} \frac {1}{4} \plus{} \ldots \equal{} 2$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Difficulty: 1\r\n\r\n$ \\sum_{n \\equal{} 1}^\\infty \\frac n{2^n}$\r\n\r\nIn other news... Audacity + La Campanella = :D",
"content_html": "Difficulty: 1<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/d/5/bd5c6f02df46111dc23ac97b2e895ebf924c6146.png\" class=\"latex\" alt=\"$ \\sum_{n = 1}^\\infty \\frac n{2^n}$\" style=\"vertical-align: -20px\" width=\"49\" height=\"48\" ><br>\n<br>\nIn other news... Audacity + La Campanella = <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4417913,
"post_number": 1,
"post_time_unix": 1225904668,
"post_time_utc": "2008-11-05 17:04:28 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I got 2....is that right?",
"content_html": "I got 2....is that right?",
"post_id": 4417914,
"post_number": 2,
"post_time_unix": 1225928512,
"post_time_utc": "2008-11-05 23:41:52 UTC",
"thanks_received": 2,
"user_id": 35887,
"username": "shentang"
},
{
"attachments": [],
"content_bbcode": "Yes it seems to be..\r\n\r\nThe sum equals $ (\\frac{1}{2} \\plus{} \\frac{1}{4} \\plus{} \\frac{1}{8} \\ldots) \\plus{} (\\frac{1}{4} \\plus{} \\frac{1}{8} \\plus{} \\frac {1}{16} \\ldots) \\plus{} \\ldots \\equal{} 1 \\plus{} \\frac {1}{2} \\plus{} \\frac {1}{4} \\plus{} \\ldots \\equal{} 2$",
"content_html": "Yes it seems to be..<br>\n<br>\nThe sum equals <img src=\"//latex.artofproblemsolving.com/8/0/2/8027cd3836dc18c9eb3dc84d1be6df68fbab66dc.png\" class=\"latex\" alt=\"$ (\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} \\ldots) + (\\frac{1}{4} + \\frac{1}{8} + \\frac {1}{16} \\ldots) + \\ldots = 1 + \\frac {1}{2} + \\frac {1}{4} + \\ldots = 2$\" style=\"vertical-align: -13px\" width=\"499\" height=\"37\" >",
"post_id": 4417915,
"post_number": 3,
"post_time_unix": 1225929995,
"post_time_utc": "2008-11-06 00:06:35 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "Very clever!!",
"content_html": "Very clever!!",
"post_id": 4417916,
"post_number": 4,
"post_time_unix": 1225933150,
"post_time_utc": "2008-11-06 00:59:10 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Yes, clever, different from the solution I had in mind (set equal to $ s$, compute $ 2s$, compute $ 2s\\minus{}s$, or something like that...)",
"content_html": "Yes, clever, different from the solution I had in mind (set equal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/b/e/dbeca274183b6a731311e1dcf290ef519365dee8.png\" class=\"latex\" alt=\"$ s$\" width=\"8\" height=\"8\" >,</span> compute <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/f/24fc2c08f088c70b6e188d3db180d23f108d2cff.png\" class=\"latex\" alt=\"$ 2s$\" width=\"17\" height=\"12\" >,</span> compute <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/9/ff917039f9bba869a3ff99df9af081a95b18d504.png\" class=\"latex\" alt=\"$ 2s-s$\" width=\"48\" height=\"12\" >,</span> or something like that...)",
"post_id": 4417917,
"post_number": 5,
"post_time_unix": 1226023464,
"post_time_utc": "2008-11-07 02:04:24 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
}
],
"source": null
}
|
Difficulty: 1
Evaluate the series:
\[
\sum_{n=1}^\infty \frac{n}{2^n}.
\]
|
[
"/Mathematics/CalculusandAnalysis/Series/GeneralSeries"
] |
Rewrite the arithmetico‑geometric series as a sum of shifted geometric series to evaluate it.
|
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aops_997563
|
Consider $ z\equal{}a\plus{}bi$. Then
$ z^z\equal{}(a\plus{}bi)^{a\plus{}bi}\\
\equal{}(a\plus{}bi)^{a}\cdot(a\plus{}bi)^{bi}\\
\equal{}\left(\displaystyle\sum^{a}_{j\equal{}0}\binom{a}{j}a^ji^{a\minus{}j}b^{a\minus{}j}\right)\cdot e^{\ln (a\plus{}bi)bi}\\
\equal{}\left(\displaystyle\sum^{a}_{j\equal{}0}\binom{a}{j}a^ji^{a\minus{}j}b^{a\minus{}j}\right)\cdot e^{(i\tan^{\minus{}1}(\frac{b}{a}) \plus{}\ln(\sqrt{a^2\plus{}b^2}))bi}\\
\equal{}\left(\displaystyle\sum^{a}_{j\equal{}0}\binom{a}{j}a^ji^{a\minus{}j}b^{a\minus{}j}\right)\cdot e^{\minus{}b\tan^{\minus{}1}(\frac{b}{a}) \plus{}bi\ln(\sqrt{a^2\plus{}b^2})}\\
\equal{}\left(\displaystyle\sum^{a}_{j\equal{}0}\binom{a}{j}a^ji^{a\minus{}j}b^{a\minus{}j}\right)\cdot \frac{e^{bi\ln(\sqrt{a^2\plus{}b^2})}}{e^{b\tan^{\minus{}1}(\frac{b}{a})}}}\\
\equal{}\left(\displaystyle\sum^{a}_{j\equal{}0}\binom{a}{j}a^ji^{a\minus{}j}b^{a\minus{}j}\right)\cdot \frac{\cos (b\ln(\sqrt{a^2\plus{}b^2}))\plus{}i\sin(b\ln(\sqrt{a^2\plus{}b^2}))}{e^{b\tan^{\minus{}1}(\frac{b}{a})}}}$
Horribly complicated!
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find all values of $ z$, $ z \\notin \\mathbb R$, such that $ z^z \\in \\mathbb R$",
"content_html": "Find all values of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/e/eaedd2e10519571924b7b1fd0fe1fbe4dfeecb0e.png\" class=\"latex\" alt=\"$ z$\" width=\"8\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/e/6/3e6800d10d109e78f130510b44f8ecf5ecf7ae55.png\" class=\"latex\" alt=\"$ z \\notin \\mathbb R$\" style=\"vertical-align: -4px\" width=\"44\" height=\"18\" >,</span> such that <img src=\"//latex.artofproblemsolving.com/a/3/5/a354b7e3e84a0d716959b573b7a3da3f8acee4af.png\" class=\"latex\" alt=\"$ z^z \\in \\mathbb R$\" style=\"vertical-align: -1px\" width=\"51\" height=\"13\" >",
"post_id": 4418003,
"post_number": 1,
"post_time_unix": 1228011437,
"post_time_utc": "2008-11-30 02:17:17 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Consider $ z\\equal{}a\\plus{}bi$. Then \r\n$ z^z\\equal{}(a\\plus{}bi)^{a\\plus{}bi}\\\\\r\n\\equal{}(a\\plus{}bi)^{a}\\cdot(a\\plus{}bi)^{bi}\\\\\r\n\\equal{}\\left(\\displaystyle\\sum^{a}_{j\\equal{}0}\\binom{a}{j}a^ji^{a\\minus{}j}b^{a\\minus{}j}\\right)\\cdot e^{\\ln (a\\plus{}bi)bi}\\\\\r\n\\equal{}\\left(\\displaystyle\\sum^{a}_{j\\equal{}0}\\binom{a}{j}a^ji^{a\\minus{}j}b^{a\\minus{}j}\\right)\\cdot e^{(i\\tan^{\\minus{}1}(\\frac{b}{a}) \\plus{}\\ln(\\sqrt{a^2\\plus{}b^2}))bi}\\\\\r\n\\equal{}\\left(\\displaystyle\\sum^{a}_{j\\equal{}0}\\binom{a}{j}a^ji^{a\\minus{}j}b^{a\\minus{}j}\\right)\\cdot e^{\\minus{}b\\tan^{\\minus{}1}(\\frac{b}{a}) \\plus{}bi\\ln(\\sqrt{a^2\\plus{}b^2})}\\\\\r\n\\equal{}\\left(\\displaystyle\\sum^{a}_{j\\equal{}0}\\binom{a}{j}a^ji^{a\\minus{}j}b^{a\\minus{}j}\\right)\\cdot \\frac{e^{bi\\ln(\\sqrt{a^2\\plus{}b^2})}}{e^{b\\tan^{\\minus{}1}(\\frac{b}{a})}}}\\\\\r\n\\equal{}\\left(\\displaystyle\\sum^{a}_{j\\equal{}0}\\binom{a}{j}a^ji^{a\\minus{}j}b^{a\\minus{}j}\\right)\\cdot \\frac{\\cos (b\\ln(\\sqrt{a^2\\plus{}b^2}))\\plus{}i\\sin(b\\ln(\\sqrt{a^2\\plus{}b^2}))}{e^{b\\tan^{\\minus{}1}(\\frac{b}{a})}}}$\r\n\r\nHorribly complicated!",
"content_html": "Consider <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/f/8/3f8d788bd0130f4724666d17f2daae145adbfdc9.png\" class=\"latex\" alt=\"$ z=a+bi$\" style=\"vertical-align: -1px\" width=\"79\" height=\"14\" >.</span> Then<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ z^z=(a+bi)^{a+bi}\\\\\n=(a+bi)^{a}\\cdot(a+bi)^{bi}\\\\\n=\\left(\\displaystyle\\sum^{a}_{j=0}\\binom{a}{j}a^ji^{a-j}b^{a-j}\\right)\\cdot e^{\\ln (a+bi)bi}\\\\\n=\\left(\\displaystyle\\sum^{a}_{j=0}\\binom{a}{j}a^ji^{a-j}b^{a-j}\\right)\\cdot e^{(i\\tan^{-1}(\\frac{b}{a}) +\\ln(\\sqrt{a^2+b^2}))bi}\\\\\n=\\left(\\displaystyle\\sum^{a}_{j=0}\\binom{a}{j}a^ji^{a-j}b^{a-j}\\right)\\cdot e^{-b\\tan^{-1}(\\frac{b}{a}) +bi\\ln(\\sqrt{a^2+b^2})}\\\\\n=\\left(\\displaystyle\\sum^{a}_{j=0}\\binom{a}{j}a^ji^{a-j}b^{a-j}\\right)\\cdot \\frac{e^{bi\\ln(\\sqrt{a^2+b^2})}}{e^{b\\tan^{-1}(\\frac{b}{a})}}}\\\\\n=\\left(\\displaystyle\\sum^{a}_{j=0}\\binom{a}{j}a^ji^{a-j}b^{a-j}\\right)\\cdot \\frac{\\cos (b\\ln(\\sqrt{a^2+b^2}))+i\\sin(b\\ln(\\sqrt{a^2+b^2}))}{e^{b\\tan^{-1}(\\frac{b}{a})}}}$</span><br>\n<br>\nHorribly complicated!",
"post_id": 4418004,
"post_number": 2,
"post_time_unix": 1228064717,
"post_time_utc": "2008-11-30 17:05:17 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
},
{
"attachments": [],
"content_bbcode": "what about $ \\mathbb{H}$ and $ \\mathbb{O}$? :wink:",
"content_html": "what about <img src=\"//latex.artofproblemsolving.com/e/0/a/e0ad3fe8e788689e4171503d440e998854b5d02d.png\" class=\"latex\" alt=\"$ \\mathbb{H}$\" width=\"13\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/2/8429383c68bcfef3643761c38589ed10eae4379f.png\" class=\"latex\" alt=\"$ \\mathbb{O}$\" style=\"vertical-align: 0px\" width=\"13\" height=\"13\" >?</span> <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />",
"post_id": 4418005,
"post_number": 3,
"post_time_unix": 1228081417,
"post_time_utc": "2008-11-30 21:43:37 UTC",
"thanks_received": 2,
"user_id": 28419,
"username": "Temperal"
},
{
"attachments": [],
"content_bbcode": "As long as $ z^z \\in \\mathbb R$ :D",
"content_html": "As long as <img src=\"//latex.artofproblemsolving.com/a/3/5/a354b7e3e84a0d716959b573b7a3da3f8acee4af.png\" class=\"latex\" alt=\"$ z^z \\in \\mathbb R$\" style=\"vertical-align: -1px\" width=\"51\" height=\"13\" > <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4418006,
"post_number": 4,
"post_time_unix": 1228086474,
"post_time_utc": "2008-11-30 23:07:54 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Yeah, I thought about including the bit where $ z \\not\\in \\mathbb{C}$, but decided that I wouldn't, because I don't know how to exponentiate on non-complex numbers.",
"content_html": "Yeah, I thought about including the bit where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/3/6/9364f5c85d4b4e2759686f4588014c7b920d0fe8.png\" class=\"latex\" alt=\"$ z \\not\\in \\mathbb{C}$\" style=\"vertical-align: -4px\" width=\"44\" height=\"17\" >,</span> but decided that I wouldn't, because I don't know how to exponentiate on non-complex numbers.",
"post_id": 4418007,
"post_number": 5,
"post_time_unix": 1228090259,
"post_time_utc": "2008-12-01 00:10:59 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
}
],
"source": null
}
|
Find all values of \(z\), \(z\notin\mathbb{R}\), such that \(z^z\in\mathbb{R}\).
|
[
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Polynomials/ComplexVariable"
] |
Express z as re^{iθ} and require the imaginary part of z·Log z to be an integer multiple of π for z^z to be real.
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aops_99757
|
$\frac{BM}{CM}=\frac{\triangle{ABM}}{\triangle{ACM}}=\frac{AB \cdot AM \cdot \sin{\alpha}}{AC \cdot AM \cdot \sin{\beta}}=\frac{AB \cdot \sin{\alpha}}{AC \cdot \sin{\beta}}$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Source: 360 PMC\r\nOn side $BC$ of a triangle $ABC$ consider points $M$ and $N$ such that $\\angle BAM=\\angle CAN$. Prove that\r\n$\\frac{MB}{MC}+\\frac{NB}{NC}\\geq 2\\frac{AB}{AC}$.\r\n\r\nfirst, on the book, it has some symbol above $BAM$ and $CAN$, it looks like a long hat...does this mean angle $BAM$ and angle $CAN$? I assumed it means angle on the above problem..",
"content_html": "Source: 360 PMC<br>\nOn side <img src=\"//latex.artofproblemsolving.com/6/c/5/6c52a41dcbd739f1d026c5d4f181438b75b76976.png\" class=\"latex\" alt=\"$BC$\" width=\"28\" height=\"12\" > of a triangle <img src=\"//latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > consider points <img src=\"//latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png\" class=\"latex\" alt=\"$M$\" width=\"19\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png\" class=\"latex\" alt=\"$N$\" width=\"16\" height=\"12\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/5/a/a5af701b9c1de5e09560a74208458c02c003b10e.png\" class=\"latex\" alt=\"$\\angle BAM=\\angle CAN$\" width=\"142\" height=\"13\" >.</span> Prove that<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/4/b/b4bd2ceabea85c949c5070bb7a5441bcd92fd802.png\" class=\"latex\" alt=\"$\\frac{MB}{MC}+\\frac{NB}{NC}\\geq 2\\frac{AB}{AC}$\" style=\"vertical-align: -12px\" width=\"159\" height=\"38\" >.</span><br>\n<br>\nfirst, on the book, it has some symbol above <img src=\"//latex.artofproblemsolving.com/c/5/d/c5de10b32441ec05aa270d8711dc2109a64455f4.png\" class=\"latex\" alt=\"$BAM$\" width=\"47\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/d/6/7d617251cc9cdaae2828921aac96aa645fbbce0b.png\" class=\"latex\" alt=\"$CAN$\" width=\"44\" height=\"13\" >,</span> it looks like a long hat...does this mean angle <img src=\"//latex.artofproblemsolving.com/c/5/d/c5de10b32441ec05aa270d8711dc2109a64455f4.png\" class=\"latex\" alt=\"$BAM$\" width=\"47\" height=\"13\" > and angle <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/d/6/7d617251cc9cdaae2828921aac96aa645fbbce0b.png\" class=\"latex\" alt=\"$CAN$\" width=\"44\" height=\"13\" >?</span> I assumed it means angle on the above problem..",
"post_id": 563374,
"post_number": 1,
"post_time_unix": 1151728076,
"post_time_utc": "2006-07-01 04:27:56 UTC",
"thanks_received": 2,
"user_id": 9197,
"username": "kimby_102"
},
{
"attachments": [],
"content_bbcode": "By mistake you have posted this twice.",
"content_html": "By mistake you have posted this twice.",
"post_id": 563501,
"post_number": 2,
"post_time_unix": 1151751240,
"post_time_utc": "2006-07-01 10:54:00 UTC",
"thanks_received": 2,
"user_id": 17435,
"username": "ashwinrk_jain"
},
{
"attachments": [],
"content_bbcode": "Let $\\angle{BAM}=\\angle{CAN}=\\alpha$ and $\\angle{BAN}=\\angle{CAM}= \\beta$. Now use the fact that: $\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$ and $\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$, we get: \\[\\frac{BM}{CM}+\\frac{BN}{CN}=\\frac{AB}{AC}\\left( \\frac{\\sin{\\alpha}}{\\sin{\\beta}}+\\frac{\\sin{\\beta}}{\\sin{\\alpha}}\\right) \\geq 2 \\cdot \\frac{AB}{AC}\\]",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/1/a/3/1a33bff147a4228913b38a08cff69ba40275565d.png\" class=\"latex\" alt=\"$\\angle{BAM}=\\angle{CAN}=\\alpha$\" width=\"177\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/a/3/9a371f4a040a91b15003f94fd5b54a073434459b.png\" class=\"latex\" alt=\"$\\angle{BAN}=\\angle{CAM}= \\beta$\" style=\"vertical-align: -3px\" width=\"177\" height=\"16\" >.</span> Now use the fact that: <img src=\"//latex.artofproblemsolving.com/a/f/1/af18d72ad56930fce99175c3a9ecd443bd6b5f28.png\" class=\"latex\" alt=\"$\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$\" style=\"vertical-align: -16px\" width=\"133\" height=\"41\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/4/8e468d2dc6dbbbaced3a5173c5037d346beea4bb.png\" class=\"latex\" alt=\"$\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$\" style=\"vertical-align: -12px\" width=\"130\" height=\"38\" >,</span> we get: <img src=\"//latex.artofproblemsolving.com/8/e/1/8e18be06357e4b1434373982a4106e9b13352ede.png\" class=\"latexcenter\" alt=\"\\[\\frac{BM}{CM}+\\frac{BN}{CN}=\\frac{AB}{AC}\\left( \\frac{\\sin{\\alpha}}{\\sin{\\beta}}+\\frac{\\sin{\\beta}}{\\sin{\\alpha}}\\right) \\geq 2 \\cdot \\frac{AB}{AC}\\]\" width=\"361\" height=\"43\" >",
"post_id": 563517,
"post_number": 3,
"post_time_unix": 1151754284,
"post_time_utc": "2006-07-01 11:44:44 UTC",
"thanks_received": 2,
"user_id": 6601,
"username": "shobber"
},
{
"attachments": [],
"content_bbcode": "[quote=\"shobber\"]Let $\\angle{BAM}=\\angle{CAN}=\\alpha$ and $\\angle{BAN}=\\angle{CAM}= \\beta$. Now use the fact that: $\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$ and $\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$, we get: \\[\\frac{BM}{CM}+\\frac{BN}{CN}=\\frac{AB}{AC}\\left( \\frac{\\sin{\\alpha}}{\\sin{\\beta}}+\\frac{\\sin{\\beta}}{\\sin{\\alpha}}\\right) \\geq 2 \\cdot \\frac{AB}{AC}\\] [/quote]\r\nWait, i still don't understand how u got\r\n$\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$ and $\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$..\r\nIs it Steiner's theorem? (that's what the solution in my book stated, but i don't know what Steiner's theorem is....)\r\nThanks,",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">shobber wrote:</div>\n<div class=\"bbcode_quote_body\">Let <img src=\"//latex.artofproblemsolving.com/1/a/3/1a33bff147a4228913b38a08cff69ba40275565d.png\" class=\"latex\" alt=\"$\\angle{BAM}=\\angle{CAN}=\\alpha$\" width=\"177\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/a/3/9a371f4a040a91b15003f94fd5b54a073434459b.png\" class=\"latex\" alt=\"$\\angle{BAN}=\\angle{CAM}= \\beta$\" style=\"vertical-align: -3px\" width=\"177\" height=\"16\" >.</span> Now use the fact that: <img src=\"//latex.artofproblemsolving.com/a/f/1/af18d72ad56930fce99175c3a9ecd443bd6b5f28.png\" class=\"latex\" alt=\"$\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$\" style=\"vertical-align: -16px\" width=\"133\" height=\"41\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/4/8e468d2dc6dbbbaced3a5173c5037d346beea4bb.png\" class=\"latex\" alt=\"$\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$\" style=\"vertical-align: -12px\" width=\"130\" height=\"38\" >,</span> we get: <img src=\"//latex.artofproblemsolving.com/8/e/1/8e18be06357e4b1434373982a4106e9b13352ede.png\" class=\"latexcenter\" alt=\"\\[\\frac{BM}{CM}+\\frac{BN}{CN}=\\frac{AB}{AC}\\left( \\frac{\\sin{\\alpha}}{\\sin{\\beta}}+\\frac{\\sin{\\beta}}{\\sin{\\alpha}}\\right) \\geq 2 \\cdot \\frac{AB}{AC}\\]\" width=\"361\" height=\"43\" ></div>\n</div>\nWait, i still don't understand how u got<br>\n<img src=\"//latex.artofproblemsolving.com/a/f/1/af18d72ad56930fce99175c3a9ecd443bd6b5f28.png\" class=\"latex\" alt=\"$\\frac{BM}{CM}=\\frac{AB \\sin{\\alpha}}{AC \\sin{\\beta}}$\" style=\"vertical-align: -16px\" width=\"133\" height=\"41\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/e/4/8e468d2dc6dbbbaced3a5173c5037d346beea4bb.png\" class=\"latex\" alt=\"$\\frac{BN}{CN}=\\frac{AB\\sin{\\beta}}{AC \\sin{\\alpha}}$\" style=\"vertical-align: -12px\" width=\"130\" height=\"38\" >.</span>.<br>\nIs it Steiner's theorem? (that's what the solution in my book stated, but i don't know what Steiner's theorem is....)<br>\nThanks,",
"post_id": 563867,
"post_number": 4,
"post_time_unix": 1151779125,
"post_time_utc": "2006-07-01 18:38:45 UTC",
"thanks_received": 2,
"user_id": 9197,
"username": "kimby_102"
},
{
"attachments": [],
"content_bbcode": "$\\frac{BM}{CM}=\\frac{\\triangle{ABM}}{\\triangle{ACM}}=\\frac{AB \\cdot AM \\cdot \\sin{\\alpha}}{AC \\cdot AM \\cdot \\sin{\\beta}}=\\frac{AB \\cdot \\sin{\\alpha}}{AC \\cdot \\sin{\\beta}}$",
"content_html": "<img src=\"//latex.artofproblemsolving.com/9/e/7/9e7faee8033ece34839c02dc658531103b20ec09.png\" class=\"latex\" alt=\"$\\frac{BM}{CM}=\\frac{\\triangle{ABM}}{\\triangle{ACM}}=\\frac{AB \\cdot AM \\cdot \\sin{\\alpha}}{AC \\cdot AM \\cdot \\sin{\\beta}}=\\frac{AB \\cdot \\sin{\\alpha}}{AC \\cdot \\sin{\\beta}}$\" style=\"vertical-align: -16px\" width=\"389\" height=\"42\" >",
"post_id": 564250,
"post_number": 5,
"post_time_unix": 1151810640,
"post_time_utc": "2006-07-02 03:24:00 UTC",
"thanks_received": 3,
"user_id": 6601,
"username": "shobber"
}
],
"source": null
}
|
On side \(BC\) of a triangle \(ABC\) consider points \(M\) and \(N\) such that \(\angle BAM=\angle CAN\). Prove that
\[
\frac{MB}{MC}+\frac{NB}{NC}\ge 2\frac{AB}{AC}.
\]
|
[
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry",
"/Mathematics/Geometry/GeometricInequalities",
"/Mathematics/Geometry/PlaneGeometry/Triangles/Cevians",
"/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleLines",
"/Mathematics/Geometry/Trigonometry/Angles/Angle",
"/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry",
"/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/LawofSines",
"/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine"
] |
Express the segment ratios on BC as ratios of triangle areas, which become side‑length times sine of the equal angles, allowing comparison via the given angle condition.
|
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aops_997581
|
194
1. 97
2. 292
3. 146
4. 73
5. 220
6. 110
7. 55
8. 166
9. 83
10. 250
11. 125
12. 376
13. 188
14. 94
15. 47
16. 142
17. 71
18. 214
19. 107
20. 322
21. 161
22. 484
23. 242
24. 121
25. 364
26. 182
27. 91
28. 274
29. 137
30. 412
31. 206
32. 103
33. 310
34. 155
35. 466
36. 233
37. 700
38. 350
39. 175
40. 526
41. 263
42. 780
43. 490
44. 245
45. 736
46. 368
47. 184
48. 92
49. 46
50. 23
51. 70
52. 35
53. 106
54. 53
55. 160
56. 80
57. 40
58. 20
59. 10
60. 5
61. 16
62. 8
63. 4
64. 2
65. 1
For anyone who wants to know, I didn't use a calculator or computer to do that.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Define $ f(x)$ to be the following:\r\n\\[ f(x) \\equal{} \\begin{cases} x/2 & , 2\\ |\\ x \\\\\r\n3x \\plus{} 1 & , 2\\not|\\ x\\end{cases}\r\n\\]\r\nMoreover, denote $ f^n(x)$ as:\r\n$ f^0(x) \\equal{} x, f^n(x) \\equal{} f(f^{n \\minus{} 1}(x))$, for $ n \\in \\mathbb Z^ \\plus{}$.\r\n\r\nDetermine the minimum value of $ n$ such that $ f^n(194)\\equal{}1$.",
"content_html": "Define <img src=\"//latex.artofproblemsolving.com/9/3/1/93151cc6bf9af56ebcb1b6e891401f06a07aefc8.png\" class=\"latex\" alt=\"$ f(x)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" > to be the following:<br>\n<img src=\"//latex.artofproblemsolving.com/5/0/8/508ae31756f030d608b3b6481eaecf4138e1590e.png\" class=\"latexcenter\" alt=\"\\[ f(x) = \\begin{cases} x/2 & , 2\\ |\\ x \\\\\n3x + 1 & , 2\\not|\\ x\\end{cases}\n\\]\" width=\"189\" height=\"53\" ><br>\nMoreover, denote <img src=\"//latex.artofproblemsolving.com/2/7/b/27bbdb994f0bbae505adc9ada94bcaf2a5d1bfca.png\" class=\"latex\" alt=\"$ f^n(x)$\" style=\"vertical-align: -4px\" width=\"43\" height=\"18\" > as:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/a/b/3abef1508b242d20885849334f638503df2ce680.png\" class=\"latex\" alt=\"$ f^0(x) = x, f^n(x) = f(f^{n - 1}(x))$\" style=\"vertical-align: -4px\" width=\"239\" height=\"19\" >,</span> for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/9/0/9902cb16fce9058bb390799b9e8719f25d9992b6.png\" class=\"latex\" alt=\"$ n \\in \\mathbb Z^ +$\" style=\"vertical-align: -1px\" width=\"54\" height=\"15\" >.</span><br>\n<br>\nDetermine the minimum value of <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/b/b/dbb2dba56a5dcad6a28e7ed0567190ee34efec1c.png\" class=\"latex\" alt=\"$ f^n(194)=1$\" style=\"vertical-align: -4px\" width=\"93\" height=\"18\" >.</span>",
"post_id": 4418063,
"post_number": 1,
"post_time_unix": 1229302428,
"post_time_utc": "2008-12-15 00:53:48 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Collatz Conjecture... sort of...",
"content_html": "Collatz Conjecture... sort of...",
"post_id": 4418064,
"post_number": 2,
"post_time_unix": 1229314306,
"post_time_utc": "2008-12-15 04:11:46 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "Yes, I know it's sort of, but the problem is not the conjecture :D\r\nIt's to find the number of recursions so that $ f^n(194)$ will reach 1.",
"content_html": "Yes, I know it's sort of, but the problem is not the conjecture <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\nIt's to find the number of recursions so that <img src=\"//latex.artofproblemsolving.com/1/7/3/1738a0dbff91ee512e32690d9bbe16ebceb3ea93.png\" class=\"latex\" alt=\"$ f^n(194)$\" style=\"vertical-align: -4px\" width=\"59\" height=\"18\" > will reach 1.",
"post_id": 4418065,
"post_number": 3,
"post_time_unix": 1229357013,
"post_time_utc": "2008-12-15 16:03:33 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "194\r\n\r\n1. 97\r\n2. 292\r\n3. 146\r\n4. 73\r\n5. 220\r\n6. 110\r\n7. 55\r\n8. 166\r\n9. 83\r\n10. 250\r\n11. 125\r\n12. 376\r\n13. 188\r\n14. 94\r\n15. 47\r\n16. 142\r\n17. 71\r\n18. 214\r\n19. 107\r\n20. 322\r\n21. 161\r\n22. 484\r\n23. 242\r\n24. 121\r\n25. 364\r\n26. 182\r\n27. 91\r\n28. 274\r\n29. 137\r\n30. 412\r\n31. 206\r\n32. 103\r\n33. 310\r\n34. 155\r\n35. 466\r\n36. 233\r\n37. 700\r\n38. 350\r\n39. 175\r\n40. 526\r\n41. 263\r\n42. 780\r\n43. 490\r\n44. 245\r\n45. 736\r\n46. 368\r\n47. 184\r\n48. 92\r\n49. 46\r\n50. 23\r\n51. 70\r\n52. 35\r\n53. 106\r\n54. 53\r\n55. 160\r\n56. 80\r\n57. 40\r\n58. 20\r\n59. 10\r\n60. 5\r\n61. 16\r\n62. 8\r\n63. 4\r\n64. 2\r\n65. 1\r\n\r\nFor anyone who wants to know, I didn't use a calculator or computer to do that.",
"content_html": "194<br>\n<br>\n1. 97<br>\n2. 292<br>\n3. 146<br>\n4. 73<br>\n5. 220<br>\n6. 110<br>\n7. 55<br>\n8. 166<br>\n9. 83<br>\n10. 250<br>\n11. 125<br>\n12. 376<br>\n13. 188<br>\n14. 94<br>\n15. 47<br>\n16. 142<br>\n17. 71<br>\n18. 214<br>\n19. 107<br>\n20. 322<br>\n21. 161<br>\n22. 484<br>\n23. 242<br>\n24. 121<br>\n25. 364<br>\n26. 182<br>\n27. 91<br>\n28. 274<br>\n29. 137<br>\n30. 412<br>\n31. 206<br>\n32. 103<br>\n33. 310<br>\n34. 155<br>\n35. 466<br>\n36. 233<br>\n37. 700<br>\n38. 350<br>\n39. 175<br>\n40. 526<br>\n41. 263<br>\n42. 780<br>\n43. 490<br>\n44. 245<br>\n45. 736<br>\n46. 368<br>\n47. 184<br>\n48. 92<br>\n49. 46<br>\n50. 23<br>\n51. 70<br>\n52. 35<br>\n53. 106<br>\n54. 53<br>\n55. 160<br>\n56. 80<br>\n57. 40<br>\n58. 20<br>\n59. 10<br>\n60. 5<br>\n61. 16<br>\n62. 8<br>\n63. 4<br>\n64. 2<br>\n65. 1<br>\n<br>\nFor anyone who wants to know, I didn't use a calculator or computer to do that.",
"post_id": 4418066,
"post_number": 4,
"post_time_unix": 1229469056,
"post_time_utc": "2008-12-16 23:10:56 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
},
{
"attachments": [],
"content_bbcode": "Seems like I made a mistake between 42 and 43...\r\n\r\nI checked this with a computer, and the real answer is 119. \r\n\r\n[code]1. 97\n2. 292\n3. 146\n4. 73\n5. 220\n6. 110\n7. 55\n8. 166\n9. 83\n10. 250\n11. 125\n12. 376\n13. 188\n14. 94\n15. 47\n16. 142\n17. 71\n18. 214\n19. 107\n20. 322\n21. 161\n22. 484\n23. 242\n24. 121\n25. 364\n26. 182\n27. 91\n28. 274\n29. 137\n30. 412\n31. 206\n32. 103\n33. 310\n34. 155\n35. 466\n36. 233\n37. 700\n38. 350\n39. 175\n40. 526\n41. 263\n42. 790\n43. 395\n44. 1186\n45. 593\n46. 1780\n47. 890\n48. 445\n49. 1336\n50. 668\n51. 334\n52. 167\n53. 502\n54. 251\n55. 754\n56. 377\n57. 1132\n58. 566\n59. 283\n60. 850\n61. 425\n62. 1276\n63. 638\n64. 319\n65. 958\n66. 479\n67. 1438\n68. 719\n69. 2158\n70. 1079\n71. 3238\n72. 1619\n73. 4858\n74. 2429\n75. 7288\n76. 3644\n77. 1822\n78. 911\n79. 2734\n80. 1367\n81. 4102\n82. 2051\n83. 6154\n84. 3077\n85. 9232\n86. 4616\n87. 2308\n88. 1154\n89. 577\n90. 1732\n91. 866\n92. 433\n93. 1300\n94. 650\n95. 325\n96. 976\n97. 488\n98. 244\n99. 122\n100. 61\n101. 184\n102. 92\n103. 46\n104. 23\n105. 70\n106. 35\n107. 106\n108. 53\n109. 160\n110. 80\n111. 40\n112. 20\n113. 10\n114. 5\n115. 16\n116. 8\n117. 4\n118. 2\n119. 1\n119[/code]\r\n\r\nNew problem: Find all values of $ n$ such that $ f^n(n) \\equal{} 1$.\r\n\r\nOh, and also: are there any non-brute-force solutions?",
"content_html": "Seems like I made a mistake between 42 and 43...<br>\n<br>\nI checked this with a computer, and the real answer is 119.<br>\n<br>\n<pre><code>1. 97\n2. 292\n3. 146\n4. 73\n5. 220\n6. 110\n7. 55\n8. 166\n9. 83\n10. 250\n11. 125\n12. 376\n13. 188\n14. 94\n15. 47\n16. 142\n17. 71\n18. 214\n19. 107\n20. 322\n21. 161\n22. 484\n23. 242\n24. 121\n25. 364\n26. 182\n27. 91\n28. 274\n29. 137\n30. 412\n31. 206\n32. 103\n33. 310\n34. 155\n35. 466\n36. 233\n37. 700\n38. 350\n39. 175\n40. 526\n41. 263\n42. 790\n43. 395\n44. 1186\n45. 593\n46. 1780\n47. 890\n48. 445\n49. 1336\n50. 668\n51. 334\n52. 167\n53. 502\n54. 251\n55. 754\n56. 377\n57. 1132\n58. 566\n59. 283\n60. 850\n61. 425\n62. 1276\n63. 638\n64. 319\n65. 958\n66. 479\n67. 1438\n68. 719\n69. 2158\n70. 1079\n71. 3238\n72. 1619\n73. 4858\n74. 2429\n75. 7288\n76. 3644\n77. 1822\n78. 911\n79. 2734\n80. 1367\n81. 4102\n82. 2051\n83. 6154\n84. 3077\n85. 9232\n86. 4616\n87. 2308\n88. 1154\n89. 577\n90. 1732\n91. 866\n92. 433\n93. 1300\n94. 650\n95. 325\n96. 976\n97. 488\n98. 244\n99. 122\n100. 61\n101. 184\n102. 92\n103. 46\n104. 23\n105. 70\n106. 35\n107. 106\n108. 53\n109. 160\n110. 80\n111. 40\n112. 20\n113. 10\n114. 5\n115. 16\n116. 8\n117. 4\n118. 2\n119. 1\n119</code></pre><br>\n<br>\nNew problem: Find all values of <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/1/da16051a87c86627f565f8807ebe046bc041848f.png\" class=\"latex\" alt=\"$ f^n(n) = 1$\" style=\"vertical-align: -4px\" width=\"77\" height=\"18\" >.</span><br>\n<br>\nOh, and also: are there any non-brute-force solutions?",
"post_id": 4418067,
"post_number": 5,
"post_time_unix": 1229469872,
"post_time_utc": "2008-12-16 23:24:32 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
}
],
"source": null
}
|
Define f(x) by
\[
f(x)=\begin{cases}
x/2, & 2\mid x,\\[4pt]
3x+1, & 2\nmid x.
\end{cases}
\]
For n∈\mathbb{Z}_{\ge0} define f^0(x)=x and f^n(x)=f(f^{\,n-1}(x)) for n\ge1.
Determine the minimum n such that f^n(194)=1.
|
[
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence",
"/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic",
"/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision",
"/Mathematics/NumberTheory/GeneralNumberTheory/ComputationalNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/EvenNumber",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/RationalInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/NumberTheoreticFunctions/IntegerFunctions",
"/Mathematics/NumberTheory/Parity/EvenNumber",
"/Mathematics/NumberTheory/Parity/OddNumber",
"/Mathematics/NumberTheory/Sequences/3nPlus1Problem",
"/Mathematics/NumberTheory/Sequences/3xPlus1Mapping",
"/Mathematics/NumberTheory/Sequences/CollatzProblem",
"/Mathematics/NumberTheory/Sequences/HailstoneNumber",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Iteratively apply the piecewise rule based on parity (the Collatz 3x+1 process) until reaching 1.
|
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aops_997582
|
No, much simpler, actually. It's actually just Pythagorean Theorem.
(This solution assumes that $ \cos$ and $ \sin$ use radian measure)
First imagine flattening out the helix.
When $ t$ increases by $ \theta$, the bug travels horizontally $ \theta$ and vertically $ \theta$ for a total distance of $ \sqrt{2}\theta$. So for this distance to equal 10, $ t$ must equal $ 5\sqrt{2}$ for final coordinates of $ (\cos(5\sqrt{2}), \sin(5\sqrt{2}), 5\sqrt{2})$.
No calculus necessary.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "A circular helix is defined by the parametric equations\r\n$ x(t) \\equal{} \\cos t$\r\n$ y(t) \\equal{} \\sin t$\r\n$ z(t) \\equal{} t$\r\nIf a bug starts at $ (1, 0, 0)$ and walks up the helix for a distance of 10 units, what are its final coordinates?",
"content_html": "A circular helix is defined by the parametric equations<br>\n<img src=\"//latex.artofproblemsolving.com/8/3/6/836c3eb35759493db87089fd9ec266b7c0aa10a3.png\" class=\"latex\" alt=\"$ x(t) = \\cos t$\" style=\"vertical-align: -4px\" width=\"88\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/2/f/c2fad4820341df1f6e3b3e221d079d3d808dbb95.png\" class=\"latex\" alt=\"$ y(t) = \\sin t$\" style=\"vertical-align: -4px\" width=\"86\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/a/f/faf0a82232ea5a4e392002078d4650bda444177a.png\" class=\"latex\" alt=\"$ z(t) = t$\" style=\"vertical-align: -4px\" width=\"60\" height=\"18\" ><br>\nIf a bug starts at <img src=\"//latex.artofproblemsolving.com/e/7/7/e775fdef02e0fa2f7c1e7278273d2895ed240830.png\" class=\"latex\" alt=\"$ (1, 0, 0)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" > and walks up the helix for a distance of 10 units, what are its final coordinates?",
"post_id": 4418068,
"post_number": 1,
"post_time_unix": 1229379552,
"post_time_utc": "2008-12-15 22:19:12 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Basically a modified form of 2-variable parametric length.",
"content_html": "Basically a modified form of 2-variable parametric length.",
"post_id": 4418069,
"post_number": 2,
"post_time_unix": 1229382658,
"post_time_utc": "2008-12-15 23:10:58 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "No, much simpler, actually. It's actually just Pythagorean Theorem.\r\n\r\n(This solution assumes that $ \\cos$ and $ \\sin$ use radian measure) \r\nFirst imagine flattening out the helix.\r\n\r\nWhen $ t$ increases by $ \\theta$, the bug travels horizontally $ \\theta$ and vertically $ \\theta$ for a total distance of $ \\sqrt{2}\\theta$. So for this distance to equal 10, $ t$ must equal $ 5\\sqrt{2}$ for final coordinates of $ (\\cos(5\\sqrt{2}), \\sin(5\\sqrt{2}), 5\\sqrt{2})$.\r\n\r\nNo calculus necessary.",
"content_html": "No, much simpler, actually. It's actually just Pythagorean Theorem.<br>\n<br>\n(This solution assumes that <img src=\"//latex.artofproblemsolving.com/d/2/8/d28e19093050d38985df05a9ab63abb2388af0fe.png\" class=\"latex\" alt=\"$ \\cos$\" width=\"24\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/6/6/5/66508fe8c47573778bf51e0a6ac4cf1f88995f8d.png\" class=\"latex\" alt=\"$ \\sin$\" width=\"22\" height=\"12\" > use radian measure)<br>\nFirst imagine flattening out the helix.<br>\n<br>\nWhen <img src=\"//latex.artofproblemsolving.com/d/0/9/d09c2835f4b6988d1b5a8c2e4783372992fc5c99.png\" class=\"latex\" alt=\"$ t$\" width=\"6\" height=\"11\" > increases by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/d/c/cdc87059950c5386ad9d0c2c42c598418eda35d3.png\" class=\"latex\" alt=\"$ \\theta$\" width=\"8\" height=\"13\" >,</span> the bug travels horizontally <img src=\"//latex.artofproblemsolving.com/c/d/c/cdc87059950c5386ad9d0c2c42c598418eda35d3.png\" class=\"latex\" alt=\"$ \\theta$\" width=\"8\" height=\"13\" > and vertically <img src=\"//latex.artofproblemsolving.com/c/d/c/cdc87059950c5386ad9d0c2c42c598418eda35d3.png\" class=\"latex\" alt=\"$ \\theta$\" width=\"8\" height=\"13\" > for a total distance of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/c/a/eca9b67c4b8f2796e07aaaf5adce77bcf953eea4.png\" class=\"latex\" alt=\"$ \\sqrt{2}\\theta$\" style=\"vertical-align: -1px\" width=\"32\" height=\"18\" >.</span> So for this distance to equal 10, <img src=\"//latex.artofproblemsolving.com/d/0/9/d09c2835f4b6988d1b5a8c2e4783372992fc5c99.png\" class=\"latex\" alt=\"$ t$\" width=\"6\" height=\"11\" > must equal <img src=\"//latex.artofproblemsolving.com/f/5/3/f536cf1d1c0e7e0dba16ef5e769148eabc2bbad4.png\" class=\"latex\" alt=\"$ 5\\sqrt{2}$\" style=\"vertical-align: -1px\" width=\"33\" height=\"18\" > for final coordinates of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/5/ce5ef9a25b5949dca8be953620892d887f4d5e2b.png\" class=\"latex\" alt=\"$ (\\cos(5\\sqrt{2}), \\sin(5\\sqrt{2}), 5\\sqrt{2})$\" style=\"vertical-align: -4px\" width=\"203\" height=\"21\" >.</span><br>\n<br>\nNo calculus necessary.",
"post_id": 4418070,
"post_number": 3,
"post_time_unix": 1229468218,
"post_time_utc": "2008-12-16 22:56:58 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
},
{
"attachments": [],
"content_bbcode": "\"Flattening it out\" changes its length. At first I though it didn't, then I realized it did.",
"content_html": ""Flattening it out" changes its length. At first I though it didn't, then I realized it did.",
"post_id": 4418071,
"post_number": 4,
"post_time_unix": 1229473615,
"post_time_utc": "2008-12-17 00:26:55 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "It does, but then he can use the Pythagorean Theorem for infinitesimally small $ dt$ with $ z$, since $ z$ increases by $ dt$.\r\n\r\nSo herefishyfishy1 is correct. :D\r\n\r\nBut this might still require calculus [i]concepts[/i], namely Pythaogrean Theorem for infinitesimally small lengths (isn't that the definition of arc length?).\r\n\r\nTo see why, look at the calculus solution.\r\n\r\nFirst, notice that the point $ (1,0,0)$ corresponds to $ t \\equal{} 0$.\r\n\r\n$ L \\equal{} \\int_0^T \\sqrt {\\left(\\frac {dx}{dt}\\right)^2 \\plus{} \\left(\\frac {dy}{dt}\\right)^2 \\plus{} \\left(\\frac {dz}{dt}\\right)^2}\\,dt$\r\n\r\nSince $ \\frac {dx}{dt} \\equal{} \\minus{} \\sin x$ and $ \\frac {dy}{dt} \\equal{} \\cos x$, the sum of their squares is just 1. Then, adding the square of $ \\frac {dz}{dt}$, which is just 1, we have\r\n$ L \\equal{} \\int_0^T \\sqrt 2\\,dt \\equal{} \\sqrt 2T$,\r\n\r\nwhere $ T$ is the final value of $ t$ such that the distance from $ t \\equal{} 0$ to $ t \\equal{} T$ is 10. This is just obtained by solving $ \\sqrt 2T \\equal{} 10$, so $ T \\equal{} \\frac {10}{\\sqrt 2} \\equal{} 5\\sqrt 2$.\r\n\r\nSo the coordinates are $ (\\cos(5\\sqrt 2), \\sin(5\\sqrt 2), 5\\sqrt 2)$.",
"content_html": "It does, but then he can use the Pythagorean Theorem for infinitesimally small <img src=\"//latex.artofproblemsolving.com/2/a/2/2a2d1645d0c05baef4ed35e83ed509005e0ad060.png\" class=\"latex\" alt=\"$ dt$\" width=\"15\" height=\"12\" > with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/e/eaedd2e10519571924b7b1fd0fe1fbe4dfeecb0e.png\" class=\"latex\" alt=\"$ z$\" width=\"8\" height=\"8\" >,</span> since <img src=\"//latex.artofproblemsolving.com/e/a/e/eaedd2e10519571924b7b1fd0fe1fbe4dfeecb0e.png\" class=\"latex\" alt=\"$ z$\" width=\"8\" height=\"8\" > increases by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/a/2/2a2d1645d0c05baef4ed35e83ed509005e0ad060.png\" class=\"latex\" alt=\"$ dt$\" width=\"15\" height=\"12\" >.</span><br>\n<br>\nSo herefishyfishy1 is correct. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /><br>\n<br>\nBut this might still require calculus <i>concepts</i>, namely Pythaogrean Theorem for infinitesimally small lengths (isn't that the definition of arc length?).<br>\n<br>\nTo see why, look at the calculus solution.<br>\n<br>\nFirst, notice that the point <img src=\"//latex.artofproblemsolving.com/3/c/c/3cc77d917d0e9ea50859e93bc765bd2c6027167a.png\" class=\"latex\" alt=\"$ (1,0,0)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" > corresponds to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/5/1/b5170298599ca7d4e9d90eb9e50ba754b7ea12c5.png\" class=\"latex\" alt=\"$ t = 0$\" width=\"39\" height=\"12\" >.</span><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/8/4/7844e715f57cf19fd73474ba2c67c3face4db15a.png\" class=\"latex\" alt=\"$ L = \\int_0^T \\sqrt {\\left(\\frac {dx}{dt}\\right)^2 + \\left(\\frac {dy}{dt}\\right)^2 + \\left(\\frac {dz}{dt}\\right)^2}\\,dt$\" style=\"vertical-align: -18px\" width=\"319\" height=\"53\" ><br>\n<br>\nSince <img src=\"//latex.artofproblemsolving.com/2/1/d/21d33d540238f830fb2340ab47492794b737f096.png\" class=\"latex\" alt=\"$ \\frac {dx}{dt} = - \\sin x$\" style=\"vertical-align: -12px\" width=\"100\" height=\"37\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/7/097d16d5f41d7bcc96a1c0e74f89b102640ff94e.png\" class=\"latex\" alt=\"$ \\frac {dy}{dt} = \\cos x$\" style=\"vertical-align: -12px\" width=\"84\" height=\"37\" >,</span> the sum of their squares is just 1. Then, adding the square of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/c/a/2ca4827acb0364d2feebb38131085188b18f3f9d.png\" class=\"latex\" alt=\"$ \\frac {dz}{dt}$\" style=\"vertical-align: -12px\" width=\"21\" height=\"37\" >,</span> which is just 1, we have<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/8/0788e87404c8312bbece5681a03976be354b1f64.png\" class=\"latex\" alt=\"$ L = \\int_0^T \\sqrt 2\\,dt = \\sqrt 2T$\" style=\"vertical-align: -16px\" width=\"172\" height=\"43\" >,</span><br>\n<br>\nwhere <img src=\"//latex.artofproblemsolving.com/6/4/5/645aa68d8b1c8cc634ce387d6321d9f9652b43cc.png\" class=\"latex\" alt=\"$ T$\" width=\"13\" height=\"12\" > is the final value of <img src=\"//latex.artofproblemsolving.com/d/0/9/d09c2835f4b6988d1b5a8c2e4783372992fc5c99.png\" class=\"latex\" alt=\"$ t$\" width=\"6\" height=\"11\" > such that the distance from <img src=\"//latex.artofproblemsolving.com/b/5/1/b5170298599ca7d4e9d90eb9e50ba754b7ea12c5.png\" class=\"latex\" alt=\"$ t = 0$\" width=\"39\" height=\"12\" > to <img src=\"//latex.artofproblemsolving.com/1/e/d/1ed467bdb23142de077f6d2b05f02a3c70dfd1b2.png\" class=\"latex\" alt=\"$ t = T$\" width=\"44\" height=\"12\" > is 10. This is just obtained by solving <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/d/b/0db11ef300810ba5d8b49d82549a3d18dfc37f97.png\" class=\"latex\" alt=\"$ \\sqrt 2T = 10$\" style=\"vertical-align: -1px\" width=\"79\" height=\"18\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/f/3/5f37adffecc3a1828f516fb094481a4b9638fb51.png\" class=\"latex\" alt=\"$ T = \\frac {10}{\\sqrt 2} = 5\\sqrt 2$\" style=\"vertical-align: -17px\" width=\"123\" height=\"42\" >.</span><br>\n<br>\nSo the coordinates are <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/1/b/51b5d4fe7d5a571b0832e77e3e20916d35d38fcd.png\" class=\"latex\" alt=\"$ (\\cos(5\\sqrt 2), \\sin(5\\sqrt 2), 5\\sqrt 2)$\" style=\"vertical-align: -4px\" width=\"203\" height=\"21\" >.</span>",
"post_id": 4418072,
"post_number": 5,
"post_time_unix": 1229476966,
"post_time_utc": "2008-12-17 01:22:46 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Well, if I felt bored, I could turn the proof that e is transcendental into an elementary proof. You can turn a huge amount of things elementary...",
"content_html": "Well, if I felt bored, I could turn the proof that e is transcendental into an elementary proof. You can turn a huge amount of things elementary...",
"post_id": 4418073,
"post_number": 6,
"post_time_unix": 1229477202,
"post_time_utc": "2008-12-17 01:26:42 UTC",
"thanks_received": 2,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "Yeah. $ e$ is the solution to a polynomial of infinite degree, so it cannot be a solution to a polynomial of finite degree :D.",
"content_html": "Yeah. <img src=\"//latex.artofproblemsolving.com/1/d/b/1dbb5e41cc7aaba3d6d19d4f507eab68e20bdaac.png\" class=\"latex\" alt=\"$ e$\" width=\"8\" height=\"8\" > is the solution to a polynomial of infinite degree, so it cannot be a solution to a polynomial of finite degree <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />.",
"post_id": 4418074,
"post_number": 7,
"post_time_unix": 1229477495,
"post_time_utc": "2008-12-17 01:31:35 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I didn't have to use calculus because the length of an interval $ t\\equal{}a \\to t\\equal{}a\\plus{}n$ is always constant in $ a$ by symmetry.\r\n\r\nIf the length of the curve were not constant for a fixed-length interval of $ t$, then that would become quite annoying.",
"content_html": "I didn't have to use calculus because the length of an interval <img src=\"//latex.artofproblemsolving.com/7/b/3/7b3af59f3f70d95b02c8682bdecd5a561461b78e.png\" class=\"latex\" alt=\"$ t=a \\to t=a+n$\" style=\"vertical-align: -1px\" width=\"142\" height=\"13\" > is always constant in <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > by symmetry.<br>\n<br>\nIf the length of the curve were not constant for a fixed-length interval of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/0/9/d09c2835f4b6988d1b5a8c2e4783372992fc5c99.png\" class=\"latex\" alt=\"$ t$\" width=\"6\" height=\"11\" >,</span> then that would become quite annoying.",
"post_id": 4418075,
"post_number": 8,
"post_time_unix": 1229556223,
"post_time_utc": "2008-12-17 23:23:43 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
}
],
"source": null
}
|
A circular helix is defined by the parametric equations
\[
x(t)=\cos t,\qquad y(t)=\sin t,\qquad z(t)=t.
\]
If a bug starts at \((1,0,0)\) and walks up the helix for a distance of \(10\) units, what are its final coordinates?
|
[
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/Coordinates",
"/Mathematics/Geometry/CoordinateGeometry/ParametricEquations",
"/Mathematics/Geometry/Curves/SpaceCurves/Helix",
"/Mathematics/Geometry/Curves/SpaceCurves/SpaceCurve",
"/Mathematics/Geometry/Distance"
] |
Recognize that the helix’s horizontal arc length and vertical rise are both equal to the parameter t, so the total traveled distance is √2·t.
|
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aops_997585
|
Hmm...
Call the answer $ S$. We will first prove that $ f(S) \equal{} S$.
Proof: $ S$ is equivalent to $ \displaystyle\lim_{n\to \infty} f(f^{n \minus{} 1}(2))$, which, since $ f$ is continuous for a positive value (which $ f^{n \minus{} 1}(2)$ always is), is equal to $ f(\displaystyle\lim_{n\to \infty} f^{n \minus{} 1}(2))$. Making the substitution $ m \equal{} n \minus{} 1$, this is simplified to $ f(\displaystyle\lim_{m\to \infty} f^{m}(2))$, or simply $ f(S)$.
So we solve $ f(S) \equal{} S$:
$ \begin{align} f(S) & \equal{} S \\
\sqrt {S} \plus{} 3 & \equal{} S \\
\sqrt {S} & \equal{} S \minus{} 3 \\
S & \equal{} S^2 \minus{} 6S \plus{} 9 \\
0 & \equal{} S^2 \minus{} 7S \plus{} 9 \\
S & \equal{} \frac {7 \pm \sqrt {13}}{2} \end{align}$
and we choose the larger value for $ S$ because the other one is extraneous. The same approach can be used to solve the harder problem.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $ f(x) \\equal{} \\sqrt {x} \\plus{} 3$. Find $ \\lim_{n\\to\\infty}f^n(2)$ where $ f^n(x)$ is defined by $ f^0(x) \\equal{} x$, $ f^n(x) \\equal{} f(f^{n \\minus{} 1}(x))$.\r\n\r\n\r\nYeah, that's a bit easy. Here's another one: Prove that for ANY $ x \\geq 0$, the limit will converge to the above answer.",
"content_html": "Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/4/2/042e22ac2fb842340fce925bf008e30068ad09e1.png\" class=\"latex\" alt=\"$ f(x) = \\sqrt {x} + 3$\" style=\"vertical-align: -4px\" width=\"116\" height=\"19\" >.</span> Find <img src=\"//latex.artofproblemsolving.com/d/1/9/d19d6ab70a150f3fa752fae83bbdca3c2b073a81.png\" class=\"latex\" alt=\"$ \\lim_{n\\to\\infty}f^n(2)$\" style=\"vertical-align: -11px\" width=\"79\" height=\"24\" > where <img src=\"//latex.artofproblemsolving.com/2/7/b/27bbdb994f0bbae505adc9ada94bcaf2a5d1bfca.png\" class=\"latex\" alt=\"$ f^n(x)$\" style=\"vertical-align: -4px\" width=\"43\" height=\"18\" > is defined by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/9/a/09a3e7343048c8029e0a4af58c2a834d0a833e8a.png\" class=\"latex\" alt=\"$ f^0(x) = x$\" style=\"vertical-align: -4px\" width=\"77\" height=\"19\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/4/a/b4a6f30fbcace35f3de8b5a60325ab895dc9392f.png\" class=\"latex\" alt=\"$ f^n(x) = f(f^{n - 1}(x))$\" style=\"vertical-align: -4px\" width=\"153\" height=\"19\" >.</span><br>\n<br>\n<br>\nYeah, that's a bit easy. Here's another one: Prove that for ANY <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/a/bbacda3a199c1f7896564efde3e69c547b98ffab.png\" class=\"latex\" alt=\"$ x \\geq 0$\" style=\"vertical-align: -2px\" width=\"43\" height=\"15\" >,</span> the limit will converge to the above answer.",
"post_id": 4418078,
"post_number": 1,
"post_time_unix": 1229572412,
"post_time_utc": "2008-12-18 03:53:32 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Hmm...\r\n\r\nCall the answer $ S$. We will first prove that $ f(S) \\equal{} S$.\r\n\r\nProof: $ S$ is equivalent to $ \\displaystyle\\lim_{n\\to \\infty} f(f^{n \\minus{} 1}(2))$, which, since $ f$ is continuous for a positive value (which $ f^{n \\minus{} 1}(2)$ always is), is equal to $ f(\\displaystyle\\lim_{n\\to \\infty} f^{n \\minus{} 1}(2))$. Making the substitution $ m \\equal{} n \\minus{} 1$, this is simplified to $ f(\\displaystyle\\lim_{m\\to \\infty} f^{m}(2))$, or simply $ f(S)$.\r\n\r\nSo we solve $ f(S) \\equal{} S$:\r\n$ \\begin{align} f(S) & \\equal{} S \\\\\r\n\\sqrt {S} \\plus{} 3 & \\equal{} S \\\\\r\n\\sqrt {S} & \\equal{} S \\minus{} 3 \\\\\r\nS & \\equal{} S^2 \\minus{} 6S \\plus{} 9 \\\\\r\n0 & \\equal{} S^2 \\minus{} 7S \\plus{} 9 \\\\\r\nS & \\equal{} \\frac {7 \\pm \\sqrt {13}}{2} \\end{align}$\r\n\r\n and we choose the larger value for $ S$ because the other one is extraneous. The same approach can be used to solve the harder problem.",
"content_html": "Hmm...<br>\n<br>\nCall the answer <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/6/c663ecbe181b4e84e91ccbac30187e24917af1ed.png\" class=\"latex\" alt=\"$ S$\" width=\"12\" height=\"12\" >.</span> We will first prove that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/d/5/8d55de89af297ea31ea8ecff5f952f7cd7e1c13e.png\" class=\"latex\" alt=\"$ f(S) = S$\" style=\"vertical-align: -4px\" width=\"73\" height=\"18\" >.</span><br>\n<br>\nProof: <img src=\"//latex.artofproblemsolving.com/c/6/6/c663ecbe181b4e84e91ccbac30187e24917af1ed.png\" class=\"latex\" alt=\"$ S$\" width=\"12\" height=\"12\" > is equivalent to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/f/4/bf42ab60a0599d8a80f01f150e5c1ad4fb601770.png\" class=\"latex\" alt=\"$ \\displaystyle\\lim_{n\\to \\infty} f(f^{n - 1}(2))$\" style=\"vertical-align: -11px\" width=\"121\" height=\"26\" >,</span> which, since <img src=\"//latex.artofproblemsolving.com/4/f/3/4f392c23dd74e27d6e51bd8ebd239249d19c16b8.png\" class=\"latex\" alt=\"$ f$\" style=\"vertical-align: -3px\" width=\"10\" height=\"16\" > is continuous for a positive value (which <img src=\"//latex.artofproblemsolving.com/a/4/0/a40087c6d3817416e065a7698bc80f02c5c607cf.png\" class=\"latex\" alt=\"$ f^{n - 1}(2)$\" style=\"vertical-align: -4px\" width=\"58\" height=\"19\" > always is), is equal to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/8/2/182cac3759d9b3d1d6a38bb41d6b2bc17ebc5eb0.png\" class=\"latex\" alt=\"$ f(\\displaystyle\\lim_{n\\to \\infty} f^{n - 1}(2))$\" style=\"vertical-align: -11px\" width=\"121\" height=\"26\" >.</span> Making the substitution <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/c/6/1c6486b05ba6fbd64bfb2db3f02c8658e91a192a.png\" class=\"latex\" alt=\"$ m = n - 1$\" style=\"vertical-align: 0px\" width=\"81\" height=\"12\" >,</span> this is simplified to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/1/e/11e1a44ee755d0122c2e5b0146e1e1fefe7fcacd.png\" class=\"latex\" alt=\"$ f(\\displaystyle\\lim_{m\\to \\infty} f^{m}(2))$\" style=\"vertical-align: -11px\" width=\"111\" height=\"24\" >,</span> or simply <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/5/ff5f695d97372077f5af4cb3c8599bffc81e69e3.png\" class=\"latex\" alt=\"$ f(S)$\" style=\"vertical-align: -4px\" width=\"35\" height=\"18\" >.</span><br>\n<br>\nSo we solve <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/d/5/8d55de89af297ea31ea8ecff5f952f7cd7e1c13e.png\" class=\"latex\" alt=\"$ f(S) = S$\" style=\"vertical-align: -4px\" width=\"73\" height=\"18\" >:</span><br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align} f(S) & = S \\\\\n\\sqrt {S} + 3 & = S \\\\\n\\sqrt {S} & = S - 3 \\\\\nS & = S^2 - 6S + 9 \\\\\n0 & = S^2 - 7S + 9 \\\\\nS & = \\frac {7 \\pm \\sqrt {13}}{2} \\end{align}$</span><br>\n<br>\nand we choose the larger value for <img src=\"//latex.artofproblemsolving.com/c/6/6/c663ecbe181b4e84e91ccbac30187e24917af1ed.png\" class=\"latex\" alt=\"$ S$\" width=\"12\" height=\"12\" > because the other one is extraneous. The same approach can be used to solve the harder problem.",
"post_id": 4418079,
"post_number": 2,
"post_time_unix": 1229642908,
"post_time_utc": "2008-12-18 23:28:28 UTC",
"thanks_received": 2,
"user_id": 45289,
"username": "dysfunctionalequations"
}
],
"source": null
}
|
Let \(f(x)=\sqrt{x}+3\). Define \(f^0(x)=x\) and \(f^n(x)=f(f^{n-1}(x))\) for \(n\ge1\).
(a) Find \(\displaystyle\lim_{n\to\infty} f^n(2)\).
(b) Prove that for any \(x\ge0\), the sequence \(\bigl(f^n(x)\bigr)_{n\ge0}\) converges to the same limit.
|
[
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula",
"/Mathematics/CalculusandAnalysis/FixedPoints/FixedPoint",
"/Mathematics/CalculusandAnalysis/Functions/Function",
"/Mathematics/CalculusandAnalysis/Functions/RealFunction",
"/Mathematics/CalculusandAnalysis/Functions/UnivariateFunction"
] |
Set the limit equal to its image under f and solve the fixed‑point equation f(L)=L.
|
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aops_997631
|
the formula for finding the square root of
$ c \plus{} di$
is
$ \pm 1 \left(\sqrt {\frac {c \plus{} \sqrt {c^2 \plus{} d^2}}{2}} \plus{} \sqrt {\frac { \minus{} c \plus{} \sqrt {c^2 \plus{} d^2}}{2}}i \right)$
Plugging in c and d gives
$ \pm 1 \left(6\sqrt3\plus{}2\sqrt3i \right)$
let's calculator check it now...
hooray!
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": ":rotfl: \r\n\r\nFirst, express $ a^2 \\plus{} b^2 \\plus{} c^2 \\equal{} a^2\\left(1 \\plus{} \\left(\\frac ba\\right)^2 \\plus{} \\left(\\frac ca\\right)^2\\right)$.\r\n\r\nNow, using the fact that $ \\frac ba \\equal{} \\minus{} (r_1 \\plus{} r_2)$ and that $ \\frac ca \\equal{} r_1r_2$, we have:\r\n\\[ a^2 \\plus{} b^2 \\plus{} c^2 \\equal{} a^2\\left(1 \\plus{} (r_1 \\plus{} r_2)^2 \\plus{} (r_1r_2)^2\\right).\r\n\\]\r\nExpanding, we get:\r\n\\[ a^2\\left(1 \\plus{} (r_1 \\plus{} r_2)^2 \\plus{} (r_1r_2)^2\\right) \\equal{} a^2 \\plus{} a^2r_1^2 \\plus{} 2a^2r_1r_2 \\plus{} a^2r_2^2 \\plus{} a^2r_1^2r_2^2.\r\n\\]\r\n\r\n[b]New problem:[/b]\r\n\r\nSimplify: $ \\sqrt {96 \\plus{} 72i}$ in the form $ a\\plus{}bi$, if $ i^2\\equal{}\\minus{}1$.",
"content_html": "<img src=\"/assets/images/smilies/rotfl.gif\" width=\"32\" height=\"20\" alt=\":rotfl:\" title=\":rotfl:\" class=\"bbcode_smiley\" /><br>\n<br>\nFirst, express <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/0/e30b87d588ba8c77ca33df6397bb34db88e9027a.png\" class=\"latex\" alt=\"$ a^2 + b^2 + c^2 = a^2\\left(1 + \\left(\\frac ba\\right)^2 + \\left(\\frac ca\\right)^2\\right)$\" style=\"vertical-align: -22px\" width=\"303\" height=\"53\" >.</span><br>\n<br>\nNow, using the fact that <img src=\"//latex.artofproblemsolving.com/d/3/0/d30dd2e1c49a55c4d6eece0e30127f1a81f64e20.png\" class=\"latex\" alt=\"$ \\frac ba = - (r_1 + r_2)$\" style=\"vertical-align: -12px\" width=\"118\" height=\"37\" > and that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/3/b/03b8f006be2c95623cb9806ff36d3ec933b66f31.png\" class=\"latex\" alt=\"$ \\frac ca = r_1r_2$\" style=\"vertical-align: -12px\" width=\"67\" height=\"33\" >,</span> we have:<br>\n<img src=\"//latex.artofproblemsolving.com/3/9/7/397b4bc0eeadd3508d09ee784a321ec05f35af78.png\" class=\"latexcenter\" alt=\"\\[ a^2 + b^2 + c^2 = a^2\\left(1 + (r_1 + r_2)^2 + (r_1r_2)^2\\right).\n\\]\" width=\"340\" height=\"22\" ><br>\nExpanding, we get:<br>\n<img src=\"//latex.artofproblemsolving.com/a/9/4/a94fa4682a89ef5d11da105bd6c9387e12f5884c.png\" class=\"latexcenter\" alt=\"\\[ a^2\\left(1 + (r_1 + r_2)^2 + (r_1r_2)^2\\right) = a^2 + a^2r_1^2 + 2a^2r_1r_2 + a^2r_2^2 + a^2r_1^2r_2^2.\n\\]\" width=\"492\" height=\"22\" ><br>\n<br>\n<b>New problem:</b><br>\n<br>\nSimplify: <img src=\"//latex.artofproblemsolving.com/f/8/e/f8e0d2b6bb4be3cd6e536bdc7ac3ac0190152986.png\" class=\"latex\" alt=\"$ \\sqrt {96 + 72i}$\" style=\"vertical-align: -2px\" width=\"80\" height=\"18\" > in the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/6/0161d4c215824596ba503c741e62e6b1240fabe7.png\" class=\"latex\" alt=\"$ a+bi$\" style=\"vertical-align: -1px\" width=\"45\" height=\"14\" >,</span> if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/d/8/3d8738c06e0205fc2b5306e110d8520557398799.png\" class=\"latex\" alt=\"$ i^2=-1$\" style=\"vertical-align: 0px\" width=\"60\" height=\"15\" >.</span>",
"post_id": 4418205,
"post_number": 1,
"post_time_unix": 1236439091,
"post_time_utc": "2009-03-07 15:18:11 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "the formula for finding the square root of\r\n$ c \\plus{} di$\r\nis \r\n$ \\pm 1 \\left(\\sqrt {\\frac {c \\plus{} \\sqrt {c^2 \\plus{} d^2}}{2}} \\plus{} \\sqrt {\\frac { \\minus{} c \\plus{} \\sqrt {c^2 \\plus{} d^2}}{2}}i \\right)$\r\nPlugging in c and d gives\r\n$ \\pm 1 \\left(6\\sqrt3\\plus{}2\\sqrt3i \\right)$\r\n\r\nlet's calculator check it now...\r\nhooray!",
"content_html": "the formula for finding the square root of<br>\n<img src=\"//latex.artofproblemsolving.com/0/8/d/08d7de279fa1d38889d0873735d2de15f0de6e49.png\" class=\"latex\" alt=\"$ c + di$\" style=\"vertical-align: -1px\" width=\"45\" height=\"14\" ><br>\nis<br>\n<img src=\"//latex.artofproblemsolving.com/0/6/e/06e061331cac62d1d3e8d7ce4caf4fc9667f1aa2.png\" class=\"latex\" alt=\"$ \\pm 1 \\left(\\sqrt {\\frac {c + \\sqrt {c^2 + d^2}}{2}} + \\sqrt {\\frac { - c + \\sqrt {c^2 + d^2}}{2}}i \\right)$\" style=\"vertical-align: -27px\" width=\"338\" height=\"64\" ><br>\nPlugging in c and d gives<br>\n<img src=\"//latex.artofproblemsolving.com/5/a/6/5a6ebf0f1f7919af4893461647dd02d6b49c453a.png\" class=\"latex\" alt=\"$ \\pm 1 \\left(6\\sqrt3+2\\sqrt3i \\right)$\" style=\"vertical-align: -11px\" width=\"140\" height=\"32\" ><br>\n<br>\nlet's calculator check it now...<br>\nhooray!",
"post_id": 4418206,
"post_number": 2,
"post_time_unix": 1236448363,
"post_time_utc": "2009-03-07 17:52:43 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "Lol, here's a comment. What if you plug in $ c \\equal{} r\\cos \\theta$ and $ d \\equal{} r\\sin \\theta$? What do you get then?\r\n\r\nIn polar coordinates, we have $ \\sqrt {r\\,\\text{cis} \\,\\theta} \\equal{} \\pm\\sqrt r\\,\\text{cis}\\,\\frac {\\theta}2$.\r\n\r\nThe formula for $ \\text{cis}\\,\\frac {\\theta}2$ is:\r\n\\[ \\cos \\frac {\\theta}2 & \\equal{} \\sqrt {\\frac {1 \\plus{} \\cos \\theta}2} \\\\\r\n\\sin \\frac {\\theta}2 & \\equal{} \\sqrt {\\frac {1 \\minus{} \\cos \\theta}2}\r\n\\]\r\nTherefore, we have:\r\n\\[ \\sqrt {\\frac {r \\plus{} r\\cos\\theta}2} \\plus{} i\\sqrt {\\frac {r \\minus{} r\\cos\\theta}2}\r\n\\]\r\nSubstituting $ r \\equal{} \\sqrt {a^2 \\plus{} b^2}$ and $ r\\cos\\theta \\equal{} a$, we obtain our desired formula.\r\n\\[ \\sqrt {a \\plus{} bi} \\equal{} \\pm\\left(\\sqrt {\\frac {\\sqrt {a^2 \\plus{} b^2} \\plus{} a}2} \\plus{} i\\sqrt {\\frac {\\sqrt {a^2 \\plus{} b^2} \\minus{} a}2}\\right)\r\n\\]\r\n\r\n[hide=\"Of course, you could also quadratic bash it...\"]\nSet $ (c\\plus{}di)^2\\equal{}a\\plus{}bi$, and equate components. You have the following system of equations:\n$ \\begin{align*}\nc^2\\minus{}d^2&\\equal{}a \\\\\ncd&\\equal{}\\frac b2\n\\end{align*}$\n[/hide]",
"content_html": "Lol, here's a comment. What if you plug in <img src=\"//latex.artofproblemsolving.com/d/e/8/de81004a4d0c351f6f1aadf627011f2ce3d72752.png\" class=\"latex\" alt=\"$ c = r\\cos \\theta$\" width=\"79\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/8/d/98d836eb8309803063462d3659ad102637970b5e.png\" class=\"latex\" alt=\"$ d = r\\sin \\theta$\" width=\"79\" height=\"13\" >?</span> What do you get then?<br>\n<br>\nIn polar coordinates, we have <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/1/c/71ceeeaf058d22fad3ad5b425b23ac9634d0fa5e.png\" class=\"latex\" alt=\"$ \\sqrt {r\\,\\text{cis} \\,\\theta} = \\pm\\sqrt r\\,\\text{cis}\\,\\frac {\\theta}2$\" style=\"vertical-align: -12px\" width=\"156\" height=\"38\" >.</span><br>\n<br>\nThe formula for <img src=\"//latex.artofproblemsolving.com/2/9/f/29fbcc6c5f0ac8f642053dceaac8f142182788a6.png\" class=\"latex\" alt=\"$ \\text{cis}\\,\\frac {\\theta}2$\" style=\"vertical-align: -12px\" width=\"33\" height=\"38\" > is:<br>\n<pre class=\"aopscode-error aopscode-latex-error\">\\[ \\cos \\frac {\\theta}2 & = \\sqrt {\\frac {1 + \\cos \\theta}2} \\\\\n\\sin \\frac {\\theta}2 & = \\sqrt {\\frac {1 - \\cos \\theta}2}\n\\]</pre><br>\nTherefore, we have:<br>\n<img src=\"//latex.artofproblemsolving.com/8/3/1/831eadc13a5a8ba2ab1977e95c9fe01dda4f3c79.png\" class=\"latexcenter\" alt=\"\\[ \\sqrt {\\frac {r + r\\cos\\theta}2} + i\\sqrt {\\frac {r - r\\cos\\theta}2}\n\\]\" width=\"229\" height=\"43\" ><br>\nSubstituting <img src=\"//latex.artofproblemsolving.com/9/8/8/9886cd8bd0cb92bc3681c89cfce177011307362a.png\" class=\"latex\" alt=\"$ r = \\sqrt {a^2 + b^2}$\" style=\"vertical-align: -2px\" width=\"103\" height=\"18\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/6/bb687c330b5ea3b9db16f26e034c4d11ecab5a35.png\" class=\"latex\" alt=\"$ r\\cos\\theta = a$\" width=\"82\" height=\"13\" >,</span> we obtain our desired formula.<br>\n<img src=\"//latex.artofproblemsolving.com/6/7/4/6748e8b1bc4956723eaa9e3b1c6477d4b1cef5ac.png\" class=\"latexcenter\" alt=\"\\[ \\sqrt {a + bi} = \\pm\\left(\\sqrt {\\frac {\\sqrt {a^2 + b^2} + a}2} + i\\sqrt {\\frac {\\sqrt {a^2 + b^2} - a}2}\\right)\n\\]\" width=\"402\" height=\"64\" ><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Of course, you could also quadratic bash it...</a><div class=\"cmty-hide-content\" style=\"display:none\">Set <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/3/a/23af4e73b9ced724c6071b2c3027594ce37a0714.png\" class=\"latex\" alt=\"$ (c+di)^2=a+bi$\" style=\"vertical-align: -4px\" width=\"137\" height=\"19\" >,</span> and equate components. You have the following system of equations:<br>\n<span class=\"aopscode-error aopscode-latex-error\">$ \\begin{align*}\nc^2-d^2&=a \\\\\ncd&=\\frac b2\n\\end{align*}$</span></div>",
"post_id": 4418207,
"post_number": 3,
"post_time_unix": 1236456201,
"post_time_utc": "2009-03-07 20:03:21 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I used the latter method.",
"content_html": "I used the latter method.",
"post_id": 4418208,
"post_number": 4,
"post_time_unix": 1236463012,
"post_time_utc": "2009-03-07 21:56:52 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
}
],
"source": null
}
|
First express
\[
a^2+b^2+c^2=a^2\!\left(1+\left(\frac ba\right)^2+\left(\frac ca\right)^2\right).
\]
Using \(\dfrac ba=-(r_1+r_2)\) and \(\dfrac ca=r_1r_2\),
\[
a^2+b^2+c^2=a^2\!\left(1+(r_1+r_2)^2+(r_1r_2)^2\right).
\]
Expanding,
\[
a^2\!\left(1+(r_1+r_2)^2+(r_1r_2)^2\right)=a^2+a^2r_1^2+2a^2r_1r_2+a^2r_2^2+a^2r_1^2r_2^2.
\]
New problem:
Simplify \(\sqrt{96+72i}\) in the form \(a+bi\), where \(i^2=-1\).
|
[
"/Mathematics/Algebra"
] |
Apply the standard formula for √(c+di) using the modulus to split into real and imaginary parts.
|
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aops_997643
|
Sets MUST have distinct elements.
Assuming you meant 'multiset'...
$ S_3$ contains both $ (a_1\plus{}a_2)\plus{}(a_3\plus{}a_4)$ and $ (a_1\plus{}a_3)\plus{}(a_2\plus{}a_4)$ which are equal...
So, thus $ S_3$ cannot have distinct elements.
We can prove by induction that none of $ S_n$ for $ n\geq 3$ have distinct elements as follows:
Let $ s$ be the repeated element in $ S_n$, and let $ x$ be any other element in the set. Then since $ s$ appears at least twice in $ S_n$, $ s\plus{}x$ appears at least twice in $ S_{n\plus{}1}$.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "This was not a problem on any USAMO, but I will not claim ownership of that problem in case it is actually well-known and found in a book like Engel...\r\n\r\nConsider the 2009-element set $ S_1 \\equal{} \\{a_1,a_2,a_3,\\dots,a_{2009}\\}$. Now consider the $ \\binom{2009}2$-element set $ S_2 \\equal{} \\{a_1 \\plus{} a_2,a_1 \\plus{} a_3,\\dots,a_{2008} \\plus{} a_{2009}\\}$, consisting of all the possible sums of $ S_1$. Let $ S_3$ be the set of all possible sums of $ S_2$, and so on. Determine, with proof, whether it is possible for $ S_{2009}$ to contain all distinct elements.\r\n\r\nIt is harder than it looks to all but pythag011. The following does not constitute a valid proof:\r\n\r\n\"Just let $ a_1$ be 0 and $ a_2$ be very large, and $ a_3$ be even larger...\"\r\n\r\n[color=darkred]Edit: Replace every occurrence of \"set\" with \"multiset\"...[/color]",
"content_html": "This was not a problem on any USAMO, but I will not claim ownership of that problem in case it is actually well-known and found in a book like Engel...<br>\n<br>\nConsider the 2009-element set <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/f/e/7fe91073d617b861bf3de8eb5537eaa245e0404c.png\" class=\"latex\" alt=\"$ S_1 = \\{a_1,a_2,a_3,\\dots,a_{2009}\\}$\" style=\"vertical-align: -4px\" width=\"204\" height=\"18\" >.</span> Now consider the <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/0/e/a0e68bda0695dcc5660573b2dc8cde07d5a940ed.png\" class=\"latex\" alt=\"$ \\binom{2009}2$\" style=\"vertical-align: -22px\" width=\"60\" height=\"53\" >-</span>element set <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/2/8/c28c30d6bd8e2061f928e27131f2cae23db69eab.png\" class=\"latex\" alt=\"$ S_2 = \\{a_1 + a_2,a_1 + a_3,\\dots,a_{2008} + a_{2009}\\}$\" style=\"vertical-align: -4px\" width=\"317\" height=\"18\" >,</span> consisting of all the possible sums of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/f/a1f3db1d32aceccf529c972c6fcccb98d8f71fd2.png\" class=\"latex\" alt=\"$ S_1$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" >.</span> Let <img src=\"//latex.artofproblemsolving.com/2/1/a/21a78d7ce310e405e9a006979449d558f3d7a364.png\" class=\"latex\" alt=\"$ S_3$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > be the set of all possible sums of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/1/1/1110d92489b6e6de1b7ed57ef6c9a127ff8b7fca.png\" class=\"latex\" alt=\"$ S_2$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" >,</span> and so on. Determine, with proof, whether it is possible for <img src=\"//latex.artofproblemsolving.com/a/7/0/a70745e040aa90e515bf13d54d483e74ee662b2f.png\" class=\"latex\" alt=\"$ S_{2009}$\" style=\"vertical-align: -2px\" width=\"36\" height=\"15\" > to contain all distinct elements.<br>\n<br>\nIt is harder than it looks to all but pythag011. The following does not constitute a valid proof:<br>\n<br>\n"Just let <img src=\"//latex.artofproblemsolving.com/d/e/7/de7b0c6ea3e31c387633fa07b2b2c195c01d58e8.png\" class=\"latex\" alt=\"$ a_1$\" style=\"vertical-align: -2px\" width=\"15\" height=\"10\" > be 0 and <img src=\"//latex.artofproblemsolving.com/e/5/2/e52558ad68514f196dece9838eabd14034b7be3a.png\" class=\"latex\" alt=\"$ a_2$\" style=\"vertical-align: -2px\" width=\"15\" height=\"10\" > be very large, and <img src=\"//latex.artofproblemsolving.com/a/a/7/aa79f5425c679dbde52e6c76fa33fc683e6579aa.png\" class=\"latex\" alt=\"$ a_3$\" style=\"vertical-align: -2px\" width=\"15\" height=\"10\" > be even larger..."<br>\n<br>\n<span style=\"color:darkred\">Edit: Replace every occurrence of "set" with "multiset"...</span>",
"post_id": 4418236,
"post_number": 1,
"post_time_unix": 1237518282,
"post_time_utc": "2009-03-20 03:04:42 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Sets MUST have distinct elements.\r\n\r\nAssuming you meant 'multiset'...\r\n\r\n$ S_3$ contains both $ (a_1\\plus{}a_2)\\plus{}(a_3\\plus{}a_4)$ and $ (a_1\\plus{}a_3)\\plus{}(a_2\\plus{}a_4)$ which are equal...\r\n\r\nSo, thus $ S_3$ cannot have distinct elements.\r\n\r\nWe can prove by induction that none of $ S_n$ for $ n\\geq 3$ have distinct elements as follows:\r\n\r\nLet $ s$ be the repeated element in $ S_n$, and let $ x$ be any other element in the set. Then since $ s$ appears at least twice in $ S_n$, $ s\\plus{}x$ appears at least twice in $ S_{n\\plus{}1}$.",
"content_html": "Sets MUST have distinct elements.<br>\n<br>\nAssuming you meant 'multiset'...<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/1/a/21a78d7ce310e405e9a006979449d558f3d7a364.png\" class=\"latex\" alt=\"$ S_3$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > contains both <img src=\"//latex.artofproblemsolving.com/7/6/c/76caa346db5b104fec0448d5015813acf5a4e048.png\" class=\"latex\" alt=\"$ (a_1+a_2)+(a_3+a_4)$\" style=\"vertical-align: -4px\" width=\"161\" height=\"18\" > and <img src=\"//latex.artofproblemsolving.com/5/1/f/51f5d4e1c4a823a40bebec8ed5f3203d34b9c074.png\" class=\"latex\" alt=\"$ (a_1+a_3)+(a_2+a_4)$\" style=\"vertical-align: -4px\" width=\"161\" height=\"18\" > which are equal...<br>\n<br>\nSo, thus <img src=\"//latex.artofproblemsolving.com/2/1/a/21a78d7ce310e405e9a006979449d558f3d7a364.png\" class=\"latex\" alt=\"$ S_3$\" style=\"vertical-align: -2px\" width=\"17\" height=\"15\" > cannot have distinct elements.<br>\n<br>\nWe can prove by induction that none of <img src=\"//latex.artofproblemsolving.com/f/a/d/fadf331ef65921140f0d8aaeefdd0c08c9b2926d.png\" class=\"latex\" alt=\"$ S_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > for <img src=\"//latex.artofproblemsolving.com/2/8/2/2826d7b2c4faf2306ad724a6c90b7df89537074f.png\" class=\"latex\" alt=\"$ n\\geq 3$\" style=\"vertical-align: -2px\" width=\"43\" height=\"15\" > have distinct elements as follows:<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/d/b/e/dbeca274183b6a731311e1dcf290ef519365dee8.png\" class=\"latex\" alt=\"$ s$\" width=\"8\" height=\"8\" > be the repeated element in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/d/fadf331ef65921140f0d8aaeefdd0c08c9b2926d.png\" class=\"latex\" alt=\"$ S_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,</span> and let <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > be any other element in the set. Then since <img src=\"//latex.artofproblemsolving.com/d/b/e/dbeca274183b6a731311e1dcf290ef519365dee8.png\" class=\"latex\" alt=\"$ s$\" width=\"8\" height=\"8\" > appears at least twice in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/d/fadf331ef65921140f0d8aaeefdd0c08c9b2926d.png\" class=\"latex\" alt=\"$ S_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,</span> <img src=\"//latex.artofproblemsolving.com/1/d/8/1d85b0851c94fc0f6778b4c12a8e69a11c156949.png\" class=\"latex\" alt=\"$ s+x$\" style=\"vertical-align: -1px\" width=\"40\" height=\"12\" > appears at least twice in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/8/3/e8347bfa92aedba6a28a247d84f0cbb6e5312d81.png\" class=\"latex\" alt=\"$ S_{n+1}$\" style=\"vertical-align: -4px\" width=\"34\" height=\"16\" >.</span>",
"post_id": 4418237,
"post_number": 2,
"post_time_unix": 1237518697,
"post_time_utc": "2009-03-20 03:11:37 UTC",
"thanks_received": 1,
"user_id": 45289,
"username": "dysfunctionalequations"
},
{
"attachments": [],
"content_bbcode": "This is an example of a problem that might overwhelm you but really has a simple solution :D",
"content_html": "This is an example of a problem that might overwhelm you but really has a simple solution <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 4418238,
"post_number": 3,
"post_time_unix": 1237519559,
"post_time_utc": "2009-03-20 03:25:59 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "I was not overwhelmed. Clearly you are ignorant of Zeb's Theorem on Equal Newton Sums of polynomials with only integer roots..",
"content_html": "I was not overwhelmed. Clearly you are ignorant of Zeb's Theorem on Equal Newton Sums of polynomials with only integer roots..",
"post_id": 4418239,
"post_number": 4,
"post_time_unix": 1237519956,
"post_time_utc": "2009-03-20 03:32:36 UTC",
"thanks_received": 1,
"user_id": 30008,
"username": "pythag011"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Yongyi781\"]... to all but pythag011.[/quote]\r\n\r\nClearly you should read my posts more.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Yongyi781 wrote:</div>\n<div class=\"bbcode_quote_body\">... to all but pythag011.</div>\n</div>\n<br>\nClearly you should read my posts more.",
"post_id": 4418240,
"post_number": 5,
"post_time_unix": 1237520449,
"post_time_utc": "2009-03-20 03:40:49 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
Consider the 2009-element multiset \(S_1=\{a_1,a_2,\dots,a_{2009}\}\). Let \(S_2\) be the multiset of all pairwise sums of elements of \(S_1\):
\[
S_2=\{a_i+a_j:\ 1\le i<j\le 2009\},
\]
and for \(k\ge2\) let \(S_{k+1}\) be the multiset of all pairwise sums of elements of \(S_k\). Determine, with proof, whether it is possible for \(S_{2009}\) to consist of all distinct elements.
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics"
] |
At the third level, different ways of pairing four original elements produce the same sum, guaranteeing a duplicate in S₃.
|
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aops_99767
|
There are three conditions:
1) Outside the circle $C_{1}$: $x^{2}+y^{2}=1$.
2) Inside or on the circle $C_{2}$: $x^{2}+y^{2}=x+y$, that is $(x-\frac{1}{2})^{2}+(y-\frac{1}{2})^{2}=\frac{1}{2}$.
3) Maximizing $k$ in $x+2y=k$, $y=\frac{-1}{2}x+\frac{k}{2}$.
There are two points on $C_{2}$ where the slope of the tangent line is $\frac{-1}{2}$, and they are the ones that optimize $k$ in $y=\frac{-1}{2}x+\frac{k}{2}$ inside or on the circle. One is the minimum and one is the maximum.
We'll see that, luckily, the maximum is outside $C_{1}$, and it will be the answer to our problem.
Using implicit differentiation with $C_{2}$, $(x-\frac{1}{2})dx+(y-\frac{1}{2})dy=0$.
We want $\frac{dy}{dx}=\frac{-1}{2}$ $\to$ $(y-\frac{1}{2})=2(x-\frac{1}{2})$.
Subst. in $C_{2}$, $5(x-\frac{1}{2})^{2}=\frac{1}{2}$.
The two solutions are $(x,y)=(\frac{1}{2}\pm\frac{1}{\sqrt{10}}, \frac{1}{2}\pm\frac{2}{\sqrt{10}})$.
The maximum $k=\frac{3}{2}+\frac{5}{\sqrt{10}}$ is at $\boxed{(x,y)=(\frac{1}{2}+\frac{1}{\sqrt{10}}, \frac{1}{2}+\frac{2}{\sqrt{10}})}$.
It's outside $C_{1}$ because $x^{2}+y^{2}=1+\frac{3}{\sqrt{10}}>1$.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $(x,y)$ be the root of system: \\begin{eqnarray*}x^{2}+y^{2}&>& 1, \\\\ x+y &\\ge& x^{2}+y^{2}. \\end{eqnarray*} Find the pair $(x,y)$ such that the sum $x+2y$ takes the greatest value.",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/e/5/3/e53b1fe25be1c679117fb44a6a886fe1247d189a.png\" class=\"latex\" alt=\"$(x,y)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > be the root of system: <img src=\"//latex.artofproblemsolving.com/6/e/4/6e4283d4147b03be7e06b809273bd4f5489f9ae6.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*}x^{2}+y^{2}&>& 1, \\\\ x+y &\\ge& x^{2}+y^{2}. \\end{eqnarray*}\" width=\"164\" height=\"43\" > Find the pair <img src=\"//latex.artofproblemsolving.com/e/5/3/e53b1fe25be1c679117fb44a6a886fe1247d189a.png\" class=\"latex\" alt=\"$(x,y)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > such that the sum <img src=\"//latex.artofproblemsolving.com/e/0/6/e06c1e4a8a63fa0a6607ea4f6c79c1ce629412c5.png\" class=\"latex\" alt=\"$x+2y$\" style=\"vertical-align: -3px\" width=\"50\" height=\"15\" > takes the greatest value.",
"post_id": 563411,
"post_number": 1,
"post_time_unix": 1151737432,
"post_time_utc": "2006-07-01 07:03:52 UTC",
"thanks_received": 2,
"user_id": 12772,
"username": "Lovasz"
},
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"attachments": [],
"content_bbcode": "There are three conditions:\r\n1) Outside the circle $C_{1}$: $x^{2}+y^{2}=1$.\r\n2) Inside or on the circle $C_{2}$: $x^{2}+y^{2}=x+y$, that is $(x-\\frac{1}{2})^{2}+(y-\\frac{1}{2})^{2}=\\frac{1}{2}$.\r\n3) Maximizing $k$ in $x+2y=k$, $y=\\frac{-1}{2}x+\\frac{k}{2}$.\r\n\r\nThere are two points on $C_{2}$ where the slope of the tangent line is $\\frac{-1}{2}$, and they are the ones that optimize $k$ in $y=\\frac{-1}{2}x+\\frac{k}{2}$ inside or on the circle. One is the minimum and one is the maximum.\r\nWe'll see that, luckily, the maximum is outside $C_{1}$, and it will be the answer to our problem.\r\n\r\nUsing implicit differentiation with $C_{2}$, $(x-\\frac{1}{2})dx+(y-\\frac{1}{2})dy=0$.\r\nWe want $\\frac{dy}{dx}=\\frac{-1}{2}$ $\\to$ $(y-\\frac{1}{2})=2(x-\\frac{1}{2})$.\r\nSubst. in $C_{2}$, $5(x-\\frac{1}{2})^{2}=\\frac{1}{2}$.\r\nThe two solutions are $(x,y)=(\\frac{1}{2}\\pm\\frac{1}{\\sqrt{10}}, \\frac{1}{2}\\pm\\frac{2}{\\sqrt{10}})$.\r\nThe maximum $k=\\frac{3}{2}+\\frac{5}{\\sqrt{10}}$ is at $\\boxed{(x,y)=(\\frac{1}{2}+\\frac{1}{\\sqrt{10}}, \\frac{1}{2}+\\frac{2}{\\sqrt{10}})}$.\r\nIt's outside $C_{1}$ because $x^{2}+y^{2}=1+\\frac{3}{\\sqrt{10}}>1$.",
"content_html": "There are three conditions:<br>\n1) Outside the circle <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/0/010dd7e68eb27f1ce5b462841ea650bffe2169f8.png\" class=\"latex\" alt=\"$C_{1}$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" >:</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/c/8/cc8bf1134a948062a604e809acaa36c0505150c9.png\" class=\"latex\" alt=\"$x^{2}+y^{2}=1$\" style=\"vertical-align: -3px\" width=\"89\" height=\"18\" >.</span><br>\n2) Inside or on the circle <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/6/8469fb9e58c21494d012cf7973ed8870b8db3de6.png\" class=\"latex\" alt=\"$C_{2}$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >:</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/a/cea3dd710b75d8fbcaa43805ad4b0b5e5f918a08.png\" class=\"latex\" alt=\"$x^{2}+y^{2}=x+y$\" style=\"vertical-align: -3px\" width=\"123\" height=\"18\" >,</span> that is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/7/a/87a04e4fa601bd626fd28c05df150cf8305b0ffc.png\" class=\"latex\" alt=\"$(x-\\frac{1}{2})^{2}+(y-\\frac{1}{2})^{2}=\\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"191\" height=\"37\" >.</span><br>\n3) Maximizing <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/5/bd5ded8681795a3aca2a7175a95a7764ac467ea5.png\" class=\"latex\" alt=\"$x+2y=k$\" style=\"vertical-align: -3px\" width=\"85\" height=\"16\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/e/a/7ea61c48366a51ffec11966dcf3c593a3b141a11.png\" class=\"latex\" alt=\"$y=\\frac{-1}{2}x+\\frac{k}{2}$\" style=\"vertical-align: -12px\" width=\"106\" height=\"37\" >.</span><br>\n<br>\nThere are two points on <img src=\"//latex.artofproblemsolving.com/8/4/6/8469fb9e58c21494d012cf7973ed8870b8db3de6.png\" class=\"latex\" alt=\"$C_{2}$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" > where the slope of the tangent line is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/8/5/c853bfb15f8cf5a622c714ab9586566175ef5a90.png\" class=\"latex\" alt=\"$\\frac{-1}{2}$\" style=\"vertical-align: -12px\" width=\"25\" height=\"37\" >,</span> and they are the ones that optimize <img src=\"//latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png\" class=\"latex\" alt=\"$k$\" width=\"9\" height=\"12\" > in <img src=\"//latex.artofproblemsolving.com/7/e/a/7ea61c48366a51ffec11966dcf3c593a3b141a11.png\" class=\"latex\" alt=\"$y=\\frac{-1}{2}x+\\frac{k}{2}$\" style=\"vertical-align: -12px\" width=\"106\" height=\"37\" > inside or on the circle. One is the minimum and one is the maximum.<br>\nWe'll see that, luckily, the maximum is outside <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/0/010dd7e68eb27f1ce5b462841ea650bffe2169f8.png\" class=\"latex\" alt=\"$C_{1}$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" >,</span> and it will be the answer to our problem.<br>\n<br>\nUsing implicit differentiation with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/6/8469fb9e58c21494d012cf7973ed8870b8db3de6.png\" class=\"latex\" alt=\"$C_{2}$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/3/2/7329239fa2714985403f818bc19427a6c785d420.png\" class=\"latex\" alt=\"$(x-\\frac{1}{2})dx+(y-\\frac{1}{2})dy=0$\" style=\"vertical-align: -12px\" width=\"212\" height=\"37\" >.</span><br>\nWe want <img src=\"//latex.artofproblemsolving.com/a/8/c/a8c1838b579b97929dcdcc80141805a98b330b62.png\" class=\"latex\" alt=\"$\\frac{dy}{dx}=\\frac{-1}{2}$\" style=\"vertical-align: -12px\" width=\"73\" height=\"37\" > <img src=\"//latex.artofproblemsolving.com/b/8/e/b8ea4e52bb41fa9c0a1f8a387ea39b5b9c993b74.png\" class=\"latex\" alt=\"$\\to$\" style=\"vertical-align: 0px\" width=\"17\" height=\"10\" > <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/3/f/d3f3ddc13a381c361f1b712bef604351fc98b08c.png\" class=\"latex\" alt=\"$(y-\\frac{1}{2})=2(x-\\frac{1}{2})$\" style=\"vertical-align: -12px\" width=\"150\" height=\"37\" >.</span><br>\nSubst. in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/6/8469fb9e58c21494d012cf7973ed8870b8db3de6.png\" class=\"latex\" alt=\"$C_{2}$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/2/a/e2af791d76088c3633a9255f2b33ac04f2f651e3.png\" class=\"latex\" alt=\"$5(x-\\frac{1}{2})^{2}=\\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"112\" height=\"37\" >.</span><br>\nThe two solutions are <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/2/4/824241e682cb3425d428724c085ab090e2a729d2.png\" class=\"latex\" alt=\"$(x,y)=(\\frac{1}{2}\\pm\\frac{1}{\\sqrt{10}}, \\frac{1}{2}\\pm\\frac{2}{\\sqrt{10}})$\" style=\"vertical-align: -17px\" width=\"232\" height=\"41\" >.</span><br>\nThe maximum <img src=\"//latex.artofproblemsolving.com/8/e/d/8edc994f3861846af8186da1b6e8dffaeea7789f.png\" class=\"latex\" alt=\"$k=\\frac{3}{2}+\\frac{5}{\\sqrt{10}}$\" style=\"vertical-align: -17px\" width=\"105\" height=\"42\" > is at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/5/245d16f49828405861ab6a7d6f50c6663bb48960.png\" class=\"latex\" alt=\"$\\boxed{(x,y)=(\\frac{1}{2}+\\frac{1}{\\sqrt{10}}, \\frac{1}{2}+\\frac{2}{\\sqrt{10}})}$\" style=\"vertical-align: -22px\" width=\"245\" height=\"52\" >.</span><br>\nIt's outside <img src=\"//latex.artofproblemsolving.com/0/1/0/010dd7e68eb27f1ce5b462841ea650bffe2169f8.png\" class=\"latex\" alt=\"$C_{1}$\" style=\"vertical-align: -2px\" width=\"18\" height=\"15\" > because <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/9/2/f92064eb81d580a3eeb4aa3cfce2bc6155ee0275.png\" class=\"latex\" alt=\"$x^{2}+y^{2}=1+\\frac{3}{\\sqrt{10}}>1$\" style=\"vertical-align: -17px\" width=\"182\" height=\"42\" >.</span>",
"post_id": 563461,
"post_number": 2,
"post_time_unix": 1151745193,
"post_time_utc": "2006-07-01 09:13:13 UTC",
"thanks_received": 2,
"user_id": 19718,
"username": "lordWings"
},
{
"attachments": [],
"content_bbcode": "Modified problem: :)\r\n\r\nBeing $0<r<2$, let $S$ be the set of points $(x,y)$ that are solutions to the system: \\begin{eqnarray*}x^{2}+y^{2}&>& r, \\\\ x+y &\\ge& x^{2}+y^{2}. \\end{eqnarray*} Find the lowest upper bound of the sum $x+2y$ for every pair $(x,y)$ in $S$.\r\n\r\nIn the last post, we've seen the solution for every $r<1+\\frac{3}{\\sqrt{10}}$.\r\nBut, what about $1+\\frac{3}{\\sqrt{10}}\\le r<2$?",
"content_html": "Modified problem: <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /><br>\n<br>\nBeing <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/d/d/1dd97a55a52dcd28fef4c9179662e36c6af1efad.png\" class=\"latex\" alt=\"$0<r<2$\" style=\"vertical-align: 0px\" width=\"75\" height=\"13\" >,</span> let <img src=\"//latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png\" class=\"latex\" alt=\"$S$\" width=\"12\" height=\"12\" > be the set of points <img src=\"//latex.artofproblemsolving.com/e/5/3/e53b1fe25be1c679117fb44a6a886fe1247d189a.png\" class=\"latex\" alt=\"$(x,y)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > that are solutions to the system: <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3956613c23c754e7f86ece4449a005222efc21.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*}x^{2}+y^{2}&>& r, \\\\ x+y &\\ge& x^{2}+y^{2}. \\end{eqnarray*}\" width=\"164\" height=\"43\" > Find the lowest upper bound of the sum <img src=\"//latex.artofproblemsolving.com/e/0/6/e06c1e4a8a63fa0a6607ea4f6c79c1ce629412c5.png\" class=\"latex\" alt=\"$x+2y$\" style=\"vertical-align: -3px\" width=\"50\" height=\"15\" > for every pair <img src=\"//latex.artofproblemsolving.com/e/5/3/e53b1fe25be1c679117fb44a6a886fe1247d189a.png\" class=\"latex\" alt=\"$(x,y)$\" style=\"vertical-align: -4px\" width=\"40\" height=\"18\" > in <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>\nIn the last post, we've seen the solution for every <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/4/0/f4075e4494d2db318634d2ccf971549d67e3cca6.png\" class=\"latex\" alt=\"$r<1+\\frac{3}{\\sqrt{10}}$\" style=\"vertical-align: -17px\" width=\"100\" height=\"42\" >.</span><br>\nBut, what about <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/c/92cfd2abec6122971f48e5c41e2a4e5092036408.png\" class=\"latex\" alt=\"$1+\\frac{3}{\\sqrt{10}}\\le r<2$\" style=\"vertical-align: -17px\" width=\"134\" height=\"42\" >?</span>",
"post_id": 563497,
"post_number": 3,
"post_time_unix": 1151750890,
"post_time_utc": "2006-07-01 10:48:10 UTC",
"thanks_received": 1,
"user_id": 19718,
"username": "lordWings"
}
],
"source": null
}
|
Let \((x,y)\) satisfy
\[
\begin{aligned}
x^{2}+y^{2}&>1,\\[4pt]
x+y&\ge x^{2}+y^{2}.
\end{aligned}
\]
Find the pair \((x,y)\) for which the sum \(x+2y\) is maximal.
|
[
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/ImplicitDifferentiation",
"/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus",
"/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/Extremum",
"/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/GlobalMaximum",
"/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/Maximum",
"/Mathematics/CalculusandAnalysis/Inequalities/Inequality",
"/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry",
"/Mathematics/Geometry/CoordinateGeometry/Cartesian",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/CartesianGeometry",
"/Mathematics/Geometry/CoordinateGeometry/CartesianPlane",
"/Mathematics/Geometry/CoordinateGeometry/CoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/Coordinates",
"/Mathematics/Geometry/Curves/PlaneCurves/ConicSections",
"/Mathematics/Geometry/Curves/PlaneCurves/ImplicitCurves",
"/Mathematics/Geometry/GeometricInequalities",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle-CircleIntersection"
] |
Locate the point where the level line x+2y = constant is tangent to the inner circle, so its slope matches -1/2.
|
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aops_997683
|
$ a_{n\plus{}1}\equal{} 2a_{n}\plus{}\sqrt{3a_{n}^{2}\minus{}2} \implies a_{n\plus{}1}^2\minus{}4a_{n}a_{n\plus{}1}\plus{}a_{n}^2\equal{}\minus{}2$
the main thing about this relation is that it is symmetric (unless I made some calculation mistake)
$ a_{1},a_{2}$ are integers. Now, consider the equation $ t^2\minus{}4ta_n\plus{}a_n^2\plus{}2\equal{}0$. It has 2 solutions: $ a_{n\plus{}1}$ and $ a_{n\minus{}1}$. $ a_{n\minus{}1}$ is integer (by induction). By Vietta, $ a_{n\plus{}1}\plus{}a_{n\minus{}1}\equal{}4a_n$, and again, since $ 4a_n$ is an integer (by induction), we get: $ a_{n\plus{}1}$ is an integer. That's it.
Note - I should have also mentioned that $ a_{n\plus{}1}$ and $ a_{n\minus{}1}$ are 2 -distinct- roots, otherwise we can't use Vietta (but $ a_{n\plus{}1} > 2a_{n} > 4a_{n\minus{}1} >a_{n\minus{}1}$ so that's ok)
| null |
{
"competition": null,
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"content_bbcode": "Prove that for every positive integer $ n$, $ |\\{n \\plus{} \\sigma \\forall \\sigma|n\\}\\cap \\{2n \\minus{} \\sigma \\forall \\sigma|2n\\}|\\leq 2$.\r\n\r\nMeanwhile, here's some contrived induction (source: bubala)\r\n\r\nLet the sequence $ \\{a_n\\}$ be defined such that $ a_1 \\equal{} 1$ and for $ n \\geq 1$, $ a_{n \\plus{} 1} \\equal{} 2a_n \\plus{} \\sqrt {3a_n^2 \\minus{} 2}$. Prove that all the terms in $ \\{a_n\\}$ are integers.\r\n\r\n[hide=\"Spoiler\"]\nInduction. Clearly, $ a_1$ is an integer, and $ a_2 \\equal{} 2 \\plus{} \\sqrt 1 \\equal{} 3$ is an integer as well. Suppose that for some $ k$, both $ a_k$ and $ a_{k \\plus{} 1}$ are integers. Then, since $ \\sqrt {3a_k^2 \\minus{} 2}$ is an integer by the hypothesis, and $ a_{k \\plus{} 2} \\equal{} 2a_{k \\plus{} 1} \\plus{} \\sqrt {3a_{k \\plus{} 1}^2 \\minus{} 2}$, it suffices to prove that $ 3a_{k \\plus{} 1}^2 \\minus{} 2$ is a perfect square, which is simple; substituting the definition of $ a_{k \\plus{} 1}$ yields $ 21a_k^2 \\plus{} 12a_k\\sqrt {3a_k^2 \\minus{} 2} \\minus{} 8 \\equal{} \\left(3a_k \\plus{} 2\\sqrt {3a_k^2 \\minus{} 2}\\right)^2$, qed.[/hide]",
"content_html": "Prove that for every positive integer <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/1/3/91367be2763bff4f072cad58916f0025cefe18ec.png\" class=\"latex\" alt=\"$ |\\{n + \\sigma \\forall \\sigma|n\\}\\cap \\{2n - \\sigma \\forall \\sigma|2n\\}|\\leq 2$\" style=\"vertical-align: -4px\" width=\"280\" height=\"18\" >.</span><br>\n<br>\nMeanwhile, here's some contrived induction (source: bubala)<br>\n<br>\nLet the sequence <img src=\"//latex.artofproblemsolving.com/d/b/e/dbedffef8959bbae1f31f40515a6a6eff7b74abb.png\" class=\"latex\" alt=\"$ \\{a_n\\}$\" style=\"vertical-align: -4px\" width=\"35\" height=\"18\" > be defined such that <img src=\"//latex.artofproblemsolving.com/4/b/e/4be795893930539766acffe8b31f75726e96b136.png\" class=\"latex\" alt=\"$ a_1 = 1$\" style=\"vertical-align: -2px\" width=\"49\" height=\"14\" > and for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/5/0e513bc23f73514d3b4f83d093e43b9d717878a1.png\" class=\"latex\" alt=\"$ n \\geq 1$\" style=\"vertical-align: -2px\" width=\"43\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/8/0/6806df5b0facea45e6f359b2a8b1696ef7d98584.png\" class=\"latex\" alt=\"$ a_{n + 1} = 2a_n + \\sqrt {3a_n^2 - 2}$\" style=\"vertical-align: -5px\" width=\"187\" height=\"22\" >.</span> Prove that all the terms in <img src=\"//latex.artofproblemsolving.com/d/b/e/dbedffef8959bbae1f31f40515a6a6eff7b74abb.png\" class=\"latex\" alt=\"$ \\{a_n\\}$\" style=\"vertical-align: -4px\" width=\"35\" height=\"18\" > are integers.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Spoiler</a><div class=\"cmty-hide-content\" style=\"display:none\">Induction. Clearly, <img src=\"//latex.artofproblemsolving.com/d/e/7/de7b0c6ea3e31c387633fa07b2b2c195c01d58e8.png\" class=\"latex\" alt=\"$ a_1$\" style=\"vertical-align: -2px\" width=\"15\" height=\"10\" > is an integer, and <img src=\"//latex.artofproblemsolving.com/4/3/6/43642a0c4c62b1e9d36daaef326de13b89838669.png\" class=\"latex\" alt=\"$ a_2 = 2 + \\sqrt 1 = 3$\" style=\"vertical-align: -2px\" width=\"129\" height=\"19\" > is an integer as well. Suppose that for some <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/0/d10af8b3fc779f307fe5c87020e433b00a41801d.png\" class=\"latex\" alt=\"$ k$\" width=\"9\" height=\"12\" >,</span> both <img src=\"//latex.artofproblemsolving.com/b/8/4/b84283800b7cb9541cba688c72a8caf5b0dd9db5.png\" class=\"latex\" alt=\"$ a_k$\" style=\"vertical-align: -2px\" width=\"16\" height=\"10\" > and <img src=\"//latex.artofproblemsolving.com/e/b/b/ebb30c524af44562030fc9fdcd7bfdeaabda8090.png\" class=\"latex\" alt=\"$ a_{k + 1}$\" style=\"vertical-align: -4px\" width=\"32\" height=\"12\" > are integers. Then, since <img src=\"//latex.artofproblemsolving.com/9/d/5/9d5f8a67c40042990e99fbbe93749d4907f9a585.png\" class=\"latex\" alt=\"$ \\sqrt {3a_k^2 - 2}$\" style=\"vertical-align: -10px\" width=\"76\" height=\"32\" > is an integer by the hypothesis, and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/6/9/269a7a782763cc362cd5feb72411e02ac5d2e4a6.png\" class=\"latex\" alt=\"$ a_{k + 2} = 2a_{k + 1} + \\sqrt {3a_{k + 1}^2 - 2}$\" style=\"vertical-align: -10px\" width=\"217\" height=\"32\" >,</span> it suffices to prove that <img src=\"//latex.artofproblemsolving.com/4/1/d/41d053625e8e14cded5ab74327cd4e28274651e3.png\" class=\"latex\" alt=\"$ 3a_{k + 1}^2 - 2$\" style=\"vertical-align: -6px\" width=\"74\" height=\"21\" > is a perfect square, which is simple; substituting the definition of <img src=\"//latex.artofproblemsolving.com/e/b/b/ebb30c524af44562030fc9fdcd7bfdeaabda8090.png\" class=\"latex\" alt=\"$ a_{k + 1}$\" style=\"vertical-align: -4px\" width=\"32\" height=\"12\" > yields <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/6/ff64acebcb48b3829feb75d0bbf47373f1659ad1.png\" class=\"latex\" alt=\"$ 21a_k^2 + 12a_k\\sqrt {3a_k^2 - 2} - 8 = \\left(3a_k + 2\\sqrt {3a_k^2 - 2}\\right)^2$\" style=\"vertical-align: -17px\" width=\"392\" height=\"46\" >,</span> qed.</div>",
"post_id": 4418380,
"post_number": 1,
"post_time_unix": 1240755066,
"post_time_utc": "2009-04-26 14:11:06 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
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{
"attachments": [],
"content_bbcode": "$ a_{n\\plus{}1}\\equal{} 2a_{n}\\plus{}\\sqrt{3a_{n}^{2}\\minus{}2} \\implies a_{n\\plus{}1}^2\\minus{}4a_{n}a_{n\\plus{}1}\\plus{}a_{n}^2\\equal{}\\minus{}2$\r\nthe main thing about this relation is that it is symmetric (unless I made some calculation mistake)\r\n\r\n$ a_{1},a_{2}$ are integers. Now, consider the equation $ t^2\\minus{}4ta_n\\plus{}a_n^2\\plus{}2\\equal{}0$. It has 2 solutions: $ a_{n\\plus{}1}$ and $ a_{n\\minus{}1}$. $ a_{n\\minus{}1}$ is integer (by induction). By Vietta, $ a_{n\\plus{}1}\\plus{}a_{n\\minus{}1}\\equal{}4a_n$, and again, since $ 4a_n$ is an integer (by induction), we get: $ a_{n\\plus{}1}$ is an integer. That's it.\r\n\r\nNote - I should have also mentioned that $ a_{n\\plus{}1}$ and $ a_{n\\minus{}1}$ are 2 -distinct- roots, otherwise we can't use Vietta (but $ a_{n\\plus{}1} > 2a_{n} > 4a_{n\\minus{}1} >a_{n\\minus{}1}$ so that's ok)",
"content_html": "<img src=\"//latex.artofproblemsolving.com/8/e/7/8e7989cc53d00ea537ff61e1c984ffd2852f813c.png\" class=\"latex\" alt=\"$ a_{n+1}= 2a_{n}+\\sqrt{3a_{n}^{2}-2} \\implies a_{n+1}^2-4a_{n}a_{n+1}+a_{n}^2=-2$\" style=\"vertical-align: -6px\" width=\"444\" height=\"23\" ><br>\nthe main thing about this relation is that it is symmetric (unless I made some calculation mistake)<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/4/0/4/404b6706c434dd7b18955ef3e1305b747e94724d.png\" class=\"latex\" alt=\"$ a_{1},a_{2}$\" style=\"vertical-align: -3px\" width=\"40\" height=\"11\" > are integers. Now, consider the equation <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/3/5/a35b14ee6f228ab4c01b3732aefac07b914e81d3.png\" class=\"latex\" alt=\"$ t^2-4ta_n+a_n^2+2=0$\" style=\"vertical-align: -4px\" width=\"175\" height=\"19\" >.</span> It has 2 solutions: <img src=\"//latex.artofproblemsolving.com/3/8/c/38c7ec679bc4640b9b676db356d17ee12d9172bc.png\" class=\"latex\" alt=\"$ a_{n+1}$\" style=\"vertical-align: -4px\" width=\"33\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/f/b/7fb8c8347edc5c8c66bba26a016c04735066dbb0.png\" class=\"latex\" alt=\"$ a_{n-1}$\" style=\"vertical-align: -2px\" width=\"33\" height=\"10\" >.</span> <img src=\"//latex.artofproblemsolving.com/7/f/b/7fb8c8347edc5c8c66bba26a016c04735066dbb0.png\" class=\"latex\" alt=\"$ a_{n-1}$\" style=\"vertical-align: -2px\" width=\"33\" height=\"10\" > is integer (by induction). By Vietta, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/9/839add00489eda5a252a537b6f39c966ebb4fa13.png\" class=\"latex\" alt=\"$ a_{n+1}+a_{n-1}=4a_n$\" style=\"vertical-align: -4px\" width=\"144\" height=\"16\" >,</span> and again, since <img src=\"//latex.artofproblemsolving.com/7/5/b/75b864532fcd7e4fe4d702c8841eeca4835673e9.png\" class=\"latex\" alt=\"$ 4a_n$\" style=\"vertical-align: -2px\" width=\"26\" height=\"14\" > is an integer (by induction), we get: <img src=\"//latex.artofproblemsolving.com/3/8/c/38c7ec679bc4640b9b676db356d17ee12d9172bc.png\" class=\"latex\" alt=\"$ a_{n+1}$\" style=\"vertical-align: -4px\" width=\"33\" height=\"12\" > is an integer. That's it.<br>\n<br>\nNote - I should have also mentioned that <img src=\"//latex.artofproblemsolving.com/3/8/c/38c7ec679bc4640b9b676db356d17ee12d9172bc.png\" class=\"latex\" alt=\"$ a_{n+1}$\" style=\"vertical-align: -4px\" width=\"33\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/7/f/b/7fb8c8347edc5c8c66bba26a016c04735066dbb0.png\" class=\"latex\" alt=\"$ a_{n-1}$\" style=\"vertical-align: -2px\" width=\"33\" height=\"10\" > are 2 -distinct- roots, otherwise we can't use Vietta (but <img src=\"//latex.artofproblemsolving.com/7/6/7/767e098196b53abd83764b9ea8425318e85b1bfc.png\" class=\"latex\" alt=\"$ a_{n+1} > 2a_{n} > 4a_{n-1} >a_{n-1}$\" style=\"vertical-align: -4px\" width=\"214\" height=\"16\" > so that's ok)",
"post_id": 4418381,
"post_number": 2,
"post_time_unix": 1240864499,
"post_time_utc": "2009-04-27 20:34:59 UTC",
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}
|
Prove that for every positive integer \(n\),
\[
\left|\{\,n+\sigma \mid \sigma\mid n\,\}\cap\{\,2n-\sigma \mid \sigma\mid 2n\,\}\right|\le 2.
\]
Let the sequence \(\{a_n\}\) be defined by \(a_1=1\) and for \(n\ge 1\),
\[
a_{n+1}=2a_n+\sqrt{3a_n^2-2}.
\]
Prove that all terms of \(\{a_n\}\) are integers.
|
[
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula",
"/Mathematics/Algebra/NumberTheory/Divisors/Divides",
"/Mathematics/Algebra/NumberTheory/Divisors/Divisor",
"/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/Algebra/NumberTheory/Integers/Integer",
"/Mathematics/Algebra/NumberTheory/Integers/N",
"/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger",
"/Mathematics/Algebra/NumberTheory/Integers/WholeNumber",
"/Mathematics/Algebra/NumberTheory/Sequences/IntegerSequence",
"/Mathematics/Algebra/NumberTheory/Sequences/Sequence",
"/Mathematics/Algebra/Polynomials/VietasFormulas",
"/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic",
"/Mathematics/NumberTheory/Divisors/Divides",
"/Mathematics/NumberTheory/Divisors/Divisor",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/IntegerRelations/IntegerRelation",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus",
"/Mathematics/NumberTheory/Numbers",
"/Mathematics/NumberTheory/Sequences/IntegerSequence",
"/Mathematics/NumberTheory/Sequences/Sequence"
] |
Rewrite the recurrence as a quadratic whose two roots are a_{n+1} and a_{n-1}, then use Vieta to show a_{n+1} is integer by induction.
|
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aops_997722
|
well the gcf of any difference of 2 of these numbers is $ 2^{5^n} \cdot \left( 2^{4 \cdot 5^n} \minus{}1 \right)$
[hide="hehe"]$ 2^{5^n} . \left( 2^{4 . 5^n} \minus{}1 \right)$[/hide]
This is provable by factoring $ x^n\minus{}x^m | n>m$
The former factor will obviously contribute the 2's to $ 10^{n\plus{}1}$, and $ 5^n>n\plus{}1 | n \ge 1$ due to derivative junk
We want to prove that the latter factor has n+1 factors of 5.
Well powers of 2 are periodic mod $ 5^{n\plus{}1}$ in cycles of totient $ 5^{n\plus{}1}$, or $ 4 \cdot 5^n$
This means that $ 2^0$ and $ 2^{4 \cdot 5^n}$ are congruent mod $ 5^{n\plus{}1}$ meaning that the latter factor is divisible by $ 5^{n\plus{}1}$
USAMO score: $ \pi$
| null |
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"content_bbcode": "Number theory!\r\n\r\nFor every positive integer $ n$, prove that the values $ 2^{5^n},2^{5^{n \\plus{} 1}},2^{5^{n \\plus{} 2}},\\dots$ are all equal mod $ 10^{n \\plus{} 1}$.\r\n\r\nIf you want, express the limit of this value as $ n$ approaches infinity, in 10-adic form.",
"content_html": "Number theory!<br>\n<br>\nFor every positive integer <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" >,</span> prove that the values <img src=\"//latex.artofproblemsolving.com/f/6/1/f61c6ab52329c1958c7332787d4f051433eb5a37.png\" class=\"latex\" alt=\"$ 2^{5^n},2^{5^{n + 1}},2^{5^{n + 2}},\\dots$\" style=\"vertical-align: -3px\" width=\"146\" height=\"22\" > are all equal mod <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/e/7/3e7c3de78fce7312e787ac6a57513dd4af0ced8f.png\" class=\"latex\" alt=\"$ 10^{n + 1}$\" style=\"vertical-align: 0px\" width=\"41\" height=\"15\" >.</span><br>\n<br>\nIf you want, express the limit of this value as <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > approaches infinity, in 10-adic form.",
"post_id": 4418524,
"post_number": 1,
"post_time_unix": 1243127665,
"post_time_utc": "2009-05-24 01:14:25 UTC",
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"content_bbcode": "well the gcf of any difference of 2 of these numbers is $ 2^{5^n} \\cdot \\left( 2^{4 \\cdot 5^n} \\minus{}1 \\right)$\r\n[hide=\"hehe\"]$ 2^{5^n} . \\left( 2^{4 . 5^n} \\minus{}1 \\right)$[/hide]\r\nThis is provable by factoring $ x^n\\minus{}x^m | n>m$\r\n\r\nThe former factor will obviously contribute the 2's to $ 10^{n\\plus{}1}$, and $ 5^n>n\\plus{}1 | n \\ge 1$ due to derivative junk\r\nWe want to prove that the latter factor has n+1 factors of 5.\r\nWell powers of 2 are periodic mod $ 5^{n\\plus{}1}$ in cycles of totient $ 5^{n\\plus{}1}$, or $ 4 \\cdot 5^n$\r\nThis means that $ 2^0$ and $ 2^{4 \\cdot 5^n}$ are congruent mod $ 5^{n\\plus{}1}$ meaning that the latter factor is divisible by $ 5^{n\\plus{}1}$\r\n\r\nUSAMO score: $ \\pi$",
"content_html": "well the gcf of any difference of 2 of these numbers is <img src=\"//latex.artofproblemsolving.com/9/f/5/9f530fa2f89fd155780a09366baef711d61824ae.png\" class=\"latex\" alt=\"$ 2^{5^n} \\cdot \\left( 2^{4 \\cdot 5^n} -1 \\right)$\" style=\"vertical-align: -6px\" width=\"118\" height=\"22\" ><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">hehe</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/0/0/a00a025608f910f03631a151bff013d12e9781c1.png\" class=\"latex\" alt=\"$ 2^{5^n} . \\left( 2^{4 . 5^n} -1 \\right)$\" style=\"vertical-align: -6px\" width=\"113\" height=\"22\" ></div><br>\nThis is provable by factoring <img src=\"//latex.artofproblemsolving.com/4/1/e/41e5a44d18a58d7fd33a1b4c66e89298b64239ab.png\" class=\"latex\" alt=\"$ x^n-x^m | n>m$\" style=\"vertical-align: -4px\" width=\"120\" height=\"18\" ><br>\n<br>\nThe former factor will obviously contribute the 2's to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/b/4/2b440fd1b37b86e1668085a0578c95bbb00c9d24.png\" class=\"latex\" alt=\"$ 10^{n+1}$\" style=\"vertical-align: 0px\" width=\"41\" height=\"15\" >,</span> and <img src=\"//latex.artofproblemsolving.com/1/5/a/15a9d43a9983fca7f3246f58c98dfd0ae6782176.png\" class=\"latex\" alt=\"$ 5^n>n+1 | n \\ge 1$\" style=\"vertical-align: -4px\" width=\"133\" height=\"18\" > due to derivative junk<br>\nWe want to prove that the latter factor has n+1 factors of 5.<br>\nWell powers of 2 are periodic mod <img src=\"//latex.artofproblemsolving.com/f/a/3/fa3f66272e70c56f7f8e201217d12726e4fcc574.png\" class=\"latex\" alt=\"$ 5^{n+1}$\" width=\"32\" height=\"15\" > in cycles of totient <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/a/3/fa3f66272e70c56f7f8e201217d12726e4fcc574.png\" class=\"latex\" alt=\"$ 5^{n+1}$\" width=\"32\" height=\"15\" >,</span> or <img src=\"//latex.artofproblemsolving.com/3/e/9/3e9be0a93df654124bb8dbfb9ef9a7301e6d7d6b.png\" class=\"latex\" alt=\"$ 4 \\cdot 5^n$\" style=\"vertical-align: 0px\" width=\"39\" height=\"13\" ><br>\nThis means that <img src=\"//latex.artofproblemsolving.com/3/e/7/3e78bd7ff2ff2d021da173dbf31e62d43570776f.png\" class=\"latex\" alt=\"$ 2^0$\" width=\"15\" height=\"15\" > and <img src=\"//latex.artofproblemsolving.com/4/d/4/4d415ad0b8859f8fb030af30458954a773b05949.png\" class=\"latex\" alt=\"$ 2^{4 \\cdot 5^n}$\" width=\"33\" height=\"16\" > are congruent mod <img src=\"//latex.artofproblemsolving.com/f/a/3/fa3f66272e70c56f7f8e201217d12726e4fcc574.png\" class=\"latex\" alt=\"$ 5^{n+1}$\" width=\"32\" height=\"15\" > meaning that the latter factor is divisible by <img src=\"//latex.artofproblemsolving.com/f/a/3/fa3f66272e70c56f7f8e201217d12726e4fcc574.png\" class=\"latex\" alt=\"$ 5^{n+1}$\" width=\"32\" height=\"15\" ><br>\n<br>\nUSAMO score: <img src=\"//latex.artofproblemsolving.com/e/e/8/ee82cbf599965b97411681a6bf57888d3b969660.png\" class=\"latex\" alt=\"$ \\pi$\" width=\"10\" height=\"8\" >",
"post_id": 4418525,
"post_number": 2,
"post_time_unix": 1243184253,
"post_time_utc": "2009-05-24 16:57:33 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "10-adics don't exist",
"content_html": "10-adics don't exist",
"post_id": 4418526,
"post_number": 3,
"post_time_unix": 1243371183,
"post_time_utc": "2009-05-26 20:53:03 UTC",
"thanks_received": 2,
"user_id": 1,
"username": "Anonymous"
},
{
"attachments": [],
"content_bbcode": "Of course 10-adics exist, they just don't have the properties that prime adics have.",
"content_html": "Of course 10-adics exist, they just don't have the properties that prime adics have.",
"post_id": 4418527,
"post_number": 4,
"post_time_unix": 1243385001,
"post_time_utc": "2009-05-27 00:43:21 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
Number theory.
For every positive integer \(n\), prove that the values
\[
2^{5^n},\ 2^{5^{n+1}},\ 2^{5^{n+2}},\dots
\]
are all congruent modulo \(10^{\,n+1}\).
If desired, express the limit of this value as \(n\) approaches infinity in 10-adic form.
|
[
"/Mathematics/NumberTheory/Congruences/Congruence",
"/Mathematics/NumberTheory/Congruences/Congruent",
"/Mathematics/NumberTheory/Congruences/EulersTotientTheorem",
"/Mathematics/NumberTheory/Congruences/Mod",
"/Mathematics/NumberTheory/Congruences/ModularArithmetic",
"/Mathematics/NumberTheory/Congruences/Modulus",
"/Mathematics/NumberTheory/Congruences/MultiplicativeOrder",
"/Mathematics/NumberTheory/Congruences/Residue",
"/Mathematics/NumberTheory/Congruences/ResidueClass",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory"
] |
Apply Euler’s theorem: 2^{4·5^n}≡1 (mod 5^{n+1}), so differences of exponents are multiples of φ(5^{n+1}) giving congruence modulo 5^{n+1}, and the 2‑power factor supplies the needed 2‑adic part.
|
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0.00885009765625
] |
aops_997731
|
in all seriousness $ \sum_{i \equal{} 1}^{\infty}\frac {\lfloor10^{2n \minus{} 1}\pi \rfloor \minus{} 10\lfloor10^{2n \minus{} 2}\pi\rfloor}{10^{2n \minus{} 1}}$. there's probably a way to simplify that (hermite's identity?) but i am feeling lazy.
also in response to your new rule, let $ x\equal{}3.45255992866\dots$. now we have a named constant. in terms of pi, your number is $ 0\pi\plus{}x$.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Consider the number $ \\pi \\equal{} 3.14159265358979323846264\\dots$\r\n\r\nLet's form another number by taking $ \\pi$ and removing every other digit starting with $ 1$, so this new number is:\r\n\\[ 3.45255992866\\dots\r\n\\]\r\nExpress this number in terms of $ \\pi$.",
"content_html": "Consider the number <img src=\"//latex.artofproblemsolving.com/4/4/0/440ff43d1af655b1f7e7c9ec06d7333980b95456.png\" class=\"latex\" alt=\"$ \\pi = 3.14159265358979323846264\\dots$\" style=\"vertical-align: 0px\" width=\"281\" height=\"13\" ><br>\n<br>\nLet's form another number by taking <img src=\"//latex.artofproblemsolving.com/e/e/8/ee82cbf599965b97411681a6bf57888d3b969660.png\" class=\"latex\" alt=\"$ \\pi$\" width=\"10\" height=\"8\" > and removing every other digit starting with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/9/0/39064bdd89b3dfa0626ca59d843d926ea072830b.png\" class=\"latex\" alt=\"$ 1$\" style=\"vertical-align: 0px\" width=\"8\" height=\"12\" >,</span> so this new number is:<br>\n<img src=\"//latex.artofproblemsolving.com/3/c/3/3c3fcbea06c4160dab1f4619589e00e7d459c0a9.png\" class=\"latexcenter\" alt=\"\\[ 3.45255992866\\dots\n\\]\" width=\"136\" height=\"13\" ><br>\nExpress this number in terms of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/8/ee82cbf599965b97411681a6bf57888d3b969660.png\" class=\"latex\" alt=\"$ \\pi$\" width=\"10\" height=\"8\" >.</span>",
"post_id": 4418548,
"post_number": 1,
"post_time_unix": 1243797530,
"post_time_utc": "2009-05-31 19:18:50 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "$ \\pi \\minus{} \\pi \\plus{} 3.45255992866\\dots$",
"content_html": "<img src=\"//latex.artofproblemsolving.com/c/4/b/c4b3710ba505bd58fc30a6f4e63466bf5912f986.png\" class=\"latex\" alt=\"$ \\pi - \\pi + 3.45255992866\\dots$\" style=\"vertical-align: -1px\" width=\"203\" height=\"14\" >",
"post_id": 4418549,
"post_number": 2,
"post_time_unix": 1243797853,
"post_time_utc": "2009-05-31 19:24:13 UTC",
"thanks_received": 2,
"user_id": 28419,
"username": "Temperal"
},
{
"attachments": [],
"content_bbcode": "Lawlz......\r\n\r\nNew rule: No constants with nonrepeating decimal forms are allowed except for named constants.",
"content_html": "Lawlz......<br>\n<br>\nNew rule: No constants with nonrepeating decimal forms are allowed except for named constants.",
"post_id": 4418550,
"post_number": 3,
"post_time_unix": 1243797914,
"post_time_utc": "2009-05-31 19:25:14 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "in all seriousness $ \\sum_{i \\equal{} 1}^{\\infty}\\frac {\\lfloor10^{2n \\minus{} 1}\\pi \\rfloor \\minus{} 10\\lfloor10^{2n \\minus{} 2}\\pi\\rfloor}{10^{2n \\minus{} 1}}$. there's probably a way to simplify that (hermite's identity?) but i am feeling lazy.\r\n\r\nalso in response to your new rule, let $ x\\equal{}3.45255992866\\dots$. now we have a named constant. in terms of pi, your number is $ 0\\pi\\plus{}x$.",
"content_html": "in all seriousness <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/1/471a75a25f7f00bf83bd069480f790fe14e25508.png\" class=\"latex\" alt=\"$ \\sum_{i = 1}^{\\infty}\\frac {\\lfloor10^{2n - 1}\\pi \\rfloor - 10\\lfloor10^{2n - 2}\\pi\\rfloor}{10^{2n - 1}}$\" style=\"vertical-align: -20px\" width=\"227\" height=\"48\" >.</span> there's probably a way to simplify that (hermite's identity?) but i am feeling lazy.<br>\n<br>\nalso in response to your new rule, let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/9/d/69dfb14bb5f79f860863aa5f4ac51d1f60cea16a.png\" class=\"latex\" alt=\"$ x=3.45255992866\\dots$\" style=\"vertical-align: 0px\" width=\"171\" height=\"13\" >.</span> now we have a named constant. in terms of pi, your number is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/e/3/4e39dddd24cf6c83f526cf9d4331735baba5f599.png\" class=\"latex\" alt=\"$ 0\\pi+x$\" style=\"vertical-align: -1px\" width=\"52\" height=\"14\" >.</span>",
"post_id": 4418551,
"post_number": 4,
"post_time_unix": 1243798464,
"post_time_utc": "2009-05-31 19:34:24 UTC",
"thanks_received": 2,
"user_id": 28419,
"username": "Temperal"
}
],
"source": null
}
|
Consider the number \(\pi = 3.14159265358979323846264\ldots\).
Form another number by taking \(\pi\) and removing every other digit starting with the first digit after the decimal point, yielding
\[
3.45255992866\ldots
\]
Express this number in terms of \(\pi\).
|
[
"/Mathematics/RecreationalMathematics/MathematicalHumor/Pi",
"/Mathematics/RecreationalMathematics/MathematicalHumor/PiWordplay",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Extract alternating decimal digits of π with floor(10^k π) and sum them as a series
|
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aops_997732
|
YAY I ACTUALLY GOT A TRIVIAL PROBLEM.
Draw OP. Q.E.D.
[hide="Slightly longer"]OP is an angle bisector, so incenter lies on it. Also, since OAPB is cyclic, and OAP is 90 degrees, the circumcenter lies on the diamter OP (midpoint actually.) So they all lie on OP yay.[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Unlike my pi problem earlier...\r\n\r\nTrivial geometry problem:\r\n\r\nCircle $ \\omega$ is drawn, and point $ P$ is drawn anywhere outside $ \\omega$. Tangents from $ P$ to $ \\omega$ are drawn at $ A$ and $ B$. Prove that the center of $ \\omega$, the circumcenter of $ \\triangle ABP$, and the incenter of $ \\triangle ABP$ are collinear.\r\n\r\nWow, geometry problems could not get more trivial than this...",
"content_html": "Unlike my pi problem earlier...<br>\n<br>\nTrivial geometry problem:<br>\n<br>\nCircle <img src=\"//latex.artofproblemsolving.com/d/a/8/da815ae262e1cda637d8659e7a754e9dbfa8a959.png\" class=\"latex\" alt=\"$ \\omega$\" width=\"11\" height=\"8\" > is drawn, and point <img src=\"//latex.artofproblemsolving.com/f/b/1/fb1b7d554ea7d9094511c084a2682639363022f5.png\" class=\"latex\" alt=\"$ P$\" width=\"14\" height=\"12\" > is drawn anywhere outside <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/8/da815ae262e1cda637d8659e7a754e9dbfa8a959.png\" class=\"latex\" alt=\"$ \\omega$\" width=\"11\" height=\"8\" >.</span> Tangents from <img src=\"//latex.artofproblemsolving.com/f/b/1/fb1b7d554ea7d9094511c084a2682639363022f5.png\" class=\"latex\" alt=\"$ P$\" width=\"14\" height=\"12\" > to <img src=\"//latex.artofproblemsolving.com/d/a/8/da815ae262e1cda637d8659e7a754e9dbfa8a959.png\" class=\"latex\" alt=\"$ \\omega$\" width=\"11\" height=\"8\" > are drawn at <img src=\"//latex.artofproblemsolving.com/a/f/a/afa1e039a54a9d6ce6141d013c997827f98f4add.png\" class=\"latex\" alt=\"$ A$\" width=\"13\" height=\"13\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/d/bcdab73bee32e76511e76ed8ff9633e3611e5df6.png\" class=\"latex\" alt=\"$ B$\" width=\"14\" height=\"12\" >.</span> Prove that the center of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/8/da815ae262e1cda637d8659e7a754e9dbfa8a959.png\" class=\"latex\" alt=\"$ \\omega$\" width=\"11\" height=\"8\" >,</span> the circumcenter of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/c/3/9c31b6aa5f1ae7b712b68c25685539a09dbc3242.png\" class=\"latex\" alt=\"$ \\triangle ABP$\" width=\"58\" height=\"13\" >,</span> and the incenter of <img src=\"//latex.artofproblemsolving.com/9/c/3/9c31b6aa5f1ae7b712b68c25685539a09dbc3242.png\" class=\"latex\" alt=\"$ \\triangle ABP$\" width=\"58\" height=\"13\" > are collinear.<br>\n<br>\nWow, geometry problems could not get more trivial than this...",
"post_id": 4418552,
"post_number": 1,
"post_time_unix": 1243798764,
"post_time_utc": "2009-05-31 19:39:24 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "fail\r\n\r\n$ \\text{fail}$",
"content_html": "fail<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/3/b/5/3b5c303e635e6c8a052f9564349a63d17e62180e.png\" class=\"latex\" alt=\"$ \\text{fail}$\" style=\"vertical-align: 0px\" width=\"22\" height=\"12\" >",
"post_id": 4418553,
"post_number": 2,
"post_time_unix": 1243799664,
"post_time_utc": "2009-05-31 19:54:24 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "YAY I ACTUALLY GOT A TRIVIAL PROBLEM. \r\n\r\nDraw OP. Q.E.D.\r\n\r\n[hide=\"Slightly longer\"]OP is an angle bisector, so incenter lies on it. Also, since OAPB is cyclic, and OAP is 90 degrees, the circumcenter lies on the diamter OP (midpoint actually.) So they all lie on OP yay.[/hide]",
"content_html": "YAY I ACTUALLY GOT A TRIVIAL PROBLEM.<br>\n<br>\nDraw OP. Q.E.D.<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Slightly longer</a><div class=\"cmty-hide-content\" style=\"display:none\">OP is an angle bisector, so incenter lies on it. Also, since OAPB is cyclic, and OAP is 90 degrees, the circumcenter lies on the diamter OP (midpoint actually.) So they all lie on OP yay.</div>",
"post_id": 4418554,
"post_number": 3,
"post_time_unix": 1243997984,
"post_time_utc": "2009-06-03 02:59:44 UTC",
"thanks_received": 2,
"user_id": 35129,
"username": "Zhero"
}
],
"source": null
}
|
Circle \(\omega\) is given, and point \(P\) is any point outside \(\omega\). Tangents from \(P\) to \(\omega\) touch \(\omega\) at \(A\) and \(B\). Prove that the center of \(\omega\), the circumcenter of triangle \(ABP\), and the incenter of triangle \(ABP\) are collinear.
|
[
"/Mathematics/Geometry/GeneralGeometry/Bisector",
"/Mathematics/Geometry/GeneralGeometry/Center",
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circumcenter",
"/Mathematics/Geometry/PlaneGeometry/Circles/Incenter",
"/Mathematics/Geometry/PlaneGeometry/CircularTriangles/CircularTriangle",
"/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters",
"/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles",
"/Mathematics/Geometry/Points/Midpoint",
"/Mathematics/Geometry/Points/Point"
] |
Use the right‑angle tangent property to prove O,A,B,P are cyclic with OP as a diameter, forcing OP to contain both the circumcenter and the angle‑bisector (incenter).
|
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aops_997748
|
Change $ 8$ to an arbitrary positive even integer and I bet this is an IMO 2/5 level problem or higher. I'm almost positive of the answer, but haven't nailed it down. (probably won't get the time to soon either)
If it's $ 2n \times 2n \times 2n$, it should be $ 2n^2$. Take two $ n \times n \times n$ cubes that share a vertex at the center and oppose each other. Put rooks in each one in the manner described in my solution to the previous rook problem (so each has $ n^2$). This will cover the entire board.
To show this is the minimum, I believe you start by supposing not and looking at a planar cross section with at most $ n\minus{}1$ rooks (which must exist). Then there are $ (n\plus{}1)^2$ spaces in that plane that must be covered by other rooks, so now we're up to $ (n\minus{}1) \plus{} (n\plus{}1)^2$. Then you do other stuff with the $ (n\minus{}1)^2 \minus{} (n\minus{}1)$ spaces where the first $ n\minus{}1$ rooks (or less) were, and good things happen. But there's so many possible configurations to account for...
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Now, instead of looking for the maximum number of rooks that can be placed without any two attacking each other, we are looking for the minimum number of rooks that can be placed on a chessboard such that [i]every[/i] cell is either occupied by a rook or attacked by a rook.\r\n\r\nFor $ 8\\times 8$, it is still $ 8$. But what about $ 8\\times 8 \\times 8$?",
"content_html": "Now, instead of looking for the maximum number of rooks that can be placed without any two attacking each other, we are looking for the minimum number of rooks that can be placed on a chessboard such that <i>every</i> cell is either occupied by a rook or attacked by a rook.<br>\n<br>\nFor <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/6/e/36e359df72d997aadb25e37c309f1ec744bc5162.png\" class=\"latex\" alt=\"$ 8\\times 8$\" width=\"40\" height=\"12\" >,</span> it is still <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/d/1/6d15585fa5b808a51e13da80401bc6fee33184d8.png\" class=\"latex\" alt=\"$ 8$\" width=\"8\" height=\"12\" >.</span> But what about <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/7/e/97ea85e6ec707d58b2cb3a81582d4b97d3fa9c17.png\" class=\"latex\" alt=\"$ 8\\times 8 \\times 8$\" width=\"71\" height=\"12\" >?</span>",
"post_id": 4418595,
"post_number": 1,
"post_time_unix": 1244754260,
"post_time_utc": "2009-06-11 21:04:20 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Change $ 8$ to an arbitrary positive even integer and I bet this is an IMO 2/5 level problem or higher. I'm almost positive of the answer, but haven't nailed it down. (probably won't get the time to soon either)\r\n\r\nIf it's $ 2n \\times 2n \\times 2n$, it should be $ 2n^2$. Take two $ n \\times n \\times n$ cubes that share a vertex at the center and oppose each other. Put rooks in each one in the manner described in my solution to the previous rook problem (so each has $ n^2$). This will cover the entire board.\r\n\r\nTo show this is the minimum, I believe you start by supposing not and looking at a planar cross section with at most $ n\\minus{}1$ rooks (which must exist). Then there are $ (n\\plus{}1)^2$ spaces in that plane that must be covered by other rooks, so now we're up to $ (n\\minus{}1) \\plus{} (n\\plus{}1)^2$. Then you do other stuff with the $ (n\\minus{}1)^2 \\minus{} (n\\minus{}1)$ spaces where the first $ n\\minus{}1$ rooks (or less) were, and good things happen. But there's so many possible configurations to account for...",
"content_html": "Change <img src=\"//latex.artofproblemsolving.com/6/d/1/6d15585fa5b808a51e13da80401bc6fee33184d8.png\" class=\"latex\" alt=\"$ 8$\" width=\"8\" height=\"12\" > to an arbitrary positive even integer and I bet this is an IMO 2/5 level problem or higher. I'm almost positive of the answer, but haven't nailed it down. (probably won't get the time to soon either)<br>\n<br>\nIf it's <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/f/a1f2f48bc544be9b7a286c8eacb2c763b8b31353.png\" class=\"latex\" alt=\"$ 2n \\times 2n \\times 2n$\" width=\"104\" height=\"12\" >,</span> it should be <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/a/5/3a5655758c6cc13cd7255f1759c2ca396b85e67b.png\" class=\"latex\" alt=\"$ 2n^2$\" width=\"26\" height=\"15\" >.</span> Take two <img src=\"//latex.artofproblemsolving.com/c/3/9/c39274a2e47507828a17590f87269bedf9d12125.png\" class=\"latex\" alt=\"$ n \\times n \\times n$\" width=\"77\" height=\"9\" > cubes that share a vertex at the center and oppose each other. Put rooks in each one in the manner described in my solution to the previous rook problem (so each has <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/f/4/bf4fac4d2c566d5728b8388b9d56c90e11496b83.png\" class=\"latex\" alt=\"$ n^2$\" width=\"17\" height=\"15\" >)</span>. This will cover the entire board.<br>\n<br>\nTo show this is the minimum, I believe you start by supposing not and looking at a planar cross section with at most <img src=\"//latex.artofproblemsolving.com/6/7/3/673149a23ba4295562d199482862afb9e118dc21.png\" class=\"latex\" alt=\"$ n-1$\" style=\"vertical-align: 0px\" width=\"41\" height=\"12\" > rooks (which must exist). Then there are <img src=\"//latex.artofproblemsolving.com/9/9/2/992306c79bc8c94d49042b085251015f5a071d08.png\" class=\"latex\" alt=\"$ (n+1)^2$\" style=\"vertical-align: -4px\" width=\"62\" height=\"19\" > spaces in that plane that must be covered by other rooks, so now we're up to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/9/e/f9e7aab6b62956400b77107479d08c4bcd403761.png\" class=\"latex\" alt=\"$ (n-1) + (n+1)^2$\" style=\"vertical-align: -4px\" width=\"141\" height=\"19\" >.</span> Then you do other stuff with the <img src=\"//latex.artofproblemsolving.com/3/d/0/3d08da171c3fe8adf1f4da3fc5d2e21ec2619958.png\" class=\"latex\" alt=\"$ (n-1)^2 - (n-1)$\" style=\"vertical-align: -4px\" width=\"140\" height=\"19\" > spaces where the first <img src=\"//latex.artofproblemsolving.com/6/7/3/673149a23ba4295562d199482862afb9e118dc21.png\" class=\"latex\" alt=\"$ n-1$\" style=\"vertical-align: 0px\" width=\"41\" height=\"12\" > rooks (or less) were, and good things happen. But there's so many possible configurations to account for...",
"post_id": 4418596,
"post_number": 2,
"post_time_unix": 1244846276,
"post_time_utc": "2009-06-12 22:37:56 UTC",
"thanks_received": 2,
"user_id": 29126,
"username": "MellowMelon"
}
],
"source": null
}
|
Now, instead of looking for the maximum number of rooks that can be placed without any two attacking each other, we are looking for the minimum number of rooks that can be placed on a chessboard such that every cell is either occupied by a rook or attacked by a rook.
For \(8\times 8\), it is still \(8\). But what about \(8\times 8\times 8\)?
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization",
"/Mathematics/DiscreteMathematics/Combinatorics/Configurations",
"/Mathematics/DiscreteMathematics/Combinatorics/Covers",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics",
"/Mathematics/DiscreteMathematics/PackingProblems/Packing",
"/Mathematics/RecreationalMathematics/Games/BoardGames/Chess",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Divide the large cube into two opposite n×n×n corner subcubes and dominate each with n² rooks, covering the whole board.
|
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aops_997757
|
Um, $ 4 \cos^3 \theta \minus{} 3 \cos \theta \equal{} \cos 3\theta$?
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Prove that\r\n\\[ 1 \\plus{} 6\\cos 20^\\circ \\equal{} 8\\cos^3 20^\\circ.\r\n\\]\r\n[hide]\nBoo! :ninja: [/hide]",
"content_html": "Prove that<br>\n<img src=\"//latex.artofproblemsolving.com/d/7/d/d7dca490c54e18444043bc9e5b5dfaefab567c6c.png\" class=\"latexcenter\" alt=\"\\[ 1 + 6\\cos 20^\\circ = 8\\cos^3 20^\\circ.\n\\]\" width=\"195\" height=\"16\" ><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(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\">Boo! <img src=\"/assets/images/smilies/ph34r.gif\" width=\"20\" height=\"20\" alt=\":ninja:\" title=\":ninja:\" class=\"bbcode_smiley\" /></div>",
"post_id": 4418624,
"post_number": 1,
"post_time_unix": 1246141749,
"post_time_utc": "2009-06-27 22:29:09 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "Um, $ 4 \\cos^3 \\theta \\minus{} 3 \\cos \\theta \\equal{} \\cos 3\\theta$?",
"content_html": "Um, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/0/470535625c1ea30161bf6848fcadeaba0886b6b8.png\" class=\"latex\" alt=\"$ 4 \\cos^3 \\theta - 3 \\cos \\theta = \\cos 3\\theta$\" style=\"vertical-align: 0px\" width=\"196\" height=\"15\" >?</span>",
"post_id": 4418625,
"post_number": 2,
"post_time_unix": 1246154025,
"post_time_utc": "2009-06-28 01:53:45 UTC",
"thanks_received": 2,
"user_id": 29126,
"username": "MellowMelon"
},
{
"attachments": [],
"content_bbcode": "The fact that 20=60/3 and 60 is a nice number screams triple angle stuffs.\r\nso we are like do cos3x and use tons of crap and we get $ \\cos 3x \\equal{} 4\\cos ^3 x \\minus{} 3 \\cos x$, so $ 2\\cos 60 \\equal{} 1 \\equal{} 8\\cos ^3 20 \\minus{} 6 \\cos 20$ so we win.\r\n\r\nEDIT: darn mellon beat me to it",
"content_html": "The fact that 20=60/3 and 60 is a nice number screams triple angle stuffs.<br>\nso we are like do cos3x and use tons of crap and we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/f/cef0ab4543e4983a2b14de7cc0c6ea62a8104654.png\" class=\"latex\" alt=\"$ \\cos 3x = 4\\cos ^3 x - 3 \\cos x$\" style=\"vertical-align: 0px\" width=\"200\" height=\"15\" >,</span> so <img src=\"//latex.artofproblemsolving.com/a/7/9/a79e59c24ae2597d7566f12426b5a3792289016a.png\" class=\"latex\" alt=\"$ 2\\cos 60 = 1 = 8\\cos ^3 20 - 6 \\cos 20$\" style=\"vertical-align: 0px\" width=\"259\" height=\"15\" > so we win.<br>\n<br>\nEDIT: darn mellon beat me to it",
"post_id": 4418626,
"post_number": 3,
"post_time_unix": 1246154067,
"post_time_utc": "2009-06-28 01:54:27 UTC",
"thanks_received": 1,
"user_id": 31435,
"username": "abacadaea"
},
{
"attachments": [],
"content_bbcode": "Haha lol [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=285259]here[/url] is a purely geometrical proof (and some algebra) :P\r\n\r\nJust equate $ b(b^{2}\\minus{}2a^{2})/a^{2}$ with $ a\\plus{}b$, and let $ a\\equal{}1$ and $ b\\equal{}2\\cos{20^\\circ}$ :P",
"content_html": "Haha lol <a href=\"http://www.artofproblemsolving.com/Forum/viewtopic.php?t=285259\" class=\"bbcode_url\" target=\"_blank\">here</a> is a purely geometrical proof (and some algebra) <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /><br>\n<br>\nJust equate <img src=\"//latex.artofproblemsolving.com/6/8/3/683e3b134a1c10f8e415138b8eda75a335bae552.png\" class=\"latex\" alt=\"$ b(b^{2}-2a^{2})/a^{2}$\" style=\"vertical-align: -4px\" width=\"110\" height=\"19\" > with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/8/6/086518282cc39fa39d9b406ffafcd5ec7d588593.png\" class=\"latex\" alt=\"$ a+b$\" style=\"vertical-align: -1px\" width=\"40\" height=\"14\" >,</span> and let <img src=\"//latex.artofproblemsolving.com/1/8/d/18d8f702345fe76500ee6591f259225e0915bc6a.png\" class=\"latex\" alt=\"$ a=1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/6/8/3/683c6773b34f99c986e83dccd1dbbb6dcafa8f1c.png\" class=\"latex\" alt=\"$ b=2\\cos{20^\\circ}$\" width=\"95\" height=\"13\" > <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4418627,
"post_number": 4,
"post_time_unix": 1246154718,
"post_time_utc": "2009-06-28 02:05:18 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "wait, can you assume that? its regular...",
"content_html": "wait, can you assume that? its regular...",
"post_id": 4418628,
"post_number": 5,
"post_time_unix": 1246155412,
"post_time_utc": "2009-06-28 02:16:52 UTC",
"thanks_received": 1,
"user_id": 31435,
"username": "abacadaea"
},
{
"attachments": [],
"content_bbcode": "Well $ a\\equal{}1$ implies $ b\\equal{}2\\cos 20^\\circ$...",
"content_html": "Well <img src=\"//latex.artofproblemsolving.com/1/8/d/18d8f702345fe76500ee6591f259225e0915bc6a.png\" class=\"latex\" alt=\"$ a=1$\" style=\"vertical-align: 0px\" width=\"42\" height=\"12\" > implies <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/3/9/f39ea221fac7ccab31778555d9d8e3c3cceac8fa.png\" class=\"latex\" alt=\"$ b=2\\cos 20^\\circ$\" width=\"95\" height=\"13\" >.</span>..",
"post_id": 4418629,
"post_number": 6,
"post_time_unix": 1246157796,
"post_time_utc": "2009-06-28 02:56:36 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "well i was being lazy, thats all :P",
"content_html": "well i was being lazy, thats all <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4418630,
"post_number": 7,
"post_time_unix": 1246227027,
"post_time_utc": "2009-06-28 22:10:27 UTC",
"thanks_received": 1,
"user_id": 31435,
"username": "abacadaea"
}
],
"source": null
}
|
Prove that
\[
1 + 6\cos 20^\circ = 8\cos^3 20^\circ.
\]
|
[
"/Mathematics/Algebra/AlgebraicEquations",
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity",
"/Mathematics/Algebra/Polynomials/CubicEquation",
"/Mathematics/Algebra/Polynomials/PolynomialIdentity",
"/Mathematics/RecreationalMathematics"
] |
Apply the cosine triple-angle formula to θ = 20° and use cos 60° = ½ to relate 1 + 6cos20° and 8cos³20°.
|
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aops_997761
|
homo....thety
so i'm assuming you mean C is the intersection point on the same side of AX as B
you see then, A,B,C are collinear iff AXC~AMB (because BM||XC and A,M,X are collinear), so AC/AB=AX/AM=2. Then B must be the midpoint of chord AC, which means that the locus of points B is the midpoints of all chords passing through A. Which is a dilation of circle O by a factor of 1/2 through A.
w/e
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
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"content_bbcode": "Circle $ O$ is given. Point $ A$ is any fixed point on $ O$. Let $ X$ be any point on the line through $ OA$ and in the interior of circle $ O$, and $ M$ be the midpoint of $ XA$. Find the locus of points $ B$ in the interior of circle $ O$ such that, if the parallel to $ BM$ through $ X$ intersects circle $ O$ at $ C$, the points $ A,B,C$ are collinear.",
"content_html": "Circle <img src=\"//latex.artofproblemsolving.com/7/b/6/7b675f3af043c2f7a4e8646bc7acaa7af544ff56.png\" class=\"latex\" alt=\"$ O$\" width=\"13\" height=\"12\" > is given. Point <img src=\"//latex.artofproblemsolving.com/a/f/a/afa1e039a54a9d6ce6141d013c997827f98f4add.png\" class=\"latex\" alt=\"$ A$\" width=\"13\" height=\"13\" > is any fixed point on <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/b/6/7b675f3af043c2f7a4e8646bc7acaa7af544ff56.png\" class=\"latex\" alt=\"$ O$\" width=\"13\" height=\"12\" >.</span> Let <img src=\"//latex.artofproblemsolving.com/1/9/2/19223fc42c69e613a4cdc9c7fddd725b28b68dc6.png\" class=\"latex\" alt=\"$ X$\" width=\"15\" height=\"12\" > be any point on the line through <img src=\"//latex.artofproblemsolving.com/2/8/b/28b6e3eb6efcca94ad8d6207fc6f3a6f377ea028.png\" class=\"latex\" alt=\"$ OA$\" width=\"28\" height=\"13\" > and in the interior of circle <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/b/6/7b675f3af043c2f7a4e8646bc7acaa7af544ff56.png\" class=\"latex\" alt=\"$ O$\" width=\"13\" height=\"12\" >,</span> and <img src=\"//latex.artofproblemsolving.com/2/3/2/23293d929381045b3b629c3bdfc5d8d61362dcda.png\" class=\"latex\" alt=\"$ M$\" width=\"19\" height=\"12\" > be the midpoint of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/7/9/9798ba7ef0e1f75639ad3a170fd89bb77e56d9ea.png\" class=\"latex\" alt=\"$ XA$\" width=\"30\" height=\"13\" >.</span> Find the locus of points <img src=\"//latex.artofproblemsolving.com/b/c/d/bcdab73bee32e76511e76ed8ff9633e3611e5df6.png\" class=\"latex\" alt=\"$ B$\" width=\"14\" height=\"12\" > in the interior of circle <img src=\"//latex.artofproblemsolving.com/7/b/6/7b675f3af043c2f7a4e8646bc7acaa7af544ff56.png\" class=\"latex\" alt=\"$ O$\" width=\"13\" height=\"12\" > such that, if the parallel to <img src=\"//latex.artofproblemsolving.com/4/1/4/414f30bd5de6703413311626a1d3a1ebc4e5f258.png\" class=\"latex\" alt=\"$ BM$\" width=\"33\" height=\"12\" > through <img src=\"//latex.artofproblemsolving.com/1/9/2/19223fc42c69e613a4cdc9c7fddd725b28b68dc6.png\" class=\"latex\" alt=\"$ X$\" width=\"15\" height=\"12\" > intersects circle <img src=\"//latex.artofproblemsolving.com/7/b/6/7b675f3af043c2f7a4e8646bc7acaa7af544ff56.png\" class=\"latex\" alt=\"$ O$\" width=\"13\" height=\"12\" > at <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/7/0e78e1fa5523edccdfff7441e3889d048aaee5f6.png\" class=\"latex\" alt=\"$ C$\" width=\"14\" height=\"12\" >,</span> the points <img src=\"//latex.artofproblemsolving.com/4/b/e/4bead2a3278240d9276d8fba07f695f4de54c3c8.png\" class=\"latex\" alt=\"$ A,B,C$\" style=\"vertical-align: -3px\" width=\"58\" height=\"16\" > are collinear.",
"post_id": 4418641,
"post_number": 1,
"post_time_unix": 1246589077,
"post_time_utc": "2009-07-03 02:44:37 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
},
{
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"content_bbcode": "homo....thety\r\nso i'm assuming you mean C is the intersection point on the same side of AX as B\r\nyou see then, A,B,C are collinear iff AXC~AMB (because BM||XC and A,M,X are collinear), so AC/AB=AX/AM=2. Then B must be the midpoint of chord AC, which means that the locus of points B is the midpoints of all chords passing through A. Which is a dilation of circle O by a factor of 1/2 through A.\r\nw/e",
"content_html": "homo....thety<br>\nso i'm assuming you mean C is the intersection point on the same side of AX as B<br>\nyou see then, A,B,C are collinear iff AXC~AMB (because BM||XC and A,M,X are collinear), so AC/AB=AX/AM=2. Then B must be the midpoint of chord AC, which means that the locus of points B is the midpoints of all chords passing through A. Which is a dilation of circle O by a factor of 1/2 through A.<br>\nw/e",
"post_id": 4418642,
"post_number": 2,
"post_time_unix": 1246731233,
"post_time_utc": "2009-07-04 18:13:53 UTC",
"thanks_received": 1,
"user_id": 47992,
"username": "Sephiroth"
}
],
"source": null
}
|
Circle \(O\) is given. Point \(A\) is a fixed point on \(O\). Let \(X\) be any point on the line \(OA\) and in the interior of circle \(O\), and let \(M\) be the midpoint of \(XA\). Find the locus of points \(B\) in the interior of circle \(O\) such that, if the line through \(X\) parallel to \(BM\) meets circle \(O\) again at \(C\), then \(A,B,C\) are collinear.
|
[
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle-PointMidpointTheorem",
"/Mathematics/Geometry/PlaneGeometry/Circles/CircleDivisionbyChords",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Homothecy",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Homothetic",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/HomotheticCenter",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/RatioofMagnification",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Scaling",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similar",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similarity",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/SimilitudeCenter",
"/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/SimilitudeRatio",
"/Mathematics/Geometry/Points/Locus",
"/Mathematics/Geometry/Points/Midpoint"
] |
Use BM ∥ XC to obtain triangle similarity, which forces B to be the midpoint of chord AC through A.
|
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aops_997774
|
[quote="stevenmeow"]or, by matching coefficients, that
$ \sum_{k \equal{} 0}^{n} \binom{n}{k} ( \minus{} 1)^k k^x \equal{} 0$[/quote]
Actually, by matching coefficients, we find the coefficient of $ x^x$ (:stink:) to be...
Let's do that again. From your quote, let's change your "x" to "y". So now it would read:
[quote="stevenmeow edited by Yongyi781"]or, by matching coefficients, that
$ \sum_{k \equal{} 0}^{n} \binom{n}{k} ( \minus{} 1)^k k^y \equal{} 0$[/quote]
Actually, by matching coefficients, we find the coefficient of $ x^y$ to be:
\[ \sum_{k \equal{} 0}^{n} \binom{n}{k} \binom{n}{y} ( \minus{} 1)^k k^{n\minus{}y} \equal{} 0.\]
(The coefficient of $ x^y$ in the expansion of $ (x\plus{}k)^n$ is $ \binom ny k^{n\minus{}y}$. Therefore, the coefficient of $ x^y$ in $ \binom nk(\minus{}1)^k(x\plus{}k)^n$ is $ \binom nk\binom ny(\minus{}1)^kk^{n\minus{}y}$.)
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "[hide=\"Hidden by Yongyi781\"]so i conjectured that the $ c_1$ are binomial coefficients times a real root of unity.\n\nhere is the proof\n\nfor any n, we only need to prove that \n$ \\sum_{k \\equal{} 0}^{n \\minus{} 1} \\binom{n}{k} ( \\minus{} 1)^{n \\minus{} k \\plus{} 1} (x \\plus{} k)^n \\equal{} (x \\plus{} n)^n$\nif that wasn't clear, the coefficients for, say, n=5 are 1, -5, 10, -10, 5 with the last term being positive always.\n\nThis is the same as proving that\n$ \\sum_{k \\equal{} 0}^{n} \\binom{n}{k} ( \\minus{} 1)^{k} (x \\plus{} k)^n \\equal{} 0$\n\nor, by matching coefficients, that\n$ \\sum_{k \\equal{} 0}^{n} \\binom{n}{k} ( \\minus{} 1)^k k^x \\equal{} 0$\nfor all $ x| 0 \\le x < n$\nBut this sum is just the $ n^{th}$ difference of the equation with degree $ x$\nSince $ n > x$, this difference is always $ 0$ $ \\blacksquare$\n[hide]\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n$ \\blacksquare$\n\n[/hide][/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Hidden by Yongyi781</a><div class=\"cmty-hide-content\" style=\"display:none\">so i conjectured that the <img src=\"//latex.artofproblemsolving.com/4/8/5/4858a14c1a16e4b864298297ad82b14a4b31e590.png\" class=\"latex\" alt=\"$ c_1$\" style=\"vertical-align: -2px\" width=\"13\" height=\"10\" > are binomial coefficients times a real root of unity.<br>\n<br>\nhere is the proof<br>\n<br>\nfor any n, we only need to prove that<br>\n<img src=\"//latex.artofproblemsolving.com/9/c/4/9c48b9529f3970efd229c64c8b360abe60d3f60c.png\" class=\"latex\" alt=\"$ \\sum_{k = 0}^{n - 1} \\binom{n}{k} ( - 1)^{n - k + 1} (x + k)^n = (x + n)^n$\" style=\"vertical-align: -22px\" width=\"304\" height=\"53\" ><br>\nif that wasn't clear, the coefficients for, say, n=5 are 1, -5, 10, -10, 5 with the last term being positive always.<br>\n<br>\nThis is the same as proving that<br>\n<img src=\"//latex.artofproblemsolving.com/9/1/5/915fcb8799148f13d4c0908e5cba970f59a95576.png\" class=\"latex\" alt=\"$ \\sum_{k = 0}^{n} \\binom{n}{k} ( - 1)^{k} (x + k)^n = 0$\" style=\"vertical-align: -22px\" width=\"212\" height=\"53\" ><br>\n<br>\nor, by matching coefficients, that<br>\n<img src=\"//latex.artofproblemsolving.com/9/1/c/91c3948b0ec01251af8ffb1d17c6f0b638de9f5a.png\" class=\"latex\" alt=\"$ \\sum_{k = 0}^{n} \\binom{n}{k} ( - 1)^k k^x = 0$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ><br>\nfor all <img src=\"//latex.artofproblemsolving.com/e/f/8/ef841846740a144ca53f665cfbc02f6930997307.png\" class=\"latex\" alt=\"$ x| 0 \\le x < n$\" style=\"vertical-align: -4px\" width=\"94\" height=\"18\" ><br>\nBut this sum is just the <img src=\"//latex.artofproblemsolving.com/3/f/2/3f2945c1f6e5dab81789b9ae6d3cf5b3bb907da1.png\" class=\"latex\" alt=\"$ n^{th}$\" width=\"23\" height=\"15\" > difference of the equation with degree <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" ><br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/f/b/0fbb7f5d7d4eb390b7a9b03a7ba8c5cbf830e80c.png\" class=\"latex\" alt=\"$ n > x$\" style=\"vertical-align: 0px\" width=\"45\" height=\"10\" >,</span> this difference is always <img src=\"//latex.artofproblemsolving.com/d/6/0/d6033a0bb0547c396d9c45af99df7a43af61eb69.png\" class=\"latex\" alt=\"$ 0$\" width=\"8\" height=\"12\" > <img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/1/c/01c79d0810e3de934a94e77682afe6d3a7fcf33c.png\" class=\"latex\" alt=\"$ \\blacksquare$\" width=\"13\" height=\"12\" ></div></div>",
"post_id": 4418672,
"post_number": 1,
"post_time_unix": 1248807866,
"post_time_utc": "2009-07-28 19:04:26 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "[quote=\"stevenmeow\"]or, by matching coefficients, that\n$ \\sum_{k \\equal{} 0}^{n} \\binom{n}{k} ( \\minus{} 1)^k k^x \\equal{} 0$[/quote]\n\nActually, by matching coefficients, we find the coefficient of $ x^x$ (:stink:) to be...\n\nLet's do that again. From your quote, let's change your \"x\" to \"y\". So now it would read:\n[quote=\"stevenmeow edited by Yongyi781\"]or, by matching coefficients, that\n$ \\sum_{k \\equal{} 0}^{n} \\binom{n}{k} ( \\minus{} 1)^k k^y \\equal{} 0$[/quote]\r\n\r\nActually, by matching coefficients, we find the coefficient of $ x^y$ to be:\r\n\\[ \\sum_{k \\equal{} 0}^{n} \\binom{n}{k} \\binom{n}{y} ( \\minus{} 1)^k k^{n\\minus{}y} \\equal{} 0.\\]\r\n(The coefficient of $ x^y$ in the expansion of $ (x\\plus{}k)^n$ is $ \\binom ny k^{n\\minus{}y}$. Therefore, the coefficient of $ x^y$ in $ \\binom nk(\\minus{}1)^k(x\\plus{}k)^n$ is $ \\binom nk\\binom ny(\\minus{}1)^kk^{n\\minus{}y}$.)",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">stevenmeow wrote:</div>\n<div class=\"bbcode_quote_body\">or, by matching coefficients, that<br>\n<img src=\"//latex.artofproblemsolving.com/9/1/c/91c3948b0ec01251af8ffb1d17c6f0b638de9f5a.png\" class=\"latex\" alt=\"$ \\sum_{k = 0}^{n} \\binom{n}{k} ( - 1)^k k^x = 0$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ></div>\n</div>\n<br>\nActually, by matching coefficients, we find the coefficient of <img src=\"//latex.artofproblemsolving.com/c/3/4/c34339ed9046fe1104a53b660be8eaf135c90fc5.png\" class=\"latex\" alt=\"$ x^x$\" width=\"17\" height=\"12\" > (<img src=\"/assets/images/smilies/stink.gif\" width=\"18\" height=\"18\" alt=\":stink:\" title=\":stink:\" class=\"bbcode_smiley\" />) to be...<br>\n<br>\nLet's do that again. From your quote, let's change your "x" to "y". So now it would read:\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">stevenmeow edited by Yongyi781 wrote:</div>\n<div class=\"bbcode_quote_body\">or, by matching coefficients, that<br>\n<img src=\"//latex.artofproblemsolving.com/5/e/5/5e517b61cd28685b6bae4aec57aa807e1f728afa.png\" class=\"latex\" alt=\"$ \\sum_{k = 0}^{n} \\binom{n}{k} ( - 1)^k k^y = 0$\" style=\"vertical-align: -22px\" width=\"164\" height=\"53\" ></div>\n</div>\n<br>\nActually, by matching coefficients, we find the coefficient of <img src=\"//latex.artofproblemsolving.com/8/e/6/8e651cb2ff30d593fc16c5ef402791fbd900c84f.png\" class=\"latex\" alt=\"$ x^y$\" width=\"17\" height=\"12\" > to be:<br>\n<img src=\"//latex.artofproblemsolving.com/3/c/9/3c911b577917ce1468efad968856c370646d22af.png\" class=\"latexcenter\" alt=\"\\[ \\sum_{k = 0}^{n} \\binom{n}{k} \\binom{n}{y} ( - 1)^k k^{n-y} = 0.\\]\" width=\"225\" height=\"53\" ><br>\n(The coefficient of <img src=\"//latex.artofproblemsolving.com/8/e/6/8e651cb2ff30d593fc16c5ef402791fbd900c84f.png\" class=\"latex\" alt=\"$ x^y$\" width=\"17\" height=\"12\" > in the expansion of <img src=\"//latex.artofproblemsolving.com/5/9/d/59d9461b61f4e8edcce2494f846716555b469c97.png\" class=\"latex\" alt=\"$ (x+k)^n$\" style=\"vertical-align: -4px\" width=\"64\" height=\"18\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/2/6/3264a5a323af011d831c75b4351f84ad9b9e5362.png\" class=\"latex\" alt=\"$ \\binom ny k^{n-y}$\" style=\"vertical-align: -22px\" width=\"74\" height=\"53\" >.</span> Therefore, the coefficient of <img src=\"//latex.artofproblemsolving.com/8/e/6/8e651cb2ff30d593fc16c5ef402791fbd900c84f.png\" class=\"latex\" alt=\"$ x^y$\" width=\"17\" height=\"12\" > in <img src=\"//latex.artofproblemsolving.com/8/1/a/81a97928262e0ca040e4a2b761609e2faa325b89.png\" class=\"latex\" alt=\"$ \\binom nk(-1)^k(x+k)^n$\" style=\"vertical-align: -22px\" width=\"149\" height=\"53\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/c/dac37336f5e574e443e0717bde37d0b1c283ef2c.png\" class=\"latex\" alt=\"$ \\binom nk\\binom ny(-1)^kk^{n-y}$\" style=\"vertical-align: -22px\" width=\"159\" height=\"53\" >.</span>)",
"post_id": 4418673,
"post_number": 2,
"post_time_unix": 1248819298,
"post_time_utc": "2009-07-28 22:14:58 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
For any integer n ≥ 1, prove that
\[
\sum_{k=0}^{n-1} \binom{n}{k}(-1)^{n-k+1}(x+k)^n=(x+n)^n.
\]
Equivalently, prove
\[
\sum_{k=0}^{n} \binom{n}{k}(-1)^{k}(x+k)^n=0.
\]
By comparing coefficients, it suffices to show that for each integer m with 0 ≤ m < n,
\[
\sum_{k=0}^{n} \binom{n}{k}(-1)^k k^m=0,
\]
which holds because this sum is the n-th forward difference of the polynomial k↦k^m of degree m<n, and the n-th difference of a polynomial of degree <n is zero.
|
[
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity",
"/Mathematics/Algebra/Polynomials/Binomial",
"/Mathematics/Algebra/Polynomials/Coefficient",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialIdentity",
"/Mathematics/Algebra/Sums/BinomialSums",
"/Mathematics/Algebra/Sums/PowerSum",
"/Mathematics/Algebra/Sums/Sum",
"/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients",
"/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/UmbralCalculus/BinomialIdentity",
"/Mathematics/DiscreteMathematics/UmbralCalculus/DeltaOperator"
] |
Interpret the alternating binomial sum as the n‑th forward difference of a degree‑m polynomial, which vanishes when m < n.
|
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aops_997778
|
induction i guess
first, let's list some numbers with the help of a calculator
1, 3, 21, 987, 2178309, too big
let's find the fibonacci values.
1/2, 4, 8, 16, etc.
It is apparent that they double.
I forgot whether $ \varphi$ is phi or reciprocal of phi, but asusme that it is phi
For induction, we only need to prove that for powers of 2 greater than 1, $ F_{2n} \equal{} F_n\sqrt {5F_n^2 \plus{} 4}$. Actually, we only need that n is even.
Base case: $ F{2 \cdot 2} \equal{} F_2\sqrt {5F_2^2 \plus{} 4}$
($ F_n$ is Fibonacci)
So, $ F_n \equal{} \frac {\varphi^n \minus{} \varphi^{ \minus{} n}}{\sqrt {5}}$ For even n.
$ 5F_n^2 \plus{} 4 \equal{} \left(\varphi^n \minus{} \varphi^{ \minus{} n}\right)^2 \plus{} 4 \equal{} \varphi^{2n} \minus{} 2 \plus{} \varphi^{ \minus{} 2n} \plus{} 4 \equal{} \varphi^{2n} \plus{} 2 \plus{} \varphi^{ \minus{} 2n}$
$ F_n\sqrt {5F_n^2 \plus{} 4} \equal{} F_n\sqrt {\varphi^{2n} \plus{} 2 \plus{} \varphi^{ \minus{} 2n}}$
$ \equal{} \frac {1}{\sqrt {5}}\left(\varphi^n \minus{} \varphi^{ \minus{} n}\right)\left(\varphi^{n} \plus{} \varphi^{ \minus{} n}\right)$
$ \equal{} \frac {\varphi^{2n} \minus{} \varphi^{ \minus{} 2n}}{\sqrt {5} } \equal{} F_{2n}$
Done.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $ a_n$ be a sequence defined by $ a_1 \\equal{} 1$ and for $ n \\geq 1$, $ a_{n \\plus{} 1} \\equal{} a_n\\sqrt {5a_n^2 \\plus{} 4}$. Prove that $ a_n$ is an integer for all positive integers $ n$.\r\n\r\nProve that if $ x$ is a positive integer and $ x\\sqrt{5x^2\\plus{}4}$ is an integer, then it is also a Fibonacci number.",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/1/4/f/14fb7898d637e7386091f58bf9d57bc24e86b4bb.png\" class=\"latex\" alt=\"$ a_n$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > be a sequence defined by <img src=\"//latex.artofproblemsolving.com/4/b/e/4be795893930539766acffe8b31f75726e96b136.png\" class=\"latex\" alt=\"$ a_1 = 1$\" style=\"vertical-align: -2px\" width=\"49\" height=\"14\" > and for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/5/0e513bc23f73514d3b4f83d093e43b9d717878a1.png\" class=\"latex\" alt=\"$ n \\geq 1$\" style=\"vertical-align: -2px\" width=\"43\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/f/c/9fca5dc31a66af7df8666eda6b331fd4e69af567.png\" class=\"latex\" alt=\"$ a_{n + 1} = a_n\\sqrt {5a_n^2 + 4}$\" style=\"vertical-align: -5px\" width=\"155\" height=\"22\" >.</span> Prove that <img src=\"//latex.artofproblemsolving.com/1/4/f/14fb7898d637e7386091f58bf9d57bc24e86b4bb.png\" class=\"latex\" alt=\"$ a_n$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > is an integer for all positive integers <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" >.</span><br>\n<br>\nProve that if <img src=\"//latex.artofproblemsolving.com/2/e/8/2e894ebb8c4e2dcb0ad8c2b8d415aaf4a0303e86.png\" class=\"latex\" alt=\"$ x$\" width=\"10\" height=\"8\" > is a positive integer and <img src=\"//latex.artofproblemsolving.com/8/e/7/8e7ec9df40eb4b4a07f4cac5ea698de0e5445a5c.png\" class=\"latex\" alt=\"$ x\\sqrt{5x^2+4}$\" style=\"vertical-align: -2px\" width=\"84\" height=\"18\" > is an integer, then it is also a Fibonacci number.",
"post_id": 4418688,
"post_number": 1,
"post_time_unix": 1249418265,
"post_time_utc": "2009-08-04 20:37:45 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
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{
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"content_bbcode": "induction i guess\r\n\r\nfirst, let's list some numbers with the help of a calculator\r\n1, 3, 21, 987, 2178309, too big\r\n\r\nlet's find the fibonacci values.\r\n1/2, 4, 8, 16, etc.\r\nIt is apparent that they double.\r\nI forgot whether $ \\varphi$ is phi or reciprocal of phi, but asusme that it is phi\r\n\r\nFor induction, we only need to prove that for powers of 2 greater than 1, $ F_{2n} \\equal{} F_n\\sqrt {5F_n^2 \\plus{} 4}$. Actually, we only need that n is even.\r\nBase case: $ F{2 \\cdot 2} \\equal{} F_2\\sqrt {5F_2^2 \\plus{} 4}$\r\n($ F_n$ is Fibonacci)\r\n\r\n\r\nSo, $ F_n \\equal{} \\frac {\\varphi^n \\minus{} \\varphi^{ \\minus{} n}}{\\sqrt {5}}$ For even n.\r\n$ 5F_n^2 \\plus{} 4 \\equal{} \\left(\\varphi^n \\minus{} \\varphi^{ \\minus{} n}\\right)^2 \\plus{} 4 \\equal{} \\varphi^{2n} \\minus{} 2 \\plus{} \\varphi^{ \\minus{} 2n} \\plus{} 4 \\equal{} \\varphi^{2n} \\plus{} 2 \\plus{} \\varphi^{ \\minus{} 2n}$\r\n$ F_n\\sqrt {5F_n^2 \\plus{} 4} \\equal{} F_n\\sqrt {\\varphi^{2n} \\plus{} 2 \\plus{} \\varphi^{ \\minus{} 2n}}$ \r\n$ \\equal{} \\frac {1}{\\sqrt {5}}\\left(\\varphi^n \\minus{} \\varphi^{ \\minus{} n}\\right)\\left(\\varphi^{n} \\plus{} \\varphi^{ \\minus{} n}\\right)$\r\n$ \\equal{} \\frac {\\varphi^{2n} \\minus{} \\varphi^{ \\minus{} 2n}}{\\sqrt {5} } \\equal{} F_{2n}$\r\n\r\nDone.",
"content_html": "induction i guess<br>\n<br>\nfirst, let's list some numbers with the help of a calculator<br>\n1, 3, 21, 987, 2178309, too big<br>\n<br>\nlet's find the fibonacci values.<br>\n1/2, 4, 8, 16, etc.<br>\nIt is apparent that they double.<br>\nI forgot whether <img src=\"//latex.artofproblemsolving.com/1/9/1/1911bcd02d56571067c923f90cc54537aa827e90.png\" class=\"latex\" alt=\"$ \\varphi$\" style=\"vertical-align: -3px\" width=\"11\" height=\"11\" > is phi or reciprocal of phi, but asusme that it is phi<br>\n<br>\nFor induction, we only need to prove that for powers of 2 greater than 1, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/4/f/24fadb3d76907025a21bcc22041c308b48c31f0e.png\" class=\"latex\" alt=\"$ F_{2n} = F_n\\sqrt {5F_n^2 + 4}$\" style=\"vertical-align: -5px\" width=\"152\" height=\"22\" >.</span> Actually, we only need that n is even.<br>\nBase case: <img src=\"//latex.artofproblemsolving.com/9/3/b/93b37388ea73a8b77343974278c923c5ba08c9f2.png\" class=\"latex\" alt=\"$ F{2 \\cdot 2} = F_2\\sqrt {5F_2^2 + 4}$\" style=\"vertical-align: -4px\" width=\"169\" height=\"22\" ><br>\n<span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/a/b/9/ab91dccc691981a6b9d488cc590b7af4b341b03d.png\" class=\"latex\" alt=\"$ F_n$\" style=\"vertical-align: -2px\" width=\"19\" height=\"15\" ></span> is Fibonacci)<br>\n<br>\n<br>\nSo, <img src=\"//latex.artofproblemsolving.com/c/3/8/c385dd3f54648b8e6a255296dd1af6d34aaba862.png\" class=\"latex\" alt=\"$ F_n = \\frac {\\varphi^n - \\varphi^{ - n}}{\\sqrt {5}}$\" style=\"vertical-align: -17px\" width=\"121\" height=\"41\" > For even n.<br>\n<img src=\"//latex.artofproblemsolving.com/8/0/e/80eea42f6deb117450228e69790f324015416854.png\" class=\"latex\" alt=\"$ 5F_n^2 + 4 = \\left(\\varphi^n - \\varphi^{ - n}\\right)^2 + 4 = \\varphi^{2n} - 2 + \\varphi^{ - 2n} + 4 = \\varphi^{2n} + 2 + \\varphi^{ - 2n}$\" style=\"vertical-align: -6px\" width=\"532\" height=\"25\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/b/f/5bfdb8a34bc033bf0ed927948115e03fd8dbf0f2.png\" class=\"latex\" alt=\"$ F_n\\sqrt {5F_n^2 + 4} = F_n\\sqrt {\\varphi^{2n} + 2 + \\varphi^{ - 2n}}$\" style=\"vertical-align: -5px\" width=\"283\" height=\"22\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/0/3/a03a49b907adfd4b5bfc1ec62e4990257df00195.png\" class=\"latex\" alt=\"$ = \\frac {1}{\\sqrt {5}}\\left(\\varphi^n - \\varphi^{ - n}\\right)\\left(\\varphi^{n} + \\varphi^{ - n}\\right)$\" style=\"vertical-align: -17px\" width=\"232\" height=\"41\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/9/6/a962b6a012fd9c3f4a221b81ebf1b6f049f05c42.png\" class=\"latex\" alt=\"$ = \\frac {\\varphi^{2n} - \\varphi^{ - 2n}}{\\sqrt {5} } = F_{2n}$\" style=\"vertical-align: -17px\" width=\"161\" height=\"43\" ><br>\n<br>\nDone.",
"post_id": 4418689,
"post_number": 2,
"post_time_unix": 1249515612,
"post_time_utc": "2009-08-05 23:40:12 UTC",
"thanks_received": 1,
"user_id": 37558,
"username": "stevenmeow"
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{
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"content_bbcode": "Algebra proof for the first part:\r\n\r\n[hide]\n(Insert base case here.)\n\nSuppose that $ a_n$ and $ a_{n\\minus{}1}$ are integers. Then we have:\n\\[ \\begin{align*} 5a_n^2 \\plus{} 4 & \\equal{} 5a_{n \\minus{} 1}^2(5a_{n \\minus{} 1}^2 \\plus{} 4) \\plus{} 4 \\\\\n& \\equal{} 25a_{n \\minus{} 1}^4 \\plus{} 20a_{n \\minus{} 1}^2 \\plus{} 4 \\\\\n& \\equal{} (5a_{n \\minus{} 1}^2 \\plus{} 2)^2, \\end{align*}\\]\nhence $ \\sqrt {5a_n^2 \\plus{} 4}$ is an integer and so is $ a_{n \\plus{} 1} \\equal{} a_n\\sqrt {5a_n^2 \\plus{} 4}$, INDUCTION COMPLETE.[/hide]",
"content_html": "Algebra proof for the first part:<br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">(Insert base case here.)<br>\n<br>\nSuppose that <img src=\"//latex.artofproblemsolving.com/1/4/f/14fb7898d637e7386091f58bf9d57bc24e86b4bb.png\" class=\"latex\" alt=\"$ a_n$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" > and <img src=\"//latex.artofproblemsolving.com/7/f/b/7fb8c8347edc5c8c66bba26a016c04735066dbb0.png\" class=\"latex\" alt=\"$ a_{n-1}$\" style=\"vertical-align: -2px\" width=\"33\" height=\"10\" > are integers. Then we have:<br>\n<pre class=\"aopscode-error aopscode-latex-error\">\\[ \\begin{align*} 5a_n^2 + 4 & = 5a_{n - 1}^2(5a_{n - 1}^2 + 4) + 4 \\\\\n& = 25a_{n - 1}^4 + 20a_{n - 1}^2 + 4 \\\\\n& = (5a_{n - 1}^2 + 2)^2, \\end{align*}\\]</pre><br>\nhence <img src=\"//latex.artofproblemsolving.com/2/5/f/25fcc5f4c20ea0172bbd503c59c6cde0f3c1bf54.png\" class=\"latex\" alt=\"$ \\sqrt {5a_n^2 + 4}$\" style=\"vertical-align: -5px\" width=\"77\" height=\"22\" > is an integer and so is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/f/c/9fca5dc31a66af7df8666eda6b331fd4e69af567.png\" class=\"latex\" alt=\"$ a_{n + 1} = a_n\\sqrt {5a_n^2 + 4}$\" style=\"vertical-align: -5px\" width=\"155\" height=\"22\" >,</span> INDUCTION COMPLETE.</div>",
"post_id": 4418690,
"post_number": 3,
"post_time_unix": 1249518956,
"post_time_utc": "2009-08-06 00:35:56 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
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"source": null
}
|
Let \(a_1=1\) and for \(n\ge1\) define
\[
a_{n+1}=a_n\sqrt{5a_n^2+4}.
\]
(a) Prove that \(a_n\) is an integer for all positive integers \(n\).
(b) Prove that if \(x\) is a positive integer and \(x\sqrt{5x^2+4}\) is an integer, then \(x\sqrt{5x^2+4}\) is a Fibonacci number.
|
[
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"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/Fibonaccin-StepNumber",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation",
"/Mathematics/DiscreteMathematics/RecurrenceEquations/RecursiveSequence",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/RationalInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus",
"/Mathematics/NumberTheory/Sequences/BinetForms",
"/Mathematics/NumberTheory/Sequences/FibonacciDualTheorem",
"/Mathematics/NumberTheory/Sequences/IntegerSequence"
] |
Use the identity F_{2k}=F_k·√(5F_k^2+4) to show the recurrence generates Fibonacci numbers at doubled indices.
|
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aops_997780
|
unless i made a typo, absolutely not.
Suppose that we want f(2)
In fact, this is an AMC 12 problem, and you just missed it
(unless i made a typo in which i'm pretty sorry)
We have $ a_{1,1}a_{1,2} \plus{} a_{2,1}a_{2,2}$
If $ a_{1,2}$ is even, the parity of $ a_{1,1}$ will not change anything.
Furthermore, the answer is $ \frac {5}{8}$
darn. i made a typo :(
i am pretty sorry
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "define the expressions $ E_n, n \\in 2\\mathbb{N}$ as follows\r\n$ E_n \\equal{} \\sum_{j \\equal{} 1}^{n} \\prod_{k \\equal{} 1}^{n} a_{j,k}$\r\nLet $ f(n), n \\in 2\\mathbb{N}$ denote the probability that $ E_n$ is even given that all the $ a_{j,k}$ are chosen among the natural numbers to be even or odd with equal probability.\r\nFind the first few values of $ f(n)$\r\nyay\r\n\r\nin case there is still a typo, f(2) is an AMC 12 problem with answer $ \\frac{5}{8}$\r\nI just realized that $ n$ could be an odd natural number. If $ f(n)$ where $ n$ is an odd natural number greater than 1 is not $ \\frac{1}{2}$, then the only restriction on n is that $ n\\minus{}1 \\in \\mathbb{N}$",
"content_html": "define the expressions <img src=\"//latex.artofproblemsolving.com/1/6/d/16df7ba7f61914ba861799e7370b20a443e39e2f.png\" class=\"latex\" alt=\"$ E_n, n \\in 2\\mathbb{N}$\" style=\"vertical-align: -3px\" width=\"85\" height=\"16\" > as follows<br>\n<img src=\"//latex.artofproblemsolving.com/d/f/e/dfe623536f4dde687cd49aec0be3408e433ead39.png\" class=\"latex\" alt=\"$ E_n = \\sum_{j = 1}^{n} \\prod_{k = 1}^{n} a_{j,k}$\" style=\"vertical-align: -22px\" width=\"128\" height=\"50\" ><br>\nLet <img src=\"//latex.artofproblemsolving.com/d/3/2/d320f28d10b41b604ea2ff7b64394cdec63a0812.png\" class=\"latex\" alt=\"$ f(n), n \\in 2\\mathbb{N}$\" style=\"vertical-align: -4px\" width=\"98\" height=\"18\" > denote the probability that <img src=\"//latex.artofproblemsolving.com/8/e/d/8edaf9bd5f9401fdfab1d0d21228fda124166f9c.png\" class=\"latex\" alt=\"$ E_n$\" style=\"vertical-align: -2px\" width=\"21\" height=\"15\" > is even given that all the <img src=\"//latex.artofproblemsolving.com/0/a/c/0acd8c197997a76f126c4bac33db63a65fb62bd4.png\" class=\"latex\" alt=\"$ a_{j,k}$\" style=\"vertical-align: -4px\" width=\"26\" height=\"12\" > are chosen among the natural numbers to be even or odd with equal probability.<br>\nFind the first few values of <img src=\"//latex.artofproblemsolving.com/2/4/7/247c5c4a02e9c1587ae084fc06cd45fc44368bdb.png\" class=\"latex\" alt=\"$ f(n)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" ><br>\nyay<br>\n<br>\nin case there is still a typo, f(2) is an AMC 12 problem with answer <img src=\"//latex.artofproblemsolving.com/d/6/6/d66912b5e78a9ac37ecb51b74c971f734e4ea72f.png\" class=\"latex\" alt=\"$ \\frac{5}{8}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ><br>\nI just realized that <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > could be an odd natural number. If <img src=\"//latex.artofproblemsolving.com/2/4/7/247c5c4a02e9c1587ae084fc06cd45fc44368bdb.png\" class=\"latex\" alt=\"$ f(n)$\" style=\"vertical-align: -4px\" width=\"34\" height=\"18\" > where <img src=\"//latex.artofproblemsolving.com/6/d/3/6d3f8b726378d5420223c5cb14b10f24b202b187.png\" class=\"latex\" alt=\"$ n$\" width=\"10\" height=\"8\" > is an odd natural number greater than 1 is not <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/0/bd07c5db06840f47e4ced4ebeb7997a4bd0fd569.png\" class=\"latex\" alt=\"$ \\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" >,</span> then the only restriction on n is that <img src=\"//latex.artofproblemsolving.com/f/7/c/f7c6df04feb757a060b54372f538fe966f07b5c9.png\" class=\"latex\" alt=\"$ n-1 \\in \\mathbb{N}$\" style=\"vertical-align: -1px\" width=\"77\" height=\"13\" >",
"post_id": 4418696,
"post_number": 1,
"post_time_unix": 1250301765,
"post_time_utc": "2009-08-15 02:02:45 UTC",
"thanks_received": 1,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "Wait what isn't this just $ \\frac{1}{2}$ by bijection (change parity of $ a_{1,1}$)?",
"content_html": "Wait what isn't this just <img src=\"//latex.artofproblemsolving.com/b/d/0/bd07c5db06840f47e4ced4ebeb7997a4bd0fd569.png\" class=\"latex\" alt=\"$ \\frac{1}{2}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" > by bijection (change parity of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/2/6/f26a4c6421482e87239fd8ff5e27ee1ba59de8bd.png\" class=\"latex\" alt=\"$ a_{1,1}$\" style=\"vertical-align: -4px\" width=\"25\" height=\"12\" >)</span>?",
"post_id": 4418697,
"post_number": 2,
"post_time_unix": 1250302664,
"post_time_utc": "2009-08-15 02:17:44 UTC",
"thanks_received": 1,
"user_id": 45289,
"username": "dysfunctionalequations"
},
{
"attachments": [],
"content_bbcode": "unless i made a typo, absolutely not.\r\nSuppose that we want f(2) \r\nIn fact, this is an AMC 12 problem, and you just missed it\r\n(unless i made a typo in which i'm pretty sorry)\r\nWe have $ a_{1,1}a_{1,2} \\plus{} a_{2,1}a_{2,2}$\r\nIf $ a_{1,2}$ is even, the parity of $ a_{1,1}$ will not change anything.\r\nFurthermore, the answer is $ \\frac {5}{8}$\r\n\r\ndarn. i made a typo :(\r\ni am pretty sorry",
"content_html": "unless i made a typo, absolutely not.<br>\nSuppose that we want f(2)<br>\nIn fact, this is an AMC 12 problem, and you just missed it<br>\n(unless i made a typo in which i'm pretty sorry)<br>\nWe have <img src=\"//latex.artofproblemsolving.com/5/7/d/57d3e59dad93598bbff3c055d31288c23bfe3974.png\" class=\"latex\" alt=\"$ a_{1,1}a_{1,2} + a_{2,1}a_{2,2}$\" style=\"vertical-align: -4px\" width=\"131\" height=\"15\" ><br>\nIf <img src=\"//latex.artofproblemsolving.com/6/3/3/633a97357b02a088d6923f0e282c37bdaf8e26b2.png\" class=\"latex\" alt=\"$ a_{1,2}$\" style=\"vertical-align: -4px\" width=\"26\" height=\"12\" > is even, the parity of <img src=\"//latex.artofproblemsolving.com/f/2/6/f26a4c6421482e87239fd8ff5e27ee1ba59de8bd.png\" class=\"latex\" alt=\"$ a_{1,1}$\" style=\"vertical-align: -4px\" width=\"25\" height=\"12\" > will not change anything.<br>\nFurthermore, the answer is <img src=\"//latex.artofproblemsolving.com/b/2/3/b23a4758dc531e1e7876ad2a382ed60beb8936f5.png\" class=\"latex\" alt=\"$ \\frac {5}{8}$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" ><br>\n<br>\ndarn. i made a typo <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /><br>\ni am pretty sorry",
"post_id": 4418698,
"post_number": 3,
"post_time_unix": 1250351242,
"post_time_utc": "2009-08-15 15:47:22 UTC",
"thanks_received": 1,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "[quote=\"stevenmeow\"]unless i made a typo in which [color=red][b]case[/b][/color] i'm pretty sorry[/quote]\r\n\r\n:P Another typo yay!",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">stevenmeow wrote:</div>\n<div class=\"bbcode_quote_body\">unless i made a typo in which <span style=\"color:red\"><b>case</b></span> i'm pretty sorry</div>\n</div>\n<br>\n<img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" /> Another typo yay!",
"post_id": 4418699,
"post_number": 4,
"post_time_unix": 1250368142,
"post_time_utc": "2009-08-15 20:29:02 UTC",
"thanks_received": 1,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
Define the expressions \(E_n\) for \(n\in 2\mathbb{N}\) by
\[
E_n=\sum_{j=1}^n\prod_{k=1}^n a_{j,k}.
\]
Let \(f(n)\) for \(n\in 2\mathbb{N}\) denote the probability that \(E_n\) is even given that each \(a_{j,k}\) is independently chosen to be even or odd with probability \(1/2\). Find the first few values of \(f(n)\).
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/ProbabilityandStatistics/Probability"
] |
Reduce the parity of E_n to the parity of the count of rows whose product is odd (i.e., rows with all odd entries) and evaluate the even‑count probability.
|
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aops_997785
|
$ a\equal{}33, b\equal{}115$
i actually knew 33 but had to research it because it produced irrational $ r_i$
anyways, $ r_1^2\plus{}r_2^2\equal{}(r_1\plus{}r_2)^2\minus{}2r_1r_2\equal{}(a)^2\minus{}2b\equal{}33^2\minus{}2(115)\equal{}1089\minus{}230\equal{}859$
tsk tsk this isn't a special number.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "A [b]trivial[/b] problem is a problem concerning [b]trivia[/b]: little, unimportant, insignificant information.\r\n\r\nThis is an example of a trivial problem:\r\n\r\n[i]Let $ a$ be the largest finite number of moves required to force checkmate with a king, a knight, and a bishop against a lone king. Let $ b$ be the largest finite number of moves required to force checkmate with a king and two knights against a king and a pawn. If the polynomial $ x^2 \\minus{} ax \\plus{} b$ has two roots $ r_1$ and $ r_2$, find $ r_1^2 \\plus{} r_2^2$.[/i]\r\n\r\nIt is a trivial problem because it requires knowledge of trivial information about chess.\r\n\r\nThis is an example of a non-trivial problem:\r\n\r\n[i]Find the value of $ 1 \\plus{} 1$. Express your answer in decimal form.[/i]\r\n\r\nIt is a non-trivial problem because it does not rely on trivial knowledge -- lesser-known or arcane pieces of information -- just mathematical truths.",
"content_html": "A <b>trivial</b> problem is a problem concerning <b>trivia</b>: little, unimportant, insignificant information.<br>\n<br>\nThis is an example of a trivial problem:<br>\n<br>\n<i>Let <img src=\"//latex.artofproblemsolving.com/2/5/5/255f65757f75ce300036173cb8e6f8f86dcfe90f.png\" class=\"latex\" alt=\"$ a$\" width=\"9\" height=\"8\" > be the largest finite number of moves required to force checkmate with a king, a knight, and a bishop against a lone king. Let <img src=\"//latex.artofproblemsolving.com/b/9/d/b9d389de6d8a8314b29faf761bb09a117e5f53c4.png\" class=\"latex\" alt=\"$ b$\" width=\"8\" height=\"12\" > be the largest finite number of moves required to force checkmate with a king and two knights against a king and a pawn. If the polynomial <img src=\"//latex.artofproblemsolving.com/d/1/f/d1f05d82d4b5de5e27d01a6286602218c8f48895.png\" class=\"latex\" alt=\"$ x^2 - ax + b$\" style=\"vertical-align: -1px\" width=\"90\" height=\"16\" > has two roots <img src=\"//latex.artofproblemsolving.com/c/4/f/c4fd3480f15ad3d8ee79e68ca1e09402a2e9e038.png\" class=\"latex\" alt=\"$ r_1$\" style=\"vertical-align: -2px\" width=\"13\" height=\"10\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/7/a/a7a9bc0938e7ac4b276ae80decad531c7d98ae2b.png\" class=\"latex\" alt=\"$ r_2$\" style=\"vertical-align: -2px\" width=\"14\" height=\"10\" >,</span> find <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/4/ba4d852518f2785edcfa2950bf7aa5895b8c47b7.png\" class=\"latex\" alt=\"$ r_1^2 + r_2^2$\" style=\"vertical-align: -4px\" width=\"53\" height=\"19\" >.</span></i><br>\n<br>\nIt is a trivial problem because it requires knowledge of trivial information about chess.<br>\n<br>\nThis is an example of a non-trivial problem:<br>\n<br>\n<i>Find the value of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/7/3/573bff8fb3f4b17ec45f38d9da5dbca697322895.png\" class=\"latex\" alt=\"$ 1 + 1$\" style=\"vertical-align: -1px\" width=\"39\" height=\"13\" >.</span> Express your answer in decimal form.</i><br>\n<br>\nIt is a non-trivial problem because it does not rely on trivial knowledge -- lesser-known or arcane pieces of information -- just mathematical truths.",
"post_id": 4418719,
"post_number": 1,
"post_time_unix": 1251333487,
"post_time_utc": "2009-08-27 00:38:07 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
},
{
"attachments": [],
"content_bbcode": "$ a\\equal{}33, b\\equal{}115$\r\ni actually knew 33 but had to research it because it produced irrational $ r_i$\r\nanyways, $ r_1^2\\plus{}r_2^2\\equal{}(r_1\\plus{}r_2)^2\\minus{}2r_1r_2\\equal{}(a)^2\\minus{}2b\\equal{}33^2\\minus{}2(115)\\equal{}1089\\minus{}230\\equal{}859$ \r\ntsk tsk this isn't a special number.",
"content_html": "<img src=\"//latex.artofproblemsolving.com/e/b/c/ebc7deb11529dcffd511612f45d755bcc08f589f.png\" class=\"latex\" alt=\"$ a=33, b=115$\" style=\"vertical-align: -3px\" width=\"118\" height=\"16\" ><br>\ni actually knew 33 but had to research it because it produced irrational <img src=\"//latex.artofproblemsolving.com/9/3/5/935573b6f8cad23f2e5583bd601f158204cfe7a8.png\" class=\"latex\" alt=\"$ r_i$\" style=\"vertical-align: -2px\" width=\"12\" height=\"10\" ><br>\nanyways, <img src=\"//latex.artofproblemsolving.com/2/4/a/24a4d93b6150eb25610de7247ce06fa66178a373.png\" class=\"latex\" alt=\"$ r_1^2+r_2^2=(r_1+r_2)^2-2r_1r_2=(a)^2-2b=33^2-2(115)=1089-230=859$\" style=\"vertical-align: -4px\" width=\"594\" height=\"19\" ><br>\ntsk tsk this isn't a special number.",
"post_id": 4418720,
"post_number": 2,
"post_time_unix": 1252191293,
"post_time_utc": "2009-09-05 22:54:53 UTC",
"thanks_received": 2,
"user_id": 37558,
"username": "stevenmeow"
},
{
"attachments": [],
"content_bbcode": "Of course it is interesting, it is approximately 5.577922 times the sum of the scores of my 6 solo records in GOP.\r\n\r\nDarn, the above will probably be false very soon.",
"content_html": "Of course it is interesting, it is approximately 5.577922 times the sum of the scores of my 6 solo records in GOP.<br>\n<br>\nDarn, the above will probably be false very soon.",
"post_id": 4418721,
"post_number": 3,
"post_time_unix": 1252380353,
"post_time_utc": "2009-09-08 03:25:53 UTC",
"thanks_received": 2,
"user_id": 40253,
"username": "Yongyi781"
}
],
"source": null
}
|
A trivial problem is a problem concerning trivia: little, unimportant, insignificant information.
This is an example of a trivial problem:
Let \(a\) be the largest finite number of moves required to force checkmate with a king, a knight, and a bishop against a lone king. Let \(b\) be the largest finite number of moves required to force checkmate with a king and two knights against a king and a pawn. If the polynomial \(x^2-ax+b\) has two roots \(r_1\) and \(r_2\), find \(r_1^2+r_2^2\).
This is an example of a non-trivial problem:
Find the value of \(1+1\). Express your answer in decimal form.
|
[
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialEquation",
"/Mathematics/Algebra/Polynomials/PolynomialRoots",
"/Mathematics/Algebra/Polynomials/QuadraticPolynomial",
"/Mathematics/Algebra/Polynomials/VietasFormulas"
] |
Use Vieta’s formulas to write r1^2+r2^2 as (r1+r2)^2−2r1r2 and substitute the coefficients a and b.
|
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aops_99780
|
Each integer n is appropriate:
take A=diag(0,1,...,n-1) and B=[Bij] with Bij=1 for all i,j.
Rank(A)=n-1,rank(B)=1,UA=AU iff U is a diagonal matrix.
If U is a diagonal matrix then BU=UB implies that U is an homothety (cf. the first row of UB-BU).
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find all $n \\in \\mathbb{N}$ for which there exist square $n \\times n$ matrices $A$ and $B$ such that\r\n$\\mathrm{rank}\\, A+\\mathrm{rank}\\, B\\leq n$ and every square real matrix $X$ which commutes with $A$ and $B$ is of the form $X=\\lambda I,\\,\\lambda \\in \\mathbb{R}.$",
"content_html": "Find all <img src=\"//latex.artofproblemsolving.com/9/6/1/961d336c96250f23ace9c62514783e30511c69cf.png\" class=\"latex\" alt=\"$n \\in \\mathbb{N}$\" style=\"vertical-align: -1px\" width=\"45\" height=\"13\" > for which there exist square <img src=\"//latex.artofproblemsolving.com/c/3/d/c3d38f82f48ef9e81c04d49354293305b0067afc.png\" class=\"latex\" alt=\"$n \\times n$\" width=\"43\" height=\"9\" > matrices <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\" > such that<br>\n<img src=\"//latex.artofproblemsolving.com/b/2/8/b28912483df2b59831308062c1985a792483eff8.png\" class=\"latex\" alt=\"$\\mathrm{rank}\\, A+\\mathrm{rank}\\, B\\leq n$\" style=\"vertical-align: -2px\" width=\"163\" height=\"15\" > and every square real matrix <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > which commutes with <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\" > is of the form <img src=\"//latex.artofproblemsolving.com/1/2/6/1262133b762487c8a72bd81785d29b3db63bc2b3.png\" class=\"latex\" alt=\"$X=\\lambda I,\\,\\lambda \\in \\mathbb{R}.$\" style=\"vertical-align: -3px\" width=\"121\" height=\"16\" >",
"post_id": 563447,
"post_number": 1,
"post_time_unix": 1151743606,
"post_time_utc": "2006-07-01 08:46:46 UTC",
"thanks_received": 1,
"user_id": 902,
"username": "eugene"
},
{
"attachments": [],
"content_bbcode": "Each integer n is appropriate:\r\ntake A=diag(0,1,...,n-1) and B=[Bij] with Bij=1 for all i,j.\r\nRank(A)=n-1,rank(B)=1,UA=AU iff U is a diagonal matrix.\r\nIf U is a diagonal matrix then BU=UB implies that U is an homothety (cf. the first row of UB-BU).",
"content_html": "Each integer n is appropriate:<br>\ntake A=diag(0,1,...,n-1) and B=[Bij] with Bij=1 for all i,j.<br>\nRank(A)=n-1,rank(B)=1,UA=AU iff U is a diagonal matrix.<br>\nIf U is a diagonal matrix then BU=UB implies that U is an homothety (cf. the first row of UB-BU).",
"post_id": 606648,
"post_number": 2,
"post_time_unix": 1156032794,
"post_time_utc": "2006-08-20 00:13:14 UTC",
"thanks_received": 1,
"user_id": 16972,
"username": "loup blanc"
}
],
"source": null
}
|
Find all \(n\in\mathbb{N}\) for which there exist square \(n\times n\) matrices \(A\) and \(B\) such that
\[
\operatorname{rank}A+\operatorname{rank}B\le n
\]
and every real \(n\times n\) matrix \(X\) which commutes with both \(A\) and \(B\) is of the form \(X=\lambda I\) for some \(\lambda\in\mathbb{R}\).
|
[
"/Mathematics/Algebra/GeneralAlgebra",
"/Mathematics/Algebra/LinearAlgebra/Matrices/MatrixProperties"
] |
Combine a diagonal matrix with distinct eigenvalues and a rank‑1 all‑ones matrix so that commuting with the first forces diagonal form and commuting with the second forces scalar.
|
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aops_997852
|
um, if the recips. of the roots are $ \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$, then the sum is
$ \frac{acd \plus{} bcd \plus{}abc \plus{} abd}{abcd}$. Right now if you factor the polynomial.
$ 154(x\minus{}a)(x\minus{}b)(x\minus{}c)(x\minus{}d)$ we know that, $ 154abcd \equal{} 1337$,
and
$ \minus{}154(acd \plus{} bcd \plus{}abc \plus{} abd) \equal{} 0$
therefore
$ acd \plus{} bcd \plus{}abc \plus{} abd \equal{} 0$ making the sum of the reciprocals ......$ \fbox{0}$ :ewpu: um okay i probly got it wrong.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Find the sum of the reciprocals of the roots of $ 154x^4 \\plus{}781x^3 \\plus{} 666x^2 \\plus{} 1337$",
"content_html": "Find the sum of the reciprocals of the roots of <img src=\"//latex.artofproblemsolving.com/5/4/e/54e0905e7f3a869d9fe1845182be374e05ff6145.png\" class=\"latex\" alt=\"$ 154x^4 +781x^3 + 666x^2 + 1337$\" style=\"vertical-align: -1px\" width=\"238\" height=\"16\" >",
"post_id": 4418917,
"post_number": 1,
"post_time_unix": 1214346421,
"post_time_utc": "2008-06-24 22:27:01 UTC",
"thanks_received": 2,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "um, if the recips. of the roots are $ \\frac{1}{a}, \\frac{1}{b}, \\frac{1}{c}, \\frac{1}{d}$, then the sum is \r\n\r\n$ \\frac{acd \\plus{} bcd \\plus{}abc \\plus{} abd}{abcd}$. Right now if you factor the polynomial.\r\n\r\n$ 154(x\\minus{}a)(x\\minus{}b)(x\\minus{}c)(x\\minus{}d)$ we know that, $ 154abcd \\equal{} 1337$, \r\n\r\nand \r\n\r\n$ \\minus{}154(acd \\plus{} bcd \\plus{}abc \\plus{} abd) \\equal{} 0$\r\n\r\ntherefore\r\n\r\n$ acd \\plus{} bcd \\plus{}abc \\plus{} abd \\equal{} 0$ making the sum of the reciprocals ......$ \\fbox{0}$ :ewpu: um okay i probly got it wrong.",
"content_html": "um, if the recips. of the roots are <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/8/7/687e4658e3832d68278acc12a5608c364d414bf6.png\" class=\"latex\" alt=\"$ \\frac{1}{a}, \\frac{1}{b}, \\frac{1}{c}, \\frac{1}{d}$\" style=\"vertical-align: -12px\" width=\"75\" height=\"37\" >,</span> then the sum is<br>\n<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/c/17c3c1bbc1102ea350b46893f4b0ba791d56293c.png\" class=\"latex\" alt=\"$ \\frac{acd + bcd +abc + abd}{abcd}$\" style=\"vertical-align: -12px\" width=\"173\" height=\"37\" >.</span> Right now if you factor the polynomial.<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/e/a/8ea739082429423824f198c65cb572963cde6b56.png\" class=\"latex\" alt=\"$ 154(x-a)(x-b)(x-c)(x-d)$\" style=\"vertical-align: -4px\" width=\"247\" height=\"18\" > we know that, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/f/8/ef8ad9030695d9fcce5c064726a7de728356028b.png\" class=\"latex\" alt=\"$ 154abcd = 1337$\" style=\"vertical-align: 0px\" width=\"122\" height=\"13\" >,</span><br>\n<br>\nand<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/8/3/e/83e0a6132a830f53fdb47d46667c22d64f8cfa2e.png\" class=\"latex\" alt=\"$ -154(acd + bcd +abc + abd) = 0$\" style=\"vertical-align: -4px\" width=\"259\" height=\"18\" ><br>\n<br>\ntherefore<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/e/3/1/e31c64392dab1764fc80e123b4b1e9596a2371ea.png\" class=\"latex\" alt=\"$ acd + bcd +abc + abd = 0$\" style=\"vertical-align: -1px\" width=\"203\" height=\"14\" > making the sum of the reciprocals .....<span style=\"white-space:nowrap;\">.<img src=\"//latex.artofproblemsolving.com/b/3/c/b3c44f72f26e1003475dafb44705d898d94e9290.png\" class=\"latex\" alt=\"$ \\fbox{0}$\" style=\"vertical-align: -5px\" width=\"20\" height=\"22\" ></span> <img src=\"/assets/images/smilies/ewpu.gif\" width=\"30\" height=\"26\" alt=\":ewpu:\" title=\":ewpu:\" class=\"bbcode_smiley\" /> um okay i probly got it wrong.",
"post_id": 4418918,
"post_number": 2,
"post_time_unix": 1214347974,
"post_time_utc": "2008-06-24 22:52:54 UTC",
"thanks_received": 2,
"user_id": 27938,
"username": "cognos599"
},
{
"attachments": [],
"content_bbcode": "correct!!!!",
"content_html": "correct!!!!",
"post_id": 4418919,
"post_number": 3,
"post_time_unix": 1214353617,
"post_time_utc": "2008-06-25 00:26:57 UTC",
"thanks_received": 2,
"user_id": 39364,
"username": "RunpengFAILS"
},
{
"attachments": [],
"content_bbcode": "hm...tinytim seems to be obsessed with 4...quartic polynomials, systems of 4 quartic equations...and a couple of 4-letter words.... :P",
"content_html": "hm...tinytim seems to be obsessed with 4...quartic polynomials, systems of 4 quartic equations...and a couple of 4-letter words.... <img src=\"/assets/images/smilies/tongue.gif\" width=\"20\" height=\"20\" alt=\":P\" title=\":P\" class=\"bbcode_smiley\" />",
"post_id": 4418920,
"post_number": 4,
"post_time_unix": 1214402324,
"post_time_utc": "2008-06-25 13:58:44 UTC",
"thanks_received": 2,
"user_id": 37259,
"username": "math154"
},
{
"attachments": [],
"content_bbcode": "I feel stupid :( :jump:",
"content_html": "I feel stupid <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/jump.gif\" width=\"22\" height=\"55\" alt=\":jump:\" title=\":jump:\" class=\"bbcode_smiley\" />",
"post_id": 4418921,
"post_number": 5,
"post_time_unix": 1214405512,
"post_time_utc": "2008-06-25 14:51:52 UTC",
"thanks_received": 2,
"user_id": 40880,
"username": "leoxnlin"
}
],
"source": null
}
|
Find the sum of the reciprocals of the roots of
\[
154x^4 + 781x^3 + 666x^2 + 1337 = 0.
\]
|
[
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/QuarticEquation",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialRoots",
"/Mathematics/Algebra/Polynomials/VietasFormulas"
] |
Apply Vieta’s formulas to express the sum of reciprocals as a symmetric sum that vanishes because the x‑term coefficient is zero.
|
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aops_99788
|
For any positive integer $n > 0$, \begin{eqnarray*}\sum_{d\mid n}\mu ( d ) & = & \{\begin{array}{l}1, n = 1\\ 0, n > 1 \end{array} \end{eqnarray*} By the Moebius inversion formula, for $x \geq 1$
\[
\sum_{n \leq x}\mu ( n ) \cdot \lfloor \frac{x}{n} \rfloor = \sum_{n \leq x}( \sum_{d\mid n}\mu ( d ) ) = 1+0+0+\ldots+0 = 1
\]
[b]The upper bound[/b]
If $\mu ( n ) \geq 0$ we have $\mu ( n ) \lfloor \frac{x}{n}\rfloor \geq \mu ( n ) \cdot \frac{x}{n}-\mu ( n )$.
If $\mu ( n ) < 0$ we have $\mu ( n ) \lfloor \frac{x}{n} \rfloor \geq \mu ( n ) \frac{x}{n}$.
Summing these relations we obtain \begin{eqnarray*} \sum_{n \leq x}\mu ( n ) \cdot \lfloor \frac{x}{n} \rfloor & \geq & \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}( \mu ( n ) \cdot \frac{x}{n}-\mu ( n ) \cdot 1 )+\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) < 0 \end{array}}}\mu ( n ) \cdot \frac{x}{n}\\ \sum_{n \leq x}\mu ( n ) \cdot \lfloor \frac{x}{n}\rfloor & \geq & \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x \end{array}}}\mu ( n ) \cdot \frac{x}{n}-\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n ) \cdot 1 \end{eqnarray*} Hence \begin{eqnarray*} \frac{1}{x}\cdot 1 = \frac{1}{x}\sum_{n \leq x}\mu ( n ) \cdot \lfloor \frac{x}{n}\rfloor & \geq & \frac{1}{x}\cdot ( \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x \end{array}}}\mu ( n ) \cdot \frac{x}{n}-\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n ) \cdot 1 )\\ \frac{1}{x}& \geq & \frac{1}{x}\cdot x \sum_{n \leq x}\frac{\mu ( n )}{n}-\frac{1}{x}\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n )\\ \frac{1}{x}\cdot ( \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n )+1 ) & \geq & \sum_{n \leq x}\frac{\mu ( n )}{n}\end{eqnarray*} We trivially have \[x-1 \geq \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n )\] Therefore \[1 = \frac{1}{x}\cdot ( x-1+1 ) \geq \frac{1}{x}( \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) \geq 0 \end{array}}}\mu ( n )+1 ) \geq \sum_{n \leq x}\frac{\mu ( n )}{n}\] [b]The lower bound.[/b]
By equation $( 1 )$, using a similar reasoning as for the uppper bound, we obtain \begin{eqnarray*}\frac{1}{x}\cdot ( \sum_{n \leq x}\mu ( n ) \frac{x}{n}-\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) < 0 \end{array}}}\mu ( n ) ) & \geq & \frac{1}{x}\sum_{n \leq x}\mu ( n ) \cdot \lfloor \frac{x}{n}\rfloor = \frac{1}{x}\cdot 1\\ \sum_{n \leq x}\frac{\mu ( n )}{n}& \geq & \frac{1}{x}+\frac{1}{x}\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) < 0 \end{array}}}\mu ( n ) \end{eqnarray*} Now clearly, \[\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) < 0 \end{array}}}\mu ( n ) \geq \sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x \end{array}}}(-1 ) \geq-( x+1 )\] Therefore \[\sum_{n \leq x}\frac{\mu ( n )}{n}\geq \frac{1}{x}+\frac{1}{x}\sum_{%Error. "tmscript" is a bad command.
{\begin{array}{c}n \leq x\\ \mu ( n ) < 0 \end{array}}}\mu ( n ) \geq \frac{1}{x}+\frac{-( x+1 )}{x}=-1\] Also note that [b]1.[/b] $\sum_{n \leq x}\mu ( n ) = o(x)$ is equivalent to the prime number theorem
[b]2.[/b] you can show that $\sum_{n \in \mathbb{N}}\frac{\mu ( n )}{n}= 0$ knowing that $1 / \zeta ( s ) = \sum_{n = 1}^{\infty}\frac{\mu ( n )}{n^{s}}$ and that $\lim_{s \rightarrow 1^{+}}\zeta ( s ) =+\infty$.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $\\mu$ is Mobius function.Prove that \r\n\r\n$\\frac{\\mu(1)}{1}+\\frac{\\mu(2)}{2}+...+\\frac{\\mu(n)}{n}\\leq 1$ forall $n>0$.\r\n\r\n@to all:I can't use LaTex,Why?\r\n\r\n[mod.: For me it works :? ]",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/2/d/8/2d8c833ed800824727cd7bd2fb9de1a12ad7e674.png\" class=\"latex\" alt=\"$\\mu$\" style=\"vertical-align: -3px\" width=\"10\" height=\"11\" > is Mobius function.Prove that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/3/3/d3387a0225ca42b37a03ce848da1b6cc409ac090.png\" class=\"latex\" alt=\"$\\frac{\\mu(1)}{1}+\\frac{\\mu(2)}{2}+...+\\frac{\\mu(n)}{n}\\leq 1$\" style=\"vertical-align: -13px\" width=\"230\" height=\"39\" > forall <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/2/4/0247db3d702b5ce6be98f50771750a2723cddcea.png\" class=\"latex\" alt=\"$n>0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >.</span><br>\n<br>\n@to all:I can't use LaTex,Why?<br>\n<br>\n[mod.: For me it works <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" /> ]",
"post_id": 563460,
"post_number": 1,
"post_time_unix": 1151745137,
"post_time_utc": "2006-07-01 09:12:17 UTC",
"thanks_received": 2,
"user_id": 5820,
"username": "N.T.TUAN"
},
{
"attachments": [],
"content_bbcode": "For any positive integer $n > 0$, \\begin{eqnarray*}\\sum_{d\\mid n}\\mu ( d ) & = & \\{\\begin{array}{l}1, n = 1\\\\ 0, n > 1 \\end{array} \\end{eqnarray*} By the Moebius inversion formula, for $x \\geq 1$\n\n\\[\n\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor = \\sum_{n \\leq x}( \\sum_{d\\mid n}\\mu ( d ) ) = 1+0+0+\\ldots+0 = 1\n\\]\n\n[b]The upper bound[/b]\n\nIf $\\mu ( n ) \\geq 0$ we have $\\mu ( n ) \\lfloor \\frac{x}{n}\\rfloor \\geq \\mu ( n ) \\cdot \\frac{x}{n}-\\mu ( n )$.\nIf $\\mu ( n ) < 0$ we have $\\mu ( n ) \\lfloor \\frac{x}{n} \\rfloor \\geq \\mu ( n ) \\frac{x}{n}$.\nSumming these relations we obtain \\begin{eqnarray*} \\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}( \\mu ( n ) \\cdot \\frac{x}{n}-\\mu ( n ) \\cdot 1 )+\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}\\\\ \\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor & \\geq & \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n ) \\cdot 1 \\end{eqnarray*} Hence \\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n ) \\cdot 1 )\\\\ \\frac{1}{x}& \\geq & \\frac{1}{x}\\cdot x \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}-\\frac{1}{x}\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )\\\\ \\frac{1}{x}\\cdot ( \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )+1 ) & \\geq & \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\end{eqnarray*} We trivially have \\[x-1 \\geq \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )\\] Therefore \\[1 = \\frac{1}{x}\\cdot ( x-1+1 ) \\geq \\frac{1}{x}( \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )+1 ) \\geq \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\] [b]The lower bound.[/b]\n\nBy equation $( 1 )$, using a similar reasoning as for the uppper bound, we obtain \\begin{eqnarray*}\\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\frac{x}{n}-\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) ) & \\geq & \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor = \\frac{1}{x}\\cdot 1\\\\ \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}& \\geq & \\frac{1}{x}+\\frac{1}{x}\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\end{eqnarray*} Now clearly, \\[\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\geq \\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}(-1 ) \\geq-( x+1 )\\] Therefore \\[\\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\geq \\frac{1}{x}+\\frac{1}{x}\\sum_{%Error. \"tmscript\" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\geq \\frac{1}{x}+\\frac{-( x+1 )}{x}=-1\\] Also note that [b]1.[/b] $\\sum_{n \\leq x}\\mu ( n ) = o(x)$ is equivalent to the prime number theorem \n[b]2.[/b] you can show that $\\sum_{n \\in \\mathbb{N}}\\frac{\\mu ( n )}{n}= 0$ knowing that $1 / \\zeta ( s ) = \\sum_{n = 1}^{\\infty}\\frac{\\mu ( n )}{n^{s}}$ and that $\\lim_{s \\rightarrow 1^{+}}\\zeta ( s ) =+\\infty$.",
"content_html": "For any positive integer <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/f/3/d/f3d1431ad346632b25a0e7b3f12ebe19723b3960.png\" class=\"latex\" alt=\"$n > 0$\" style=\"vertical-align: 0px\" width=\"43\" height=\"13\" >,</span> <img src=\"//latex.artofproblemsolving.com/3/1/3/313f93578c004966769156fc83464cfd18177048.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*}\\sum_{d\\mid n}\\mu ( d ) & = & \\{\\begin{array}{l}1, n = 1\\\\ 0, n > 1 \\end{array} \\end{eqnarray*}\" width=\"188\" height=\"47\" >By the Moebius inversion formula, for <img src=\"//latex.artofproblemsolving.com/7/9/1/791aa0ff911a09ac656cd5ce7638ca8e64805d8b.png\" class=\"latex\" alt=\"$x \\geq 1$\" style=\"vertical-align: -2px\" width=\"42\" height=\"14\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/a/6/8/a683ad05bb7cde1bd7a1c9a7c70c1b12ad4ff929.png\" class=\"latexcenter\" alt=\"\\[\n\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor = \\sum_{n \\leq x}( \\sum_{d\\mid n}\\mu ( d ) ) = 1+0+0+\\ldots+0 = 1\n\\]\" width=\"439\" height=\"44\" ><br>\n<b>The upper bound</b><br>\n<br>\nIf <img src=\"//latex.artofproblemsolving.com/6/5/8/65890c8e6d96c4e47f892584bf25892bc0260f1d.png\" class=\"latex\" alt=\"$\\mu ( n ) \\geq 0$\" style=\"vertical-align: -4px\" width=\"69\" height=\"18\" > we have <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/a/0/d/a0de5ff77d3db1b2600eae3f4ff14b26bd525afb.png\" class=\"latex\" alt=\"$\\mu ( n ) \\lfloor \\frac{x}{n}\\rfloor \\geq \\mu ( n ) \\cdot \\frac{x}{n}-\\mu ( n )$\" style=\"vertical-align: -12px\" width=\"212\" height=\"33\" >.</span><br>\nIf <img src=\"//latex.artofproblemsolving.com/b/b/a/bbabaa3fab52cc00906c44967e8a762960684eda.png\" class=\"latex\" alt=\"$\\mu ( n ) < 0$\" style=\"vertical-align: -4px\" width=\"69\" height=\"18\" > we have <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/d/0/8/d08d2920f612d52ac79706370be4d76cc8269741.png\" class=\"latex\" alt=\"$\\mu ( n ) \\lfloor \\frac{x}{n} \\rfloor \\geq \\mu ( n ) \\frac{x}{n}$\" style=\"vertical-align: -12px\" width=\"140\" height=\"33\" >.</span><br>\nSumming these relations we obtain <img src=\"//latex.artofproblemsolving.com/3/9/0/39010ec63744d42ac6a52d9f66a86a9ea67839a7.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*} \\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}( \\mu ( n ) \\cdot \\frac{x}{n}-\\mu ( n ) \\cdot 1 )+\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}\\\\ \\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor & \\geq & \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n ) \\cdot 1 \\end{eqnarray*}\" width=\"574\" height=\"152\" >Hence <img src=\"//latex.artofproblemsolving.com/2/4/1/2411eeb74c4f444ec987c49f3f67b05e2a6aa234.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n ) \\cdot 1 )\\\\ \\frac{1}{x}& \\geq & \\frac{1}{x}\\cdot x \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}-\\frac{1}{x}\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )\\\\ \\frac{1}{x}\\cdot ( \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )+1 ) & \\geq & \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\end{eqnarray*}\" width=\"582\" height=\"249\" >We trivially have <img src=\"//latex.artofproblemsolving.com/4/0/7/407bf5281ea00d7b2c14dd80d226c9c907220073.png\" class=\"latexcenter\" alt=\"\\[x-1 \\geq \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )\\]\" width=\"189\" height=\"68\" >Therefore <img src=\"//latex.artofproblemsolving.com/9/9/9/999b8c2d7d4b21f49ad5552459ad326a314bcf45.png\" class=\"latexcenter\" alt=\"\\[1 = \\frac{1}{x}\\cdot ( x-1+1 ) \\geq \\frac{1}{x}( \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) \\geq 0 \\end{array}}}\\mu ( n )+1 ) \\geq \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\]\" width=\"447\" height=\"77\" ><b>The lower bound.</b><br>\n<br>\nBy equation <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/9/9/f/99fdff8f69513cc63a303b5b7d88c749721e96d1.png\" class=\"latex\" alt=\"$( 1 )$\" style=\"vertical-align: -4px\" width=\"21\" height=\"18\" >,</span> using a similar reasoning as for the uppper bound, we obtain <img src=\"//latex.artofproblemsolving.com/8/c/a/8ca26ef1de2ec7d1ae8149236bd9f02d954be745.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*}\\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\frac{x}{n}-\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) ) & \\geq & \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n}\\rfloor = \\frac{1}{x}\\cdot 1\\\\ \\sum_{n \\leq x}\\frac{\\mu ( n )}{n}& \\geq & \\frac{1}{x}+\\frac{1}{x}\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\end{eqnarray*}\" width=\"500\" height=\"162\" >Now clearly, <img src=\"//latex.artofproblemsolving.com/c/e/b/ceb15f424b11e403359a02ee63cec9f7b318b6b5.png\" class=\"latexcenter\" alt=\"\\[\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\geq \\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x \\end{array}}}(-1 ) \\geq-( x+1 )\\]\" width=\"332\" height=\"68\" >Therefore <img src=\"//latex.artofproblemsolving.com/c/9/2/c92ea4c7dde88862a1351375a358cf2c9def72d4.png\" class=\"latexcenter\" alt=\"\\[\\sum_{n \\leq x}\\frac{\\mu ( n )}{n}\\geq \\frac{1}{x}+\\frac{1}{x}\\sum_{%Error. "tmscript" is a bad command.\n{\\begin{array}{c}n \\leq x\\\\ \\mu ( n ) < 0 \\end{array}}}\\mu ( n ) \\geq \\frac{1}{x}+\\frac{-( x+1 )}{x}=-1\\]\" width=\"453\" height=\"77\" >Also note that <b>1.</b> <img src=\"//latex.artofproblemsolving.com/6/5/9/659595fd2ec06d42a1f099782457d0ed3bbd13ad.png\" class=\"latex\" alt=\"$\\sum_{n \\leq x}\\mu ( n ) = o(x)$\" style=\"vertical-align: -23px\" width=\"121\" height=\"40\" > is equivalent to the prime number theorem<br>\n<b>2.</b> you can show that <img src=\"//latex.artofproblemsolving.com/1/a/3/1a3edfe927c3a80988ad3397936d7b6e0643a317.png\" class=\"latex\" alt=\"$\\sum_{n \\in \\mathbb{N}}\\frac{\\mu ( n )}{n}= 0$\" style=\"vertical-align: -22px\" width=\"102\" height=\"48\" > knowing that <img src=\"//latex.artofproblemsolving.com/6/4/1/641cc40f59fced957d1044a0bbace18046e8538b.png\" class=\"latex\" alt=\"$1 / \\zeta ( s ) = \\sum_{n = 1}^{\\infty}\\frac{\\mu ( n )}{n^{s}}$\" style=\"vertical-align: -20px\" width=\"141\" height=\"48\" > and that <span style=\"white-space:pre;\"><img src=\"//latex.artofproblemsolving.com/3/1/c/31ceacf18f5e7a7a0c1a373be9e802bad3fe2c95.png\" class=\"latex\" alt=\"$\\lim_{s \\rightarrow 1^{+}}\\zeta ( s ) =+\\infty$\" style=\"vertical-align: -12px\" width=\"126\" height=\"26\" >.</span>",
"post_id": 563924,
"post_number": 2,
"post_time_unix": 1151783033,
"post_time_utc": "2006-07-01 19:43:53 UTC",
"thanks_received": 2,
"user_id": 12380,
"username": "{x}"
},
{
"attachments": [],
"content_bbcode": "[quote=\"{x}\"] \\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{n \\leq x}\\mu ( n ) \\cdot 1 )\\\\ \\end{eqnarray*} [/quote]\r\nWhy? :roll: if $\\mu (n)<0$\r\n\r\nSorry,because probem is -1<=f(1)/1+f(2)/2+...+f(n)/n<=1 :(",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">{x} wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/2/7/2/2727b3af3150df317fd4fb46702f13f6c17ce374.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{n \\leq x}\\mu ( n ) \\cdot 1 )\\\\ \\end{eqnarray*}\" width=\"476\" height=\"47\" ></div>\n</div>\nWhy? <img src=\"/assets/images/smilies/rolleyes.gif\" width=\"20\" height=\"20\" alt=\":roll:\" title=\":roll:\" class=\"bbcode_smiley\" /> if <img src=\"//latex.artofproblemsolving.com/a/f/3/af321f09ace7e55116bb58f2984d809fe21060d8.png\" class=\"latex\" alt=\"$\\mu (n)<0$\" style=\"vertical-align: -4px\" width=\"69\" height=\"18\" ><br>\n<br>\nSorry,because probem is -1<=f(1)/1+f(2)/2+...+f(n)/n<=1 <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 564275,
"post_number": 3,
"post_time_unix": 1151813366,
"post_time_utc": "2006-07-02 04:09:26 UTC",
"thanks_received": 2,
"user_id": 5820,
"username": "N.T.TUAN"
},
{
"attachments": [],
"content_bbcode": "[quote=\"N.T.TUAN\"][quote=\"{x}\"] \\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{n \\leq x}\\mu ( n ) \\cdot 1 )\\\\ \\end{eqnarray*} [/quote]\nWhy? :roll: if $\\mu (n)<0$\n\nSorry,because probem is -1<=f(1)/1+f(2)/2+...+f(n)/n<=1 :([/quote]\r\nI also don't know why.\r\nBut I think we can modify the solution of {x}'s.\r\n$\\frac{1}{x}*1=\\frac{1}{x}*\\sum_{x\\leq n}{\\mu (n)*[\\frac{x}{n}]}\\geq \\frac{1}{x}(\\sum_{n\\leq x}{\\mu(n)\\frac{x}{n}}-\\sum_{n\\leq x,\\mu (n)>0}\\mu(n)*1)$\r\nNotice that$\\sum_{n\\leq x,\\mu (n)>0}\\mu(n)*1\\leq x-1$\r\nthen$\\sum_{n\\leq x}{\\mu(n)\\frac{1}{n}}\\leq 1$",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">N.T.TUAN wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">{x} wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/2/7/2/2727b3af3150df317fd4fb46702f13f6c17ce374.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*} \\frac{1}{x}\\cdot 1 = \\frac{1}{x}\\sum_{n \\leq x}\\mu ( n ) \\cdot \\lfloor \\frac{x}{n} \\rfloor & \\geq & \\frac{1}{x}\\cdot ( \\sum_{n \\leq x}\\mu ( n ) \\cdot \\frac{x}{n}-\\sum_{n \\leq x}\\mu ( n ) \\cdot 1 )\\\\ \\end{eqnarray*}\" width=\"476\" height=\"47\" ></div>\n</div>\nWhy? <img src=\"/assets/images/smilies/rolleyes.gif\" width=\"20\" height=\"20\" alt=\":roll:\" title=\":roll:\" class=\"bbcode_smiley\" /> if <img src=\"//latex.artofproblemsolving.com/a/f/3/af321f09ace7e55116bb58f2984d809fe21060d8.png\" class=\"latex\" alt=\"$\\mu (n)<0$\" style=\"vertical-align: -4px\" width=\"69\" height=\"18\" ><br>\n<br>\nSorry,because probem is -1<=f(1)/1+f(2)/2+...+f(n)/n<=1 <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /></div>\n</div>\nI also don't know why.<br>\nBut I think we can modify the solution of {x}'s.<br>\n<img src=\"//latex.artofproblemsolving.com/4/f/7/4f7ec2501857d699e97331317ee3f236a1b407f1.png\" class=\"latex\" alt=\"$\\frac{1}{x}*1=\\frac{1}{x}*\\sum_{x\\leq n}{\\mu (n)*[\\frac{x}{n}]}\\geq \\frac{1}{x}(\\sum_{n\\leq x}{\\mu(n)\\frac{x}{n}}-\\sum_{n\\leq x,\\mu (n)>0}\\mu(n)*1)$\" style=\"vertical-align: -24px\" width=\"496\" height=\"49\" ><br>\nNotice tha<span style=\"white-space:nowrap;\">t<img src=\"//latex.artofproblemsolving.com/0/0/3/003ec46de6ef1762ad80770cb4674c6b1a2d91df.png\" class=\"latex\" alt=\"$\\sum_{n\\leq x,\\mu (n)>0}\\mu(n)*1\\leq x-1$\" style=\"vertical-align: -24px\" width=\"204\" height=\"42\" ></span><br>\nthe<span style=\"white-space:nowrap;\">n<img src=\"//latex.artofproblemsolving.com/a/f/b/afbd6ef891265946cfc98ebf02e5c620c882a2af.png\" class=\"latex\" alt=\"$\\sum_{n\\leq x}{\\mu(n)\\frac{1}{n}}\\leq 1$\" style=\"vertical-align: -23px\" width=\"112\" height=\"47\" ></span>",
"post_id": 564357,
"post_number": 4,
"post_time_unix": 1151833844,
"post_time_utc": "2006-07-02 09:50:44 UTC",
"thanks_received": 2,
"user_id": 14130,
"username": "Hawk Tiger"
},
{
"attachments": [],
"content_bbcode": "Hawk Tiger: Thank you. I corrected using your idea :) \r\nN.T.TUAN: I simply made a mistake. I'm sorry. Now it's corrected and I proved the lower bound too :) (which is actually more or less the same thing) Enjoy !",
"content_html": "Hawk Tiger: Thank you. I corrected using your idea <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /><br>\nN.T.TUAN: I simply made a mistake. I'm sorry. Now it's corrected and I proved the lower bound too <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" /> (which is actually more or less the same thing) Enjoy !",
"post_id": 564381,
"post_number": 5,
"post_time_unix": 1151839249,
"post_time_utc": "2006-07-02 11:20:49 UTC",
"thanks_received": 1,
"user_id": 12380,
"username": "{x}"
}
],
"source": null
}
|
Let \(\mu\) be the Möbius function. Prove that
\[
\sum_{k=1}^n \frac{\mu(k)}{k} \le 1
\]
for all integers \(n>0\).
|
[
"/Mathematics/NumberTheory/GeneralNumberTheory/AnalyticNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/NumberTheoreticFunctions/GeneralNumberTheoreticFunctions",
"/Mathematics/NumberTheory/NumberTheoreticFunctions/NumberTheoreticSums"
] |
Apply Möbius inversion to obtain ∑ₙ≤x μ(n)⌊x/n⌋ = 1 and compare ⌊x/n⌋ with x/n using sign‑dependent inequalities to bound ∑ₙ≤x μ(n)/n.
|
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0.0234222412109375,
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aops_9979
|
[hide]
like JBL said, the sum cannot be the sum of two primes, so it must be from the list 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, etc. Now, we must look at each possible sum, in a guess and check method to see what works. We know that P is able to tell what the two numbers are, which means of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a*b = product, all but one pair has the property that a+b can be written as the sum of two primes (thus he can eliminate all but one). When S now knows the two numbers, this tells us that of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a+b = sum, if we consider a*b, only one of the pairs would have led P to know what the two numbers are.
I'll leave this up to you, but it is easy to show that 11 does not work, and the next possible sum is 17.
17 = (2+15),(3+14),(4+13),(5+12),(6+11),(7+10),(8+9), so we must go through all of the products, which are 30, 42, 52, 60, 66, 70, 72 respectively. Now, 30 = (2*15), (3*10), (5*6), but note that 2+15 = 17, and 5+6 = 11 both cannot be written as the sum of two primes, so 30 is not possible. 42 = (2*21),(3*14),(6*7), but 2+21=23 and 3+14=17 both cannot be written as the sum of two primes. If i continue in this fashion, 30,42,60,66,70,72 will not work, but 52 will. 52 = (2*26),(4*13), and (4,13) is the only pair with sum that cannot be written as the sum of two primes, so that means that P would get it, since only one pair of (a*b) couldn't be written as the sum of two primes, and S would get it because only 52 worked.
Therefore, the two numbers are 4 and 13.
[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "I have two nephews Sanjay (Nickname can be S) and Pratik (nick name can be P). You can take it as given that both are good in math. I once selected 2 positive integers, both greater than 1 (equal or greater than 2), and less than 50. Let the sum of these numbers be S and Product of these numbers be P. I gave sum of these numbers (S) to S(anjay), and product of these numbers (P) to P(ratik).\r\n\r\nNow S calls P on phone and tells; \"You can never guess the sum S\"\r\nLittle later P Calls S , \" Yes, now I do know the value of S\"\r\nLittle later S calls P . \" Big deal, now I also know the value of P\" \r\n\r\nWhat were the numbers I selected?",
"content_html": "I have two nephews Sanjay (Nickname can be S) and Pratik (nick name can be P). You can take it as given that both are good in math. I once selected 2 positive integers, both greater than 1 (equal or greater than 2), and less than 50. Let the sum of these numbers be S and Product of these numbers be P. I gave sum of these numbers (S) to S(anjay), and product of these numbers (P) to P(ratik).<br>\n<br>\nNow S calls P on phone and tells; "You can never guess the sum S"<br>\nLittle later P Calls S , " Yes, now I do know the value of S"<br>\nLittle later S calls P . " Big deal, now I also know the value of P"<br>\n<br>\nWhat were the numbers I selected?",
"post_id": 62608,
"post_number": 1,
"post_time_unix": 1074042671,
"post_time_utc": "2004-01-14 01:11:11 UTC",
"thanks_received": 2,
"user_id": 1944,
"username": "Gyan"
},
{
"attachments": [],
"content_bbcode": "Cute -- I'll need a little bit of time, though. This is a lot like the \"how old are my children?\" question, only tougher.",
"content_html": "Cute -- I'll need a little bit of time, though. This is a lot like the "how old are my children?" question, only tougher.",
"post_id": 62647,
"post_number": 2,
"post_time_unix": 1074047181,
"post_time_utc": "2004-01-14 02:26:21 UTC",
"thanks_received": 2,
"user_id": 1430,
"username": "JBL"
},
{
"attachments": [],
"content_bbcode": "Okay, so, from the first statement I get [hide]that we know that S is not the sum of two prime numbers. So in particular we know that it is odd.[/hide] I'll keep working from there.",
"content_html": "Okay, so, from the first statement I get <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\">that we know that S is not the sum of two prime numbers. So in particular we know that it is odd.</div> I'll keep working from there.",
"post_id": 62839,
"post_number": 3,
"post_time_unix": 1074144355,
"post_time_utc": "2004-01-15 05:25:55 UTC",
"thanks_received": 2,
"user_id": 1430,
"username": "JBL"
},
{
"attachments": [],
"content_bbcode": "you actually know a tiny tiny bit stronger than that eg [hide] 2 and 4 aren't both prime, but those two adding together to 6 won't work either (since 1 and 8 isn't a possibility)[/hide]",
"content_html": "you actually know a tiny tiny bit stronger than that eg <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\">2 and 4 aren't both prime, but those two adding together to 6 won't work either (since 1 and 8 isn't a possibility)</div>",
"post_id": 62853,
"post_number": 4,
"post_time_unix": 1074150476,
"post_time_utc": "2004-01-15 07:07:56 UTC",
"thanks_received": 2,
"user_id": 1219,
"username": "TripleM"
},
{
"attachments": [],
"content_bbcode": "[hide]\n\n\n\nlike JBL said, the sum cannot be the sum of two primes, so it must be from the list 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, etc. Now, we must look at each possible sum, in a guess and check method to see what works. We know that P is able to tell what the two numbers are, which means of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a*b = product, all but one pair has the property that a+b can be written as the sum of two primes (thus he can eliminate all but one). When S now knows the two numbers, this tells us that of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a+b = sum, if we consider a*b, only one of the pairs would have led P to know what the two numbers are.\n\n\n\nI'll leave this up to you, but it is easy to show that 11 does not work, and the next possible sum is 17.\n\n\n\n17 = (2+15),(3+14),(4+13),(5+12),(6+11),(7+10),(8+9), so we must go through all of the products, which are 30, 42, 52, 60, 66, 70, 72 respectively. Now, 30 = (2*15), (3*10), (5*6), but note that 2+15 = 17, and 5+6 = 11 both cannot be written as the sum of two primes, so 30 is not possible. 42 = (2*21),(3*14),(6*7), but 2+21=23 and 3+14=17 both cannot be written as the sum of two primes. If i continue in this fashion, 30,42,60,66,70,72 will not work, but 52 will. 52 = (2*26),(4*13), and (4,13) is the only pair with sum that cannot be written as the sum of two primes, so that means that P would get it, since only one pair of (a*b) couldn't be written as the sum of two primes, and S would get it because only 52 worked.\n\n\n\nTherefore, the two numbers are 4 and 13.\n\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\">like JBL said, the sum cannot be the sum of two primes, so it must be from the list 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, etc. Now, we must look at each possible sum, in a guess and check method to see what works. We know that P is able to tell what the two numbers are, which means of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a*b = product, all but one pair has the property that a+b can be written as the sum of two primes (thus he can eliminate all but one). When S now knows the two numbers, this tells us that of all pairs (a,b) a and b positive integers > 1 and a :ge: b, with a+b = sum, if we consider a*b, only one of the pairs would have led P to know what the two numbers are.<br>\n<br>\n<br>\n<br>\nI'll leave this up to you, but it is easy to show that 11 does not work, and the next possible sum is 17.<br>\n<br>\n<br>\n<br>\n17 = (2+15),(3+14),(4+13),(5+12),(6+11),(7+10),(8+9), so we must go through all of the products, which are 30, 42, 52, 60, 66, 70, 72 respectively. Now, 30 = (2*15), (3*10), (5*6), but note that 2+15 = 17, and 5+6 = 11 both cannot be written as the sum of two primes, so 30 is not possible. 42 = (2*21),(3*14),(6*7), but 2+21=23 and 3+14=17 both cannot be written as the sum of two primes. If i continue in this fashion, 30,42,60,66,70,72 will not work, but 52 will. 52 = (2*26),(4*13), and (4,13) is the only pair with sum that cannot be written as the sum of two primes, so that means that P would get it, since only one pair of (a*b) couldn't be written as the sum of two primes, and S would get it because only 52 worked.<br>\n<br>\n<br>\n<br>\nTherefore, the two numbers are 4 and 13.</div>",
"post_id": 62874,
"post_number": 5,
"post_time_unix": 1074183366,
"post_time_utc": "2004-01-15 16:16:06 UTC",
"thanks_received": 2,
"user_id": 1362,
"username": "zscool"
}
],
"source": null
}
|
I selected two integers x and y such that 2 \le x,y < 50. Let S = x + y and P = xy. Sanjay is given S and Pratik is given P.
Sanjay to Pratik: "You can never guess the sum S."
Pratik to Sanjay: "Yes, now I do know the value of S."
Sanjay to Pratik: "Big deal, now I also know the value of P."
Find the integers x and y.
|
[
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Numbers/SmallNumbers",
"/Mathematics/NumberTheory/PrimeNumbers/PrimeRepresentations",
"/Mathematics/NumberTheory/PrimeNumbers/PrimeSumsandProducts",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Find a sum that cannot be expressed as a sum of two primes and whose partitions give exactly one product that is uniquely identifiable because all other factorizations correspond to sums that are sums of two primes.
|
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aops_99803
|
[hide]
This isn't as much an actual proof as just reasoning. I don't know if it works for all closed curves, either.
Let $P =$ the arbitrary point. Now locate the point on the curve that is closest to $P$; call it point $A$. Draw the chord through $P$ and $A$, and call the point on the other side of the chord point $B$. Clearly, since $A$ is the point closest to $P$, $AP \leq BP$. Now here's the tricky part. What if we redifined point $A$, "sliding" it along the curve and redifining $B$ in the process? This will make $AP$ decrease, until finally, $A$ and $B$ are switched so that $AP \geq BP$. But since the curve is continuous, in the process of going from $AP \leq BP$ to $AP \geq BP$ there must be at least one instance where $AP = BP$.
[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
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"content_bbcode": "You are given a closed curve(can be absolutely any closed curve). Prove that if you take any arbitrary point inside the curve, there always exists a chord of the curve with that point as its mid-point. \r\n\r\nNote that a chord is a line segment joining two points on a curve.\r\n??",
"content_html": "You are given a closed curve(can be absolutely any closed curve). Prove that if you take any arbitrary point inside the curve, there always exists a chord of the curve with that point as its mid-point.<br>\n<br>\nNote that a chord is a line segment joining two points on a curve.<br>\n??",
"post_id": 563508,
"post_number": 1,
"post_time_unix": 1151752363,
"post_time_utc": "2006-07-01 11:12:43 UTC",
"thanks_received": 2,
"user_id": 16952,
"username": "xxxyyyy"
},
{
"attachments": [],
"content_bbcode": "[hide]\nThis isn't as much an actual proof as just reasoning. I don't know if it works for all closed curves, either.\n\nLet $P =$ the arbitrary point. Now locate the point on the curve that is closest to $P$; call it point $A$. Draw the chord through $P$ and $A$, and call the point on the other side of the chord point $B$. Clearly, since $A$ is the point closest to $P$, $AP \\leq BP$. Now here's the tricky part. What if we redifined point $A$, \"sliding\" it along the curve and redifining $B$ in the process? This will make $AP$ decrease, until finally, $A$ and $B$ are switched so that $AP \\geq BP$. But since the curve is continuous, in the process of going from $AP \\leq BP$ to $AP \\geq BP$ there must be at least one instance where $AP = BP$.\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\">This isn't as much an actual proof as just reasoning. I don't know if it works for all closed curves, either.<br>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/3/0/4/304bd5180deaa10843d6a51fa5ad2077be3b038b.png\" class=\"latex\" alt=\"$P =$\" width=\"32\" height=\"12\" > the arbitrary point. Now locate the point on the curve that is closest to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >;</span> call it point <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >.</span> Draw the chord through <img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,</span> and call the point on the other side of the chord point <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" >.</span> Clearly, since <img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" > is the point closest to <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png\" class=\"latex\" alt=\"$P$\" width=\"14\" height=\"12\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/7/d87b5a4d41ffdee4c523db8a4c2e77dacb174a8d.png\" class=\"latex\" alt=\"$AP \\leq BP$\" style=\"vertical-align: -2px\" width=\"81\" height=\"15\" >.</span> Now here's the tricky part. What if we redifined point <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png\" class=\"latex\" alt=\"$A$\" width=\"13\" height=\"13\" >,</span> "sliding" it along the curve and redifining <img src=\"//latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png\" class=\"latex\" alt=\"$B$\" width=\"14\" height=\"12\" > in the process? This will make <img src=\"//latex.artofproblemsolving.com/e/3/b/e3bb4eeb347e74e025fd175c22f1f7275af0555e.png\" class=\"latex\" alt=\"$AP$\" width=\"27\" height=\"13\" > decrease, until finally, <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 switched so that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/3/2/f32b44dcc2d742ff04ff7504d8bb6d8f006959ba.png\" class=\"latex\" alt=\"$AP \\geq BP$\" style=\"vertical-align: -2px\" width=\"81\" height=\"15\" >.</span> But since the curve is continuous, in the process of going from <img src=\"//latex.artofproblemsolving.com/d/8/7/d87b5a4d41ffdee4c523db8a4c2e77dacb174a8d.png\" class=\"latex\" alt=\"$AP \\leq BP$\" style=\"vertical-align: -2px\" width=\"81\" height=\"15\" > to <img src=\"//latex.artofproblemsolving.com/f/3/2/f32b44dcc2d742ff04ff7504d8bb6d8f006959ba.png\" class=\"latex\" alt=\"$AP \\geq BP$\" style=\"vertical-align: -2px\" width=\"81\" height=\"15\" > there must be at least one instance where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/1/30194aecfd79b7ac409ee684149d9de91546c204.png\" class=\"latex\" alt=\"$AP = BP$\" width=\"81\" height=\"13\" >.</span></div>",
"post_id": 563641,
"post_number": 2,
"post_time_unix": 1151764010,
"post_time_utc": "2006-07-01 14:26:50 UTC",
"thanks_received": 2,
"user_id": 12328,
"username": "TheAmazingOne"
},
{
"attachments": [],
"content_bbcode": "what if at such an instance you intersect the curve more than once??? He didn't mention that the curve is \"convex'\".",
"content_html": "what if at such an instance you intersect the curve more than once??? He didn't mention that the curve is "convex'".",
"post_id": 563657,
"post_number": 3,
"post_time_unix": 1151764686,
"post_time_utc": "2006-07-01 14:38:06 UTC",
"thanks_received": 2,
"user_id": 10705,
"username": "pkerichang"
}
],
"source": null
}
|
You are given a closed curve (any closed curve). Prove that if you take any arbitrary point inside the curve, there always exists a chord of the curve with that point as its midpoint.
Note: a chord is a line segment joining two points on the curve.
|
[
"/Mathematics/Geometry/ContinuityPrinciple",
"/Mathematics/Geometry/Curves/GeneralCurves/ClosedCurve",
"/Mathematics/Geometry/Curves/GeneralCurves/ClosedCurveProblem",
"/Mathematics/Geometry/Curves/GeneralCurves/Curve",
"/Mathematics/Geometry/Curves/PlaneCurves/GeneralPlaneCurves"
] |
Track the signed distance difference between two opposite points on the curve as one slides continuously, and apply the Intermediate Value Theorem to obtain equality.
|
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0.0197906494140625,
0.0029201507568359375
] |
aops_99812
|
[quote="Treething"]Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.[/quote]
[hide]$\frac{\binom{4}{3}}{\binom{5}{3}}=\frac{4}{10}=\boxed{40}$percent[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.",
"content_html": "Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.",
"post_id": 563528,
"post_number": 1,
"post_time_unix": 1151755814,
"post_time_utc": "2006-07-01 12:10:14 UTC",
"thanks_received": 2,
"user_id": 5336,
"username": "Treething"
},
{
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"content_bbcode": "[quote=\"Treething\"]Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.[/quote]\r\n\r\n[hide]$\\frac{\\binom{4}{3}}{\\binom{5}{3}}=\\frac{4}{10}=\\boxed{40}$percent[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Treething wrote:</div>\n<div class=\"bbcode_quote_body\">Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.</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/a/d/9/ad9a620eeeb49e942bf6be6e180fe9d6c0c0746e.png\" class=\"latex\" alt=\"$\\frac{\\binom{4}{3}}{\\binom{5}{3}}=\\frac{4}{10}=\\boxed{40}$\" style=\"vertical-align: -20px\" width=\"127\" height=\"49\" >p</span>ercent</div>",
"post_id": 563535,
"post_number": 2,
"post_time_unix": 1151756324,
"post_time_utc": "2006-07-01 12:18:44 UTC",
"thanks_received": 2,
"user_id": 18270,
"username": "SplashD"
},
{
"attachments": [],
"content_bbcode": "SplashD, your answer is \"[Unparseable or potentially dangerous latex formula. Error 6]\" ;)\r\n\r\n[hide]\nThe answer is $\\frac{{4 \\choose 3}}{{5 \\choose 3}}=\\frac{2}{5}=40 \\%$\n[/hide]\r\n\r\nOh, and why can't I quote SplashD?\r\n\r\n[size=75][color=darkred]Fixed your LaTeX\n-nebula42[/color][/size]",
"content_html": "SplashD, your answer is "[Unparseable or potentially dangerous latex formula. Error 6]" <img src=\"/assets/images/smilies/wink.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=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">The answer is <img src=\"//latex.artofproblemsolving.com/3/1/4/3142fec6da000f3a6a2079eeee85cd913d2b4f56.png\" class=\"latex\" alt=\"$\\frac{{4 \\choose 3}}{{5 \\choose 3}}=\\frac{2}{5}=40 \\%$\" style=\"vertical-align: -20px\" width=\"121\" height=\"49\" ></div><br>\n<br>\nOh, and why can't I quote SplashD?<br>\n<br>\n<span class=\"bbfont-three-q\"><span style=\"color:darkred\">Fixed your LaTeX<br>\n-nebula42</span></span>",
"post_id": 563585,
"post_number": 3,
"post_time_unix": 1151760134,
"post_time_utc": "2006-07-01 13:22:14 UTC",
"thanks_received": 2,
"user_id": 9152,
"username": "mathnerd314"
},
{
"attachments": [],
"content_bbcode": "The quote option is hidden under one of the hide tags in his signature.\r\n\r\n[hide=\"answer\"]There are $\\binom43$ ways to choose 3 boys and $\\binom53$ total ways to choose them, making the answer $\\frac{\\binom43}{\\binom53}=\\frac{4}{10}=\\frac25=40\\%$[/hide]\r\n\r\nmathnerd, you need to have \\frac{{4\\choose3}{{5\\choose3}} or what I have :)",
"content_html": "The quote option is hidden under one of the hide tags in his signature.<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\">There are <img src=\"//latex.artofproblemsolving.com/0/0/8/0082835dadc0fdac5fba119a48d8fd27ec86133d.png\" class=\"latex\" alt=\"$\\binom43$\" style=\"vertical-align: -22px\" width=\"33\" height=\"53\" > ways to choose 3 boys and <img src=\"//latex.artofproblemsolving.com/0/b/4/0b43070e51d1f17999c62ae6174f82c69422f48f.png\" class=\"latex\" alt=\"$\\binom53$\" style=\"vertical-align: -22px\" width=\"33\" height=\"53\" > total ways to choose them, making the answer <img src=\"//latex.artofproblemsolving.com/f/c/6/fc6174bf7ef6154629c95e0440dd535c4c4999c6.png\" class=\"latex\" alt=\"$\\frac{\\binom43}{\\binom53}=\\frac{4}{10}=\\frac25=40\\%$\" style=\"vertical-align: -20px\" width=\"167\" height=\"49\" ></div><br>\n<br>\nmathnerd, you need to have \\frac{{4\\choose3}{{5\\choose3}} or what I have <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 563590,
"post_number": 4,
"post_time_unix": 1151760474,
"post_time_utc": "2006-07-01 13:27:54 UTC",
"thanks_received": 2,
"user_id": 8949,
"username": "b-flat"
},
{
"attachments": [],
"content_bbcode": "[hide=\"Assuming mary is female and jerry, harry larry, and steve are male\"]$\\frac{\\binom43}{\\binom53}=\\frac25=40\\%$[/hide]Treething said [b]%[/b]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Assuming mary is female and jerry, harry larry, and steve are male</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/e/d/3/ed35c197b3e11dbb2bff6735acd29a18b42e646c.png\" class=\"latex\" alt=\"$\\frac{\\binom43}{\\binom53}=\\frac25=40\\%$\" style=\"vertical-align: -20px\" width=\"121\" height=\"49\" ></div>Treething said <b>%</b>",
"post_id": 563627,
"post_number": 5,
"post_time_unix": 1151763051,
"post_time_utc": "2006-07-01 14:10:51 UTC",
"thanks_received": 1,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "[hide]The answer is $\\frac{\\binom{4}{3}}{\\binom{5}{3}}=\\frac{2}{5}=40\\%$.\n\n$40\\%$[/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\">The answer is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/1/1/01103845f7fb1ee43f99fe05ec9bc591dc87e57e.png\" class=\"latex\" alt=\"$\\frac{\\binom{4}{3}}{\\binom{5}{3}}=\\frac{2}{5}=40\\%$\" style=\"vertical-align: -20px\" width=\"121\" height=\"49\" >.</span><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/4/e/f4e4e5c6d549ce47918b94180cad5c1644d9ea88.png\" class=\"latex\" alt=\"$40\\%$\" style=\"vertical-align: -1px\" width=\"32\" height=\"14\" ></div>",
"post_id": 563761,
"post_number": 6,
"post_time_unix": 1151769704,
"post_time_utc": "2006-07-01 16:01:44 UTC",
"thanks_received": 2,
"user_id": 11714,
"username": "mathgeniuse^ln(x)"
},
{
"attachments": [],
"content_bbcode": "[quote=\"mathnerd314\"]SplashD, your answer is \"[Unparseable or potentially dangerous latex formula. Error 6]\" ;)\n[/quote]\r\n\r\nYeah, I put % in $LaTeX$ instead of \\%.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">mathnerd314 wrote:</div>\n<div class=\"bbcode_quote_body\">SplashD, your answer is "[Unparseable or potentially dangerous latex formula. Error 6]" <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\";)\" title=\";)\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\nYeah, I put % in <img src=\"//latex.artofproblemsolving.com/b/a/0/ba0b91b5dd1bb1d4e59e6943e2a422e44cd39474.png\" class=\"latex\" alt=\"$LaTeX$\" width=\"59\" height=\"12\" > instead of \\%.",
"post_id": 563777,
"post_number": 7,
"post_time_unix": 1151771018,
"post_time_utc": "2006-07-01 16:23:38 UTC",
"thanks_received": 1,
"user_id": 18270,
"username": "SplashD"
},
{
"attachments": [],
"content_bbcode": "[quote=\"mathnerd314\"]SplashD, your answer is \"[Unparseable or potentially dangerous latex formula. Error 6]\" ;)\n\n[hide]\nThe answer is $\\frac{4 \\choose 3}{5 \\choose 3}=\\frac{2}{5}$\n[/hide]\n\nOh, and why can't I quote SplashD?[/quote]\r\n\r\nAnd your answer has an error.\r\n\r\nYou can quote SplashD, just click in his signature, \"Click here to reveal hidden content\", or something like that, and then the quote will come.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">mathnerd314 wrote:</div>\n<div class=\"bbcode_quote_body\">SplashD, your answer is "[Unparseable or potentially dangerous latex formula. Error 6]" <img src=\"/assets/images/smilies/wink.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=\"#\">Click to reveal hidden text</a><div class=\"cmty-hide-content\" style=\"display:none\">The answer is <span class=\"aopscode-error aopscode-latex-error\">$\\frac{4 \\choose 3}{5 \\choose 3}=\\frac{2}{5}$</span></div><br>\n<br>\nOh, and why can't I quote SplashD?</div>\n</div>\n<br>\nAnd your answer has an error.<br>\n<br>\nYou can quote SplashD, just click in his signature, "Click here to reveal hidden content", or something like that, and then the quote will come.",
"post_id": 563799,
"post_number": 8,
"post_time_unix": 1151772917,
"post_time_utc": "2006-07-01 16:55:17 UTC",
"thanks_received": 1,
"user_id": 11714,
"username": "mathgeniuse^ln(x)"
}
],
"source": null
}
|
Mary and Jerry have three kids: Harry, Larry, and Steve. If three family members are picked at random, what is the probability that they are all male? Express your answer as a percentage.
|
[
"/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients",
"/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities",
"/Mathematics/DiscreteMathematics/Combinatorics/Enumeration",
"/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics",
"/Mathematics/ProbabilityandStatistics/Probability/SampleSpace"
] |
Count favorable and total selections using combinations (hypergeometric probability).
|
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0.043609619140625,
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0.0176239013671875,
0.0660400390625,
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0.0084228515625,
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0.0210723876953125,
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0.00115966796875,
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0.009185791015625,
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0.0169677734375,
0.0036468505859375,
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0.01158905029296875,
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0.0038967132568359375,
0.0182342529296875,
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0.018646240234375,
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0.009674072265625,
0.00939178466796875,
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0.002162933349609375,
0.044586181640625,
0.0007781982421875,
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0.004146575927734375,
0.00365447998046875,
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0.031341552734375,
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0.0283203125,
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0.047393798828125,
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0.0439453125,
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0.007137298583984375,
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0.01371002197265625,
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0.006519317626953125,
0.037353515625,
0.0182037353515625,
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0.038360595703125,
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0.007785797119140625,
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0.02227783203125,
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0.01282501220703125,
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0.0462646484375,
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0.0086212158203125,
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0.008026123046875,
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0.0101470947265625,
0.005428314208984375,
0.0103759765625,
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0.01788330078125,
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0.01018524169921875,
0.0021076202392578125,
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0.0151519775390625,
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0.00514984130859375,
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aops_99823
|
it's easy to show that $a,b,c\equiv 0(mod 9)$ then $a=9k_{1}, b=9k_{2}, c=9k_{3}$ where $k_{1}, k_{2},k_{3}\in \mathbb{N}$. Follow this method,(infinit decrease) so that the equation has not answer over $\mathbb{N}$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "solve the equation $a^{3}+b^{3}=22c^{3}$ in positive integers.",
"content_html": "solve the equation <img src=\"//latex.artofproblemsolving.com/0/1/e/01e98aec55c6461161b9e10c634bebbc0ea13c82.png\" class=\"latex\" alt=\"$a^{3}+b^{3}=22c^{3}$\" style=\"vertical-align: -1px\" width=\"111\" height=\"16\" > in positive integers.",
"post_id": 563560,
"post_number": 1,
"post_time_unix": 1151757797,
"post_time_utc": "2006-07-01 12:43:17 UTC",
"thanks_received": 2,
"user_id": 9056,
"username": "suweijie"
},
{
"attachments": [],
"content_bbcode": "it's easy to show that $a,b,c\\equiv 0(mod 9)$ then $a=9k_{1}, b=9k_{2}, c=9k_{3}$ where $k_{1}, k_{2},k_{3}\\in \\mathbb{N}$. Follow this method,(infinit decrease) so that the equation has not answer over $\\mathbb{N}$",
"content_html": "it's easy to show that <img src=\"//latex.artofproblemsolving.com/e/5/e/e5eba5d427dd6cf61eeadd372b1ca3122c45601e.png\" class=\"latex\" alt=\"$a,b,c\\equiv 0(mod 9)$\" style=\"vertical-align: -4px\" width=\"130\" height=\"18\" > then <img src=\"//latex.artofproblemsolving.com/2/4/5/245acfbbf86267c113faf426e6c6d12faab39d85.png\" class=\"latex\" alt=\"$a=9k_{1}, b=9k_{2}, c=9k_{3}$\" style=\"vertical-align: -3px\" width=\"190\" height=\"16\" > where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/1/7/b1716ad6d1c828ca8d8f930a60332711afca6cb2.png\" class=\"latex\" alt=\"$k_{1}, k_{2},k_{3}\\in \\mathbb{N}$\" style=\"vertical-align: -3px\" width=\"101\" height=\"16\" >.</span> Follow this method,(infinit decrease) so that the equation has not answer over <img src=\"//latex.artofproblemsolving.com/a/1/e/a1e6cd0e4103015efba12e0fc4291f012b467eaa.png\" class=\"latex\" alt=\"$\\mathbb{N}$\" style=\"vertical-align: 0px\" width=\"12\" height=\"12\" >",
"post_id": 564325,
"post_number": 2,
"post_time_unix": 1151824651,
"post_time_utc": "2006-07-02 07:17:31 UTC",
"thanks_received": 2,
"user_id": 5729,
"username": "ehsan2004"
},
{
"attachments": [],
"content_bbcode": "Sorry,I can't agree with you,ehsan2004.\r\nIf $a\\equiv-1,b\\equiv+1(\\mod 9)$\r\n$c\\equiv 0 \\mod 9$\r\nthen there is not a contraduction.\r\nSo I don't think we can prove it easily.",
"content_html": "Sorry,I can't agree with you,ehsan2004.<br>\nIf <img src=\"//latex.artofproblemsolving.com/7/7/1/771eb81f274a27bcb6b362e8e0e94ec03433415d.png\" class=\"latex\" alt=\"$a\\equiv-1,b\\equiv+1(\\mod 9)$\" style=\"vertical-align: -4px\" width=\"195\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/0/a/6/0a63dd0789c0c90230e2a5cbc349acadc83c98a6.png\" class=\"latex\" alt=\"$c\\equiv 0 \\mod 9$\" width=\"102\" height=\"12\" ><br>\nthen there is not a contraduction.<br>\nSo I don't think we can prove it easily.",
"post_id": 564456,
"post_number": 3,
"post_time_unix": 1151850101,
"post_time_utc": "2006-07-02 14:21:41 UTC",
"thanks_received": 2,
"user_id": 14130,
"username": "Hawk Tiger"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Hawk Tiger\"]Sorry,I can't agree with you,ehsan2004.\nIf $a\\equiv-1,b\\equiv+1(\\mod 9)$\n$c\\equiv 0 \\mod 9$\nthen there is not a contraduction.\nSo I don't think we can prove it easily.[/quote]\r\n\r\noops :oops: \r\n\r\nI forgot it easily. Well, you can solve it only for this manner :blush:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Hawk Tiger wrote:</div>\n<div class=\"bbcode_quote_body\">Sorry,I can't agree with you,ehsan2004.<br>\nIf <img src=\"//latex.artofproblemsolving.com/7/7/1/771eb81f274a27bcb6b362e8e0e94ec03433415d.png\" class=\"latex\" alt=\"$a\\equiv-1,b\\equiv+1(\\mod 9)$\" style=\"vertical-align: -4px\" width=\"195\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/0/a/6/0a63dd0789c0c90230e2a5cbc349acadc83c98a6.png\" class=\"latex\" alt=\"$c\\equiv 0 \\mod 9$\" width=\"102\" height=\"12\" ><br>\nthen there is not a contraduction.<br>\nSo I don't think we can prove it easily.</div>\n</div>\n<br>\noops <img src=\"/assets/images/smilies/blush.gif\" width=\"20\" height=\"20\" alt=\":oops:\" title=\":oops:\" class=\"bbcode_smiley\" /><br>\n<br>\nI forgot it easily. Well, you can solve it only for this manner <img src=\"/assets/images/smilies/redface_anim.gif\" width=\"19\" height=\"19\" alt=\":blush:\" title=\":blush:\" class=\"bbcode_smiley\" />",
"post_id": 565048,
"post_number": 4,
"post_time_unix": 1151915525,
"post_time_utc": "2006-07-03 08:32:05 UTC",
"thanks_received": 2,
"user_id": 5729,
"username": "ehsan2004"
},
{
"attachments": [],
"content_bbcode": "with infinite decrease we can show that $a,b$ have to be odd...",
"content_html": "with infinite decrease we can show that <img src=\"//latex.artofproblemsolving.com/e/9/b/e9bdf3e496517e8f8742f81dd774be1f93cf6e8c.png\" class=\"latex\" alt=\"$a,b$\" style=\"vertical-align: -3px\" width=\"25\" height=\"16\" > have to be odd...",
"post_id": 565175,
"post_number": 5,
"post_time_unix": 1151935044,
"post_time_utc": "2006-07-03 13:57:24 UTC",
"thanks_received": 2,
"user_id": 17962,
"username": "hydro"
},
{
"attachments": [],
"content_bbcode": "by the same reasoning a and b can't be multiples of 11\r\nAlso 7 divides (abc) and 9 divides c",
"content_html": "by the same reasoning a and b can't be multiples of 11<br>\nAlso 7 divides (abc) and 9 divides c",
"post_id": 565293,
"post_number": 6,
"post_time_unix": 1151944023,
"post_time_utc": "2006-07-03 16:27:03 UTC",
"thanks_received": 1,
"user_id": 20129,
"username": "clinkmath"
},
{
"attachments": [],
"content_bbcode": "I want to ask a question to suweijie,Have you solved this hard problem?\r\nOr are you sure this problem can be solved by using elementary Number theory?",
"content_html": "I want to ask a question to suweijie,Have you solved this hard problem?<br>\nOr are you sure this problem can be solved by using elementary Number theory?",
"post_id": 565804,
"post_number": 7,
"post_time_unix": 1151989622,
"post_time_utc": "2006-07-04 05:07:02 UTC",
"thanks_received": 2,
"user_id": 14130,
"username": "Hawk Tiger"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Hawk Tiger\"]I want to ask a question to suweijie,Have you solved this hard problem?\nOr are you sure this problem can be solved by using elementary Number theory?[/quote]\r\nSorry that I haven't solve it yet.This problem is from the book called Dr.Riemann's Zeros.The author of it solved the problem in half a term's time.Surely it's a hard problem,but I think there must exist an elementary method to solve it. :lol:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Hawk Tiger wrote:</div>\n<div class=\"bbcode_quote_body\">I want to ask a question to suweijie,Have you solved this hard problem?<br>\nOr are you sure this problem can be solved by using elementary Number theory?</div>\n</div>\nSorry that I haven't solve it yet.This problem is from the book called Dr.Riemann's Zeros.The author of it solved the problem in half a term's time.Surely it's a hard problem,but I think there must exist an elementary method to solve it. <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />",
"post_id": 568063,
"post_number": 8,
"post_time_unix": 1152283956,
"post_time_utc": "2006-07-07 14:52:36 UTC",
"thanks_received": 2,
"user_id": 9056,
"username": "suweijie"
},
{
"attachments": [],
"content_bbcode": "Thank you.",
"content_html": "Thank you.",
"post_id": 568105,
"post_number": 9,
"post_time_unix": 1152286847,
"post_time_utc": "2006-07-07 15:40:47 UTC",
"thanks_received": 2,
"user_id": 14130,
"username": "Hawk Tiger"
},
{
"attachments": [],
"content_bbcode": "no one? :maybe:",
"content_html": "no one? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" />",
"post_id": 582097,
"post_number": 10,
"post_time_unix": 1153833752,
"post_time_utc": "2006-07-25 13:22:32 UTC",
"thanks_received": 2,
"user_id": 9056,
"username": "suweijie"
},
{
"attachments": [],
"content_bbcode": "[quote=\"suweijie\"]no one? :maybe:[/quote]\r\n\r\nI think it is really hard,please help me!",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">suweijie wrote:</div>\n<div class=\"bbcode_quote_body\">no one? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" /></div>\n</div>\n<br>\nI think it is really hard,please help me!",
"post_id": 1028011,
"post_number": 11,
"post_time_unix": 1202379729,
"post_time_utc": "2008-02-07 10:22:09 UTC",
"thanks_received": 2,
"user_id": 26604,
"username": "shfdfzhjj"
},
{
"attachments": [],
"content_bbcode": "If you accept \"nonelementary\" solutions, it's not that hard (see your book :wink: ). And it's quite obvious that it's rather hard if you needlessly restrict to \"elementary\" methods.",
"content_html": "If you accept "nonelementary" solutions, it's not that hard (see your book <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" /> ). And it's quite obvious that it's rather hard if you needlessly restrict to "elementary" methods.",
"post_id": 1028022,
"post_number": 12,
"post_time_unix": 1202383024,
"post_time_utc": "2008-02-07 11:17:04 UTC",
"thanks_received": 2,
"user_id": 5787,
"username": "ZetaX"
},
{
"attachments": [],
"content_bbcode": "[quote=\"ZetaX\"]If you accept \"nonelementary\" solutions, it's not that hard (see your book :wink: ). And it's quite obvious that it's rather hard if you needlessly restrict to \"elementary\" methods.[/quote]\r\n\r\nBut i don't have this book :( \r\nCan you tell me \"nonelementary\" solutions? :wink:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ZetaX wrote:</div>\n<div class=\"bbcode_quote_body\">If you accept "nonelementary" solutions, it's not that hard (see your book <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" /> ). And it's quite obvious that it's rather hard if you needlessly restrict to "elementary" methods.</div>\n</div>\n<br>\nBut i don't have this book <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" /><br>\nCan you tell me "nonelementary" solutions? <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />",
"post_id": 1028082,
"post_number": 13,
"post_time_unix": 1202391208,
"post_time_utc": "2008-02-07 13:33:28 UTC",
"thanks_received": 2,
"user_id": 26604,
"username": "shfdfzhjj"
}
],
"source": null
}
|
Solve the equation
\[
a^{3}+b^{3}=22c^{3}
\]
in positive integers.
|
[
"/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation",
"/Mathematics/Algebra/AlgebraicEquations/CubicEquation",
"/Mathematics/Algebra/NumberTheory/Congruences/CastingOutNines",
"/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/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/Z",
"/Mathematics/Algebra/NumberTheory/Integers/Z-Plus",
"/Mathematics/NumberTheory/Congruences/CastingOutNines",
"/Mathematics/NumberTheory/Congruences/Congruence",
"/Mathematics/NumberTheory/Congruences/ModularArithmetic",
"/Mathematics/NumberTheory/DiophantineEquations/Diophantine",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation3rdPowers",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory"
] |
Show all three numbers are divisible by 9 and then use infinite descent to rule out any positive integer solutions.
|
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aops_9983
|
Here's my solution
[hide]
We want to build up (digit by digit) a set of special numbers that will sum to 1=.99999... So:
.7 = 1x0.7
.98=.7+.07x4
.994=.98+.007x2
.9996=.994+.0007x8
.99995=.9996+.00007x5
.999999=.99995+.000007x7
Since the highest coefficient was 8, it is easy to piece together 8 special numbers that sum to .999999. Then if we turn each of those numbers into repeating decimals with period 6, we have 8 special numbers that sum to .999999...=1.
It isn't clear to me how to prove that 8 is the minimum necessary.
One more thought. For some reason, when I saw this problem, I almost immediately thought to make use of the fact that 1=.999999... After that, everything is relatively straightforward. If I'm trying to teach someone how to do this, how do I explain the jump to the idea of using the fact that 1=.999999...?
[/hide]
--Dan
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "this is from 1997- \n\n\n\nCall a positive real number special if it has a decimal representation that consists entirely of digits 0 and 7. For example, 700/99=7.070707070707....and 77.007 are special numbers. What is the smallest n such that 1 can be written as the sum of n special numbers?\n\n\n\n(A) 7 (B) 8 (C) 9 (D) 10 (E) 1 can't be represented as the sum of finitely many special numbers\n\n\n\nhow do you do this? btw the answer is [hide] B [/hide]",
"content_html": "this is from 1997-<br>\n<br>\n<br>\n<br>\nCall a positive real number special if it has a decimal representation that consists entirely of digits 0 and 7. For example, 700/99=7.070707070707....and 77.007 are special numbers. What is the smallest n such that 1 can be written as the sum of n special numbers?<br>\n<br>\n<br>\n<br>\n(A) 7 (B) 8 (C) 9 (D) 10 (E) 1 can't be represented as the sum of finitely many special numbers<br>\n<br>\n<br>\n<br>\nhow do you do this? btw the 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\">B</div>",
"post_id": 62719,
"post_number": 1,
"post_time_unix": 1074115767,
"post_time_utc": "2004-01-14 21:29:27 UTC",
"thanks_received": 2,
"user_id": 1337,
"username": "s0mp"
},
{
"attachments": [],
"content_bbcode": "Oooh, this was a fun one :). Just try and make 1 from lots of 7's! (Although 1 will have to look slightly different.)",
"content_html": "Oooh, this was a fun one <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />. Just try and make 1 from lots of 7's! (Although 1 will have to look slightly different.)",
"post_id": 62720,
"post_number": 2,
"post_time_unix": 1074116039,
"post_time_utc": "2004-01-14 21:33:59 UTC",
"thanks_received": 2,
"user_id": 1430,
"username": "JBL"
},
{
"attachments": [],
"content_bbcode": "ah thanks, i was thinking there was a better way than just writing it out.",
"content_html": "ah thanks, i was thinking there was a better way than just writing it out.",
"post_id": 62723,
"post_number": 3,
"post_time_unix": 1074117017,
"post_time_utc": "2004-01-14 21:50:17 UTC",
"thanks_received": 2,
"user_id": 1337,
"username": "s0mp"
},
{
"attachments": [],
"content_bbcode": "Nope, pretty much not.",
"content_html": "Nope, pretty much not.",
"post_id": 62934,
"post_number": 4,
"post_time_unix": 1074213628,
"post_time_utc": "2004-01-16 00:40:28 UTC",
"thanks_received": 2,
"user_id": 1430,
"username": "JBL"
},
{
"attachments": [],
"content_bbcode": "Here's my solution\n\n[hide]\n\nWe want to build up (digit by digit) a set of special numbers that will sum to 1=.99999... So:\n\n\n\n.7 = 1x0.7\n\n.98=.7+.07x4\n\n.994=.98+.007x2\n\n.9996=.994+.0007x8\n\n.99995=.9996+.00007x5\n\n.999999=.99995+.000007x7\n\n\n\nSince the highest coefficient was 8, it is easy to piece together 8 special numbers that sum to .999999. Then if we turn each of those numbers into repeating decimals with period 6, we have 8 special numbers that sum to .999999...=1.\n\n\n\nIt isn't clear to me how to prove that 8 is the minimum necessary.\n\n\n\nOne more thought. For some reason, when I saw this problem, I almost immediately thought to make use of the fact that 1=.999999... After that, everything is relatively straightforward. If I'm trying to teach someone how to do this, how do I explain the jump to the idea of using the fact that 1=.999999...?\n\n[/hide]\n\n\n\n--Dan",
"content_html": "Here's my solution<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 want to build up (digit by digit) a set of special numbers that will sum to 1=.99999... So:<br>\n<br>\n<br>\n<br>\n.7 = 1x0.7<br>\n<br>\n.98=.7+.07x4<br>\n<br>\n.994=.98+.007x2<br>\n<br>\n.9996=.994+.0007x8<br>\n<br>\n.99995=.9996+.00007x5<br>\n<br>\n.999999=.99995+.000007x7<br>\n<br>\n<br>\n<br>\nSince the highest coefficient was 8, it is easy to piece together 8 special numbers that sum to .999999. Then if we turn each of those numbers into repeating decimals with period 6, we have 8 special numbers that sum to .999999...=1.<br>\n<br>\n<br>\n<br>\nIt isn't clear to me how to prove that 8 is the minimum necessary.<br>\n<br>\n<br>\n<br>\nOne more thought. For some reason, when I saw this problem, I almost immediately thought to make use of the fact that 1=.999999... After that, everything is relatively straightforward. If I'm trying to teach someone how to do this, how do I explain the jump to the idea of using the fact that 1=.999999...?</div><br>\n<br>\n<br>\n<br>\n--Dan",
"post_id": 63020,
"post_number": 5,
"post_time_unix": 1074238215,
"post_time_utc": "2004-01-16 07:30:15 UTC",
"thanks_received": 2,
"user_id": 1973,
"username": "petra"
},
{
"attachments": [],
"content_bbcode": "Your proof includes the fact that it's a minimum: if you were to do it in fewer than 8, you'd need to move some 7 up one place. Basically, there isn't any room for larger numbers. This is terribly explained, but it's true. The fact that you need 8 7's in one place won't change no matter what and there's no way you can make it change.\r\n\r\nYou explain it based on the fact that you want to get as close to 1 as you can, and .9999... is as close to 1 as you can get (that is, it's equal). Your algorithm works fine, even if you start out trying to get 1.000..., you just have to recognize that once you have .999999, you slap down a repeating decimal and are done with it.",
"content_html": "Your proof includes the fact that it's a minimum: if you were to do it in fewer than 8, you'd need to move some 7 up one place. Basically, there isn't any room for larger numbers. This is terribly explained, but it's true. The fact that you need 8 7's in one place won't change no matter what and there's no way you can make it change.<br>\n<br>\nYou explain it based on the fact that you want to get as close to 1 as you can, and .9999... is as close to 1 as you can get (that is, it's equal). Your algorithm works fine, even if you start out trying to get 1.000..., you just have to recognize that once you have .999999, you slap down a repeating decimal and are done with it.",
"post_id": 63024,
"post_number": 6,
"post_time_unix": 1074270519,
"post_time_utc": "2004-01-16 16:28:39 UTC",
"thanks_received": 2,
"user_id": 1430,
"username": "JBL"
},
{
"attachments": [],
"content_bbcode": "The way I learned .999... = 1 was by writing out fractions:\r\n\r\n1/9 = .111...\r\n2/9 = .222...\r\n...\r\n9/9 = .999... = 1",
"content_html": "The way I learned .999... = 1 was by writing out fractions:<br>\n<br>\n1/9 = .111...<br>\n2/9 = .222...<br>\n...<br>\n9/9 = .999... = 1",
"post_id": 63213,
"post_number": 7,
"post_time_unix": 1074349936,
"post_time_utc": "2004-01-17 14:32:16 UTC",
"thanks_received": 2,
"user_id": 2001,
"username": "wyra"
},
{
"attachments": [],
"content_bbcode": "[quote=\"wyra\"]The way I learned .999... = 1 was by writing out fractions:\n\n1/9 = .111...\n2/9 = .222...\n...\n9/9 = .999... = 1[/quote]\r\nThere are other cool ways. Such as x=.99999... 10x=9.99999 9x=9 x=1",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">wyra wrote:</div>\n<div class=\"bbcode_quote_body\">The way I learned .999... = 1 was by writing out fractions:<br>\n<br>\n1/9 = .111...<br>\n2/9 = .222...<br>\n...<br>\n9/9 = .999... = 1</div>\n</div>\nThere are other cool ways. Such as x=.99999... 10x=9.99999 9x=9 x=1",
"post_id": 63217,
"post_number": 8,
"post_time_unix": 1074353299,
"post_time_utc": "2004-01-17 15:28:19 UTC",
"thanks_received": 2,
"user_id": 1503,
"username": "confuted"
},
{
"attachments": [],
"content_bbcode": "Hmm... I don't get it... why are we multiplying .07 and .007 and such.",
"content_html": "Hmm... I don't get it... why are we multiplying .07 and .007 and such.",
"post_id": 63258,
"post_number": 9,
"post_time_unix": 1074371472,
"post_time_utc": "2004-01-17 20:31:12 UTC",
"thanks_received": 2,
"user_id": 1449,
"username": "Ragingg"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Ragingg\"]Hmm... I don't get it... why are we multiplying .07 and .007 and such.[/quote]\r\n\r\nAny special number between 0 and 1 is of the form .7a+.07b+.007c+.0007d+..., where each of a,b,c,d,... is either 0 or 1 depending on whether that digit in the special number is 0 or 7.\r\n\r\nSay we have k special numbers that sum to 1: \r\n.7a_1 +.07b_1+.007c_1+..., .7a_2+.07b_2+.007c_2 +..., up to\r\n.7a_k+.07b_k+.007c_k+...\r\n\r\nThen if we let a = a_1 + a_2 + ... +a_k, b = b_1 + b_2 + ... + b_k, etc,\r\nwe get 1 = .7a + .07b + .007c + .0007d + ...\r\nwhere a, b, c, d, ... are all integers between 0 and k.\r\n\r\nSo what I did above was find values for a, b, c, d, ...\r\nAfter that it is easy to pick values for a_i, b_i, c_i, d_i, ... and reconstruct the k special numbers (but we don't need to do that for the problem). The answer to the problem is max(a,b,c,d,...) = 8.\r\n\r\nHope that is a bit clearer.\r\n--Dan",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Ragingg wrote:</div>\n<div class=\"bbcode_quote_body\">Hmm... I don't get it... why are we multiplying .07 and .007 and such.</div>\n</div>\n<br>\nAny special number between 0 and 1 is of the form .7a+.07b+.007c+.0007d+..., where each of a,b,c,d,... is either 0 or 1 depending on whether that digit in the special number is 0 or 7.<br>\n<br>\nSay we have k special numbers that sum to 1:<br>\n.7a_1 +.07b_1+.007c_1+..., .7a_2+.07b_2+.007c_2 +..., up to<br>\n.7a_k+.07b_k+.007c_k+...<br>\n<br>\nThen if we let a = a_1 + a_2 + ... +a_k, b = b_1 + b_2 + ... + b_k, etc,<br>\nwe get 1 = .7a + .07b + .007c + .0007d + ...<br>\nwhere a, b, c, d, ... are all integers between 0 and k.<br>\n<br>\nSo what I did above was find values for a, b, c, d, ...<br>\nAfter that it is easy to pick values for a_i, b_i, c_i, d_i, ... and reconstruct the k special numbers (but we don't need to do that for the problem). The answer to the problem is max(a,b,c,d,...) = 8.<br>\n<br>\nHope that is a bit clearer.<br>\n--Dan",
"post_id": 63283,
"post_number": 10,
"post_time_unix": 1074376153,
"post_time_utc": "2004-01-17 21:49:13 UTC",
"thanks_received": 2,
"user_id": 1973,
"username": "petra"
},
{
"attachments": [],
"content_bbcode": "Take a look at the decimal expansion of 1/7. See if that will get you a solution.",
"content_html": "Take a look at the decimal expansion of 1/7. See if that will get you a solution.",
"post_id": 63288,
"post_number": 11,
"post_time_unix": 1074378192,
"post_time_utc": "2004-01-17 22:23:12 UTC",
"thanks_received": 2,
"user_id": 1163,
"username": "rrusczyk"
}
],
"source": null
}
|
Call a positive real number special if its decimal representation consists entirely of the digits 0 and 7. For example, 700/99 = 7.0707070707... and 77.007 are special numbers. What is the smallest n such that 1 can be written as the sum of n special numbers?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 1 cannot be represented as the sum of finitely many special numbers
|
[
"/Mathematics/NumberTheory/SpecialNumbers/Digit-RelatedNumbers",
"/Mathematics/NumberTheory/SpecialNumbers/MiscellaneousSpecialNumbers",
"/Mathematics/RecreationalMathematics/Puzzles/Puzzle"
] |
Express 1 as 0.999… and decompose each 9 digit into a sum of special numbers using carries, showing eight such numbers suffice.
|
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aops_99830
|
[quote="math92"]Joe had a really big afro which was hard to keep. He goes to a store and buys 3 combs each which cost 1.25 dollars. He also buys 2 shampoo bottles each which cost 5.05 dollars. In the end, Joe had 20 dollars left over. How much money did he start with?[/quote]
[hide] He pays, in total, $3* \$ 1.25+2* \$ 5.05= \$ 13.85$
So, he started with $\$ 13.85+\$ 20= \boxed{\boxed{\$ 33.85}}$[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Joe had a really big afro which was hard to keep. He goes to a store and buys 3 combs each which cost 1.25 dollars. He also buys 2 shampoo bottles each which cost 5.05 dollars. In the end, Joe had 20 dollars left over. How much money did he start with?",
"content_html": "Joe had a really big afro which was hard to keep. He goes to a store and buys 3 combs each which cost 1.25 dollars. He also buys 2 shampoo bottles each which cost 5.05 dollars. In the end, Joe had 20 dollars left over. How much money did he start with?",
"post_id": 563605,
"post_number": 1,
"post_time_unix": 1151761711,
"post_time_utc": "2006-07-01 13:48:31 UTC",
"thanks_received": 2,
"user_id": 8131,
"username": "math92"
},
{
"attachments": [],
"content_bbcode": "[hide]$20+3(1.25)+2(5.05)=\\boxed{\\$33.85}$[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(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/8/8/0/880c89d9e804659a13a32d28fc8c9dd7adf037b5.png\" class=\"latex\" alt=\"$20+3(1.25)+2(5.05)=\\boxed{\\$33.85}$\" style=\"vertical-align: -6px\" width=\"260\" height=\"26\" ></div>",
"post_id": 563614,
"post_number": 2,
"post_time_unix": 1151762169,
"post_time_utc": "2006-07-01 13:56:09 UTC",
"thanks_received": 2,
"user_id": 18270,
"username": "SplashD"
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{
"attachments": [],
"content_bbcode": "[quote=\"math92\"]Joe had a really big afro which was hard to keep. He goes to a store and buys 3 combs each which cost 1.25 dollars. He also buys 2 shampoo bottles each which cost 5.05 dollars. In the end, Joe had 20 dollars left over. How much money did he start with?[/quote]\r\n\r\n[hide] He pays, in total, $3* \\$ 1.25+2* \\$ 5.05= \\$ 13.85$\nSo, he started with $\\$ 13.85+\\$ 20= \\boxed{\\boxed{\\$ 33.85}}$[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">math92 wrote:</div>\n<div class=\"bbcode_quote_body\">Joe had a really big afro which was hard to keep. He goes to a store and buys 3 combs each which cost 1.25 dollars. He also buys 2 shampoo bottles each which cost 5.05 dollars. In the end, Joe had 20 dollars left over. How much money did he start with?</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(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\">He pays, in total, <img src=\"//latex.artofproblemsolving.com/7/c/1/7c1223db261fc0ce7989cfe71660bfdcc507a6a1.png\" class=\"latex\" alt=\"$3* \\$ 1.25+2* \\$ 5.05= \\$ 13.85$\" style=\"vertical-align: -1px\" width=\"231\" height=\"15\" ><br>\nSo, he started with <img src=\"//latex.artofproblemsolving.com/4/2/2/4228f3686b86c4f35bf295893209b9af323deeb3.png\" class=\"latex\" alt=\"$\\$ 13.85+\\$ 20= \\boxed{\\boxed{\\$ 33.85}}$\" style=\"vertical-align: -12px\" width=\"197\" height=\"38\" ></div>",
"post_id": 563617,
"post_number": 3,
"post_time_unix": 1151762506,
"post_time_utc": "2006-07-01 14:01:46 UTC",
"thanks_received": 2,
"user_id": 15534,
"username": "José"
},
{
"attachments": [],
"content_bbcode": "[hide]The answer is $20+3*1.25+2*5.05=33.85$\n\n$33.85$[/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\">The answer is <img src=\"//latex.artofproblemsolving.com/4/0/6/406433c76af0121a1f3c57e2070cc19300f0d0fc.png\" class=\"latex\" alt=\"$20+3*1.25+2*5.05=33.85$\" style=\"vertical-align: -1px\" width=\"244\" height=\"14\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/7/e/2/7e21aee95fc5af0350371ab5d15a939c31f2bcaa.png\" class=\"latex\" alt=\"$33.85$\" width=\"40\" height=\"12\" ></div>",
"post_id": 563814,
"post_number": 4,
"post_time_unix": 1151774680,
"post_time_utc": "2006-07-01 17:24:40 UTC",
"thanks_received": 2,
"user_id": 11714,
"username": "mathgeniuse^ln(x)"
},
{
"attachments": [],
"content_bbcode": "[hide]$33.85$[/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/7/e/2/7e21aee95fc5af0350371ab5d15a939c31f2bcaa.png\" class=\"latex\" alt=\"$33.85$\" width=\"40\" height=\"12\" ></div>",
"post_id": 564271,
"post_number": 5,
"post_time_unix": 1151813209,
"post_time_utc": "2006-07-02 04:06:49 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "[quote=\"bpms\"][hide]$33.85$[/hide][/quote]\r\n\r\nPlease, remember to put how you get that result",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">bpms 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\"><img src=\"//latex.artofproblemsolving.com/7/e/2/7e21aee95fc5af0350371ab5d15a939c31f2bcaa.png\" class=\"latex\" alt=\"$33.85$\" width=\"40\" height=\"12\" ></div></div>\n</div>\n<br>\nPlease, remember to put how you get that result",
"post_id": 564423,
"post_number": 6,
"post_time_unix": 1151845984,
"post_time_utc": "2006-07-02 13:13:04 UTC",
"thanks_received": 2,
"user_id": 15534,
"username": "José"
}
],
"source": null
}
|
Joe had a really big afro which was hard to keep. He goes to a store and buys \(3\) combs, each of which costs \$1.25. He also buys \(2\) shampoo bottles, each of which costs \$5.05. In the end, Joe had \$20 left over. How much money did he start with?
|
[
"/Mathematics/Algebra/RateProblems"
] |
Add the total amount spent to the remaining money to find the initial amount.
|
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aops_99839
|
Brute force method:
[hide]He must mess up on 6 or less lines. The probability of messing up on exactly 6 lines is $\left(\frac13\right)^{6}\cdot\left(\frac23\right)^{14}\cdot\binom{20}{6}$.
Five lines:
$\left(\frac13\right)^{5}\cdot\left(\frac23\right)^{15}\cdot\binom{20}{5}$
Four lines:
$\left(\frac13\right)^{4}\cdot\left(\frac23\right)^{16}\cdot\binom{20}{4}$
Three lines:
$\left(\frac13\right)^{3}\cdot\left(\frac23\right)^{17}\cdot\binom{20}{3}$
Two lines:
$\left(\frac13\right)^{2}\cdot\left(\frac23\right)^{18}\cdot\binom{20}{2}$
One line:
$\frac13\cdot\left(\frac23\right)^{19}\cdot20$
Zero lines:
$\left(\frac23\right)^{20}$
Add them up to get some big fraction I will figure out later :D [/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "There is a $\\frac13$ chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.",
"content_html": "There is a <img src=\"//latex.artofproblemsolving.com/e/b/a/ebafcc60a28d8759185d7808c60ea53efc73938e.png\" class=\"latex\" alt=\"$\\frac13$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" > chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.",
"post_id": 563650,
"post_number": 1,
"post_time_unix": 1151764315,
"post_time_utc": "2006-07-01 14:31:55 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "Brute force method:\r\n\r\n[hide]He must mess up on 6 or less lines. The probability of messing up on exactly 6 lines is $\\left(\\frac13\\right)^{6}\\cdot\\left(\\frac23\\right)^{14}\\cdot\\binom{20}{6}$.\nFive lines:\n$\\left(\\frac13\\right)^{5}\\cdot\\left(\\frac23\\right)^{15}\\cdot\\binom{20}{5}$\nFour lines:\n$\\left(\\frac13\\right)^{4}\\cdot\\left(\\frac23\\right)^{16}\\cdot\\binom{20}{4}$\nThree lines:\n$\\left(\\frac13\\right)^{3}\\cdot\\left(\\frac23\\right)^{17}\\cdot\\binom{20}{3}$\nTwo lines:\n$\\left(\\frac13\\right)^{2}\\cdot\\left(\\frac23\\right)^{18}\\cdot\\binom{20}{2}$\nOne line:\n$\\frac13\\cdot\\left(\\frac23\\right)^{19}\\cdot20$\nZero lines:\n$\\left(\\frac23\\right)^{20}$\n\nAdd them up to get some big fraction I will figure out later :D [/hide]",
"content_html": "Brute force method:<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\">He must mess up on 6 or less lines. The probability of messing up on exactly 6 lines is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/a/3/6a3ed7366c3570a6997416bd360fb1dd11c9aa78.png\" class=\"latex\" alt=\"$\\left(\\frac13\\right)^{6}\\cdot\\left(\\frac23\\right)^{14}\\cdot\\binom{20}{6}$\" style=\"vertical-align: -22px\" width=\"168\" height=\"53\" >.</span><br>\nFive lines:<br>\n<img src=\"//latex.artofproblemsolving.com/4/2/3/42317843273eabe7b5e3047f9b1ff3f74e3d7e38.png\" class=\"latex\" alt=\"$\\left(\\frac13\\right)^{5}\\cdot\\left(\\frac23\\right)^{15}\\cdot\\binom{20}{5}$\" style=\"vertical-align: -22px\" width=\"168\" height=\"53\" ><br>\nFour lines:<br>\n<img src=\"//latex.artofproblemsolving.com/d/0/9/d09e7a43c725443cfed1a4dd1f86c0d97fa70676.png\" class=\"latex\" alt=\"$\\left(\\frac13\\right)^{4}\\cdot\\left(\\frac23\\right)^{16}\\cdot\\binom{20}{4}$\" style=\"vertical-align: -22px\" width=\"168\" height=\"53\" ><br>\nThree lines:<br>\n<img src=\"//latex.artofproblemsolving.com/3/a/9/3a96d2e1700cfda7c752ae660fd72046f123f7b1.png\" class=\"latex\" alt=\"$\\left(\\frac13\\right)^{3}\\cdot\\left(\\frac23\\right)^{17}\\cdot\\binom{20}{3}$\" style=\"vertical-align: -22px\" width=\"168\" height=\"53\" ><br>\nTwo lines:<br>\n<img src=\"//latex.artofproblemsolving.com/b/f/f/bffc28d24465b2c8244be58d94c75cc6fa940d64.png\" class=\"latex\" alt=\"$\\left(\\frac13\\right)^{2}\\cdot\\left(\\frac23\\right)^{18}\\cdot\\binom{20}{2}$\" style=\"vertical-align: -22px\" width=\"168\" height=\"53\" ><br>\nOne line:<br>\n<img src=\"//latex.artofproblemsolving.com/c/e/4/ce48c0ac0148fec0e0c0e35643f20b38f90ee2b3.png\" class=\"latex\" alt=\"$\\frac13\\cdot\\left(\\frac23\\right)^{19}\\cdot20$\" style=\"vertical-align: -17px\" width=\"110\" height=\"46\" ><br>\nZero lines:<br>\n<img src=\"//latex.artofproblemsolving.com/4/a/f/4af0c3455a222eb2dabd5da07107037b7d31e0db.png\" class=\"latex\" alt=\"$\\left(\\frac23\\right)^{20}$\" style=\"vertical-align: -17px\" width=\"51\" height=\"46\" ><br>\n<br>\nAdd them up to get some big fraction I will figure out later <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div>",
"post_id": 563666,
"post_number": 2,
"post_time_unix": 1151764923,
"post_time_utc": "2006-07-01 14:42:03 UTC",
"thanks_received": 2,
"user_id": 8949,
"username": "b-flat"
},
{
"attachments": [],
"content_bbcode": "[quote=\"bpms\"]There is a $\\frac13$ chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.[/quote][hide] You would have to mess up only 0, 1, 2, 3, 4, 5, or 6 lines. The probability of this is $\\binom{20}{0}*(\\frac{1}{3})^{0}*(\\frac{2}{3})^{20}+\\binom{20}{1}*(\\frac{1}{3})^{1}*(\\frac{2}{3})^{19}+\\binom{20}{2}*(\\frac{1}{3})^{2}*(\\frac{2}{3})^{18}+\\binom{20}{3}*(\\frac{1}{3})^{3}*(\\frac{2}{3})^{17}+\\binom{20}{4}*(\\frac{1}{3})^{4}*(\\frac{2}{3})^{16}+\\binom{20}{5}*(\\frac{1}{3})^{5}*(\\frac{2}{3})^{15}+\\binom{20}{6}*(\\frac{1}{3})^{6}*(\\frac{2}{3})^{14}$, which I don't feel like figuring out. :D[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">bpms wrote:</div>\n<div class=\"bbcode_quote_body\">There is a <img src=\"//latex.artofproblemsolving.com/e/b/a/ebafcc60a28d8759185d7808c60ea53efc73938e.png\" class=\"latex\" alt=\"$\\frac13$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" > chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.</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\">You would have to mess up only 0, 1, 2, 3, 4, 5, or 6 lines. The probability of this is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/2/5/d25a70d90aca507bf59c690e1ea95c3f433240f0.png\" class=\"latex\" alt=\"$\\binom{20}{0}*(\\frac{1}{3})^{0}*(\\frac{2}{3})^{20}+\\binom{20}{1}*(\\frac{1}{3})^{1}*(\\frac{2}{3})^{19}+\\binom{20}{2}*(\\frac{1}{3})^{2}*(\\frac{2}{3})^{18}+\\binom{20}{3}*(\\frac{1}{3})^{3}*(\\frac{2}{3})^{17}+\\binom{20}{4}*(\\frac{1}{3})^{4}*(\\frac{2}{3})^{16}+\\binom{20}{5}*(\\frac{1}{3})^{5}*(\\frac{2}{3})^{15}+\\binom{20}{6}*(\\frac{1}{3})^{6}*(\\frac{2}{3})^{14}$\" style=\"vertical-align: -22px\" width=\"1049\" height=\"108\" >,</span> which I don't feel like figuring out. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /></div>",
"post_id": 563674,
"post_number": 3,
"post_time_unix": 1151765135,
"post_time_utc": "2006-07-01 14:45:35 UTC",
"thanks_received": 2,
"user_id": 18001,
"username": "lotrgreengrapes7926"
},
{
"attachments": [],
"content_bbcode": "Um, unless someone finds a clever solution, this is officially a target round problem :D.",
"content_html": "Um, unless someone finds a clever solution, this is officially a target round problem <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />.",
"post_id": 563675,
"post_number": 4,
"post_time_unix": 1151765149,
"post_time_utc": "2006-07-01 14:45:49 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "[quote=\"lotrgreengrapes7926\"][quote=\"bpms\"]There is a $\\frac13$ chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.[/quote][hide] You would have to mess up only 0, 1, 2, 3, 4, 5, or 6 lines. The probability of this is $\\binom{20}{0}*(\\frac{1}{3})^{0}*(\\frac{2}{3})^{2}0+\\binom{20}{1}*(\\frac{1}{3})^{1}*(\\frac{2}{3})^{1}9+\\binom{20}{2}*(\\frac{1}{3})^{2}*(\\frac{2}{3})^{1}8+\\binom{20}{3}*(\\frac{1}{3})^{3}*(\\frac{2}{3})^{1}7+\\binom{20}{4}*(\\frac{1}{3})^{4}*(\\frac{2}{3})^{1}6+\\binom{20}{5}*(\\frac{1}{3})^{5}*(\\frac{2}{3})^{1}5+\\binom{20}{6}*(\\frac{1}{3})^{6}*(\\frac{2}{3})^{1}4$, which I don't feel like figuring out. [/hide][/quote]\r\n\r\nWhen using exponents with more than one digit, put braces around them like \\frac13^{14}. Also, use \\cdot or \\times for * because it looks much nicer :)",
"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\">bpms wrote:</div>\n<div class=\"bbcode_quote_body\">There is a <img src=\"//latex.artofproblemsolving.com/e/b/a/ebafcc60a28d8759185d7808c60ea53efc73938e.png\" class=\"latex\" alt=\"$\\frac13$\" style=\"vertical-align: -12px\" width=\"11\" height=\"37\" > chance I mess one line of latex. If I do twenty lines, what is the probability that I mess up less then one third of the lines?Express your answer as a percent.</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\">You would have to mess up only 0, 1, 2, 3, 4, 5, or 6 lines. The probability of this is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/c/0/5c00a46f7809c7bd9d2d2d867cce6e9ef1e4d50c.png\" class=\"latex\" alt=\"$\\binom{20}{0}*(\\frac{1}{3})^{0}*(\\frac{2}{3})^{2}0+\\binom{20}{1}*(\\frac{1}{3})^{1}*(\\frac{2}{3})^{1}9+\\binom{20}{2}*(\\frac{1}{3})^{2}*(\\frac{2}{3})^{1}8+\\binom{20}{3}*(\\frac{1}{3})^{3}*(\\frac{2}{3})^{1}7+\\binom{20}{4}*(\\frac{1}{3})^{4}*(\\frac{2}{3})^{1}6+\\binom{20}{5}*(\\frac{1}{3})^{5}*(\\frac{2}{3})^{1}5+\\binom{20}{6}*(\\frac{1}{3})^{6}*(\\frac{2}{3})^{1}4$\" style=\"vertical-align: -22px\" width=\"1049\" height=\"108\" >,</span> which I don't feel like figuring out.</div></div>\n</div>\n<br>\nWhen using exponents with more than one digit, put braces around them like \\frac13^{14}. Also, use \\cdot or \\times for * because it looks much nicer <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 563685,
"post_number": 5,
"post_time_unix": 1151765445,
"post_time_utc": "2006-07-01 14:50:45 UTC",
"thanks_received": 1,
"user_id": 8949,
"username": "b-flat"
},
{
"attachments": [],
"content_bbcode": "This is too complicated. It requires too much finger pressing on the Calculator. Someone else can do it.\r\n\r\nIf the problem was, what is the probability that I mess up on an even number of lines or an odd number of lines for Latex, I would get it right without using a calculator.",
"content_html": "This is too complicated. It requires too much finger pressing on the Calculator. Someone else can do it.<br>\n<br>\nIf the problem was, what is the probability that I mess up on an even number of lines or an odd number of lines for Latex, I would get it right without using a calculator.",
"post_id": 563765,
"post_number": 6,
"post_time_unix": 1151769912,
"post_time_utc": "2006-07-01 16:05:12 UTC",
"thanks_received": 2,
"user_id": 11714,
"username": "mathgeniuse^ln(x)"
},
{
"attachments": [],
"content_bbcode": "Let me make it team then. That way you have 8 minutes not 3. :D",
"content_html": "Let me make it team then. That way you have 8 minutes not 3. <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" />",
"post_id": 564287,
"post_number": 7,
"post_time_unix": 1151814152,
"post_time_utc": "2006-07-02 04:22:32 UTC",
"thanks_received": 1,
"user_id": 8960,
"username": "bpms"
}
],
"source": null
}
|
There is a \(\tfrac{1}{3}\) chance I mess one line of LaTeX. If I do twenty lines, what is the probability that I mess up less than one third of the lines? Express your answer as a percent.
|
[
"/Mathematics/ProbabilityandStatistics/Probability/ProbabilityMeasure",
"/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace",
"/Mathematics/ProbabilityandStatistics/Probability/SampleSpace",
"/Mathematics/ProbabilityandStatistics/StatisticalDistributions/DiscreteDistributions/BernoulliDistribution",
"/Mathematics/ProbabilityandStatistics/StatisticalDistributions/DiscreteDistributions/BinomialDistribution",
"/Mathematics/ProbabilityandStatistics/StatisticalDistributions/DiscreteDistributions/DiscreteDistribution",
"/Mathematics/ProbabilityandStatistics/Trials/BernoulliTrial",
"/Mathematics/ProbabilityandStatistics/Trials/Experiment",
"/Mathematics/ProbabilityandStatistics/Trials/IndependentEvents",
"/Mathematics/ProbabilityandStatistics/Trials/Trial"
] |
Treat the mess‑ups as a binomial random variable and sum the probabilities for 0 through 6 errors.
|
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aops_99843
|
The only solution is 2^3+1=3^2. Assume 2^n+1=x^a where a>1. Then 2^n=(x-1)(x^(a-1)+x^(a-2)+...+x+1) Then x=2^k+1. As a result a must be even for otherwise the second expression would be a sum of an odd number of odd numbers which can't be a power of two. Set a=2b. Then one has 2^n=(x^b-1)(x^b+1). This results in n=3, x=3, a=2 as the only solution for a>1.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "find all natural numbers $n$ such that $2^{n}+1$is the power of natural number.",
"content_html": "find all natural numbers <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/0/b/10b323870a9542120000ae5b9f8e8d61efce43c2.png\" class=\"latex\" alt=\"$2^{n}+1$\" style=\"vertical-align: -1px\" width=\"48\" height=\"14\" >i</span>s the power of natural number.",
"post_id": 563684,
"post_number": 1,
"post_time_unix": 1151765398,
"post_time_utc": "2006-07-01 14:49:58 UTC",
"thanks_received": 1,
"user_id": 16000,
"username": "tchebytchev"
},
{
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"content_bbcode": "The only solution is 2^3+1=3^2. Assume 2^n+1=x^a where a>1. Then 2^n=(x-1)(x^(a-1)+x^(a-2)+...+x+1) Then x=2^k+1. As a result a must be even for otherwise the second expression would be a sum of an odd number of odd numbers which can't be a power of two. Set a=2b. Then one has 2^n=(x^b-1)(x^b+1). This results in n=3, x=3, a=2 as the only solution for a>1.",
"content_html": "The only solution is 2^3+1=3^2. Assume 2^n+1=x^a where a>1. Then 2^n=(x-1)(x^(a-1)+x^(a-2)+...+x+1) Then x=2^k+1. As a result a must be even for otherwise the second expression would be a sum of an odd number of odd numbers which can't be a power of two. Set a=2b. Then one has 2^n=(x^b-1)(x^b+1). This results in n=3, x=3, a=2 as the only solution for a>1.",
"post_id": 563718,
"post_number": 2,
"post_time_unix": 1151767240,
"post_time_utc": "2006-07-01 15:20:40 UTC",
"thanks_received": 2,
"user_id": 20129,
"username": "clinkmath"
}
],
"source": null
}
|
Find all natural numbers \(n\) such that \(2^{n} + 1\) is a perfect power of a natural number (i.e., equals \(a^{k}\) for integers \(a\ge 1\), \(k\ge 2\)).
|
[
"/Mathematics/NumberTheory/DiophantineEquations/CatalansConjecture",
"/Mathematics/NumberTheory/DiophantineEquations/CatalansDiophantineProblem",
"/Mathematics/NumberTheory/DiophantineEquations/Diophantine",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/Integers/Integer",
"/Mathematics/NumberTheory/Integers/N",
"/Mathematics/NumberTheory/Integers/PositiveInteger",
"/Mathematics/NumberTheory/Integers/Z",
"/Mathematics/NumberTheory/Integers/Z-Plus"
] |
Factor 2^n = (x-1)(x^{a-1}+…+1) and use parity to force a even, then rewrite as (x^b-1)(x^b+1) giving only small solution.
|
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aops_99869
|
[hide="Solution"]Use $a$, $b$, and $c$. Clearly, $c=\sqrt{a^{2}+b^{2}}$.
Set up the equation $ab=3a+3b+3\sqrt{a^{2}+b^{2}}$. Solve for the radical and we get $3\sqrt{a^{2}+b^{2}}=ab-3a-3b$. Square both sides to get $a^{2}b^{2}-6a^{2}b-6ab^{2}+9a^{2}+18ab+9b^{2}=9a^{2}+9b^{2}$, or $a^{2}b^{2}-6a^{2}b-6ab^{2}+18ab=0$.
Factor to get $ab(ab-6a-6b+18)=0$. Clearly, $a$ and $b$ must be positive, so $ab\neq0$. Divide by it to get $ab-6a-6b+18=0$. Add 18 to both sides and factor to get $(a-6)(b-6)=18$.
Since $a,b\in\mathbb{Z}$, $a-6$ and $b-6$ are integers as well. We have the following possibilities for $a-6$ and $b-6$:
$\pm1\text{ and }\pm18$
$\pm2\text{ and }\pm9$
$\pm3\text{ and }\pm6$
No negative solution works, because if we have -6, -9, or -18, either $a$ or $b$ will not be a positive integer.
Trying out the three that remain, we get $\boxed{7,24,25}$, $\boxed{8,15,17}$, and $\boxed{9,12,15}$. And guess what: THEY ALL WORK![/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Source: Romania 1999\r\n7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.",
"content_html": "Source: Romania 1999<br>\n7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.",
"post_id": 563877,
"post_number": 1,
"post_time_unix": 1151779833,
"post_time_utc": "2006-07-01 18:50:33 UTC",
"thanks_received": 2,
"user_id": 9197,
"username": "kimby_102"
},
{
"attachments": [],
"content_bbcode": "[hide=\"Solution\"]Use $a$, $b$, and $c$. Clearly, $c=\\sqrt{a^{2}+b^{2}}$.\nSet up the equation $ab=3a+3b+3\\sqrt{a^{2}+b^{2}}$. Solve for the radical and we get $3\\sqrt{a^{2}+b^{2}}=ab-3a-3b$. Square both sides to get $a^{2}b^{2}-6a^{2}b-6ab^{2}+9a^{2}+18ab+9b^{2}=9a^{2}+9b^{2}$, or $a^{2}b^{2}-6a^{2}b-6ab^{2}+18ab=0$.\nFactor to get $ab(ab-6a-6b+18)=0$. Clearly, $a$ and $b$ must be positive, so $ab\\neq0$. Divide by it to get $ab-6a-6b+18=0$. Add 18 to both sides and factor to get $(a-6)(b-6)=18$.\nSince $a,b\\in\\mathbb{Z}$, $a-6$ and $b-6$ are integers as well. We have the following possibilities for $a-6$ and $b-6$:\n$\\pm1\\text{ and }\\pm18$\n$\\pm2\\text{ and }\\pm9$\n$\\pm3\\text{ and }\\pm6$\nNo negative solution works, because if we have -6, -9, or -18, either $a$ or $b$ will not be a positive integer.\nTrying out the three that remain, we get $\\boxed{7,24,25}$, $\\boxed{8,15,17}$, and $\\boxed{9,12,15}$. And guess what: THEY ALL WORK![/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 <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" width=\"8\" height=\"12\" >,</span> 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> Clearly, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/e/c/aece792c9982d67fd51b827896143316509138e3.png\" class=\"latex\" alt=\"$c=\\sqrt{a^{2}+b^{2}}$\" style=\"vertical-align: -2px\" width=\"102\" height=\"18\" >.</span><br>\nSet up the equation <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/f/8/3f880791b392863a3d12366efb3bbc6758896dfe.png\" class=\"latex\" alt=\"$ab=3a+3b+3\\sqrt{a^{2}+b^{2}}$\" style=\"vertical-align: -2px\" width=\"201\" height=\"18\" >.</span> Solve for the radical and we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/4/a/84a78655edc331474f07754369703e44b5a700fe.png\" class=\"latex\" alt=\"$3\\sqrt{a^{2}+b^{2}}=ab-3a-3b$\" style=\"vertical-align: -2px\" width=\"200\" height=\"18\" >.</span> Square both sides to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/0/d805713bc75108e2438f2293e098ccf7284400a6.png\" class=\"latex\" alt=\"$a^{2}b^{2}-6a^{2}b-6ab^{2}+9a^{2}+18ab+9b^{2}=9a^{2}+9b^{2}$\" style=\"vertical-align: -1px\" width=\"393\" height=\"16\" >,</span> or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/1/9/d193c93242f31f856d73e9603064ff5aa2cab730.png\" class=\"latex\" alt=\"$a^{2}b^{2}-6a^{2}b-6ab^{2}+18ab=0$\" style=\"vertical-align: -1px\" width=\"235\" height=\"16\" >.</span><br>\nFactor to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/2/5/525735277b8171f7541325cefb55af6dfd1f57c5.png\" class=\"latex\" alt=\"$ab(ab-6a-6b+18)=0$\" style=\"vertical-align: -4px\" width=\"202\" height=\"18\" >.</span> Clearly, <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\" > must be positive, so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/3/2/e323bf7ac9870b03206d4cf6bace7458ed3ec541.png\" class=\"latex\" alt=\"$ab\\neq0$\" style=\"vertical-align: -4px\" width=\"50\" height=\"17\" >.</span> Divide by it to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/c/7/ec7a1ba7150ae9e986960fe9b6edafd1f638f7c1.png\" class=\"latex\" alt=\"$ab-6a-6b+18=0$\" style=\"vertical-align: -1px\" width=\"170\" height=\"14\" >.</span> Add 18 to both sides and factor to get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/d/1/7d1f909f3146588e96e9f4cca0759aaf04cff10f.png\" class=\"latex\" alt=\"$(a-6)(b-6)=18$\" style=\"vertical-align: -4px\" width=\"150\" height=\"18\" >.</span><br>\nSince <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/b/8/bb877000a178e818df845055f9e8fd9e698dafe9.png\" class=\"latex\" alt=\"$a,b\\in\\mathbb{Z}$\" style=\"vertical-align: -3px\" width=\"59\" height=\"16\" >,</span> <img src=\"//latex.artofproblemsolving.com/c/7/7/c77a014ae2d4fe50b097188540824969dfbb1a17.png\" class=\"latex\" alt=\"$a-6$\" width=\"40\" height=\"12\" > and <img src=\"//latex.artofproblemsolving.com/f/b/6/fb6ae3097c21e1275412bb23f3637f56d6cd6b05.png\" class=\"latex\" alt=\"$b-6$\" width=\"38\" height=\"12\" > are integers as well. We have the following possibilities for <img src=\"//latex.artofproblemsolving.com/c/7/7/c77a014ae2d4fe50b097188540824969dfbb1a17.png\" class=\"latex\" alt=\"$a-6$\" width=\"40\" height=\"12\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/b/6/fb6ae3097c21e1275412bb23f3637f56d6cd6b05.png\" class=\"latex\" alt=\"$b-6$\" width=\"38\" height=\"12\" >:</span><br>\n<img src=\"//latex.artofproblemsolving.com/7/4/5/7453763360e6cb8061bc6b2ae3bd3cd4cebd64aa.png\" class=\"latex\" alt=\"$\\pm1\\text{ and }\\pm18$\" style=\"vertical-align: 0px\" width=\"101\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/e/d/5/ed5664f12881e90f581776b8d1494f535d862171.png\" class=\"latex\" alt=\"$\\pm2\\text{ and }\\pm9$\" style=\"vertical-align: 0px\" width=\"92\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/f/2/0/f20a481049143f44841f82515a5756aa63d23810.png\" class=\"latex\" alt=\"$\\pm3\\text{ and }\\pm6$\" style=\"vertical-align: 0px\" width=\"92\" height=\"13\" ><br>\nNo negative solution works, because if we have -6, -9, or -18, either <img src=\"//latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png\" class=\"latex\" alt=\"$a$\" width=\"9\" height=\"8\" > or <img src=\"//latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" width=\"8\" height=\"12\" > will not be a positive integer.<br>\nTrying out the three that remain, we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/c/92ce68d48052eb1f5c3c8a485bf6c91f8e8c6a73.png\" class=\"latex\" alt=\"$\\boxed{7,24,25}$\" style=\"vertical-align: -9px\" width=\"73\" height=\"26\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/3/1/a3139a7f61b1b85cd1a2246e1154853c8751dffd.png\" class=\"latex\" alt=\"$\\boxed{8,15,17}$\" style=\"vertical-align: -9px\" width=\"73\" height=\"26\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/f/6/cf69493ccf58b44a39da62288454ea77a30e1867.png\" class=\"latex\" alt=\"$\\boxed{9,12,15}$\" style=\"vertical-align: -9px\" width=\"73\" height=\"26\" >.</span> And guess what: THEY ALL WORK!</div>",
"post_id": 563921,
"post_number": 2,
"post_time_unix": 1151782965,
"post_time_utc": "2006-07-01 19:42:45 UTC",
"thanks_received": 2,
"user_id": 18322,
"username": "lingomaniac88"
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{
"attachments": [],
"content_bbcode": "[hide=\"my way\"]\nin other words we have:\n$2\\triangle = 6s$\n$\\triangle = 3s$\n$rs = 3s$\n$r = 3$.\n\nNow draw the diagram, since $r = 3$, and tangent lines from the same point are congruent, we deduce that $a+b-6 = c = \\sqrt{a^{2}+b^{2}}$. Squaring and using Simon's favorite trick we get $(a-6)(b-6) = 18$ and get the sides lengths:\n$(9, 12, 15)$\n$(7, 24, 25)$\n$(8, 15, 17)$\n[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">my way</a><div class=\"cmty-hide-content\" style=\"display:none\">in other words we have:<br>\n<img src=\"//latex.artofproblemsolving.com/6/6/4/66482dfcdece4d2decb2d82d598f14e878e9c6ae.png\" class=\"latex\" alt=\"$2\\triangle = 6s$\" width=\"66\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/8/7/3/873e56a439326e5ba193fff7004bfa7893c78f47.png\" class=\"latex\" alt=\"$\\triangle = 3s$\" width=\"57\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/e/0/ae0d8792573e36e6e7caa5d1dedf5e673f1c699f.png\" class=\"latex\" alt=\"$rs = 3s$\" width=\"58\" height=\"12\" ><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/6/ce6e5c8d137537d69469ace3cfd5743592e1bf30.png\" class=\"latex\" alt=\"$r = 3$\" width=\"41\" height=\"12\" >.</span><br>\n<br>\nNow draw the diagram, since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/e/6/ce6e5c8d137537d69469ace3cfd5743592e1bf30.png\" class=\"latex\" alt=\"$r = 3$\" width=\"41\" height=\"12\" >,</span> and tangent lines from the same point are congruent, we deduce that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/f/7/cf72a1014a883dc86eca2658fa2a4ffecef3639e.png\" class=\"latex\" alt=\"$a+b-6 = c = \\sqrt{a^{2}+b^{2}}$\" style=\"vertical-align: -2px\" width=\"197\" height=\"18\" >.</span> Squaring and using Simon's favorite trick we get <img src=\"//latex.artofproblemsolving.com/f/c/1/fc1da1ca75b443736207bd88095beca47a0650f4.png\" class=\"latex\" alt=\"$(a-6)(b-6) = 18$\" style=\"vertical-align: -4px\" width=\"150\" height=\"18\" > and get the sides lengths:<br>\n<img src=\"//latex.artofproblemsolving.com/9/f/b/9fb3f8129de93ef8cc43195f8bfb591d055a6fd9.png\" class=\"latex\" alt=\"$(9, 12, 15)$\" style=\"vertical-align: -4px\" width=\"74\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/f/2/cf204f5bffc555354bd2ee5b35b8661ea2bb25bb.png\" class=\"latex\" alt=\"$(7, 24, 25)$\" style=\"vertical-align: -4px\" width=\"74\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/1/3/8/138e3c954652e8e96153c1e75b1bbdc0355579a5.png\" class=\"latex\" alt=\"$(8, 15, 17)$\" style=\"vertical-align: -4px\" width=\"74\" height=\"18\" ></div>",
"post_id": 563981,
"post_number": 3,
"post_time_unix": 1151787498,
"post_time_utc": "2006-07-01 20:58:18 UTC",
"thanks_received": 1,
"user_id": 10705,
"username": "pkerichang"
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],
"source": null
}
|
Source: Romania 1999
7.1 Determine the side lengths of a right triangle if they are integers and the product of the legs is equal to three times the perimeter.
|
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Rewrite the condition so that (a‑6)(b‑6)=18, turning it into a simple integer factorisation problem.
|
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aops_99881
|
$f(\sin\theta,\cos\theta)$ is a trigonometric polynomial. That limit will turn out to be equal to the constant term of that particular polynomial. I know I've posted that proof somewhere here (probably in a problem to show that $\{e^{ik}\}$ is dense in the unit circle), but I don't have time to look right now.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $f ( X, Y ) \\in \\mathbb{Q}[ X, Y ] .$ Prove that \\begin{eqnarray*}\\lim_{x \\longrightarrow \\infty}\\frac{1}{x}\\sum_{k \\leq x}f ( \\sin k, \\cos k ) & \\in & \\mathbb{Q}\\end{eqnarray*}",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/5/d/8/5d8f340f89e6c91f444888471d62a2f9cabb2ca5.png\" class=\"latex\" alt=\"$f ( X, Y ) \\in \\mathbb{Q}[ X, Y ] .$\" style=\"vertical-align: -5px\" width=\"151\" height=\"18\" > Prove that <img src=\"//latex.artofproblemsolving.com/5/0/2/50279913cb33f7d4c46d47f98a00b2ad87ef428f.png\" class=\"latexcenter\" alt=\"\\begin{eqnarray*}\\lim_{x \\longrightarrow \\infty}\\frac{1}{x}\\sum_{k \\leq x}f ( \\sin k, \\cos k ) & \\in & \\mathbb{Q}\\end{eqnarray*}\" width=\"255\" height=\"47\" >",
"post_id": 563940,
"post_number": 1,
"post_time_unix": 1151783662,
"post_time_utc": "2006-07-01 19:54:22 UTC",
"thanks_received": 1,
"user_id": 12380,
"username": "{x}"
},
{
"attachments": [],
"content_bbcode": "$f(\\sin\\theta,\\cos\\theta)$ is a trigonometric polynomial. That limit will turn out to be equal to the constant term of that particular polynomial. I know I've posted that proof somewhere here (probably in a problem to show that $\\{e^{ik}\\}$ is dense in the unit circle), but I don't have time to look right now.",
"content_html": "<img src=\"//latex.artofproblemsolving.com/2/f/d/2fd35b84fa506f71e039de005e3f3d246fc0cefb.png\" class=\"latex\" alt=\"$f(\\sin\\theta,\\cos\\theta)$\" style=\"vertical-align: -4px\" width=\"102\" height=\"18\" > is a trigonometric polynomial. That limit will turn out to be equal to the constant term of that particular polynomial. I know I've posted that proof somewhere here (probably in a problem to show that <img src=\"//latex.artofproblemsolving.com/1/c/4/1c488e902fb0b12f9177436e6b9d77a3b5ffabff.png\" class=\"latex\" alt=\"$\\{e^{ik}\\}$\" style=\"vertical-align: -4px\" width=\"38\" height=\"20\" > is dense in the unit circle), but I don't have time to look right now.",
"post_id": 564029,
"post_number": 2,
"post_time_unix": 1151790808,
"post_time_utc": "2006-07-01 21:53:28 UTC",
"thanks_received": 2,
"user_id": 2948,
"username": "Kent Merryfield"
}
],
"source": null
}
|
Let \(f(X,Y)\in\mathbb{Q}[X,Y]\). Prove that
\[
\lim_{x\to\infty}\frac{1}{x}\sum_{k\le x} f(\sin k,\cos k)\in\mathbb{Q}.
\]
|
[
"/Mathematics/NumberTheory/Constants",
"/Mathematics/NumberTheory/ErgodicTheory/Kronecker-WeylTheorem",
"/Mathematics/NumberTheory/RationalNumbers/FieldofRationals",
"/Mathematics/NumberTheory/RationalNumbers/Q",
"/Mathematics/NumberTheory/RationalNumbers/RationalNumber"
] |
Average the trigonometric polynomial over integer arguments to isolate its constant term via equidistribution of k (mod 2π).
|
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aops_99907
|
[quote="Inspired By Nature"]What is the image of $\left( \begin{array}{c}3 \\ 1 \\ 2 \end{array}\right)$ under the mapping $\left( \begin{array}{ccc}1 & 4 & 1 \\-2 & 0 & 0 \\ 3 & 2 &-3\ \end{array}\right)$?[/quote]
Don't you multiply them or something...
[hide]$\left( \begin{array}{ccc}1 & 4 & 1 \\-2 & 0 & 0 \\ 3 & 2 &-3\ \end{array}\right) \cdot \left( \begin{array}{c}3 \\ 1 \\ 2 \end{array}\right)=\left( \begin{array}{c}9 \\-6 \\ 5 \end{array}\right)$
[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "What is the image of $\\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)$ under the mapping $\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right)$?",
"content_html": "What is the image of <img src=\"//latex.artofproblemsolving.com/6/8/e/68e9bc9c1893ad7fc2fe1f0ab835160dfa869413.png\" class=\"latex\" alt=\"$\\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)$\" style=\"vertical-align: -27px\" width=\"51\" height=\"64\" > under the mapping <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/4/f/f4fa096ca6445665213bc7e602f50b23562f3e43.png\" class=\"latex\" alt=\"$\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right)$\" style=\"vertical-align: -27px\" width=\"137\" height=\"64\" >?</span>",
"post_id": 564130,
"post_number": 1,
"post_time_unix": 1151798036,
"post_time_utc": "2006-07-01 23:53:56 UTC",
"thanks_received": 2,
"user_id": 18217,
"username": "Inspired By Nature"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Inspired By Nature\"]What is the image of $\\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)$ under the mapping $\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right)$?[/quote]\r\n\r\nDon't you multiply them or something...\r\n\r\n[hide]$\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right) \\cdot \\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)=\\left( \\begin{array}{c}9 \\\\-6 \\\\ 5 \\end{array}\\right)$\n[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Inspired By Nature wrote:</div>\n<div class=\"bbcode_quote_body\">What is the image of <img src=\"//latex.artofproblemsolving.com/6/8/e/68e9bc9c1893ad7fc2fe1f0ab835160dfa869413.png\" class=\"latex\" alt=\"$\\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)$\" style=\"vertical-align: -27px\" width=\"51\" height=\"64\" > under the mapping <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/4/f/f4fa096ca6445665213bc7e602f50b23562f3e43.png\" class=\"latex\" alt=\"$\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right)$\" style=\"vertical-align: -27px\" width=\"137\" height=\"64\" >?</span></div>\n</div>\n<br>\nDon't you multiply them or something...<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/9/1/9/91962bb361615840541168c40001cab0698b6466.png\" class=\"latex\" alt=\"$\\left( \\begin{array}{ccc}1 & 4 & 1 \\\\-2 & 0 & 0 \\\\ 3 & 2 &-3\\ \\end{array}\\right) \\cdot \\left( \\begin{array}{c}3 \\\\ 1 \\\\ 2 \\end{array}\\right)=\\left( \\begin{array}{c}9 \\\\-6 \\\\ 5 \\end{array}\\right)$\" style=\"vertical-align: -27px\" width=\"302\" height=\"64\" ></div>",
"post_id": 564140,
"post_number": 2,
"post_time_unix": 1151798964,
"post_time_utc": "2006-07-02 00:09:24 UTC",
"thanks_received": 2,
"user_id": 11029,
"username": "ch1n353ch3s54a1l"
},
{
"attachments": [],
"content_bbcode": "oh, ofcourse, I posted the wrong question. :blush: \r\nThanks ch1n353ch3s54a1l\r\nwasted all that time... :(",
"content_html": "oh, ofcourse, I posted the wrong question. <img src=\"/assets/images/smilies/redface_anim.gif\" width=\"19\" height=\"19\" alt=\":blush:\" title=\":blush:\" class=\"bbcode_smiley\" /><br>\nThanks ch1n353ch3s54a1l<br>\nwasted all that time... <img src=\"/assets/images/smilies/sad.gif\" width=\"20\" height=\"20\" alt=\":(\" title=\":(\" class=\"bbcode_smiley\" />",
"post_id": 564204,
"post_number": 3,
"post_time_unix": 1151806407,
"post_time_utc": "2006-07-02 02:13:27 UTC",
"thanks_received": 2,
"user_id": 18217,
"username": "Inspired By Nature"
}
],
"source": null
}
|
What is the image of \(\begin{pmatrix}3\\1\\2\end{pmatrix}\) under the mapping given by the matrix \(\begin{pmatrix}1 & 4 & 1\\ -2 & 0 & 0\\ 3 & 2 & -3\end{pmatrix}\)?
|
[
"/Mathematics/Algebra/LinearAlgebra/Matrices/MatrixOperations",
"/Mathematics/Algebra/VectorAlgebra/ColumnVector",
"/Mathematics/Algebra/VectorAlgebra/RealVector",
"/Mathematics/Algebra/VectorAlgebra/Vector"
] |
Multiply the given matrix by the column vector to obtain its image.
|
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aops_99915
|
[quote="236factorial"]Prove that
$\lim_{x\to0}(1+rx)^{\frac{1}{x}}= e^{r}$
given that $\lim_{x\to0}(1+x)^{\frac{1}{x}}= e$.
[Do not use L'Hopital's rule.][/quote]
Let $u = rx$
$x = u/r$
$\lim_{u\to0}(1+u)^{\frac{1}{u/r}}=\lim_{u\to0}((1+u)^{\frac{1}{u})^{r}}= e^{r}$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Prove that \r\n\r\n$\\lim_{x\\to0}(1+rx)^{\\frac{1}{x}}= e^{r}$ \r\n\r\ngiven that $\\lim_{x\\to0}(1+x)^{\\frac{1}{x}}= e$. \r\n\r\n[Do not use L'Hopital's rule.]",
"content_html": "Prove that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/2/9/02995ab72d1b7900e5240567692ba0deda42b3c8.png\" class=\"latex\" alt=\"$\\lim_{x\\to0}(1+rx)^{\\frac{1}{x}}= e^{r}$\" style=\"vertical-align: -11px\" width=\"142\" height=\"29\" ><br>\n<br>\ngiven that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/3/463063869421356bc39539588842e5d036a0e65e.png\" class=\"latex\" alt=\"$\\lim_{x\\to0}(1+x)^{\\frac{1}{x}}= e$\" style=\"vertical-align: -11px\" width=\"127\" height=\"29\" >.</span><br>\n<br>\n[Do not use L'Hopital's rule.]",
"post_id": 564183,
"post_number": 1,
"post_time_unix": 1151804240,
"post_time_utc": "2006-07-02 01:37:20 UTC",
"thanks_received": 2,
"user_id": 7080,
"username": "236factorial"
},
{
"attachments": [],
"content_bbcode": "[quote=\"236factorial\"]Prove that \n\n$\\lim_{x\\to0}(1+rx)^{\\frac{1}{x}}= e^{r}$ \n\ngiven that $\\lim_{x\\to0}(1+x)^{\\frac{1}{x}}= e$. \n\n[Do not use L'Hopital's rule.][/quote]\r\n\r\nLet $u = rx$\r\n$x = u/r$\r\n$\\lim_{u\\to0}(1+u)^{\\frac{1}{u/r}}=\\lim_{u\\to0}((1+u)^{\\frac{1}{u})^{r}}= e^{r}$",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">236factorial wrote:</div>\n<div class=\"bbcode_quote_body\">Prove that<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/2/9/02995ab72d1b7900e5240567692ba0deda42b3c8.png\" class=\"latex\" alt=\"$\\lim_{x\\to0}(1+rx)^{\\frac{1}{x}}= e^{r}$\" style=\"vertical-align: -11px\" width=\"142\" height=\"29\" ><br>\n<br>\ngiven that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/6/3/463063869421356bc39539588842e5d036a0e65e.png\" class=\"latex\" alt=\"$\\lim_{x\\to0}(1+x)^{\\frac{1}{x}}= e$\" style=\"vertical-align: -11px\" width=\"127\" height=\"29\" >.</span><br>\n<br>\n[Do not use L'Hopital's rule.]</div>\n</div>\n<br>\nLet <img src=\"//latex.artofproblemsolving.com/7/8/a/78a4eab2b4a075cb1227907d85b145ee2a7caf16.png\" class=\"latex\" alt=\"$u = rx$\" width=\"53\" height=\"8\" ><br>\n<img src=\"//latex.artofproblemsolving.com/6/9/6/6968465fe6ade11b7e15e2030dd6372e40a1a45b.png\" class=\"latex\" alt=\"$x = u/r$\" style=\"vertical-align: -4px\" width=\"62\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/a/0/d/a0d69497ad5713be302e80ea6e1b93616488206f.png\" class=\"latex\" alt=\"$\\lim_{u\\to0}(1+u)^{\\frac{1}{u/r}}=\\lim_{u\\to0}((1+u)^{\\frac{1}{u})^{r}}= e^{r}$\" style=\"vertical-align: -11px\" width=\"284\" height=\"29\" >",
"post_id": 564294,
"post_number": 2,
"post_time_unix": 1151816137,
"post_time_utc": "2006-07-02 04:55:37 UTC",
"thanks_received": 1,
"user_id": 15749,
"username": "ShoeFactory"
}
],
"source": null
}
|
Prove that
\[
\lim_{x\to 0}\bigl(1+rx\bigr)^{\frac{1}{x}}=e^{r}
\]
given that \(\displaystyle\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e\). Do not use L'Hôpital's rule.
|
[
"/Mathematics/CalculusandAnalysis/Calculus/Limits/Limit",
"/Mathematics/CalculusandAnalysis/GeneralAnalysis/Analysis",
"/Mathematics/CalculusandAnalysis/GeneralAnalysis/RealAnalysis"
] |
Substitute u = r x to express the limit as ((1+u)^{1/u})^{r} and use the known limit (1+u)^{1/u} → e.
|
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aops_99916
|
[quote="236factorial"]1. Let $f(x)=\int^{x}_{1}\sqrt[3]{1+t^{2}}dt$
Find $\frac{d}{dx}\left[f^{-1}(x)\right]\mid_{x=0}$.
2. Given points A(2,1) and B(5,4), find the point P in the interval [2,5] on the x-axis that maximizes angle APB.
This problem looks easy, but the method I used turned out to be extremely ugly.[/quote]
1.
Denote $g(x)=f^{-1}(x)$.
$f(g(x))=x$, thus $f'(g(x))\times{g'(x)}=1$.
And $f'(x)=\sqrt[3]{1+x^{2}}$, $g(0)=1$, so $f'(g(0))=\sqrt[3]{2}$.
Therefore $g'(0)=\frac{1}{f'(g(0))}=\frac{1}{\sqrt[3]{2}}$.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "1. Let $f(x)=\\int^{x}_{1}\\sqrt[3]{1+t^{2}}dt$ \r\n\r\nFind $\\frac{d}{dx}\\left[f^{-1}(x)\\right]\\mid _{x=0}$. \r\n\r\n\r\n2. Given points A(2,1) and B(5,4), find the point P in the interval [2,5] on the x-axis that maximizes angle APB. \r\nThis problem looks easy, but the method I used turned out to be extremely ugly.",
"content_html": "1. Let <img src=\"//latex.artofproblemsolving.com/f/f/e/ffe66aba83c16bbf6a7307903e085658e40f27ad.png\" class=\"latex\" alt=\"$f(x)=\\int^{x}_{1}\\sqrt[3]{1+t^{2}}dt$\" style=\"vertical-align: -16px\" width=\"165\" height=\"41\" ><br>\n<br>\nFind <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/7/bc77c06e105b093d13b4bf757fbaa23074546762.png\" class=\"latex\" alt=\"$\\frac{d}{dx}\\left[f^{-1}(x)\\right]\\mid _{x=0}$\" style=\"vertical-align: -12px\" width=\"128\" height=\"37\" >.</span><br>\n<br>\n<br>\n2. Given points A(2,1) and B(5,4), find the point P in the interval [2,5] on the x-axis that maximizes angle APB.<br>\nThis problem looks easy, but the method I used turned out to be extremely ugly.",
"post_id": 564188,
"post_number": 1,
"post_time_unix": 1151804580,
"post_time_utc": "2006-07-02 01:43:00 UTC",
"thanks_received": 2,
"user_id": 7080,
"username": "236factorial"
},
{
"attachments": [],
"content_bbcode": "2. Look for a tangent circle or something; no calculus...",
"content_html": "2. Look for a tangent circle or something; no calculus...",
"post_id": 564206,
"post_number": 2,
"post_time_unix": 1151806797,
"post_time_utc": "2006-07-02 02:19:57 UTC",
"thanks_received": 2,
"user_id": 1253,
"username": "MysticTerminator"
},
{
"attachments": [],
"content_bbcode": "[quote=\"236factorial\"]1. Let $f(x)=\\int^{x}_{1}\\sqrt[3]{1+t^{2}}dt$ \n\nFind $\\frac{d}{dx}\\left[f^{-1}(x)\\right]\\mid_{x=0}$. \n\n\n2. Given points A(2,1) and B(5,4), find the point P in the interval [2,5] on the x-axis that maximizes angle APB. \nThis problem looks easy, but the method I used turned out to be extremely ugly.[/quote]\r\n\r\n1.\r\nDenote $g(x)=f^{-1}(x)$.\r\n$f(g(x))=x$, thus $f'(g(x))\\times{g'(x)}=1$.\r\nAnd $f'(x)=\\sqrt[3]{1+x^{2}}$, $g(0)=1$, so $f'(g(0))=\\sqrt[3]{2}$.\r\nTherefore $g'(0)=\\frac{1}{f'(g(0))}=\\frac{1}{\\sqrt[3]{2}}$.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">236factorial wrote:</div>\n<div class=\"bbcode_quote_body\">1. Let <img src=\"//latex.artofproblemsolving.com/f/f/e/ffe66aba83c16bbf6a7307903e085658e40f27ad.png\" class=\"latex\" alt=\"$f(x)=\\int^{x}_{1}\\sqrt[3]{1+t^{2}}dt$\" style=\"vertical-align: -16px\" width=\"165\" height=\"41\" ><br>\n<br>\nFind <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/3/e/13e0454e5f61d9fcd8228b641ab93cdcb399ba2e.png\" class=\"latex\" alt=\"$\\frac{d}{dx}\\left[f^{-1}(x)\\right]\\mid_{x=0}$\" style=\"vertical-align: -12px\" width=\"128\" height=\"37\" >.</span><br>\n<br>\n<br>\n2. Given points A(2,1) and B(5,4), find the point P in the interval [2,5] on the x-axis that maximizes angle APB.<br>\nThis problem looks easy, but the method I used turned out to be extremely ugly.</div>\n</div>\n<br>\n1.<br>\nDenote <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/6/9268fb7d6ab44f7532a2a9ab14da974ccde2a3c9.png\" class=\"latex\" alt=\"$g(x)=f^{-1}(x)$\" style=\"vertical-align: -4px\" width=\"110\" height=\"19\" >.</span><br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/4/d/94d90ceb992cd531b57ab8947223c9c94b07f82a.png\" class=\"latex\" alt=\"$f(g(x))=x$\" style=\"vertical-align: -4px\" width=\"93\" height=\"18\" >,</span> thus <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/f/7/af7bcee142c979dcd5a9575929f88c03c2deb9b3.png\" class=\"latex\" alt=\"$f'(g(x))\\times{g'(x)}=1$\" style=\"vertical-align: -4px\" width=\"156\" height=\"18\" >.</span><br>\nAnd <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/1/0/b10552623d7dc8303008fed226f21743a1b734c3.png\" class=\"latex\" alt=\"$f'(x)=\\sqrt[3]{1+x^{2}}$\" style=\"vertical-align: -4px\" width=\"129\" height=\"21\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/f/4/5f42ff6e02f0df88b17dbab5135423850dce5e0f.png\" class=\"latex\" alt=\"$g(0)=1$\" style=\"vertical-align: -4px\" width=\"65\" height=\"18\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/3/4/7340a905ecb83135a7e46eff1d3e82a92cd3ba59.png\" class=\"latex\" alt=\"$f'(g(0))=\\sqrt[3]{2}$\" style=\"vertical-align: -4px\" width=\"111\" height=\"21\" >.</span><br>\nTherefore <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/e/1/5e14a0019c88153b7f45150961b1ee0d85bcfb6d.png\" class=\"latex\" alt=\"$g'(0)=\\frac{1}{f'(g(0))}=\\frac{1}{\\sqrt[3]{2}}$\" style=\"vertical-align: -17px\" width=\"179\" height=\"41\" >.</span>",
"post_id": 565039,
"post_number": 3,
"post_time_unix": 1151912944,
"post_time_utc": "2006-07-03 07:49:04 UTC",
"thanks_received": 2,
"user_id": 20595,
"username": "AlvaroRecoba"
},
{
"attachments": [],
"content_bbcode": "[quote=\"MysticTerminator\"]2. Look for a tangent circle or something; no calculus...[/quote]\r\n\r\ntangent circle? I finally did figure it out with calculus, although there was more algebra involved than calculus.... :maybe:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">MysticTerminator wrote:</div>\n<div class=\"bbcode_quote_body\">2. Look for a tangent circle or something; no calculus...</div>\n</div>\n<br>\ntangent circle? I finally did figure it out with calculus, although there was more algebra involved than calculus.... <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" />",
"post_id": 566521,
"post_number": 4,
"post_time_unix": 1152104041,
"post_time_utc": "2006-07-05 12:54:01 UTC",
"thanks_received": 2,
"user_id": 7080,
"username": "236factorial"
}
],
"source": null
}
|
1. Let \(f(x)=\displaystyle\int_{1}^{x}\sqrt[3]{1+t^{2}}\,dt.\) Find \(\left.\dfrac{d}{dx}\bigl[f^{-1}(x)\bigr]\right|_{x=0}.\)
2. Given points \(A(2,1)\) and \(B(5,4)\), find the point \(P\) on the x-axis with \(x\in[2,5]\) that maximizes the angle \(\angle APB\).
|
[
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/ChainRule",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative",
"/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstFundamentalTheoremofCalculus",
"/Mathematics/CalculusandAnalysis/Calculus/GeneralCalculus/Calculus",
"/Mathematics/CalculusandAnalysis/Calculus/IntegralCalculus",
"/Mathematics/CalculusandAnalysis/Functions/InverseFunction",
"/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry",
"/Mathematics/Geometry/CoordinateGeometry/Axis",
"/Mathematics/Geometry/CoordinateGeometry/Cartesian",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/CartesianGeometry",
"/Mathematics/Geometry/CoordinateGeometry/CartesianPlane",
"/Mathematics/Geometry/CoordinateGeometry/CoordinatePlane",
"/Mathematics/Geometry/CoordinateGeometry/CoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/Coordinates",
"/Mathematics/Geometry/CoordinateGeometry/RealLine",
"/Mathematics/Geometry/CoordinateGeometry/x-Axis",
"/Mathematics/Geometry/PlaneGeometry/Triangles",
"/Mathematics/Geometry/Points/Point",
"/Mathematics/Geometry/Trigonometry/Angles/Angle",
"/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry"
] |
Apply the inverse function derivative formula: (f^{-1})'(x)=1/ f'(f^{-1}(x)) and evaluate at the given point.
|
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aops_99919
|
[quote="Soarer"]After that we are to prove $\frac{81}{2}t^{2}-54t^{3}+\frac{8}{3t-3t^{2}}\ge 14$ for $t \in [\frac{1}{2}, \frac{3}{4}]$. Expanding we get
$324t^{5}-567t^{4}+243t^{3}+84t^{2}-84t+16 \ge 0$ which is equivalent to
$(3t-2)^{2}(36t^{3}-15t^{2}-9t+4)\ge 0$ which is true.[/quote]
I see, and it is nice, but you only verify that 14 is right after he tells us 14. How would we find the value?
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $a,b,c>0$ such that $a+b+c=\\frac{3}{2}$ And\r\n$F=27abc+\\frac{8}{ab+bc+ca}$\r\nHere is my question:$minF=?$ :wink: :wink: \r\nWhen the equality occur? :wink:",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/c/7/a/c7a0dc1d53202f0003849a81d76e5555b1ee251a.png\" class=\"latex\" alt=\"$a,b,c>0$\" style=\"vertical-align: -3px\" width=\"74\" height=\"16\" > such that <img src=\"//latex.artofproblemsolving.com/0/c/3/0c366eb33fa90e3be647a61924d4f57130ace05c.png\" class=\"latex\" alt=\"$a+b+c=\\frac{3}{2}$\" style=\"vertical-align: -12px\" width=\"105\" height=\"37\" > And<br>\n<img src=\"//latex.artofproblemsolving.com/5/e/4/5e4c78e3867cf1b0467211204541c0fbb69aca02.png\" class=\"latex\" alt=\"$F=27abc+\\frac{8}{ab+bc+ca}$\" style=\"vertical-align: -14px\" width=\"201\" height=\"39\" ><br>\nHere is my question<span style=\"white-space:nowrap;\">:<img src=\"//latex.artofproblemsolving.com/8/d/f/8dfe1c9b473474bd127dee9e33d16d77585fc388.png\" class=\"latex\" alt=\"$minF=?$\" width=\"74\" height=\"12\" ></span> <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" /> <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" /><br>\nWhen the equality occur? <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />",
"post_id": 564239,
"post_number": 1,
"post_time_unix": 1151809032,
"post_time_utc": "2006-07-02 02:57:12 UTC",
"thanks_received": 1,
"user_id": 6788,
"username": "nthd"
},
{
"attachments": [],
"content_bbcode": "How do you think? :blush:",
"content_html": "How do you think? <img src=\"/assets/images/smilies/redface_anim.gif\" width=\"19\" height=\"19\" alt=\":blush:\" title=\":blush:\" class=\"bbcode_smiley\" />",
"post_id": 565778,
"post_number": 2,
"post_time_unix": 1151984958,
"post_time_utc": "2006-07-04 03:49:18 UTC",
"thanks_received": 1,
"user_id": 6788,
"username": "nthd"
},
{
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"content_bbcode": "It is easy to find minimum of this function if you replace 27 by 256/27 but for 27???",
"content_html": "It is easy to find minimum of this function if you replace 27 by 256/27 but for 27???",
"post_id": 567222,
"post_number": 3,
"post_time_unix": 1152183067,
"post_time_utc": "2006-07-06 10:51:07 UTC",
"thanks_received": 2,
"user_id": 19941,
"username": "delegat"
},
{
"attachments": [],
"content_bbcode": "[quote=\"delegat\"]It is easy to find minimum of this function if you replace 27 by 256/27 but for 27???[/quote]\r\nYes, :wink: but with 27, I had a nice pro.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">delegat wrote:</div>\n<div class=\"bbcode_quote_body\">It is easy to find minimum of this function if you replace 27 by 256/27 but for 27???</div>\n</div>\nYes, <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" /> but with 27, I had a nice pro.",
"post_id": 569320,
"post_number": 4,
"post_time_unix": 1152409705,
"post_time_utc": "2006-07-09 01:48:25 UTC",
"thanks_received": 2,
"user_id": 6788,
"username": "nthd"
},
{
"attachments": [],
"content_bbcode": "q: a,b,c > 0 and a+b+c = 1, find minimum value of f(a,b,c) = 27abc + 8/(ab+bc+ca).\r\na: not yet a full solution but some thought about it leads me to say that the minimum value is m = f(1/2,1/2,1/2). as you can see if you let a --> 0 then f(0,b,c) attains a minimum at b = c = 3/4. but m <= f(0,3/4,3/4) as you can verify it yourself. the strategy is perhaps the AC method. \r\n-------------------------------------------------------------------------------------",
"content_html": "q: a,b,c > 0 and a+b+c = 1, find minimum value of f(a,b,c) = 27abc + 8/(ab+bc+ca).<br>\na: not yet a full solution but some thought about it leads me to say that the minimum value is m = f(1/2,1/2,1/2). as you can see if you let a --> 0 then f(0,b,c) attains a minimum at b = c = 3/4. but m <= f(0,3/4,3/4) as you can verify it yourself. the strategy is perhaps the AC method.\n<hr class=\"bbcode_rule\" />\n",
"post_id": 569433,
"post_number": 5,
"post_time_unix": 1152426375,
"post_time_utc": "2006-07-09 06:26:15 UTC",
"thanks_received": 1,
"user_id": 3921,
"username": "Minh Can"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Minh Can\"]q: a,b,c > 0 and a+b+c = 1, find minimum value of f(a,b,c) = 27abc + 8/(ab+bc+ca).\na: not yet a full solution but some thought about it leads me to say that the minimum value is m = f(1/2,1/2,1/2). as you can see if you let a --> 0 then f(0,b,c) attains a minimum at b = c = 3/4. but m <= f(0,3/4,3/4) as you can verify it yourself. the strategy is perhaps the AC method. \n-------------------------------------------------------------------------------------[/quote]\r\nThe equality don't occur when $a=b=c$ or $a=0$ :wink:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Minh Can wrote:</div>\n<div class=\"bbcode_quote_body\">q: a,b,c > 0 and a+b+c = 1, find minimum value of f(a,b,c) = 27abc + 8/(ab+bc+ca).<br>\na: not yet a full solution but some thought about it leads me to say that the minimum value is m = f(1/2,1/2,1/2). as you can see if you let a --> 0 then f(0,b,c) attains a minimum at b = c = 3/4. but m <= f(0,3/4,3/4) as you can verify it yourself. the strategy is perhaps the AC method.\n<hr class=\"bbcode_rule\" /></div>\n</div>\nThe equality don't occur when <img src=\"//latex.artofproblemsolving.com/5/6/6/566b9136144a62fac113ecbc8065793a74123238.png\" class=\"latex\" alt=\"$a=b=c$\" width=\"74\" height=\"12\" > or <img src=\"//latex.artofproblemsolving.com/6/9/c/69cefa6855d2216036df094012f7ad2a7e4e204f.png\" class=\"latex\" alt=\"$a=0$\" width=\"42\" height=\"12\" > <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />",
"post_id": 571800,
"post_number": 6,
"post_time_unix": 1152675569,
"post_time_utc": "2006-07-12 03:39:29 UTC",
"thanks_received": 1,
"user_id": 6788,
"username": "nthd"
},
{
"attachments": [],
"content_bbcode": "$MinF=14$ when $(a,b,c)=(\\frac{1}{6},\\frac{2}{3},\\frac{2}{3}) or (\\frac{2}{3},\\frac{1}{6},\\frac{2}{3}) or (\\frac{2}{3},\\frac{2}{3},\\frac{1}{6})$\r\n :lol:",
"content_html": "<img src=\"//latex.artofproblemsolving.com/a/7/2/a7225c5a9c418021089381fcdda1a57d055a3754.png\" class=\"latex\" alt=\"$MinF=14$\" style=\"vertical-align: 0px\" width=\"93\" height=\"13\" > when <img src=\"//latex.artofproblemsolving.com/7/e/8/7e833faa2decedb30b005bf7735c6e3c25e37347.png\" class=\"latex\" alt=\"$(a,b,c)=(\\frac{1}{6},\\frac{2}{3},\\frac{2}{3}) or (\\frac{2}{3},\\frac{1}{6},\\frac{2}{3}) or (\\frac{2}{3},\\frac{2}{3},\\frac{1}{6})$\" style=\"vertical-align: -12px\" width=\"322\" height=\"37\" ><br>\n<img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />",
"post_id": 589795,
"post_number": 7,
"post_time_unix": 1154484567,
"post_time_utc": "2006-08-02 02:09:27 UTC",
"thanks_received": 1,
"user_id": 6788,
"username": "nthd"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Minh Can\"] The strategy is perhaps the AC method. \n-------------------------------------------------------------------------------------[/quote]No AC method, but certainly EV Method.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Minh Can wrote:</div>\n<div class=\"bbcode_quote_body\">The strategy is perhaps the AC method.\n<hr class=\"bbcode_rule\" /></div>\n</div>\nNo AC method, but certainly EV Method.",
"post_id": 590557,
"post_number": 8,
"post_time_unix": 1154514857,
"post_time_utc": "2006-08-02 10:34:17 UTC",
"thanks_received": 2,
"user_id": 908,
"username": "Vasc"
},
{
"attachments": [],
"content_bbcode": "Can we see a solution? :)",
"content_html": "Can we see a solution? <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 590823,
"post_number": 9,
"post_time_unix": 1154538033,
"post_time_utc": "2006-08-02 17:00:33 UTC",
"thanks_received": 2,
"user_id": 6233,
"username": "Xevarion"
},
{
"attachments": [],
"content_bbcode": "Mixing variables works, assume c is the smallest, then we can check that $f(a,b,c) \\ge f(t,t,c)$ where $t = \\frac{a+b}{2}$ After that we are to prove $\\frac{81}{2}t^{2}-54t^{3}+\\frac{8}{3t-3t^{2}}\\ge 14$ for $t \\in [\\frac{1}{2}, \\frac{3}{4}]$. Expanding we get\r\n$324t^{5}-567t^{4}+243t^{3}+84t^{2}-84t+16 \\ge 0$ which is equivalent to\r\n$(3t-2)^{2}(36t^{3}-15t^{2}-9t+4)\\ge 0$ which is true.",
"content_html": "Mixing variables works, assume c is the smallest, then we can check that <img src=\"//latex.artofproblemsolving.com/d/5/5/d5528f30ec5e9ad027d7aa9733d5b2dec4cb9a27.png\" class=\"latex\" alt=\"$f(a,b,c) \\ge f(t,t,c)$\" style=\"vertical-align: -4px\" width=\"151\" height=\"18\" > where <img src=\"//latex.artofproblemsolving.com/4/f/d/4fdfd32fe022a3fe66dc7ff2ca40d68f828ae357.png\" class=\"latex\" alt=\"$t = \\frac{a+b}{2}$\" style=\"vertical-align: -12px\" width=\"73\" height=\"37\" > After that we are to prove <img src=\"//latex.artofproblemsolving.com/1/f/6/1f6c66a9aeeb31b2a490392f13cc636bfab1955c.png\" class=\"latex\" alt=\"$\\frac{81}{2}t^{2}-54t^{3}+\\frac{8}{3t-3t^{2}}\\ge 14$\" style=\"vertical-align: -12px\" width=\"220\" height=\"37\" > for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/9/e/49e3a7cee319ce1d6796f6e3f8a09838f03a12f9.png\" class=\"latex\" alt=\"$t \\in [\\frac{1}{2}, \\frac{3}{4}]$\" style=\"vertical-align: -13px\" width=\"71\" height=\"38\" >.</span> Expanding we get<br>\n<img src=\"//latex.artofproblemsolving.com/a/f/e/afe2514e5f240ebc9fb95ade56e27e397d77110e.png\" class=\"latex\" alt=\"$324t^{5}-567t^{4}+243t^{3}+84t^{2}-84t+16 \\ge 0$\" style=\"vertical-align: -2px\" width=\"343\" height=\"17\" > which is equivalent to<br>\n<img src=\"//latex.artofproblemsolving.com/b/7/d/b7d714fc127a6eab511a4927d0671dd2bb99b775.png\" class=\"latex\" alt=\"$(3t-2)^{2}(36t^{3}-15t^{2}-9t+4)\\ge 0$\" style=\"vertical-align: -4px\" width=\"271\" height=\"19\" > which is true.",
"post_id": 590980,
"post_number": 10,
"post_time_unix": 1154545547,
"post_time_utc": "2006-08-02 19:05:47 UTC",
"thanks_received": 1,
"user_id": 141,
"username": "Soarer"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Soarer\"]After that we are to prove $\\frac{81}{2}t^{2}-54t^{3}+\\frac{8}{3t-3t^{2}}\\ge 14$ for $t \\in [\\frac{1}{2}, \\frac{3}{4}]$. Expanding we get\n$324t^{5}-567t^{4}+243t^{3}+84t^{2}-84t+16 \\ge 0$ which is equivalent to\n$(3t-2)^{2}(36t^{3}-15t^{2}-9t+4)\\ge 0$ which is true.[/quote]\r\n\r\nI see, and it is nice, but you only verify that 14 is right after he tells us 14. How would we find the value?",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Soarer wrote:</div>\n<div class=\"bbcode_quote_body\">After that we are to prove <img src=\"//latex.artofproblemsolving.com/1/f/6/1f6c66a9aeeb31b2a490392f13cc636bfab1955c.png\" class=\"latex\" alt=\"$\\frac{81}{2}t^{2}-54t^{3}+\\frac{8}{3t-3t^{2}}\\ge 14$\" style=\"vertical-align: -12px\" width=\"220\" height=\"37\" > for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/9/e/49e3a7cee319ce1d6796f6e3f8a09838f03a12f9.png\" class=\"latex\" alt=\"$t \\in [\\frac{1}{2}, \\frac{3}{4}]$\" style=\"vertical-align: -13px\" width=\"71\" height=\"38\" >.</span> Expanding we get<br>\n<img src=\"//latex.artofproblemsolving.com/a/f/e/afe2514e5f240ebc9fb95ade56e27e397d77110e.png\" class=\"latex\" alt=\"$324t^{5}-567t^{4}+243t^{3}+84t^{2}-84t+16 \\ge 0$\" style=\"vertical-align: -2px\" width=\"343\" height=\"17\" > which is equivalent to<br>\n<img src=\"//latex.artofproblemsolving.com/b/7/d/b7d714fc127a6eab511a4927d0671dd2bb99b775.png\" class=\"latex\" alt=\"$(3t-2)^{2}(36t^{3}-15t^{2}-9t+4)\\ge 0$\" style=\"vertical-align: -4px\" width=\"271\" height=\"19\" > which is true.</div>\n</div>\n<br>\nI see, and it is nice, but you only verify that 14 is right after he tells us 14. How would we find the value?",
"post_id": 591279,
"post_number": 11,
"post_time_unix": 1154557989,
"post_time_utc": "2006-08-02 22:33:09 UTC",
"thanks_received": 2,
"user_id": 6233,
"username": "Xevarion"
},
{
"attachments": [],
"content_bbcode": "After mixing variables, you are left with an expression of one variable. At least calculus works.",
"content_html": "After mixing variables, you are left with an expression of one variable. At least calculus works.",
"post_id": 591312,
"post_number": 12,
"post_time_unix": 1154559776,
"post_time_utc": "2006-08-02 23:02:56 UTC",
"thanks_received": 2,
"user_id": 141,
"username": "Soarer"
},
{
"attachments": [],
"content_bbcode": "By EV-Theorem, for $a+b+c=3/2$ and $abc=const$, the sum $a^{2}+b^{2}+c^{2}$ (hence E) is minimal when $0<a\\leq b=c$.\r\nTherefore, we can consider only this case. \r\nSimilarly, we can prove the inequality\r\n$3abc+\\frac{30}{ab+bc+ca}\\geq 13$\r\nfor any positive numbers $a,b,c$ such that $a+b+c=3$.",
"content_html": "By EV-Theorem, for <img src=\"//latex.artofproblemsolving.com/e/c/7/ec74ad80e09f656e9f7eb19e11e627eaa5eb7581.png\" class=\"latex\" alt=\"$a+b+c=3/2$\" style=\"vertical-align: -4px\" width=\"120\" height=\"18\" > and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/6/6/c6667d67a10430147666c23948c451d3d0330fa0.png\" class=\"latex\" alt=\"$abc=const$\" width=\"92\" height=\"12\" >,</span> the sum <img src=\"//latex.artofproblemsolving.com/0/3/5/03563388fbe63fb70e2f8af2869c2ef03673f430.png\" class=\"latex\" alt=\"$a^{2}+b^{2}+c^{2}$\" style=\"vertical-align: -1px\" width=\"91\" height=\"16\" > (hence E) is minimal when <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/4/5/d45237ab34672aee5fd977936ecd71424c4804c0.png\" class=\"latex\" alt=\"$0<a\\leq b=c$\" style=\"vertical-align: -2px\" width=\"107\" height=\"15\" >.</span><br>\nTherefore, we can consider only this case.<br>\nSimilarly, we can prove the inequality<br>\n<img src=\"//latex.artofproblemsolving.com/8/5/b/85bedc599a72fb0dcad662ab9a5407af14aafd31.png\" class=\"latex\" alt=\"$3abc+\\frac{30}{ab+bc+ca}\\geq 13$\" style=\"vertical-align: -14px\" width=\"197\" height=\"39\" ><br>\nfor any positive numbers <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 <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/e/1/de12e8a2b7ad3124b609b10d033c65b8a479130f.png\" class=\"latex\" alt=\"$a+b+c=3$\" style=\"vertical-align: -1px\" width=\"102\" height=\"14\" >.</span>",
"post_id": 591685,
"post_number": 13,
"post_time_unix": 1154605767,
"post_time_utc": "2006-08-03 11:49:27 UTC",
"thanks_received": 1,
"user_id": 908,
"username": "Vasc"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Vasc\"]$abc=const$[/quote]\r\n\r\nWhy??\r\n :?:",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Vasc wrote:</div>\n<div class=\"bbcode_quote_body\"><img src=\"//latex.artofproblemsolving.com/c/6/6/c6667d67a10430147666c23948c451d3d0330fa0.png\" class=\"latex\" alt=\"$abc=const$\" width=\"92\" height=\"12\" ></div>\n</div>\n<br>\nWhy??<br>\n<img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":?\" title=\":?\" class=\"bbcode_smiley\" />:",
"post_id": 591817,
"post_number": 14,
"post_time_unix": 1154614600,
"post_time_utc": "2006-08-03 14:16:40 UTC",
"thanks_received": 2,
"user_id": 10088,
"username": "silouan"
},
{
"attachments": [],
"content_bbcode": "Because for $abc=const$ we may apply EV-Theorem. :wink:",
"content_html": "Because for <img src=\"//latex.artofproblemsolving.com/c/6/6/c6667d67a10430147666c23948c451d3d0330fa0.png\" class=\"latex\" alt=\"$abc=const$\" width=\"92\" height=\"12\" > we may apply EV-Theorem. <img src=\"/assets/images/smilies/wink.gif\" width=\"20\" height=\"20\" alt=\":wink:\" title=\":wink:\" class=\"bbcode_smiley\" />",
"post_id": 592345,
"post_number": 15,
"post_time_unix": 1154633700,
"post_time_utc": "2006-08-03 19:35:00 UTC",
"thanks_received": 1,
"user_id": 908,
"username": "Vasc"
},
{
"attachments": [],
"content_bbcode": "EV theorem...What's this ,Vasc?\r\nWas it proved?\r\nCould you tell us about the history of that problem? :)",
"content_html": "EV theorem...What's this ,Vasc?<br>\nWas it proved?<br>\nCould you tell us about the history of that problem? <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 597416,
"post_number": 16,
"post_time_unix": 1155112297,
"post_time_utc": "2006-08-09 08:31:37 UTC",
"thanks_received": 2,
"user_id": 18006,
"username": "Augustus Caesar"
},
{
"attachments": [],
"content_bbcode": "Take a look at\r\nhttp://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160 \r\n\r\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.",
"content_html": "Take a look at<br>\n<a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160\">http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160</a><br>\n<br>\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.",
"post_id": 599764,
"post_number": 17,
"post_time_unix": 1155327953,
"post_time_utc": "2006-08-11 20:25:53 UTC",
"thanks_received": 2,
"user_id": 908,
"username": "Vasc"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Vasc\"]Take a look at\nhttp://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160\n\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.\n[/quote]\r\nI'm sorry but can you post it here?",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Vasc wrote:</div>\n<div class=\"bbcode_quote_body\">Take a look at<br>\n<a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160\">http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160</a><br>\n<br>\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.</div>\n</div>\nI'm sorry but can you post it here?",
"post_id": 600723,
"post_number": 18,
"post_time_unix": 1155441675,
"post_time_utc": "2006-08-13 04:01:15 UTC",
"thanks_received": 2,
"user_id": 13068,
"username": "The soul of rock"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Vasc\"]Take a look at\nhttp://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160 \n\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.[/quote]\r\nThank you,but where can I find this book?\r\n[b]a little problem[/b]:Do we can use EV theorem with $ab+bc+ca=const$?",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Vasc wrote:</div>\n<div class=\"bbcode_quote_body\">Take a look at<br>\n<a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160\">http://www.mathlinks.ro/Forum/viewtopic.php?t=7942&start=160</a><br>\n<br>\nI published a complete proof for Equal Variable Theorem in the last issue (no 2, 2006) of Gazeta Matematica A, in English. In Algebraic Inequalities there is a chapter which treats this method, with a partial proof, 7 corolarries and 34 applications.</div>\n</div>\nThank you,but where can I find this book?<br>\n<b>a little problem</b>:Do we can use EV theorem with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/3/9/239c0eab717396dbcb62173baf16f5078f167b9f.png\" class=\"latex\" alt=\"$ab+bc+ca=const$\" style=\"vertical-align: -1px\" width=\"161\" height=\"14\" >?</span>",
"post_id": 601325,
"post_number": 19,
"post_time_unix": 1155519621,
"post_time_utc": "2006-08-14 01:40:21 UTC",
"thanks_received": 2,
"user_id": 18006,
"username": "Augustus Caesar"
},
{
"attachments": [],
"content_bbcode": "See\r\nhttp://www.mathlinks.ro/Forum/viewtopic.php?t=100399.\r\n\r\nAnd, \r\n$2(ab+bc+ca)=(a+b+c)^{2}-(a^{2}+b^{2}+c^{2})$.",
"content_html": "See<br>\n<a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=100399.\">http://www.mathlinks.ro/Forum/viewtopic.php?t=100399.</a><br>\n<br>\nAnd,<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/7/e/27ebba73d96f3f5caa3ef89eee899beb8a1855a0.png\" class=\"latex\" alt=\"$2(ab+bc+ca)=(a+b+c)^{2}-(a^{2}+b^{2}+c^{2})$\" style=\"vertical-align: -4px\" width=\"360\" height=\"19\" >.</span>",
"post_id": 601418,
"post_number": 20,
"post_time_unix": 1155529529,
"post_time_utc": "2006-08-14 04:25:29 UTC",
"thanks_received": 2,
"user_id": 908,
"username": "Vasc"
}
],
"source": null
}
|
Let \(a,b,c>0\) such that \(a+b+c=\tfrac{3}{2}\). Define
\[
F=27abc+\frac{8}{ab+bc+ca}.
\]
Find \(\min F\) and determine when equality occurs.
|
[
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/Algebra/Polynomials/Factorization",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialFactorization",
"/Mathematics/Algebra/Polynomials/PolynomialIdentity",
"/Mathematics/Algebra/Polynomials/UnivariatePolynomial"
] |
Exploit symmetry to set two variables equal, reducing the problem to a one‑variable inequality.
|
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aops_99923
|
im still learning latex. so here is my solution. hope someone can convert it into latex.
by definition of torque and its characteristics,
t(torque)= sigma(F*r)=I(moment of inertia)*alpha(angular acceleration)
I = sigma(m*r^2) = 0.68m
Sigma(F*r) = 0.6 mg (since the two forces are opposite in direction, one clockwise and one counterclockwise)
thus,
alpha*0.68m = 0.6mg
alpha = 8.647 rad/sec^2
for linear acceleration, multiply alpha by r
a_1 = 8.647*0.2 = 1.729 m/sec^2
a_2 = 8.647*0.8 = 6.918 m/sec^2
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [
{
"name": "TwoBlockRodProblem.GIF",
"url": "https://cdn.artofproblemsolving.com/attachments/e/3/4263807736b2637da89a421de91af43d21a55b.gif"
}
],
"content_bbcode": "The attached figure shows two blocks, each of mass $m$, suspended From the ends of a rigid massless rod of length $L_{1}+L_{2}$, with $L_{1}= 20$cm, and $L_{2}= 80$cm. The rod is held horizontally on the flcrum and then released. What are the magnitudes of the initial accelerations of the blocks?\r\n\r\nAny help would be appreciated. :)",
"content_html": "The attached figure shows two blocks, each of mass <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" >,</span> suspended From the ends of a rigid massless rod of length <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/7/4/9742801cc56912cfff2f89ce72fd6b218a7d16f9.png\" class=\"latex\" alt=\"$L_{1}+L_{2}$\" style=\"vertical-align: -2px\" width=\"60\" height=\"15\" >,</span> with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/d/8/3d8d6bf1fe29689e2a2f27f236ea1f7c2a573c56.png\" class=\"latex\" alt=\"$L_{1}= 20$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" >c</span>m, and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/5/8/55866cb7bb4c8a51407fadad2502a97a3a7b2d1a.png\" class=\"latex\" alt=\"$L_{2}= 80$\" style=\"vertical-align: -2px\" width=\"61\" height=\"15\" >c</span>m. The rod is held horizontally on the flcrum and then released. What are the magnitudes of the initial accelerations of the blocks?<br>\n<br>\nAny help would be appreciated. <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 564261,
"post_number": 1,
"post_time_unix": 1151812147,
"post_time_utc": "2006-07-02 03:49:07 UTC",
"thanks_received": 2,
"user_id": 8441,
"username": "ExterAY"
},
{
"attachments": [],
"content_bbcode": "Consider the force on the objects when release.",
"content_html": "Consider the force on the objects when release.",
"post_id": 565766,
"post_number": 2,
"post_time_unix": 1151983188,
"post_time_utc": "2006-07-04 03:19:48 UTC",
"thanks_received": 2,
"user_id": 6601,
"username": "shobber"
},
{
"attachments": [],
"content_bbcode": "Can you be more specific?\r\nCan anyone provide the entire solution? PLZ? :maybe:",
"content_html": "Can you be more specific?<br>\nCan anyone provide the entire solution? PLZ? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" />",
"post_id": 566368,
"post_number": 3,
"post_time_unix": 1152062037,
"post_time_utc": "2006-07-05 01:13:57 UTC",
"thanks_received": 2,
"user_id": 8441,
"username": "ExterAY"
},
{
"attachments": [],
"content_bbcode": "im still learning latex. so here is my solution. hope someone can convert it into latex.\r\nby definition of torque and its characteristics,\r\nt(torque)= sigma(F*r)=I(moment of inertia)*alpha(angular acceleration)\r\nI = sigma(m*r^2) = 0.68m\r\nSigma(F*r) = 0.6 mg (since the two forces are opposite in direction, one clockwise and one counterclockwise)\r\nthus,\r\nalpha*0.68m = 0.6mg\r\nalpha = 8.647 rad/sec^2\r\nfor linear acceleration, multiply alpha by r\r\na_1 = 8.647*0.2 = 1.729 m/sec^2\r\na_2 = 8.647*0.8 = 6.918 m/sec^2",
"content_html": "im still learning latex. so here is my solution. hope someone can convert it into latex.<br>\nby definition of torque and its characteristics,<br>\nt(torque)= sigma(F*r)=I(moment of inertia)*alpha(angular acceleration)<br>\nI = sigma(m*r^2) = 0.68m<br>\nSigma(F*r) = 0.6 mg (since the two forces are opposite in direction, one clockwise and one counterclockwise)<br>\nthus,<br>\nalpha*0.68m = 0.6mg<br>\nalpha = 8.647 rad/sec^2<br>\nfor linear acceleration, multiply alpha by r<br>\na_1 = 8.647*0.2 = 1.729 m/sec^2<br>\na_2 = 8.647*0.8 = 6.918 m/sec^2",
"post_id": 572623,
"post_number": 4,
"post_time_unix": 1152757552,
"post_time_utc": "2006-07-13 02:25:52 UTC",
"thanks_received": 2,
"user_id": 5155,
"username": "tongchen1226"
},
{
"attachments": [],
"content_bbcode": "Thanks Tong! I never expected the solution to be such a simple one. \r\n\r\nSolution in $LaTeX$:\r\n\r\nBy definition of torque and its characteristics, \r\n$\\tau$ (torque) $= \\Sigma Fr = I$ (Inertia) $\\cdot \\alpha$ (angular acceleration) \\[I = \\Sigma mr^{2}= 0.68m\\] Since the two forces are opposite in direction, one clockwise and one counterclockwise: \\[\\Sigma Fr = 0.6mg\\] \\[0.68\\alpha m = 0.6mg\\] \\[\\alpha = 8.647 rad/sec^{2}\\] For linear acceleration, multiply $\\alpha$ by $r$ \\[a_{1}= 8.647 \\cdot 0.2 = 1.729 m/sec^{2}\\] \\[a_{2}= 8.647 \\cdot 0.8 = 6.918 m/sec^{2}\\]",
"content_html": "Thanks Tong! I never expected the solution to be such a simple one.<br>\n<br>\nSolution in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/a/0/ba0b91b5dd1bb1d4e59e6943e2a422e44cd39474.png\" class=\"latex\" alt=\"$LaTeX$\" width=\"59\" height=\"12\" >:</span><br>\n<br>\nBy definition of torque and its characteristics,<br>\n<img src=\"//latex.artofproblemsolving.com/1/d/c/1dc1c0119a604b91be9142370dc3159b6a9bbcb9.png\" class=\"latex\" alt=\"$\\tau$\" width=\"9\" height=\"8\" > (torque) <img src=\"//latex.artofproblemsolving.com/d/b/d/dbde563c929e077db7fb026f1cd60c2307480155.png\" class=\"latex\" alt=\"$= \\Sigma Fr = I$\" width=\"89\" height=\"12\" > (Inertia) <img src=\"//latex.artofproblemsolving.com/7/a/b/7abef6a4b8d0149f590383147e60229705f221b2.png\" class=\"latex\" alt=\"$\\cdot \\alpha$\" width=\"16\" height=\"8\" > (angular acceleration) <img src=\"//latex.artofproblemsolving.com/7/6/0/760c527df8ecd8fdf440a8ba0fbd5938e5145625.png\" class=\"latexcenter\" alt=\"\\[I = \\Sigma mr^{2}= 0.68m\\]\" width=\"150\" height=\"15\" > Since the two forces are opposite in direction, one clockwise and one counterclockwise: <img src=\"//latex.artofproblemsolving.com/4/2/5/42507f0a62373d3cbdbdea4d2372ab61b5656855.png\" class=\"latexcenter\" alt=\"\\[\\Sigma Fr = 0.6mg\\]\" width=\"107\" height=\"16\" > <img src=\"//latex.artofproblemsolving.com/8/8/e/88eacea18b01731fe842d09bfcd2e168d4e2b178.png\" class=\"latexcenter\" alt=\"\\[0.68\\alpha m = 0.6mg\\]\" width=\"132\" height=\"16\" > <img src=\"//latex.artofproblemsolving.com/2/a/7/2a705ce3dc5ad810fd3e6557a92052858ff26b7f.png\" class=\"latexcenter\" alt=\"\\[\\alpha = 8.647 rad/sec^{2}\\]\" width=\"144\" height=\"19\" > For linear acceleration, multiply <img src=\"//latex.artofproblemsolving.com/1/0/f/10f32377ac67d94f764f12a15ea987e88c85d3e1.png\" class=\"latex\" alt=\"$\\alpha$\" width=\"11\" height=\"8\" > by <img src=\"//latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png\" class=\"latex\" alt=\"$r$\" width=\"8\" height=\"8\" > <img src=\"//latex.artofproblemsolving.com/4/c/3/4c393bff79b22411428ecd01f1c90e6e8ea1f4bc.png\" class=\"latexcenter\" alt=\"\\[a_{1}= 8.647 \\cdot 0.2 = 1.729 m/sec^{2}\\]\" width=\"239\" height=\"19\" > <img src=\"//latex.artofproblemsolving.com/7/4/7/747f8c82e1f978cbfd766abe2e62151814e05612.png\" class=\"latexcenter\" alt=\"\\[a_{2}= 8.647 \\cdot 0.8 = 6.918 m/sec^{2}\\]\" width=\"239\" height=\"19\" >",
"post_id": 572644,
"post_number": 5,
"post_time_unix": 1152759899,
"post_time_utc": "2006-07-13 03:04:59 UTC",
"thanks_received": 2,
"user_id": 8441,
"username": "ExterAY"
}
],
"source": null
}
|
The attached figure shows two blocks, each of mass \(m\), suspended from the ends of a rigid massless rod of length \(L_{1}+L_{2}\), with \(L_{1}=20\text{ cm}\) and \(L_{2}=80\text{ cm}\). The rod is held horizontally on a fulcrum and then released. What are the magnitudes of the initial accelerations of the blocks?
|
[
"/Mathematics/AppliedMathematics/DynamicalSystems",
"/Mathematics/AppliedMathematics/Engineering/MechanicalDevices",
"/Mathematics/CalculusandAnalysis/Calculus"
] |
Use τ = Iα about the fulcrum to find angular acceleration, then convert to linear acceleration via a = α·r for each mass.
|
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aops_99925
|
just think this way:
if the probability of head is 1 out of 50 (1/50) then obviously you would expect one to come up in 50 flips... so, it's 1/p
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "A coin is weighted so that it comes up heads with probability p and tails with probability 1-p, for some value p such that 0<p<1. I keep flipping the coin until head comes up. wat is the expected value of flips necessary?",
"content_html": "A coin is weighted so that it comes up heads with probability p and tails with probability 1-p, for some value p such that 0<p<1. I keep flipping the coin until head comes up. wat is the expected value of flips necessary?",
"post_id": 564276,
"post_number": 1,
"post_time_unix": 1151813384,
"post_time_utc": "2006-07-02 04:09:44 UTC",
"thanks_received": 1,
"user_id": 17107,
"username": "Rikimaru"
},
{
"attachments": [],
"content_bbcode": "[hide]\nLet $N$ be the expected value we seek. Then we get the equation:\n\n$N = (1-p)(1+N)+p$\n\nreason: Consider the first flip, if we get a tail, then we have to flip $N$ times more, plus the first flip. if we get a head, we're done.\n\nso solving we get $N = \\frac{1}{p}$.\n[/hide]\r\nWow...your first post. welcome to AoPS!!! :)",
"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/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png\" class=\"latex\" alt=\"$N$\" width=\"16\" height=\"12\" > be the expected value we seek. Then we get the equation:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/3/c/2/3c2d585b2f540db00b152d10ee0b793804358f31.png\" class=\"latex\" alt=\"$N = (1-p)(1+N)+p$\" style=\"vertical-align: -4px\" width=\"188\" height=\"18\" ><br>\n<br>\nreason: Consider the first flip, if we get a tail, then we have to flip <img src=\"//latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png\" class=\"latex\" alt=\"$N$\" width=\"16\" height=\"12\" > times more, plus the first flip. if we get a head, we're done.<br>\n<br>\nso solving we get <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/0/7/007628edfbf13c42c00bcaf8a0f83e77bd2d9fd6.png\" class=\"latex\" alt=\"$N = \\frac{1}{p}$\" style=\"vertical-align: -16px\" width=\"52\" height=\"40\" >.</span></div><br>\nWow...your first post. welcome to AoPS!!! <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 564282,
"post_number": 2,
"post_time_unix": 1151813785,
"post_time_utc": "2006-07-02 04:16:25 UTC",
"thanks_received": 1,
"user_id": 10705,
"username": "pkerichang"
},
{
"attachments": [],
"content_bbcode": "thanks for the reply =D\r\n\r\nwhere did you get the +p though??",
"content_html": "thanks for the reply =D<br>\n<br>\nwhere did you get the +p though??",
"post_id": 566162,
"post_number": 3,
"post_time_unix": 1152038367,
"post_time_utc": "2006-07-04 18:39:27 UTC",
"thanks_received": 2,
"user_id": 17107,
"username": "Rikimaru"
},
{
"attachments": [],
"content_bbcode": "Think of that $+p$ as being $+p\\cdot 1.$ With probability $p,$ $N=1.$\r\n\r\npkerichang has given the [b]conditional expectation[/b] argument for the expected value of a well-known probability distribution called the geometric distribution.\r\n\r\nThe key theorem of conditional expectation: If $X$ is a random variable, and $Y$ is another random variable that we choose to condition on, then \\[E(X)=E(E(X|Y))\\] The right hand side of what pkerichang has written is $E(E(N|Y))$ where $Y$ is the outcome of the first flip.\r\n\r\n---\r\n\r\nBy the way, the two of you met on Saturday - or at least you were in the same room.",
"content_html": "Think of that <img src=\"//latex.artofproblemsolving.com/e/3/e/e3ee64059c775e2490373e18e39d1838e2b56e44.png\" class=\"latex\" alt=\"$+p$\" style=\"vertical-align: -3px\" width=\"23\" height=\"14\" > as being <img src=\"//latex.artofproblemsolving.com/5/4/9/5490f904b73b898f9a4692b4f25a060f16c114c3.png\" class=\"latex\" alt=\"$+p\\cdot 1.$\" style=\"vertical-align: -3px\" width=\"49\" height=\"15\" > With probability <img src=\"//latex.artofproblemsolving.com/9/c/6/9c699725218958e044a4f271042b0b40b9ed713f.png\" class=\"latex\" alt=\"$p,$\" style=\"vertical-align: -3px\" width=\"14\" height=\"11\" > <img src=\"//latex.artofproblemsolving.com/d/b/2/db2a2dfcf2d0823cbf9b028503b67bc90cf673fb.png\" class=\"latex\" alt=\"$N=1.$\" style=\"vertical-align: 0px\" width=\"53\" height=\"13\" ><br>\n<br>\npkerichang has given the <b>conditional expectation</b> argument for the expected value of a well-known probability distribution called the geometric distribution.<br>\n<br>\nThe key theorem of conditional expectation: If <img src=\"//latex.artofproblemsolving.com/6/a/4/6a47ca0fe7cb276abc022af6ac88ddae1a9d6894.png\" class=\"latex\" alt=\"$X$\" width=\"15\" height=\"12\" > is a random variable, and <img src=\"//latex.artofproblemsolving.com/c/e/5/ce58e4af225c93d08606c26554caaa5ae32edeba.png\" class=\"latex\" alt=\"$Y$\" width=\"14\" height=\"12\" > is another random variable that we choose to condition on, then <img src=\"//latex.artofproblemsolving.com/2/0/4/204a723aceced9ef2d66c70f8cbd889558bcc346.png\" class=\"latexcenter\" alt=\"\\[E(X)=E(E(X|Y))\\]\" width=\"160\" height=\"18\" > The right hand side of what pkerichang has written is <img src=\"//latex.artofproblemsolving.com/5/9/7/597bf7e3f6c528b003d15b3baf587e7fe5f849dd.png\" class=\"latex\" alt=\"$E(E(N|Y))$\" style=\"vertical-align: -4px\" width=\"92\" height=\"18\" > where <img src=\"//latex.artofproblemsolving.com/c/e/5/ce58e4af225c93d08606c26554caaa5ae32edeba.png\" class=\"latex\" alt=\"$Y$\" width=\"14\" height=\"12\" > is the outcome of the first flip.<br>\n<br>\n---<br>\n<br>\nBy the way, the two of you met on Saturday - or at least you were in the same room.",
"post_id": 631730,
"post_number": 4,
"post_time_unix": 1158543184,
"post_time_utc": "2006-09-18 01:33:04 UTC",
"thanks_received": 1,
"user_id": 2948,
"username": "Kent Merryfield"
},
{
"attachments": [],
"content_bbcode": "just think this way:\r\nif the probability of head is 1 out of 50 (1/50) then obviously you would expect one to come up in 50 flips... so, it's 1/p",
"content_html": "just think this way:<br>\nif the probability of head is 1 out of 50 (1/50) then obviously you would expect one to come up in 50 flips... so, it's 1/p",
"post_id": 631826,
"post_number": 5,
"post_time_unix": 1158552435,
"post_time_utc": "2006-09-18 04:07:15 UTC",
"thanks_received": 2,
"user_id": 4162,
"username": "Tim_Lou"
}
],
"source": null
}
|
A coin is weighted so that it comes up heads with probability \(p\) and tails with probability \(1-p\), for some value \(p\) with \(0<p<1\). I keep flipping the coin until a head appears. What is the expected number of flips required?
|
[
"/Mathematics/ProbabilityandStatistics/Probability/CoinFlipping",
"/Mathematics/ProbabilityandStatistics/Probability/CoinTossing",
"/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace",
"/Mathematics/ProbabilityandStatistics/Probability/SampleSpace",
"/Mathematics/ProbabilityandStatistics/StatisticalDistributions/DiscreteDistributions/GeometricDistribution"
] |
Recognize the process as a geometric distribution whose expectation equals 1⁄p.
|
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aops_99929
|
[quote="Astronum"]The frequency of Bendan's movement is:
$f = \frac{150}{60}= 2.5 s^{-1}$
$f = \frac{times}{\Delta t}$
$2.5 = \frac{1000}{\Delta t}$
$\Delta t = 400s$[/quote]
I think thats a little too hard for classroom math.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Bendan can snap his fingers 150 times in a minute. At this rate, how many seconds will it take for him to snap his fingers 1000 times?",
"content_html": "Bendan can snap his fingers 150 times in a minute. At this rate, how many seconds will it take for him to snap his fingers 1000 times?",
"post_id": 564313,
"post_number": 1,
"post_time_unix": 1151820443,
"post_time_utc": "2006-07-02 06:07:23 UTC",
"thanks_received": 2,
"user_id": 17514,
"username": "Ignite168"
},
{
"attachments": [],
"content_bbcode": "The frequency of Bendan's movement is:\r\n\r\n$f = \\frac{150}{60}= 2.5 s^{-1}$\r\n\r\n$f = \\frac{times}{\\Delta t}$\r\n\r\n$2.5 = \\frac{1000}{\\Delta t}$\r\n\r\n$\\Delta t = 400s$",
"content_html": "The frequency of Bendan's movement is:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/a/3/da3ab2aedac3bf6e424496fed61bf75f24632e6d.png\" class=\"latex\" alt=\"$f = \\frac{150}{60}= 2.5 s^{-1}$\" style=\"vertical-align: -12px\" width=\"138\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/f/4/bf4f5f93998843f5b4edeecd582f39c8a41d8c6d.png\" class=\"latex\" alt=\"$f = \\frac{times}{\\Delta t}$\" style=\"vertical-align: -12px\" width=\"83\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/f/4/2f4375db3fa74b11c41e46a5997c665c642a3165.png\" class=\"latex\" alt=\"$2.5 = \\frac{1000}{\\Delta t}$\" style=\"vertical-align: -12px\" width=\"86\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/a/e/fae2b8af7670b34ace84d6f70002b72abfa27eac.png\" class=\"latex\" alt=\"$\\Delta t = 400s$\" style=\"vertical-align: 0px\" width=\"81\" height=\"13\" >",
"post_id": 564431,
"post_number": 2,
"post_time_unix": 1151847085,
"post_time_utc": "2006-07-02 13:31:25 UTC",
"thanks_received": 1,
"user_id": 19956,
"username": "Astronum"
},
{
"attachments": [],
"content_bbcode": "[hide] $\\frac{1000\\ snaps}{\\frac{150\\ snaps}{60\\ seconds}}=\\frac{60000}{150}\\ seconds= \\boxed{400\\ seconds}$ [/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/6/3/4/634c5e551b3b336d753838d31a7d1e7efd526f94.png\" class=\"latex\" alt=\"$\\frac{1000\\ snaps}{\\frac{150\\ snaps}{60\\ seconds}}=\\frac{60000}{150}\\ seconds= \\boxed{400\\ seconds}$\" style=\"vertical-align: -19px\" width=\"365\" height=\"44\" ></div>",
"post_id": 564825,
"post_number": 3,
"post_time_unix": 1151884732,
"post_time_utc": "2006-07-02 23:58:52 UTC",
"thanks_received": 2,
"user_id": 18001,
"username": "lotrgreengrapes7926"
},
{
"attachments": [],
"content_bbcode": "Sean Paul and E-40 say.\r\n$\\frac{1000}{150}\\cdot60=400$",
"content_html": "Sean Paul and E-40 say.<br>\n<img src=\"//latex.artofproblemsolving.com/4/4/7/44709cf5b788b8c97e1ed815435d6b0d4796560c.png\" class=\"latex\" alt=\"$\\frac{1000}{150}\\cdot60=400$\" style=\"vertical-align: -13px\" width=\"122\" height=\"38\" >",
"post_id": 564962,
"post_number": 4,
"post_time_unix": 1151900683,
"post_time_utc": "2006-07-03 04:24:43 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "Another way is: \r\n[hide] $150 times \\Longrightarrow 60 seconds$\n$1000 times \\Longrightarrow x=\\frac{1000*60}{150}=\\boxed{\\boxed{400 seconds}}$[/hide]\r\n\r\nEDIT: Sorry bpms, I didn't see your post",
"content_html": "Another way 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\"><img src=\"//latex.artofproblemsolving.com/7/2/b/72beb5c7248502dc813fccd396bf0f7e476a7d4f.png\" class=\"latex\" alt=\"$150 times \\Longrightarrow 60 seconds$\" style=\"vertical-align: 0px\" width=\"193\" height=\"13\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/6/7/56758e8f9d4b8d6a5d92f5994e2414ac7915758c.png\" class=\"latex\" alt=\"$1000 times \\Longrightarrow x=\\frac{1000*60}{150}=\\boxed{\\boxed{400 seconds}}$\" style=\"vertical-align: -13px\" width=\"369\" height=\"38\" ></div><br>\n<br>\nEDIT: Sorry bpms, I didn't see your post",
"post_id": 565905,
"post_number": 5,
"post_time_unix": 1152014189,
"post_time_utc": "2006-07-04 11:56:29 UTC",
"thanks_received": 2,
"user_id": 15534,
"username": "José"
},
{
"attachments": [],
"content_bbcode": "[hide=\"solution\"]He snaps 50 times every 20 seconds. He snaps 100 every 40 seconds. He snaps for 400 sec.[/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\">He snaps 50 times every 20 seconds. He snaps 100 every 40 seconds. He snaps for 400 sec.</div>",
"post_id": 574504,
"post_number": 6,
"post_time_unix": 1152926550,
"post_time_utc": "2006-07-15 01:22:30 UTC",
"thanks_received": 2,
"user_id": 15223,
"username": "1=2"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Astronum\"]The frequency of Bendan's movement is:\n\n$f = \\frac{150}{60}= 2.5 s^{-1}$\n\n$f = \\frac{times}{\\Delta t}$\n\n$2.5 = \\frac{1000}{\\Delta t}$\n\n$\\Delta t = 400s$[/quote]\r\nI think thats a little too hard for classroom math.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Astronum wrote:</div>\n<div class=\"bbcode_quote_body\">The frequency of Bendan's movement is:<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/d/a/3/da3ab2aedac3bf6e424496fed61bf75f24632e6d.png\" class=\"latex\" alt=\"$f = \\frac{150}{60}= 2.5 s^{-1}$\" style=\"vertical-align: -12px\" width=\"138\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/b/f/4/bf4f5f93998843f5b4edeecd582f39c8a41d8c6d.png\" class=\"latex\" alt=\"$f = \\frac{times}{\\Delta t}$\" style=\"vertical-align: -12px\" width=\"83\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/2/f/4/2f4375db3fa74b11c41e46a5997c665c642a3165.png\" class=\"latex\" alt=\"$2.5 = \\frac{1000}{\\Delta t}$\" style=\"vertical-align: -12px\" width=\"86\" height=\"37\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/f/a/e/fae2b8af7670b34ace84d6f70002b72abfa27eac.png\" class=\"latex\" alt=\"$\\Delta t = 400s$\" style=\"vertical-align: 0px\" width=\"81\" height=\"13\" ></div>\n</div>\nI think thats a little too hard for classroom math.",
"post_id": 574812,
"post_number": 7,
"post_time_unix": 1152970930,
"post_time_utc": "2006-07-15 13:42:10 UTC",
"thanks_received": 2,
"user_id": 8131,
"username": "math92"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Ignite168\"]Bendan can snap his fingers 150 times in a minute. At this rate, how many seconds will it take for him to snap his fingers 1000 times?[/quote]\r\n\r\n[hide]150/6=each second (really fast)\n1000/(150/6)\n6000/150\n400[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Ignite168 wrote:</div>\n<div class=\"bbcode_quote_body\">Bendan can snap his fingers 150 times in a minute. At this rate, how many seconds will it take for him to snap his fingers 1000 times?</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\">150/6=each second (really fast)<br>\n1000/(150/6)<br>\n6000/150<br>\n400</div>",
"post_id": 615269,
"post_number": 8,
"post_time_unix": 1156823487,
"post_time_utc": "2006-08-29 03:51:27 UTC",
"thanks_received": 2,
"user_id": 11899,
"username": "moogra"
}
],
"source": null
}
|
Bendan can snap his fingers 150 times in a minute. At this rate, how many seconds will it take for him to snap his fingers 1000 times?
|
[
"/Mathematics/Algebra/RateProblems",
"/Mathematics/AppliedMathematics"
] |
Convert the snap rate to snaps per second and use time = quantity ÷ rate.
|
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aops_999297
|
You are right, you should use the change of base property. If you rewrite both sides of the equation in terms of ln you get $ ln(a)/ln(b)\equal{}1/(ln(b)/ln(a))$ and since the right-side of the equation is an inverted way of expressing $ ln(a)/ln(b)$ this statement is true
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "We're supposed to prove that logb(a) = 1/loga(b), does anyone know how we go about proving this? I know something with change of base is involved, I just don't know where to start.",
"content_html": "We're supposed to prove that logb(a) = 1/loga(b), does anyone know how we go about proving this? I know something with change of base is involved, I just don't know where to start.",
"post_id": 4421359,
"post_number": 1,
"post_time_unix": 1226628405,
"post_time_utc": "2008-11-14 02:06:45 UTC",
"thanks_received": 2,
"user_id": 47720,
"username": "sziegler"
},
{
"attachments": [],
"content_bbcode": "im sure this proof is missing something, but this is what i have:\r\nlogb(a)\r\n=lna/lnb (by change of base)\r\n=1/[lnb/lna]\r\n=1/loga(b) (by change of base again)",
"content_html": "im sure this proof is missing something, but this is what i have:<br>\nlogb(a)<br>\n=lna/lnb (by change of base)<br>\n=1/[lnb/lna]<br>\n=1/loga(b) (by change of base again)",
"post_id": 4421360,
"post_number": 2,
"post_time_unix": 1226633104,
"post_time_utc": "2008-11-14 03:25:04 UTC",
"thanks_received": 2,
"user_id": 47244,
"username": "ajiang"
},
{
"attachments": [],
"content_bbcode": "You are right, you should use the change of base property. If you rewrite both sides of the equation in terms of ln you get $ ln(a)/ln(b)\\equal{}1/(ln(b)/ln(a))$ and since the right-side of the equation is an inverted way of expressing $ ln(a)/ln(b)$ this statement is true",
"content_html": "You are right, you should use the change of base property. If you rewrite both sides of the equation in terms of ln you get <img src=\"//latex.artofproblemsolving.com/1/c/5/1c51920a3053a19841c37059567ef17bb44a2193.png\" class=\"latex\" alt=\"$ ln(a)/ln(b)=1/(ln(b)/ln(a))$\" style=\"vertical-align: -4px\" width=\"231\" height=\"18\" > and since the right-side of the equation is an inverted way of expressing <img src=\"//latex.artofproblemsolving.com/4/1/4/41444401dc6728f94e7e252fb3d93802b34d14da.png\" class=\"latex\" alt=\"$ ln(a)/ln(b)$\" style=\"vertical-align: -4px\" width=\"87\" height=\"18\" > this statement is true",
"post_id": 4421361,
"post_number": 3,
"post_time_unix": 1226633116,
"post_time_utc": "2008-11-14 03:25:16 UTC",
"thanks_received": 2,
"user_id": 47652,
"username": "eramos"
}
],
"source": null
}
|
Prove that \(\log_b a = \dfrac{1}{\log_a b}\).
|
[
"/Mathematics/Algebra/AlgebraicProperties",
"/Mathematics/Algebra/GeneralAlgebra/Algebra"
] |
Use the change‑of‑base formula to rewrite both logarithms with a common base, showing they are reciprocals.
|
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aops_99930
|
Yeah, you were close.
[hide]If the diameter of the circle is $10$, then the circumference is $10\pi$. Now he will make $\frac{600}{10\pi}\approx19.099$ revolutions, so he will make $\left\lfloor19.099\right\rfloor=19$ complete revolutions.[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Feipang is very heavy. He is standing a 13 foot pole in the air but the place it was planted in was very unstable so it leans and rocks around in a perfect circle. (Chinese people should know why) However, the pole is very sturdy so it remains straight the whole time. If the farthest distance between two spots Feipang was in is 10 ft. What is the number of complete revolutions he will make before traveling 600ft?",
"content_html": "Feipang is very heavy. He is standing a 13 foot pole in the air but the place it was planted in was very unstable so it leans and rocks around in a perfect circle. (Chinese people should know why) However, the pole is very sturdy so it remains straight the whole time. If the farthest distance between two spots Feipang was in is 10 ft. What is the number of complete revolutions he will make before traveling 600ft?",
"post_id": 564314,
"post_number": 1,
"post_time_unix": 1151820652,
"post_time_utc": "2006-07-02 06:10:52 UTC",
"thanks_received": 2,
"user_id": 17514,
"username": "Ignite168"
},
{
"attachments": [],
"content_bbcode": "[hide]The diameter of the circle is 10, so the circumference is $10\\pi$. $\\frac{600}{10\\pi}$ simplifies to 19.0985931710274, which, to the next largest integer, is $20$.[/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\">The diameter of the circle is 10, so the circumference is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/b/7/3b7268bb19460f2612fbe8082a521a6cef925cae.png\" class=\"latex\" alt=\"$10\\pi$\" style=\"vertical-align: 0px\" width=\"28\" height=\"13\" >.</span> <img src=\"//latex.artofproblemsolving.com/d/8/3/d8379cf5e353b4c2e77c81231be2fc404becb53c.png\" class=\"latex\" alt=\"$\\frac{600}{10\\pi}$\" style=\"vertical-align: -13px\" width=\"31\" height=\"38\" > simplifies to 19.0985931710274, which, to the next largest integer, is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/8/3/f8366cd6196dd8a9da6d38a3e9eafb109e99d53e.png\" class=\"latex\" alt=\"$20$\" width=\"17\" height=\"12\" >.</span></div>",
"post_id": 564525,
"post_number": 2,
"post_time_unix": 1151856510,
"post_time_utc": "2006-07-02 16:08:30 UTC",
"thanks_received": 2,
"user_id": 17780,
"username": "redcomet46"
},
{
"attachments": [],
"content_bbcode": "[quote=\"mtms5467\"][hide]The diameter of the circle is 10, so the circumference is $10\\pi$. $\\frac{600}{10\\pi}$ simplifies to 19.0985931710274, which, to the next largest integer, is $20$.[/hide][/quote]\nClose, but incorrect.\n[quote][b]complete[/b] revolutions[/quote]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">mtms5467 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\">The diameter of the circle is 10, so the circumference is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/b/7/3b7268bb19460f2612fbe8082a521a6cef925cae.png\" class=\"latex\" alt=\"$10\\pi$\" style=\"vertical-align: 0px\" width=\"28\" height=\"13\" >.</span> <img src=\"//latex.artofproblemsolving.com/d/8/3/d8379cf5e353b4c2e77c81231be2fc404becb53c.png\" class=\"latex\" alt=\"$\\frac{600}{10\\pi}$\" style=\"vertical-align: -13px\" width=\"31\" height=\"38\" > simplifies to 19.0985931710274, which, to the next largest integer, is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/8/3/f8366cd6196dd8a9da6d38a3e9eafb109e99d53e.png\" class=\"latex\" alt=\"$20$\" width=\"17\" height=\"12\" >.</span></div></div>\n</div>\nClose, but incorrect.\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Quote:</div>\n<div class=\"bbcode_quote_body\"><b>complete</b> revolutions</div>\n</div>\n",
"post_id": 564744,
"post_number": 3,
"post_time_unix": 1151878503,
"post_time_utc": "2006-07-02 22:15:03 UTC",
"thanks_received": 1,
"user_id": 17514,
"username": "Ignite168"
},
{
"attachments": [],
"content_bbcode": "Yeah, you were close.\r\n\r\n[hide]If the diameter of the circle is $10$, then the circumference is $10\\pi$. Now he will make $\\frac{600}{10\\pi}\\approx19.099$ revolutions, so he will make $\\left\\lfloor19.099\\right\\rfloor=19$ complete revolutions.[/hide]",
"content_html": "Yeah, you were close.<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\">If the diameter of the circle is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/c/6/fc606f7f1e530731ab4f1cc364c01dc64a4455ee.png\" class=\"latex\" alt=\"$10$\" style=\"vertical-align: 0px\" width=\"17\" height=\"13\" >,</span> then the circumference is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/b/7/3b7268bb19460f2612fbe8082a521a6cef925cae.png\" class=\"latex\" alt=\"$10\\pi$\" style=\"vertical-align: 0px\" width=\"28\" height=\"13\" >.</span> Now he will make <img src=\"//latex.artofproblemsolving.com/9/1/e/91eeac3a8e23f9fe035c8e165b3df8d43c305402.png\" class=\"latex\" alt=\"$\\frac{600}{10\\pi}\\approx19.099$\" style=\"vertical-align: -13px\" width=\"107\" height=\"38\" > revolutions, so he will make <img src=\"//latex.artofproblemsolving.com/3/1/3/3130b2dcddf96515291bb9e8891f790d4f4afcf6.png\" class=\"latex\" alt=\"$\\left\\lfloor19.099\\right\\rfloor=19$\" style=\"vertical-align: -5px\" width=\"108\" height=\"19\" > complete revolutions.</div>",
"post_id": 564780,
"post_number": 4,
"post_time_unix": 1151881277,
"post_time_utc": "2006-07-02 23:01:17 UTC",
"thanks_received": 2,
"user_id": 8949,
"username": "b-flat"
},
{
"attachments": [],
"content_bbcode": "Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.",
"content_html": "Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.",
"post_id": 564802,
"post_number": 5,
"post_time_unix": 1151882558,
"post_time_utc": "2006-07-02 23:22:38 UTC",
"thanks_received": 2,
"user_id": 17514,
"username": "Ignite168"
},
{
"attachments": [],
"content_bbcode": "Yes. I shall move it to mathcounts.",
"content_html": "Yes. I shall move it to mathcounts.",
"post_id": 564808,
"post_number": 6,
"post_time_unix": 1151883371,
"post_time_utc": "2006-07-02 23:36:11 UTC",
"thanks_received": 2,
"user_id": 8949,
"username": "b-flat"
},
{
"attachments": [],
"content_bbcode": "[quote=\"Ignite168\"]Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.[/quote]\r\nWhere was this originally?",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Ignite168 wrote:</div>\n<div class=\"bbcode_quote_body\">Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.</div>\n</div>\nWhere was this originally?",
"post_id": 564834,
"post_number": 7,
"post_time_unix": 1151885107,
"post_time_utc": "2006-07-03 00:05:07 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "[quote=\"bpms\"][quote=\"Ignite168\"]Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.[/quote]\nWhere was this originally?[/quote]\r\n\r\nClassroom",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">bpms wrote:</div>\n<div class=\"bbcode_quote_body\"><div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">Ignite168 wrote:</div>\n<div class=\"bbcode_quote_body\">Umm.. why was this moved to Getting Started? It's formula for diameter of a circle and division. Calculators are permitted on this btw.</div>\n</div>\nWhere was this originally?</div>\n</div>\n<br>\nClassroom",
"post_id": 564856,
"post_number": 8,
"post_time_unix": 1151887023,
"post_time_utc": "2006-07-03 00:37:03 UTC",
"thanks_received": 2,
"user_id": 17514,
"username": "Ignite168"
},
{
"attachments": [],
"content_bbcode": "[hide]10pi is about 31.4, and 31.4 * 19 is about 600. so i guess it's 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\">10pi is about 31.4, and 31.4 * 19 is about 600. so i guess it's 19.</div>",
"post_id": 565204,
"post_number": 9,
"post_time_unix": 1151937552,
"post_time_utc": "2006-07-03 14:39:12 UTC",
"thanks_received": 2,
"user_id": 16591,
"username": "Captain Sugar"
},
{
"attachments": [],
"content_bbcode": "Once again, I must ask, what's the secret Chinese meaning?",
"content_html": "Once again, I must ask, what's the secret Chinese meaning?",
"post_id": 569612,
"post_number": 10,
"post_time_unix": 1152458769,
"post_time_utc": "2006-07-09 15:26:09 UTC",
"thanks_received": 2,
"user_id": 6133,
"username": "mysmartmouth"
},
{
"attachments": [],
"content_bbcode": "Doesn't fei pang mean fat and heavy, or something?",
"content_html": "Doesn't fei pang mean fat and heavy, or something?",
"post_id": 569630,
"post_number": 11,
"post_time_unix": 1152460191,
"post_time_utc": "2006-07-09 15:49:51 UTC",
"thanks_received": 1,
"user_id": 18001,
"username": "lotrgreengrapes7926"
},
{
"attachments": [],
"content_bbcode": "[quote=\"lotrgreengrapes7926\"]Doesn't fei pang mean fat and heavy, or something?[/quote]\r\n\r\nYeah, it means fat but they're two words that mean different kind of fat. It's hard to explain, maybe Ignite can do it himself.",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">lotrgreengrapes7926 wrote:</div>\n<div class=\"bbcode_quote_body\">Doesn't fei pang mean fat and heavy, or something?</div>\n</div>\n<br>\nYeah, it means fat but they're two words that mean different kind of fat. It's hard to explain, maybe Ignite can do it himself.",
"post_id": 569636,
"post_number": 12,
"post_time_unix": 1152460783,
"post_time_utc": "2006-07-09 15:59:43 UTC",
"thanks_received": 1,
"user_id": 18270,
"username": "SplashD"
}
],
"source": null
}
|
Feipang is very heavy. He is standing on a 13-foot pole in the air, but the place it was planted in is very unstable so it leans and rocks around in a perfect circle. However, the pole is very sturdy so it remains straight the whole time. If the farthest distance between two spots Feipang was in is 10 ft, what is the number of complete revolutions he will make before traveling 600 ft?
|
[
"/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circle",
"/Mathematics/Geometry/PlaneGeometry/Circles/Circumference",
"/Mathematics/Geometry/PlaneGeometry/Circles/Diameter"
] |
Use the farthest distance as the circle’s diameter to get the circumference, then divide the total travel distance by that circumference.
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aops_99944
|
[hide="Proof for 1?"] $n!+(n+a)!=(n+b)!$, where $0<a<b$. $n!=(n+b)!-(n+a)!=(n+a)!(\frac{(n+b)!}{(n+a)!}-1) \longrightarrow \frac{n!}{(n+a)!}=(\frac{(n+b)!}{(n+a)!}-1)$. Since $n!<(n+a)!$, $\frac{n!}{(n+a)!}<1$. Therefore, $\frac{(n+b)!}{(n+a)!}-1<1 \longrightarrow \frac{(n+b)!}{(n+a)!}<2$. This never happens if $0<n+a<n+b$, because even if $b=a+1$, $\frac{(n+a+1)!}{(n+a)!}<2 \longrightarrow (n+a+1)<2 \longrightarrow n+a<1$. And $n+a$ would have to be 0.
So the solutions are $(0,0,2)$, $(0,1,2)$, $(1,0,2)$, or $(1,1,2)$.[/hide]
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "[b]question 1[/b]find the solutions to $a!+b!=c!$\r\n[b]question 2[/b] difficulter $a!+b!+c!=d!$\r\nare there any generalization here?\r\n\r\n[size=75][color=red]b-flat: please use either bold font, or \\text{stuff} for text to make it stand out.[/color][/size]",
"content_html": "<b>question 1</b>find the solutions to <img src=\"//latex.artofproblemsolving.com/0/0/5/0053c77b51a10ef6641dcc6dcf1c564219fa279d.png\" class=\"latex\" alt=\"$a!+b!=c!$\" style=\"vertical-align: -1px\" width=\"85\" height=\"14\" ><br>\n<b>question 2</b> difficulter <img src=\"//latex.artofproblemsolving.com/d/0/6/d0654831ecb9d5839098ebf8e38d3c21b42cb008.png\" class=\"latex\" alt=\"$a!+b!+c!=d!$\" style=\"vertical-align: -1px\" width=\"122\" height=\"14\" ><br>\nare there any generalization here?<br>\n<br>\n<span class=\"bbfont-three-q\"><span style=\"color:red\">b-flat: please use either bold font, or \\text{stuff} for text to make it stand out.</span></span>",
"post_id": 564353,
"post_number": 1,
"post_time_unix": 1151832063,
"post_time_utc": "2006-07-02 09:21:03 UTC",
"thanks_received": 2,
"user_id": 19927,
"username": "srulikbd"
},
{
"attachments": [],
"content_bbcode": "[quote=\"srulikbd\"]$question 1$find the solutions to $a!+b!=c!$\n$question 2$ difficulter $a!+b!+c!=d!$\nare there any generalization here?[/quote]\r\n[hide=\"ones I found\"]$1!+1!=2!$\n$2!+2!+2!=3!$\n[/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">srulikbd wrote:</div>\n<div class=\"bbcode_quote_body\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/8/6/086bc3f677b4ebf5ad1e7620f1949badd4993e4a.png\" class=\"latex\" alt=\"$question 1$\" style=\"vertical-align: -3px\" width=\"77\" height=\"16\" >f</span>ind the solutions to <img src=\"//latex.artofproblemsolving.com/0/0/5/0053c77b51a10ef6641dcc6dcf1c564219fa279d.png\" class=\"latex\" alt=\"$a!+b!=c!$\" style=\"vertical-align: -1px\" width=\"85\" height=\"14\" ><br>\n<img src=\"//latex.artofproblemsolving.com/8/a/a/8aaf989e246604eb6d48d4be623288e2e94fac52.png\" class=\"latex\" alt=\"$question 2$\" style=\"vertical-align: -3px\" width=\"77\" height=\"16\" > difficulter <img src=\"//latex.artofproblemsolving.com/d/0/6/d0654831ecb9d5839098ebf8e38d3c21b42cb008.png\" class=\"latex\" alt=\"$a!+b!+c!=d!$\" style=\"vertical-align: -1px\" width=\"122\" height=\"14\" ><br>\nare there any generalization here?</div>\n</div>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">ones I found</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/a/4/8/a48e86282a89cd7d5c4ab3b7ab381b5042dc3094.png\" class=\"latex\" alt=\"$1!+1!=2!$\" style=\"vertical-align: -1px\" width=\"87\" height=\"14\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/2/3/c236aff5a859728d6104a0b7e2653cf1faa82f32.png\" class=\"latex\" alt=\"$2!+2!+2!=3!$\" style=\"vertical-align: -1px\" width=\"123\" height=\"14\" ></div>",
"post_id": 564471,
"post_number": 2,
"post_time_unix": 1151850798,
"post_time_utc": "2006-07-02 14:33:18 UTC",
"thanks_received": 2,
"user_id": 8960,
"username": "bpms"
},
{
"attachments": [],
"content_bbcode": "[hide=\"Proof for 1?\"] $n!+(n+a)!=(n+b)!$, where $0<a<b$. $n!=(n+b)!-(n+a)!=(n+a)!(\\frac{(n+b)!}{(n+a)!}-1) \\longrightarrow \\frac{n!}{(n+a)!}=(\\frac{(n+b)!}{(n+a)!}-1)$. Since $n!<(n+a)!$, $\\frac{n!}{(n+a)!}<1$. Therefore, $\\frac{(n+b)!}{(n+a)!}-1<1 \\longrightarrow \\frac{(n+b)!}{(n+a)!}<2$. This never happens if $0<n+a<n+b$, because even if $b=a+1$, $\\frac{(n+a+1)!}{(n+a)!}<2 \\longrightarrow (n+a+1)<2 \\longrightarrow n+a<1$. And $n+a$ would have to be 0.\n\nSo the solutions are $(0,0,2)$, $(0,1,2)$, $(1,0,2)$, or $(1,1,2)$.[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Proof for 1?</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/a/c/dacf7a0e255b7dc5ac8bd534c7b7269030a295fb.png\" class=\"latex\" alt=\"$n!+(n+a)!=(n+b)!$\" style=\"vertical-align: -4px\" width=\"183\" height=\"18\" >,</span> where <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/a/a/eaa9031768c2fb7d820d8b5f22c6cd2cc4c77d90.png\" class=\"latex\" alt=\"$0<a<b$\" style=\"vertical-align: 0px\" width=\"75\" height=\"13\" >.</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/f/c/5fc96653cc390f4a3142ee0459a37a50b5e8b55d.png\" class=\"latex\" alt=\"$n!=(n+b)!-(n+a)!=(n+a)!(\\frac{(n+b)!}{(n+a)!}-1) \\longrightarrow \\frac{n!}{(n+a)!}=(\\frac{(n+b)!}{(n+a)!}-1)$\" style=\"vertical-align: -17px\" width=\"621\" height=\"43\" >.</span> Since <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/c/5/9c5fe619a97baf91c44d9ec4cd2d18f255190e12.png\" class=\"latex\" alt=\"$n!<(n+a)!$\" style=\"vertical-align: -4px\" width=\"100\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/0/6/f0663876d1a4c2fc31b028a833aad88932f5bc75.png\" class=\"latex\" alt=\"$\\frac{n!}{(n+a)!}<1$\" style=\"vertical-align: -17px\" width=\"98\" height=\"42\" >.</span> Therefore, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/b/9/6b926b71154a36134dbddf361018cdeb0019e5c4.png\" class=\"latex\" alt=\"$\\frac{(n+b)!}{(n+a)!}-1<1 \\longrightarrow \\frac{(n+b)!}{(n+a)!}<2$\" style=\"vertical-align: -17px\" width=\"268\" height=\"43\" >.</span> This never happens if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/7/c/d7c39968d63aa5f30f4e80fa2b53123196f6e495.png\" class=\"latex\" alt=\"$0<n+a<n+b$\" style=\"vertical-align: -1px\" width=\"141\" height=\"14\" >,</span> because even if <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/5/a/55ad8a88d8a48fdac91df9369c93c61f204cdbc4.png\" class=\"latex\" alt=\"$b=a+1$\" style=\"vertical-align: -1px\" width=\"72\" height=\"14\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/2/a/b2a38190a74afa030db494c5dd6a673babb64f85.png\" class=\"latex\" alt=\"$\\frac{(n+a+1)!}{(n+a)!}<2 \\longrightarrow (n+a+1)<2 \\longrightarrow n+a<1$\" style=\"vertical-align: -17px\" width=\"406\" height=\"43\" >.</span> And <img src=\"//latex.artofproblemsolving.com/5/a/a/5aa16b166227643381b65714d6f31d82d432bfb4.png\" class=\"latex\" alt=\"$n+a$\" style=\"vertical-align: -1px\" width=\"42\" height=\"12\" > would have to be 0.<br>\n<br>\nSo the solutions are <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/0/b/30b960b5af7479bfc23ff99af850ebe4b58c16f3.png\" class=\"latex\" alt=\"$(0,0,2)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/d/8/5/d8541e5d01bc5652a301ead355b2f68b77413495.png\" class=\"latex\" alt=\"$(0,1,2)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/1/3/f133c41581ac67b80531375a488d709e9adeaa93.png\" class=\"latex\" alt=\"$(1,0,2)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" >,</span> or <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/3/1/331794d7011f976b755400372594feded29a75aa.png\" class=\"latex\" alt=\"$(1,1,2)$\" style=\"vertical-align: -4px\" width=\"56\" height=\"18\" >.</span></div>",
"post_id": 564575,
"post_number": 3,
"post_time_unix": 1151860885,
"post_time_utc": "2006-07-02 17:21:25 UTC",
"thanks_received": 2,
"user_id": 18001,
"username": "lotrgreengrapes7926"
},
{
"attachments": [],
"content_bbcode": "[quote=\"srulikbd\"][b]question 1[/b]find the solutions to $a!+b!=c!$\n[b]question 2[/b] difficulter $a!+b!+c!=d!$\nare there any generalization here?\n\n[/quote]\r\n\r\n[hide=\"answers\"]$\\text{for question 1: }$\n1!+1!=2!\n\n$\\text{for question 2: }$\n2!+2!+2!=3!\n[hide]\n[/hide][/hide]",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">srulikbd wrote:</div>\n<div class=\"bbcode_quote_body\"><b>question 1</b>find the solutions to <img src=\"//latex.artofproblemsolving.com/0/0/5/0053c77b51a10ef6641dcc6dcf1c564219fa279d.png\" class=\"latex\" alt=\"$a!+b!=c!$\" style=\"vertical-align: -1px\" width=\"85\" height=\"14\" ><br>\n<b>question 2</b> difficulter <img src=\"//latex.artofproblemsolving.com/d/0/6/d0654831ecb9d5839098ebf8e38d3c21b42cb008.png\" class=\"latex\" alt=\"$a!+b!+c!=d!$\" style=\"vertical-align: -1px\" width=\"122\" height=\"14\" ><br>\nare there any generalization here?</div>\n</div>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">answers</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/6/8/e/68e543b7c406d96e5e276152afd23c78e3c5cb00.png\" class=\"latex\" alt=\"$\\text{for question 1: }$\" style=\"vertical-align: -3px\" width=\"103\" height=\"15\" ><br>\n1!+1!=2!<br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/c/0/7/c07eb435f733e46291d188439bee52775469eb0b.png\" class=\"latex\" alt=\"$\\text{for question 2: }$\" style=\"vertical-align: -3px\" width=\"103\" height=\"15\" ><br>\n2!+2!+2!=3!<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\"></div></div>",
"post_id": 564902,
"post_number": 4,
"post_time_unix": 1151891486,
"post_time_utc": "2006-07-03 01:51:26 UTC",
"thanks_received": 2,
"user_id": 9401,
"username": "#H34N1"
},
{
"attachments": [],
"content_bbcode": "Please avoid latexing texts, its pointless and requires graphics loading (bad for dial-ups like me). If you want to make it stand out, just use bold or italics or underline or change font color or font size but no latexing text. THANK YOU.",
"content_html": "Please avoid latexing texts, its pointless and requires graphics loading (bad for dial-ups like me). If you want to make it stand out, just use bold or italics or underline or change font color or font size but no latexing text. THANK YOU.",
"post_id": 564904,
"post_number": 5,
"post_time_unix": 1151891920,
"post_time_utc": "2006-07-03 01:58:40 UTC",
"thanks_received": 2,
"user_id": 10153,
"username": "10000th User"
}
],
"source": null
}
|
\textbf{Question 1.} Find the integer solutions to
\[
a! + b! = c!.
\]
\textbf{Question 2.} Find the integer solutions to
\[
a! + b! + c! = d!.
\]
Are there any generalizations?
|
[
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/NumberTheory/DiophantineEquations/Diophantine",
"/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/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"
] |
Use the rapid growth of factorials to bound the ratio of successive factorials below 2, forcing the numbers to be very small.
|
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aops_999485
|
k is the value by which one would have to translate the graph up or down. In the case of the problem on the practice test, another way of phrasing the question would be as such:
"What translation of this graph (up or down) will yield exactly three x-intercepts (zeros)?"
I hope this helps.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "I didn't understand the problem where it asked what does k have to equal for f(x)=w(x)+k to have three zeroes. How would I figure that out?",
"content_html": "I didn't understand the problem where it asked what does k have to equal for f(x)=w(x)+k to have three zeroes. How would I figure that out?",
"post_id": 4421629,
"post_number": 1,
"post_time_unix": 1232319116,
"post_time_utc": "2009-01-18 22:51:56 UTC",
"thanks_received": 1,
"user_id": 48183,
"username": "amybonaners"
},
{
"attachments": [],
"content_bbcode": "k is the value by which one would have to translate the graph up or down. In the case of the problem on the practice test, another way of phrasing the question would be as such:\r\n\r\n\"What translation of this graph (up or down) will yield exactly three x-intercepts (zeros)?\"\r\n\r\nI hope this helps.",
"content_html": "k is the value by which one would have to translate the graph up or down. In the case of the problem on the practice test, another way of phrasing the question would be as such:<br>\n<br>\n"What translation of this graph (up or down) will yield exactly three x-intercepts (zeros)?"<br>\n<br>\nI hope this helps.",
"post_id": 4421630,
"post_number": 2,
"post_time_unix": 1232323287,
"post_time_utc": "2009-01-19 00:01:27 UTC",
"thanks_received": 1,
"user_id": 46257,
"username": "pmorgan"
}
],
"source": null
}
|
I didn't understand the problem where it asked what does \(k\) have to equal for \(f(x)=w(x)+k\) to have three zeroes. How would I figure that out?
|
[
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/Polynomials/PolynomialEquation",
"/Mathematics/Algebra/Polynomials/PolynomialRoots"
] |
Choose k so that the vertical translation of w(x) creates exactly three intersections with the x‑axis.
|
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aops_99953
|
[hide="1"]Every prime $p > 3$ is expressible as $6m \pm 1$. So $p^{2}-1 = 36m^{2}+1 \pm 12m-1 = 12m(3m \pm 1)$. Clearly one btw $m$ and $3m \pm 1$ is even, and the result follows.[/hide]
[hide="2"]$n+1$, $n+11$, and $n+111$ belong to different classes $\bmod 3$, so one of them is divisible by $3$. Clearly it must be $n+1 = 3$ from which $n = 2$. $3, 13, 113$ are all primes.[/hide]
[hide="hint for 3"]$(x-y)(x+y) = 2 \cdot 17 \cdot 59$. Now, several cases.[/hide]
| null |
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"content_bbcode": "$Question 1$ show that for every prime $p>3,24\\mid p^{2}-1$\r\n$Question 2$ for which $n$, $(n+1,n+11,n+111)$ are all primes?\r\n$Question 3$ how many natural solutions have the equation $x^{2}-y^{2}=2006$?\r\n$Question 4$ prove: for every natural numbers there is a multiple that is made of only zeroes and ones (in any base), for example: 3-111,4-100, 5-10. are there any generalization here?\r\n$Question 5$ find all natural numbers that are squares, and all their digits are odd.\r\n$Question 6$ you need to complete a magic square 3X3 with natural numbers such that in every row, column and diagnosal, the sum of the numbers is 100. generalise for other sums! what about other magic squares?\r\n$Question 7$ the letters A-I represent the natural numbers 1-9, so that ABC is a represent of 3 digit number. prove that if ABC+DEF=GHI, $9\\mid GHI$.\r\n$Question 8$ find how many solutions does the equation $\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$ have (for constant $z$,and natural numbers).\r\n$Question 9$ find the numbers between 20 to 60 that aren't the sum of $n$ consecutive numbers",
"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\" > show that for every prime <img src=\"//latex.artofproblemsolving.com/0/6/f/06f9a03d918694efbecd1a3c6c8c03d79679f2e9.png\" class=\"latex\" alt=\"$p>3,24\\mid p^{2}-1$\" style=\"vertical-align: -4px\" width=\"131\" height=\"19\" ><br>\n<img src=\"//latex.artofproblemsolving.com/e/e/1/ee1a290540e1f27cdde089a4aa1e5c938a21906f.png\" class=\"latex\" alt=\"$Question 2$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > for which <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >,</span> <img src=\"//latex.artofproblemsolving.com/a/8/d/a8d81533a220e981e50dc7f0e6c55b0e601fc029.png\" class=\"latex\" alt=\"$(n+1,n+11,n+111)$\" style=\"vertical-align: -4px\" width=\"182\" height=\"18\" > are all primes?<br>\n<img src=\"//latex.artofproblemsolving.com/0/9/4/094a21fc144227a2dad6a319e0520d2252152658.png\" class=\"latex\" alt=\"$Question 3$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > how many natural solutions have the equation <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/8/6/b868f9f54c7772a426460e8649f467ca07b4b0e2.png\" class=\"latex\" alt=\"$x^{2}-y^{2}=2006$\" style=\"vertical-align: -3px\" width=\"117\" height=\"18\" >?</span><br>\n<img src=\"//latex.artofproblemsolving.com/b/1/6/b1695cbf9007ddcdbae76a39160405092c9163a6.png\" class=\"latex\" alt=\"$Question 4$\" style=\"vertical-align: -3px\" width=\"84\" height=\"16\" > prove: for every natural numbers there is a multiple that is made of only zeroes and ones (in any base), for example: 3-111,4-100, 5-10. are there any generalization here?<br>\n<img src=\"//latex.artofproblemsolving.com/7/a/0/7a0847ddef5e6a7728dfc25c3853205f3ef9d117.png\" class=\"latex\" alt=\"$Question 5$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > find all natural numbers that are squares, and all their digits are odd.<br>\n<img src=\"//latex.artofproblemsolving.com/1/3/c/13c2fcc5ab887472eedbf0488221924a64ca1e53.png\" class=\"latex\" alt=\"$Question 6$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > you need to complete a magic square 3X3 with natural numbers such that in every row, column and diagnosal, the sum of the numbers is 100. generalise for other sums! what about other magic squares?<br>\n<img src=\"//latex.artofproblemsolving.com/1/9/6/196ed5a611f032547e00cd2fa32645769e26152a.png\" class=\"latex\" alt=\"$Question 7$\" style=\"vertical-align: -3px\" width=\"84\" height=\"16\" > the letters A-I represent the natural numbers 1-9, so that ABC is a represent of 3 digit number. prove that if ABC+DEF=GHI, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/9/c/e9cc3bdaf2460428c58cddea96833268a387fe9d.png\" class=\"latex\" alt=\"$9\\mid GHI$\" style=\"vertical-align: -4px\" width=\"64\" height=\"18\" >.</span><br>\n<img src=\"//latex.artofproblemsolving.com/2/3/c/23c736180dc82b3925b59f24a988b6c9771c0ac7.png\" class=\"latex\" alt=\"$Question 8$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > find how many solutions does the equation <img src=\"//latex.artofproblemsolving.com/0/4/f/04f3233cb3316a43a4f56d64e6c28bb8dc7bf5d1.png\" class=\"latex\" alt=\"$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$\" style=\"vertical-align: -16px\" width=\"86\" height=\"40\" > have (for constant <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png\" class=\"latex\" alt=\"$z$\" width=\"8\" height=\"8\" >,</span>and natural numbers).<br>\n<img src=\"//latex.artofproblemsolving.com/6/4/1/641b10b64ef7875b811d4935fc4c1b07e7174f5e.png\" class=\"latex\" alt=\"$Question 9$\" style=\"vertical-align: -3px\" width=\"83\" height=\"16\" > find the numbers between 20 to 60 that aren't the sum of <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > consecutive numbers",
"post_id": 564382,
"post_number": 1,
"post_time_unix": 1151839333,
"post_time_utc": "2006-07-02 11:22:13 UTC",
"thanks_received": 1,
"user_id": 19927,
"username": "srulikbd"
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{
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"content_bbcode": "[hide=\"1\"]Every prime $p > 3$ is expressible as $6m \\pm 1$. So $p^{2}-1 = 36m^{2}+1 \\pm 12m-1 = 12m(3m \\pm 1)$. Clearly one btw $m$ and $3m \\pm 1$ is even, and the result follows.[/hide]\n\n[hide=\"2\"]$n+1$, $n+11$, and $n+111$ belong to different classes $\\bmod 3$, so one of them is divisible by $3$. Clearly it must be $n+1 = 3$ from which $n = 2$. $3, 13, 113$ are all primes.[/hide]\n\n[hide=\"hint for 3\"]$(x-y)(x+y) = 2 \\cdot 17 \\cdot 59$. Now, several cases.[/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\">Every prime <img src=\"//latex.artofproblemsolving.com/8/4/3/8437e1b1294378b7f0a667ffc66215648662ec71.png\" class=\"latex\" alt=\"$p > 3$\" style=\"vertical-align: -3px\" width=\"43\" height=\"16\" > is expressible as <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/4/4/4444310acb93d9b421560d5a8dba81641eedab19.png\" class=\"latex\" alt=\"$6m \\pm 1$\" style=\"vertical-align: 0px\" width=\"55\" height=\"13\" >.</span> So <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/7/8/0780c92037ae6590ec861d437d606f55bd957a98.png\" class=\"latex\" alt=\"$p^{2}-1 = 36m^{2}+1 \\pm 12m-1 = 12m(3m \\pm 1)$\" style=\"vertical-align: -4px\" width=\"361\" height=\"19\" >.</span> Clearly one btw <img src=\"//latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png\" class=\"latex\" alt=\"$m$\" width=\"15\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/c/7/0/c7090cc83dbf00c475d7940f7ff2cd5699d35228.png\" class=\"latex\" alt=\"$3m \\pm 1$\" style=\"vertical-align: 0px\" width=\"55\" height=\"13\" > is even, and the result follows.</div><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">2</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/e/3/0e3efd9b14723a92c2ae891fe27780d5f8e2b215.png\" class=\"latex\" alt=\"$n+1$\" style=\"vertical-align: -1px\" width=\"41\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/b/d/7bd2246f2aaaed04376c00063cfa9b26ad9c47f0.png\" class=\"latex\" alt=\"$n+11$\" style=\"vertical-align: -1px\" width=\"50\" height=\"13\" >,</span> and <img src=\"//latex.artofproblemsolving.com/8/e/6/8e6c5c7ab493ba5f307d905f4647d6b45bbfd8a6.png\" class=\"latex\" alt=\"$n+111$\" style=\"vertical-align: -1px\" width=\"59\" height=\"13\" > belong to different classes <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/6/1/e61a4b28c9c90ae7fb88b7fdc10bd9b5ca57cc16.png\" class=\"latex\" alt=\"$\\bmod 3$\" width=\"45\" height=\"12\" >,</span> so one of them is divisible by <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" >.</span> Clearly it must be <img src=\"//latex.artofproblemsolving.com/6/6/4/664d00b041a6462edb44b2b9389b16cf2c4a7cca.png\" class=\"latex\" alt=\"$n+1 = 3$\" style=\"vertical-align: -1px\" width=\"75\" height=\"14\" > from which <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/9/8/598a16d4b7dde3dec3a390109882d54bb0308ea5.png\" class=\"latex\" alt=\"$n = 2$\" width=\"43\" height=\"12\" >.</span> <img src=\"//latex.artofproblemsolving.com/3/5/1/35130891e7a8c6a72ce85774e0c7d7d8354ea378.png\" class=\"latex\" alt=\"$3, 13, 113$\" style=\"vertical-align: -3px\" width=\"70\" height=\"16\" > are all primes.</div><br>\n<br>\n<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">hint for 3</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/5/c/05cf60a152dc2dce16d20e3dfd291f7bf06f0bbe.png\" class=\"latex\" alt=\"$(x-y)(x+y) = 2 \\cdot 17 \\cdot 59$\" style=\"vertical-align: -4px\" width=\"208\" height=\"18\" >.</span> Now, several cases.</div>",
"post_id": 564406,
"post_number": 2,
"post_time_unix": 1151843229,
"post_time_utc": "2006-07-02 12:27:09 UTC",
"thanks_received": 2,
"user_id": 5839,
"username": "Andreas"
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{
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"content_bbcode": "[hide=\"5\"]Let the number be $(10^{n}a_{n}+10^{n-1}a_{n-1}+...+10a_{1}+a_{0})^{2}$ For $n=0$, $a_{0}=1, 3$ works. For all $n\\geq1$, when we expand it, the tenth digit of the number is also the tenth digit of $20a_{1}a_{0}+a_{0}^{2}$. It's easy to see that the tenth digit will always be even. The only numbers are 1 and 9.[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">5</a><div class=\"cmty-hide-content\" style=\"display:none\">Let the number be <img src=\"//latex.artofproblemsolving.com/c/5/b/c5bfc43bd551b112907cd4556b0ffc4f8e9fb356.png\" class=\"latex\" alt=\"$(10^{n}a_{n}+10^{n-1}a_{n-1}+...+10a_{1}+a_{0})^{2}$\" style=\"vertical-align: -4px\" width=\"302\" height=\"19\" > For <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/2/3/02349576cee613512cdf301a365f06c0760acab5.png\" class=\"latex\" alt=\"$n=0$\" width=\"43\" height=\"12\" >,</span> <img src=\"//latex.artofproblemsolving.com/7/1/2/7128489137466ab29d48e4e7eb7e2937c33482a9.png\" class=\"latex\" alt=\"$a_{0}=1, 3$\" style=\"vertical-align: -3px\" width=\"67\" height=\"16\" > works. For all <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/2/8/b28324eb1d52d541982598613d2d7ac4cb4e526e.png\" class=\"latex\" alt=\"$n\\geq1$\" style=\"vertical-align: -2px\" width=\"43\" height=\"14\" >,</span> when we expand it, the tenth digit of the number is also the tenth digit of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/9/3/f9342e10e96207c998f33b97c7f1c05c27207e86.png\" class=\"latex\" alt=\"$20a_{1}a_{0}+a_{0}^{2}$\" style=\"vertical-align: -4px\" width=\"90\" height=\"19\" >.</span> It's easy to see that the tenth digit will always be even. The only numbers are 1 and 9.</div>",
"post_id": 564502,
"post_number": 3,
"post_time_unix": 1151853737,
"post_time_utc": "2006-07-02 15:22:17 UTC",
"thanks_received": 2,
"user_id": 17438,
"username": "Jiang"
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{
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"content_bbcode": "[hide=\"For 3rd\"]\n$(x-y)(x+y)=2\\cdot17\\cdot59$\nSince this is not possible for natural numbers, hence no natural solutions\n[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">For 3rd</a><div class=\"cmty-hide-content\" style=\"display:none\"><img src=\"//latex.artofproblemsolving.com/0/8/f/08faa8ce7313b6d36925cae6458f193caf5742db.png\" class=\"latex\" alt=\"$(x-y)(x+y)=2\\cdot17\\cdot59$\" style=\"vertical-align: -4px\" width=\"208\" height=\"18\" ><br>\nSince this is not possible for natural numbers, hence no natural solutions</div>",
"post_id": 564521,
"post_number": 4,
"post_time_unix": 1151856200,
"post_time_utc": "2006-07-02 16:03:20 UTC",
"thanks_received": 2,
"user_id": 11111,
"username": "varun"
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{
"attachments": [],
"content_bbcode": "[hide=\"4\"]\nThere are two cases here:\n\ncase 1: if the number $n$ has only the factors that the base $b$ has, then a sufficiently large power of this base will be divisible by this number $n$. So $100000\\dots$ can be divisible by $n$, given enough zeros.\n\ncase 2: other numbers. Then take $1, b, b^{2}, \\dots$ mod by the number $n$, and we form a sequence. Since there's only $n$ possible values, it must repeat somewhere. Then let's say the repeating block has total sum of $S$. We know that $nS$ must be divisible by $n$, then we take bunch of $1$s on those repeating blocks $n$ times and put $0$ in the rest of them.\n\nIn case you don't get what I'm doing in case 2, remember how we get the divisibility of 3 in base ten: the number $10^{n}a_{n}+10^{n-1}a_{n-1}+\\dots$ mod by $3$ is $a_{n}+a_{n-1}+\\dots$. So if the latter is divisible by 3, then the original number will be divisible by 3. I just use this method and generalize to any number $n$ and any base $b$. \n[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">4</a><div class=\"cmty-hide-content\" style=\"display:none\">There are two cases here:<br>\n<br>\ncase 1: if the number <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > has only the factors that the base <img src=\"//latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" width=\"8\" height=\"12\" > has, then a sufficiently large power of this base will be divisible by this number <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >.</span> So <img src=\"//latex.artofproblemsolving.com/4/d/8/4d83d39f32a8669634b53864aa8301b021ceef62.png\" class=\"latex\" alt=\"$100000\\dots$\" style=\"vertical-align: 0px\" width=\"77\" height=\"13\" > can be divisible 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> given enough zeros.<br>\n<br>\ncase 2: other numbers. Then take <img src=\"//latex.artofproblemsolving.com/2/8/7/287018b8d8c208fad87b7ac73d6d0f460259eaa8.png\" class=\"latex\" alt=\"$1, b, b^{2}, \\dots$\" style=\"vertical-align: -3px\" width=\"76\" height=\"18\" > mod by the number <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" >,</span> and we form a sequence. Since there's only <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > possible values, it must repeat somewhere. Then let's say the repeating block has total sum 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> We know that <img src=\"//latex.artofproblemsolving.com/c/d/0/cd09579342b09a4c67f1d270964bb6b4c2dcc5b8.png\" class=\"latex\" alt=\"$nS$\" width=\"23\" height=\"12\" > must be divisible 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> then we take bunch of <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\" >s</span> on those repeating blocks <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > times and put <img src=\"//latex.artofproblemsolving.com/b/c/1/bc1f9d9bf8a1b606a4188b5ce9a2af1809e27a89.png\" class=\"latex\" alt=\"$0$\" width=\"8\" height=\"12\" > in the rest of them.<br>\n<br>\nIn case you don't get what I'm doing in case 2, remember how we get the divisibility of 3 in base ten: the number <img src=\"//latex.artofproblemsolving.com/c/4/e/c4ec1cd42fabcffa43fdda805020a544c1715085.png\" class=\"latex\" alt=\"$10^{n}a_{n}+10^{n-1}a_{n-1}+\\dots$\" style=\"vertical-align: -2px\" width=\"190\" height=\"17\" > mod by <img src=\"//latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png\" class=\"latex\" alt=\"$3$\" width=\"8\" height=\"12\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/5/7/e57f686e7b7fd2ecb9a033165bea3edb29798b95.png\" class=\"latex\" alt=\"$a_{n}+a_{n-1}+\\dots$\" style=\"vertical-align: -2px\" width=\"118\" height=\"13\" >.</span> So if the latter is divisible by 3, then the original number will be divisible by 3. I just use this method and generalize to any number <img src=\"//latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png\" class=\"latex\" alt=\"$n$\" width=\"10\" height=\"8\" > and any base <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png\" class=\"latex\" alt=\"$b$\" width=\"8\" height=\"12\" >.</span></div>",
"post_id": 564524,
"post_number": 5,
"post_time_unix": 1151856435,
"post_time_utc": "2006-07-02 16:07:15 UTC",
"thanks_received": 2,
"user_id": 10705,
"username": "pkerichang"
},
{
"attachments": [],
"content_bbcode": "[hide=\"2\"]Since the only consecutive primes are 2 and 3, $\\boxed{\\boxed{\\boxed{\\boxed{\\boxed{2}}}}}$ is the answer.[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">2</a><div class=\"cmty-hide-content\" style=\"display:none\">Since the only consecutive primes are 2 and 3, <img src=\"//latex.artofproblemsolving.com/6/e/1/6e1a189e617fffa0be5b28d393631721a5eb141b.png\" class=\"latex\" alt=\"$\\boxed{\\boxed{\\boxed{\\boxed{\\boxed{2}}}}}$\" style=\"vertical-align: -28px\" width=\"66\" height=\"68\" > is the answer.</div>",
"post_id": 564632,
"post_number": 6,
"post_time_unix": 1151866043,
"post_time_utc": "2006-07-02 18:47:23 UTC",
"thanks_received": 1,
"user_id": 15223,
"username": "1=2"
},
{
"attachments": [],
"content_bbcode": "[hide=\"8\"]I think it was posted before, but I can't find where. \nAnyway, the number of solutions equals the number of factors of $z^{2}$.[/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">8</a><div class=\"cmty-hide-content\" style=\"display:none\">I think it was posted before, but I can't find where.<br>\nAnyway, the number of solutions equals the number of factors of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/c/6cc465fd1aa32bb3b899cf7819576e9bf6d8822f.png\" class=\"latex\" alt=\"$z^{2}$\" width=\"15\" height=\"15\" >.</span></div>",
"post_id": 564655,
"post_number": 7,
"post_time_unix": 1151868969,
"post_time_utc": "2006-07-02 19:36:09 UTC",
"thanks_received": 2,
"user_id": 5839,
"username": "Andreas"
},
{
"attachments": [],
"content_bbcode": "[hide=\"Full Solution to 3\"] $(x+y)(x-y)=2\\cdot17\\cdot59$.\nThe 2 numbers $x+y$ and $x-y$ must both be either even or odd. Because $x+y+x-y=2x\\equiv0\\pmod2$. Clearly, $x+y$ cannot be $0\\pmod2$ while $x-y$ is $1\\pmod2$, and vice versa. $2\\cdot17\\cdot59$ can be expressed as the product of 2 numbers 3 ways: $34\\cdot59$, $118\\cdot17$, and $2\\cdot1003$. In none of these pairs do the numbers have the same parity. [/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">Full Solution to 3</a><div class=\"cmty-hide-content\" style=\"display:none\"><span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/2/5/125776a5cbd20428706581ae0aa651af7285e982.png\" class=\"latex\" alt=\"$(x+y)(x-y)=2\\cdot17\\cdot59$\" style=\"vertical-align: -4px\" width=\"208\" height=\"18\" >.</span><br>\nThe 2 numbers <img src=\"//latex.artofproblemsolving.com/d/3/d/d3da9f8c7fe0473fdd6f8597eff61d71fedda430.png\" class=\"latex\" alt=\"$x+y$\" style=\"vertical-align: -3px\" width=\"41\" height=\"14\" > and <img src=\"//latex.artofproblemsolving.com/6/4/3/643e1765342cf3807fd7d130ee679b71ecd4789b.png\" class=\"latex\" alt=\"$x-y$\" style=\"vertical-align: -3px\" width=\"41\" height=\"11\" > must both be either even or odd. Because <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/a/1/5/a1522e7683803b66f4428799508f102255dc23cc.png\" class=\"latex\" alt=\"$x+y+x-y=2x\\equiv0\\pmod2$\" style=\"vertical-align: -4px\" width=\"254\" height=\"18\" >.</span> Clearly, <img src=\"//latex.artofproblemsolving.com/d/3/d/d3da9f8c7fe0473fdd6f8597eff61d71fedda430.png\" class=\"latex\" alt=\"$x+y$\" style=\"vertical-align: -3px\" width=\"41\" height=\"14\" > cannot be <img src=\"//latex.artofproblemsolving.com/2/7/f/27f3dbebf22689737aaab3e0d6af8777a0381a8b.png\" class=\"latex\" alt=\"$0\\pmod2$\" style=\"vertical-align: -4px\" width=\"79\" height=\"18\" > while <img src=\"//latex.artofproblemsolving.com/6/4/3/643e1765342cf3807fd7d130ee679b71ecd4789b.png\" class=\"latex\" alt=\"$x-y$\" style=\"vertical-align: -3px\" width=\"41\" height=\"11\" > is <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/9/3/39313fc8e4b0f4de7e1c528c60ea0c2ea840051a.png\" class=\"latex\" alt=\"$1\\pmod2$\" style=\"vertical-align: -4px\" width=\"79\" height=\"18\" >,</span> and vice versa. <img src=\"//latex.artofproblemsolving.com/0/3/3/0333fcef4ca5fa9d340aa0d4f00f615b5d8a858d.png\" class=\"latex\" alt=\"$2\\cdot17\\cdot59$\" style=\"vertical-align: 0px\" width=\"71\" height=\"13\" > can be expressed as the product of 2 numbers 3 ways: <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/7/1/471a32be3bc9556ea34e3bf27cb59b88737260de.png\" class=\"latex\" alt=\"$34\\cdot59$\" style=\"vertical-align: 0px\" width=\"49\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/0/e/00e2e05259a437f6bae7c112448f2b649d4ea0c1.png\" class=\"latex\" alt=\"$118\\cdot17$\" style=\"vertical-align: 0px\" width=\"58\" height=\"13\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/5/c/f5c1a114810b73e3ddd6c374d0a21d45ae9db29e.png\" class=\"latex\" alt=\"$2\\cdot1003$\" style=\"vertical-align: 0px\" width=\"57\" height=\"13\" >.</span> In none of these pairs do the numbers have the same parity.</div>",
"post_id": 564776,
"post_number": 8,
"post_time_unix": 1151880981,
"post_time_utc": "2006-07-02 22:56:21 UTC",
"thanks_received": 2,
"user_id": 18001,
"username": "lotrgreengrapes7926"
},
{
"attachments": [],
"content_bbcode": "you have'nt finish everything! I think all the solutions until now are right :)",
"content_html": "you have'nt finish everything! I think all the solutions until now are right <img src=\"/assets/images/smilies/smile.gif\" width=\"20\" height=\"20\" alt=\":)\" title=\":)\" class=\"bbcode_smiley\" />",
"post_id": 566052,
"post_number": 9,
"post_time_unix": 1152026434,
"post_time_utc": "2006-07-04 15:20:34 UTC",
"thanks_received": 2,
"user_id": 19927,
"username": "srulikbd"
},
{
"attachments": [],
"content_bbcode": "[hide=\"#8\"]The given equation $\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$ is equivalent with $yz+xz=xy$, so $xy-xz-yz+z^{2}= z^{2}$, so $x(y-z)-z(y-z)=z^{2}$, i.e. $(x-z)(y-z)=z^{2}$. Clearly, $x>z$, otherwise we would have that $\\frac{1}{x}+\\frac{1}{y}= \\frac{1}{z}\\leq\\frac{1}{x}$, which would mean that $\\frac{1}{y}\\leq 0$, which is impossible. Same for $y>z$. Therefore, $x-z$ and $y-z$ must both be non-zero positive divisors of $z^{2}$. Therefore, there are $d(z^{2})$ solutions, where $d(n)$ denotes the number of divisors of $n$. [/hide]",
"content_html": "<a class=\"cmty-hide-heading\" onclick=\"$(this).next().toggle();$(this).toggleClass('cmty-hide-open');return false;\" href=\"#\">#8</a><div class=\"cmty-hide-content\" style=\"display:none\">The given equation <img src=\"//latex.artofproblemsolving.com/0/4/f/04f3233cb3316a43a4f56d64e6c28bb8dc7bf5d1.png\" class=\"latex\" alt=\"$\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$\" style=\"vertical-align: -16px\" width=\"86\" height=\"40\" > is equivalent with <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/c/b/bcb290e8779b770a2ad961a61602a0ad9f15d8f1.png\" class=\"latex\" alt=\"$yz+xz=xy$\" style=\"vertical-align: -3px\" width=\"104\" height=\"14\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/2/4/9249ab0aa2e2c6da8607423633f6f5dee394a9a4.png\" class=\"latex\" alt=\"$xy-xz-yz+z^{2}= z^{2}$\" style=\"vertical-align: -3px\" width=\"181\" height=\"18\" >,</span> so <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/e/a/1ea8049434bd21dfa102debf33f6a8e3d2f72c24.png\" class=\"latex\" alt=\"$x(y-z)-z(y-z)=z^{2}$\" style=\"vertical-align: -4px\" width=\"191\" height=\"19\" >,</span> i.e. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/0/6/2/062c710b9ee580a8f2220c87968b86d92fc83a0a.png\" class=\"latex\" alt=\"$(x-z)(y-z)=z^{2}$\" style=\"vertical-align: -4px\" width=\"150\" height=\"19\" >.</span> Clearly, <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/3/4/834e36e7eb6f0864e57bef50912bd599f6853636.png\" class=\"latex\" alt=\"$x>z$\" style=\"vertical-align: 0px\" width=\"43\" height=\"10\" >,</span> otherwise we would have that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/1/4/b/14b6fcf8238776a0d5feffc86e5234ed07d0dc00.png\" class=\"latex\" alt=\"$\\frac{1}{x}+\\frac{1}{y}= \\frac{1}{z}\\leq\\frac{1}{x}$\" style=\"vertical-align: -16px\" width=\"124\" height=\"40\" >,</span> which would mean that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/7/f/c7fe1dfb24144db9b0fae414d0dd57a61604e13c.png\" class=\"latex\" alt=\"$\\frac{1}{y}\\leq 0$\" style=\"vertical-align: -16px\" width=\"46\" height=\"40\" >,</span> which is impossible. Same for <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/b/7/9b762aaad3d31ac8c8168d399342418e9c9282d0.png\" class=\"latex\" alt=\"$y>z$\" style=\"vertical-align: -3px\" width=\"42\" height=\"13\" >.</span> Therefore, <img src=\"//latex.artofproblemsolving.com/3/3/d/33d7af37b081eb59a5a444341be7872f5977813d.png\" class=\"latex\" alt=\"$x-z$\" width=\"41\" height=\"8\" > and <img src=\"//latex.artofproblemsolving.com/3/9/a/39ab37330d446da2dc4086a36dd5a7b9f4ca57ab.png\" class=\"latex\" alt=\"$y-z$\" style=\"vertical-align: -3px\" width=\"40\" height=\"11\" > must both be non-zero positive divisors of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/6/c/c/6cc465fd1aa32bb3b899cf7819576e9bf6d8822f.png\" class=\"latex\" alt=\"$z^{2}$\" width=\"15\" height=\"15\" >.</span> Therefore, there are <img src=\"//latex.artofproblemsolving.com/4/8/9/48980618b5625d053e4ca6ebc79f996928dbfc93.png\" class=\"latex\" alt=\"$d(z^{2})$\" style=\"vertical-align: -4px\" width=\"38\" height=\"19\" > solutions, where <img src=\"//latex.artofproblemsolving.com/b/0/5/b05725a6508fe8df761c6e91d2735b697ebd87d9.png\" class=\"latex\" alt=\"$d(n)$\" style=\"vertical-align: -4px\" width=\"33\" height=\"18\" > denotes the number of divisors 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></div>",
"post_id": 576010,
"post_number": 10,
"post_time_unix": 1153098965,
"post_time_utc": "2006-07-17 01:16:05 UTC",
"thanks_received": 2,
"user_id": 13881,
"username": "Kurt Gödel"
}
],
"source": null
}
|
\begin{enumerate}
\item Show that for every prime \(p>3\), \(24\mid p^{2}-1\).
\item For which \(n\) are all of \(n+1,\;n+11,\;n+111\) prime?
\item How many natural solutions does the equation \(x^{2}-y^{2}=2006\) have?
\item Prove: for every natural number \(b\) and every positive integer \(n\) there exists a multiple of \(n\) whose base-\(b\) representation consists only of the digits \(0\) and \(1\).
\item Find all natural numbers that are perfect squares and all of whose digits are odd.
\item Complete a \(3\times3\) magic square with natural numbers such that every row, column, and diagonal sums to \(100\). Generalize for other sums and other sizes of magic squares.
\item The letters \(A\)–\(I\) represent the natural numbers \(1\)–\(9\) so that \(ABC\) denotes the three-digit number with digits \(A,B,C\). Prove that if \(ABC+DEF=GHI\), then \(9\mid GHI\).
\item For fixed positive integer \(z\), how many solutions in natural numbers does \(\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{z}\) have?
\item Find the numbers between \(20\) and \(60\) that are not representable as a sum of \(n\) consecutive integers (for some \(n\)).
\end{enumerate}
|
[
"/Mathematics/DiscreteMathematics/DivisionProblems",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath",
"/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory",
"/Mathematics/NumberTheory/PrimeNumbers/MiscellaneousPrimes",
"/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties"
] |
Apply simple modular classifications and elementary factorisations to reveal forced divisibility or parity constraints.
|
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aops_99964
|
Since $\mathbb{K}[X]$ is a euclidean domain,ca'nt one adapt the proof of euclid for existence of infinitely many primes in $\mathbb{Z}$ here and conclude that its not possible to have only finitely many primes? :maybe:
| null |
{
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{
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"content_bbcode": "If $K$ is a field, can the number of irreducible polynomials in $K[X]$ be finite ?",
"content_html": "If <img src=\"//latex.artofproblemsolving.com/d/f/b/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" width=\"16\" height=\"12\" > is a field, can the number of irreducible polynomials in <img src=\"//latex.artofproblemsolving.com/b/8/f/b8f4ddba89151fa31ddf44f6c22015162f2b58b0.png\" class=\"latex\" alt=\"$K[X]$\" style=\"vertical-align: -5px\" width=\"41\" height=\"18\" > be finite ?",
"post_id": 564470,
"post_number": 1,
"post_time_unix": 1151850725,
"post_time_utc": "2006-07-02 14:32:05 UTC",
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"user_id": 8980,
"username": "julien_santini"
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"content_bbcode": "Since $\\mathbb{K}[X]$ is a euclidean domain,ca'nt one adapt the proof of euclid for existence of infinitely many primes in $\\mathbb{Z}$ here and conclude that its not possible to have only finitely many primes? :maybe:",
"content_html": "Since <img src=\"//latex.artofproblemsolving.com/9/3/f/93f06163b7b490e0be9fd7e85cd4763cd0fdc67a.png\" class=\"latex\" alt=\"$\\mathbb{K}[X]$\" style=\"vertical-align: -5px\" width=\"38\" height=\"18\" > is a euclidean domain,ca'nt one adapt the proof of euclid for existence of infinitely many primes in <img src=\"//latex.artofproblemsolving.com/d/4/0/d4023589a374e5b4a1bc6063d887aec205e0d3e0.png\" class=\"latex\" alt=\"$\\mathbb{Z}$\" width=\"11\" height=\"12\" > here and conclude that its not possible to have only finitely many primes? <img src=\"/assets/images/smilies/unsure.gif\" width=\"20\" height=\"20\" alt=\":maybe:\" title=\":maybe:\" class=\"bbcode_smiley\" />",
"post_id": 564494,
"post_number": 2,
"post_time_unix": 1151852855,
"post_time_utc": "2006-07-02 15:07:35 UTC",
"thanks_received": 4,
"user_id": 20747,
"username": "Enigma1984"
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"content_bbcode": "Yes, this works.",
"content_html": "Yes, this works.",
"post_id": 564560,
"post_number": 3,
"post_time_unix": 1151859711,
"post_time_utc": "2006-07-02 17:01:51 UTC",
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"user_id": 18458,
"username": "-oo-"
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"content_bbcode": "Yes, but what is happening if our field is supposed to be non commutative ? In this case it seems that we cannot keep the Euclid-like proof, since the $P_{i}$ in $1+\\prod uct_{i=1}^{n}{P_{i}}$ cannot commute.\r\n\r\nIt seems like it can be handled this way.\r\n\r\nAssume the result is false.\r\n\r\nFirst, $K$ must be assumed finite since every $X+\\alpha$ ($\\alpha \\in K$) is irreducible.\r\n\r\nNow, since there are only finitely many irreducible polynomials in $K[X]$, one of them must have a maximal degree $d$. Then consider the polynomials of $K[X]$ with degree $d+1$ and leading coefficient $1$. Since every such polynomial can be factored into a finite product of irreducible polynomials (the factorization is not necessarily unique, but it doesn't matter), we could count the maximal number of such distinct factorizations and show that there are less than $\\mid K\\mid ^{d}$, which would solve the question.\r\n\r\nIt seems like I don't know how to get a decent upper bound for that (if we work with simple roots polynomials only, this is easy; otherwise, it seems like we have to give bound on certain partitions). I'm not even sure the result will be true eventually, by the way.",
"content_html": "Yes, but what is happening if our field is supposed to be non commutative ? In this case it seems that we cannot keep the Euclid-like proof, since the <img src=\"//latex.artofproblemsolving.com/a/2/b/a2b1f2c010f1a4e57ee7d93b6f982d9b3876f767.png\" class=\"latex\" alt=\"$P_{i}$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > in <img src=\"//latex.artofproblemsolving.com/5/2/4/524d0eb467fc7af8cf707f25865bf8b31e8eedd0.png\" class=\"latex\" alt=\"$1+\\prod uct_{i=1}^{n}{P_{i}}$\" style=\"vertical-align: -8px\" width=\"120\" height=\"25\" > cannot commute.<br>\n<br>\nIt seems like it can be handled this way.<br>\n<br>\nAssume the result is false.<br>\n<br>\nFirst, <img src=\"//latex.artofproblemsolving.com/d/f/b/dfb064112b6c94470339f6571f69d07afc1c024c.png\" class=\"latex\" alt=\"$K$\" width=\"16\" height=\"12\" > must be assumed finite since every <img src=\"//latex.artofproblemsolving.com/5/d/5/5d5b30e2352ffc713c3a9d43e08c71efe8ba8d90.png\" class=\"latex\" alt=\"$X+\\alpha$\" style=\"vertical-align: -1px\" width=\"49\" height=\"14\" > <span style=\"white-space:nowrap;\">(<img src=\"//latex.artofproblemsolving.com/5/f/6/5f68b84bcab052aebb353f5bb280c52ba63ba54a.png\" class=\"latex\" alt=\"$\\alpha \\in K$\" style=\"vertical-align: -1px\" width=\"50\" height=\"13\" >)</span> is irreducible.<br>\n<br>\nNow, since there are only finitely many irreducible polynomials in <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/8/f/b8f4ddba89151fa31ddf44f6c22015162f2b58b0.png\" class=\"latex\" alt=\"$K[X]$\" style=\"vertical-align: -5px\" width=\"41\" height=\"18\" >,</span> one of them must have a maximal degree <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png\" class=\"latex\" alt=\"$d$\" width=\"9\" height=\"12\" >.</span> Then consider the polynomials of <img src=\"//latex.artofproblemsolving.com/b/8/f/b8f4ddba89151fa31ddf44f6c22015162f2b58b0.png\" class=\"latex\" alt=\"$K[X]$\" style=\"vertical-align: -5px\" width=\"41\" height=\"18\" > with degree <img src=\"//latex.artofproblemsolving.com/0/d/1/0d1a966b7f9f2cea85dc923e34961cc1ccd08cee.png\" class=\"latex\" alt=\"$d+1$\" style=\"vertical-align: -1px\" width=\"39\" height=\"14\" > and leading coefficient <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> Since every such polynomial can be factored into a finite product of irreducible polynomials (the factorization is not necessarily unique, but it doesn't matter), we could count the maximal number of such distinct factorizations and show that there are less than <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/c/d/a/cdac495f7f78939e052610bae6b1a7edaf511199.png\" class=\"latex\" alt=\"$\\mid K\\mid ^{d}$\" style=\"vertical-align: -4px\" width=\"44\" height=\"20\" >,</span> which would solve the question.<br>\n<br>\nIt seems like I don't know how to get a decent upper bound for that (if we work with simple roots polynomials only, this is easy; otherwise, it seems like we have to give bound on certain partitions). I'm not even sure the result will be true eventually, by the way.",
"post_id": 564891,
"post_number": 4,
"post_time_unix": 1151890010,
"post_time_utc": "2006-07-03 01:26:50 UTC",
"thanks_received": 2,
"user_id": 8980,
"username": "julien_santini"
},
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"attachments": [],
"content_bbcode": "(Use \\prod, not \\product)\r\n\r\nIf it's not commutative, we don't call it a field. We would call the object a skew field or a division ring.\r\n\r\nAlso, since every finite division ring is actually a field, we don't have a problem. If the division ring has infinitely many elements, we can exhibit infinitely many irreducibles of the form $X-a$.",
"content_html": "(Use \\prod, not \\product)<br>\n<br>\nIf it's not commutative, we don't call it a field. We would call the object a skew field or a division ring.<br>\n<br>\nAlso, since every finite division ring is actually a field, we don't have a problem. If the division ring has infinitely many elements, we can exhibit infinitely many irreducibles of the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/2/b/82b4e9734f3540cc6b1e375973bb68abef0d962b.png\" class=\"latex\" alt=\"$X-a$\" width=\"48\" height=\"12\" >.</span>",
"post_id": 564910,
"post_number": 5,
"post_time_unix": 1151892699,
"post_time_utc": "2006-07-03 02:11:39 UTC",
"thanks_received": 2,
"user_id": 2975,
"username": "jmerry"
},
{
"attachments": [],
"content_bbcode": "Yep, the worst is that I thought about it yesterday ... :D thanks",
"content_html": "Yep, the worst is that I thought about it yesterday ... <img src=\"/assets/images/smilies/icon_mrgreen.gif\" width=\"19\" height=\"19\" alt=\":D\" title=\":D\" class=\"bbcode_smiley\" /> thanks",
"post_id": 564911,
"post_number": 6,
"post_time_unix": 1151892946,
"post_time_utc": "2006-07-03 02:15:46 UTC",
"thanks_received": 2,
"user_id": 8980,
"username": "julien_santini"
},
{
"attachments": [],
"content_bbcode": "[quote=\"jmerry\"](Use \\prod, not \\product)\n\nIf it's not commutative, we don't call it a field. We would call the object a skew field or a division ring.\n\nAlso, since every finite division ring is actually a field, we don't have a problem. If the division ring has infinitely many elements, we can exhibit infinitely many irreducibles of the form $X-a$.[/quote]\r\n\r\nOn second thought, conjuring Wedderburn theorem is far less elementary than trying to build up a more elementary solution (in the case of the division ring). Anybody can go further with my idea ?",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">jmerry wrote:</div>\n<div class=\"bbcode_quote_body\">(Use \\prod, not \\product)<br>\n<br>\nIf it's not commutative, we don't call it a field. We would call the object a skew field or a division ring.<br>\n<br>\nAlso, since every finite division ring is actually a field, we don't have a problem. If the division ring has infinitely many elements, we can exhibit infinitely many irreducibles of the form <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/2/b/82b4e9734f3540cc6b1e375973bb68abef0d962b.png\" class=\"latex\" alt=\"$X-a$\" width=\"48\" height=\"12\" >.</span></div>\n</div>\n<br>\nOn second thought, conjuring Wedderburn theorem is far less elementary than trying to build up a more elementary solution (in the case of the division ring). Anybody can go further with my idea ?",
"post_id": 565756,
"post_number": 7,
"post_time_unix": 1151981769,
"post_time_utc": "2006-07-04 02:56:09 UTC",
"thanks_received": 2,
"user_id": 8980,
"username": "julien_santini"
}
],
"source": null
}
|
If \(K\) is a field, can the number of irreducible polynomials in \(K[X]\) be finite?
|
[
"/Mathematics/Algebra/FieldTheory/Field",
"/Mathematics/Algebra/FieldTheory/IrreduciblePolynomial",
"/Mathematics/Algebra/Polynomials/IrreduciblePolynomial",
"/Mathematics/Algebra/Polynomials/Polynomial",
"/Mathematics/Algebra/RingTheory/EuclideanDomain",
"/Mathematics/Algebra/RingTheory/EuclideanRing",
"/Mathematics/Algebra/RingTheory/IrreducibleElement",
"/Mathematics/Algebra/RingTheory/PolynomialRing",
"/Mathematics/Algebra/RingTheory/UniqueFactorization",
"/Mathematics/Algebra/RingTheory/UniqueFactorizationDomain"
] |
Apply Euclid's argument: the product of all assumed irreducibles plus 1 yields a polynomial with a new irreducible factor.
|
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aops_999658
|
For the first problem, I'm pretty sure the answer would be 61. I just used guess and check to find the first number going down from 100 that was divisible by 2, 3, 4 ,5 and 6. Once I found that 60 was divisible by all these, I just added one to get 61.
I'm not sure where to start on the second problem - i'm also confused on if the gallery and the museum are the same thing.
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "UGH i wrote this yesterday but it didnt submit cause of the dollar signs\r\n\r\nThe teacher noticed there were fewer than 100 students on the playground. When she counted them by 2s, there was 1 stident left over. In fact when she counted them by 3s, 4s, 5s, 6s there was always one student left over. How many students were on the playground?\r\n\r\nA private art collector sold a canvas at a 10% loss during a financial depression. She bought it years later at an auction and sold it to a museum for 37,000 dollars more than the auction price.\r\n\r\nIf the gallery made a 25% profit and the collector made a 39.5% net profit on her original investment, for what price did she originally bought the picture.",
"content_html": "UGH i wrote this yesterday but it didnt submit cause of the dollar signs<br>\n<br>\nThe teacher noticed there were fewer than 100 students on the playground. When she counted them by 2s, there was 1 stident left over. In fact when she counted them by 3s, 4s, 5s, 6s there was always one student left over. How many students were on the playground?<br>\n<br>\nA private art collector sold a canvas at a 10% loss during a financial depression. She bought it years later at an auction and sold it to a museum for 37,000 dollars more than the auction price.<br>\n<br>\nIf the gallery made a 25% profit and the collector made a 39.5% net profit on her original investment, for what price did she originally bought the picture.",
"post_id": 4421877,
"post_number": 1,
"post_time_unix": 1236032010,
"post_time_utc": "2009-03-02 22:13:30 UTC",
"thanks_received": 2,
"user_id": 47135,
"username": "JMaguire"
},
{
"attachments": [],
"content_bbcode": "For the first problem, I'm pretty sure the answer would be 61. I just used guess and check to find the first number going down from 100 that was divisible by 2, 3, 4 ,5 and 6. Once I found that 60 was divisible by all these, I just added one to get 61.\r\nI'm not sure where to start on the second problem - i'm also confused on if the gallery and the museum are the same thing.",
"content_html": "For the first problem, I'm pretty sure the answer would be 61. I just used guess and check to find the first number going down from 100 that was divisible by 2, 3, 4 ,5 and 6. Once I found that 60 was divisible by all these, I just added one to get 61.<br>\nI'm not sure where to start on the second problem - i'm also confused on if the gallery and the museum are the same thing.",
"post_id": 4421878,
"post_number": 2,
"post_time_unix": 1236038256,
"post_time_utc": "2009-03-02 23:57:36 UTC",
"thanks_received": 2,
"user_id": 47644,
"username": "LindsayA"
}
],
"source": null
}
|
UGH i wrote this yesterday but it didnt submit cause of the dollar signs
The teacher noticed there were fewer than 100 students on the playground. When she counted them by 2s, there was 1 stident left over. In fact when she counted them by 3s, 4s, 5s, 6s there was always one student left over. How many students were on the playground?
A private art collector sold a canvas at a 10% loss during a financial depression. She bought it years later at an auction and sold it to a museum for 37,000 dollars more than the auction price.
If the gallery made a 25% profit and the collector made a 39.5% net profit on her original investment, for what price did she originally bought the picture.
|
[
"/Mathematics/Algebra/GeneralAlgebra/Algebra",
"/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem",
"/Mathematics/NumberTheory/Congruences/Congruence",
"/Mathematics/NumberTheory/Congruences/ModularArithmetic",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods",
"/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory",
"/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic",
"/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory"
] |
Find the least common multiple of 2,3,4,5,6 and add 1 to get the number of students.
|
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aops_999771
|
Yeah, the simplest way to look at it is as the vector addition of:
<c,d> + <0,0> and <0,0> + <c,d> which still works out as <c,d>, therefore, it's all equal.
This shows commutativity of vector addition, which is kind of neat.
<a,b> + <c,d> = <c,d> + <a,b>
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "It asked you to prove v+0=v and 0+v=v. It obviously true but with vectors it would be like this. Since v is <c,d> and 0 is <0,0>, v+0 is <c+0,d+0> and 0+v is <0+c,0+d>. Either way it will end up as <c,d> so 0+v=v=v+0",
"content_html": "It asked you to prove v+0=v and 0+v=v. It obviously true but with vectors it would be like this. Since v is <c,d> and 0 is <0,0>, v+0 is <c+0,d+0> and 0+v is <0+c,0+d>. Either way it will end up as <c,d> so 0+v=v=v+0",
"post_id": 4422040,
"post_number": 1,
"post_time_unix": 1238370470,
"post_time_utc": "2009-03-29 23:47:50 UTC",
"thanks_received": 2,
"user_id": 46181,
"username": "junc"
},
{
"attachments": [],
"content_bbcode": "Yeah, the simplest way to look at it is as the vector addition of:\r\n\r\n<c,d> + <0,0> and <0,0> + <c,d> which still works out as <c,d>, therefore, it's all equal. \r\n\r\nThis shows commutativity of vector addition, which is kind of neat. \r\n\r\n<a,b> + <c,d> = <c,d> + <a,b>",
"content_html": "Yeah, the simplest way to look at it is as the vector addition of:<br>\n<br>\n<c,d> + <0,0> and <0,0> + <c,d> which still works out as <c,d>, therefore, it's all equal.<br>\n<br>\nThis shows commutativity of vector addition, which is kind of neat.<br>\n<br>\n<a,b> + <c,d> = <c,d> + <a,b>",
"post_id": 4422041,
"post_number": 2,
"post_time_unix": 1238433017,
"post_time_utc": "2009-03-30 17:10:17 UTC",
"thanks_received": 2,
"user_id": 47581,
"username": "amaliszewski"
}
],
"source": null
}
|
Prove \(v+0=v\) and \(0+v=v\) for vectors.
Let \(v=\langle c,d\rangle\) and \(0=\langle 0,0\rangle\). Then
\[
v+0=\langle c,d\rangle+\langle 0,0\rangle=\langle c+0,\,d+0\rangle=\langle c,d\rangle=v,
\]
and
\[
0+v=\langle 0,0\rangle+\langle c,d\rangle=\langle 0+c,\,0+d\rangle=\langle c,d\rangle=v.
\]
|
[
"/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations/BinaryOperation",
"/Mathematics/Algebra/AlgebraicOperations/GeneralAlgebraicOperations/BinaryOperator",
"/Mathematics/Algebra/AlgebraicProperties/Commutative",
"/Mathematics/Algebra/AlgebraicProperties/Commute",
"/Mathematics/Algebra/LinearAlgebra",
"/Mathematics/Algebra/Sums/Sum",
"/Mathematics/Algebra/VectorAlgebra/NullVector",
"/Mathematics/Algebra/VectorAlgebra/RealVector",
"/Mathematics/Algebra/VectorAlgebra/Vector",
"/Mathematics/Algebra/VectorAlgebra/VectorAddition",
"/Mathematics/Algebra/VectorAlgebra/ZeroVector"
] |
Apply the componentwise definition of vector addition with the zero vector to see each coordinate stays the same.
|
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] |
aops_99978
|
[quote="ashwath.rabindranath"]Let $a_{n}=\frac{n^{2}+1}{\sqrt{n^{4}+4}}$ for $n = 1,2,3...$
Define $b_{n}$ as $a_{1}a_{2}...a_{n}$ for some natural $n$.
Prove that $\frac{b_{n}}{\sqrt{2}}= \frac{\sqrt{n^{2}+1}}{\sqrt{n^{2}+2n+2}}$[/quote]
factoring the denominator of $a_{n}$, we get:
$a_{n}=\frac{n^{2}+1}{\sqrt{((n+1)^{2}+1)\cdot((n-1)^{2}+1)}}$.
and i think $a_{1}a_{2}\cdot \cdot\cdot a_{n}$ telescopes....
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Let $a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$ for $n = 1,2,3...$\r\n Define $b_{n}$ as $a_{1}a_{2}...a_{n}$ for some natural $n$.\r\n Prove that $\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$",
"content_html": "Let <img src=\"//latex.artofproblemsolving.com/b/c/a/bca01c6c99de24d1c5502ec32f29e3fe06651a26.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$\" style=\"vertical-align: -17px\" width=\"110\" height=\"43\" > for <img src=\"//latex.artofproblemsolving.com/a/4/e/a4e5d961963306587d6c0c8528b38d7dde737990.png\" class=\"latex\" alt=\"$n = 1,2,3...$\" style=\"vertical-align: -3px\" width=\"92\" height=\"16\" ><br>\nDefine <img src=\"//latex.artofproblemsolving.com/d/1/3/d13ccce8e3bdf63685bfdfcc7ba98eea349996ff.png\" class=\"latex\" alt=\"$b_{n}$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > as <img src=\"//latex.artofproblemsolving.com/0/b/d/0bd6fcbbaa9f7c837ff79dff1cfdda2216711aee.png\" class=\"latex\" alt=\"$a_{1}a_{2}...a_{n}$\" style=\"vertical-align: -2px\" width=\"66\" height=\"10\" > for some natural <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>\nProve that <img src=\"//latex.artofproblemsolving.com/8/2/c/82c5f7fa10430759ae01ef3fc0218607b84a88c7.png\" class=\"latex\" alt=\"$\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$\" style=\"vertical-align: -17px\" width=\"162\" height=\"45\" >",
"post_id": 564529,
"post_number": 1,
"post_time_unix": 1151857499,
"post_time_utc": "2006-07-02 16:24:59 UTC",
"thanks_received": 2,
"user_id": 8875,
"username": "ashwath.rabindranath"
},
{
"attachments": [],
"content_bbcode": "I think there is a statement's mistake .. \r\nI guess $a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$\r\n :cool:",
"content_html": "I think there is a statement's mistake ..<br>\nI guess <img src=\"//latex.artofproblemsolving.com/b/c/a/bca01c6c99de24d1c5502ec32f29e3fe06651a26.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$\" style=\"vertical-align: -17px\" width=\"110\" height=\"43\" ><br>\n<img src=\"/assets/images/smilies/cool.gif\" width=\"20\" height=\"20\" alt=\":cool:\" title=\":cool:\" class=\"bbcode_smiley\" />",
"post_id": 564787,
"post_number": 2,
"post_time_unix": 1151881491,
"post_time_utc": "2006-07-02 23:04:51 UTC",
"thanks_received": 2,
"user_id": 112,
"username": "Diogene"
},
{
"attachments": [],
"content_bbcode": "I think so. :lol:",
"content_html": "I think so. <img src=\"/assets/images/smilies/biggrin.gif\" width=\"20\" height=\"20\" alt=\":lol:\" title=\":lol:\" class=\"bbcode_smiley\" />",
"post_id": 564788,
"post_number": 3,
"post_time_unix": 1151881557,
"post_time_utc": "2006-07-02 23:05:57 UTC",
"thanks_received": 2,
"user_id": 3182,
"username": "Kunihiko_Chikaya"
},
{
"attachments": [],
"content_bbcode": "[quote=\"ashwath.rabindranath\"]Let $a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{2}+4}}$ for $n = 1,2,3...$\n Define $b_{n}$ as $a_{1}a_{2}...a_{n}$ for some natural $n$.\n Prove that $\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$[/quote]\r\nif $\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$\r\nthen $a_{n}$must be$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ashwath.rabindranath wrote:</div>\n<div class=\"bbcode_quote_body\">Let <img src=\"//latex.artofproblemsolving.com/a/e/8/ae81d145ef20eb3b0d58affcb216d8fbe9ac2a71.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{2}+4}}$\" style=\"vertical-align: -17px\" width=\"110\" height=\"43\" > for <img src=\"//latex.artofproblemsolving.com/a/4/e/a4e5d961963306587d6c0c8528b38d7dde737990.png\" class=\"latex\" alt=\"$n = 1,2,3...$\" style=\"vertical-align: -3px\" width=\"92\" height=\"16\" ><br>\nDefine <img src=\"//latex.artofproblemsolving.com/d/1/3/d13ccce8e3bdf63685bfdfcc7ba98eea349996ff.png\" class=\"latex\" alt=\"$b_{n}$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > as <img src=\"//latex.artofproblemsolving.com/0/b/d/0bd6fcbbaa9f7c837ff79dff1cfdda2216711aee.png\" class=\"latex\" alt=\"$a_{1}a_{2}...a_{n}$\" style=\"vertical-align: -2px\" width=\"66\" height=\"10\" > for some natural <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>\nProve that <img src=\"//latex.artofproblemsolving.com/8/2/c/82c5f7fa10430759ae01ef3fc0218607b84a88c7.png\" class=\"latex\" alt=\"$\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$\" style=\"vertical-align: -17px\" width=\"162\" height=\"45\" ></div>\n</div>\nif <img src=\"//latex.artofproblemsolving.com/8/2/c/82c5f7fa10430759ae01ef3fc0218607b84a88c7.png\" class=\"latex\" alt=\"$\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$\" style=\"vertical-align: -17px\" width=\"162\" height=\"45\" ><br>\nthen <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/2/bd28b278a07bda0ea4d08737743613980c6aa6a3.png\" class=\"latex\" alt=\"$a_{n}$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" >m</span>ust b<span style=\"white-space:nowrap;\">e<img src=\"//latex.artofproblemsolving.com/b/c/a/bca01c6c99de24d1c5502ec32f29e3fe06651a26.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$\" style=\"vertical-align: -17px\" width=\"110\" height=\"43\" ></span>",
"post_id": 564999,
"post_number": 4,
"post_time_unix": 1151905987,
"post_time_utc": "2006-07-03 05:53:07 UTC",
"thanks_received": 1,
"user_id": 14130,
"username": "Hawk Tiger"
},
{
"attachments": [],
"content_bbcode": "I'm sorry the error has been corrected",
"content_html": "I'm sorry the error has been corrected",
"post_id": 565956,
"post_number": 5,
"post_time_unix": 1152020467,
"post_time_utc": "2006-07-04 13:41:07 UTC",
"thanks_received": 2,
"user_id": 8875,
"username": "ashwath.rabindranath"
},
{
"attachments": [],
"content_bbcode": "[quote=\"ashwath.rabindranath\"]Let $a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$ for $n = 1,2,3...$\n Define $b_{n}$ as $a_{1}a_{2}...a_{n}$ for some natural $n$.\n Prove that $\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$[/quote]\r\n\r\nfactoring the denominator of $a_{n}$, we get:\r\n$a_{n}=\\frac{n^{2}+1}{\\sqrt{((n+1)^{2}+1)\\cdot((n-1)^{2}+1)}}$.\r\nand i think $a_{1}a_{2}\\cdot \\cdot\\cdot a_{n}$ telescopes....",
"content_html": "\n<div class=\"bbcode_quote\">\n<div class=\"bbcode_quote_head\">ashwath.rabindranath wrote:</div>\n<div class=\"bbcode_quote_body\">Let <img src=\"//latex.artofproblemsolving.com/b/c/a/bca01c6c99de24d1c5502ec32f29e3fe06651a26.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{n^{4}+4}}$\" style=\"vertical-align: -17px\" width=\"110\" height=\"43\" > for <img src=\"//latex.artofproblemsolving.com/a/4/e/a4e5d961963306587d6c0c8528b38d7dde737990.png\" class=\"latex\" alt=\"$n = 1,2,3...$\" style=\"vertical-align: -3px\" width=\"92\" height=\"16\" ><br>\nDefine <img src=\"//latex.artofproblemsolving.com/d/1/3/d13ccce8e3bdf63685bfdfcc7ba98eea349996ff.png\" class=\"latex\" alt=\"$b_{n}$\" style=\"vertical-align: -2px\" width=\"16\" height=\"15\" > as <img src=\"//latex.artofproblemsolving.com/0/b/d/0bd6fcbbaa9f7c837ff79dff1cfdda2216711aee.png\" class=\"latex\" alt=\"$a_{1}a_{2}...a_{n}$\" style=\"vertical-align: -2px\" width=\"66\" height=\"10\" > for some natural <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>\nProve that <img src=\"//latex.artofproblemsolving.com/8/2/c/82c5f7fa10430759ae01ef3fc0218607b84a88c7.png\" class=\"latex\" alt=\"$\\frac{b_{n}}{\\sqrt{2}}= \\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}$\" style=\"vertical-align: -17px\" width=\"162\" height=\"45\" ></div>\n</div>\n<br>\nfactoring the denominator of <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/b/d/2/bd28b278a07bda0ea4d08737743613980c6aa6a3.png\" class=\"latex\" alt=\"$a_{n}$\" style=\"vertical-align: -2px\" width=\"17\" height=\"10\" >,</span> we get:<br>\n<span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/9/8/1/98166e114f75ae868762812a68d64516911b4457.png\" class=\"latex\" alt=\"$a_{n}=\\frac{n^{2}+1}{\\sqrt{((n+1)^{2}+1)\\cdot((n-1)^{2}+1)}}$\" style=\"vertical-align: -20px\" width=\"294\" height=\"47\" >.</span><br>\nand i think <img src=\"//latex.artofproblemsolving.com/0/e/a/0eafc27029ecd4c1bf6c51afb9f15491179760fc.png\" class=\"latex\" alt=\"$a_{1}a_{2}\\cdot \\cdot\\cdot a_{n}$\" style=\"vertical-align: -2px\" width=\"83\" height=\"10\" > telescopes....",
"post_id": 566144,
"post_number": 6,
"post_time_unix": 1152036984,
"post_time_utc": "2006-07-04 18:16:24 UTC",
"thanks_received": 2,
"user_id": 9197,
"username": "kimby_102"
},
{
"attachments": [],
"content_bbcode": "This can be solved with induction:\r\n$b_{n+1}=b_{n}* a_{n+1}$\r\n$=\\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}* \\sqrt{2}* \\frac{(\\sqrt{(n+1)^{2}+1})^{2}}{\\sqrt{(n+1)^{4}+1}}$\r\n$=\\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}* \\sqrt{2}* \\frac{(\\sqrt{n^{2}+2n+2})^{2}}{\\sqrt{(n+2)^{2}+1}\\sqrt{n^{2}+1}}$\r\n$=\\frac{(n+1)^{2}+1}{(n+1)^{2}+2(n+1)+1}*\\sqrt{2}$\r\nand the result follows",
"content_html": "This can be solved with induction:<br>\n<img src=\"//latex.artofproblemsolving.com/a/d/5/ad5119a7879a54f2e58cb7d17542b37b314cbed8.png\" class=\"latex\" alt=\"$b_{n+1}=b_{n}* a_{n+1}$\" style=\"vertical-align: -4px\" width=\"125\" height=\"16\" ><br>\n<img src=\"//latex.artofproblemsolving.com/4/b/6/4b6a3f8d5e3f6330be5eafe9e90da7769afbf02f.png\" class=\"latex\" alt=\"$=\\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}* \\sqrt{2}* \\frac{(\\sqrt{(n+1)^{2}+1})^{2}}{\\sqrt{(n+1)^{4}+1}}$\" style=\"vertical-align: -20px\" width=\"325\" height=\"49\" ><br>\n<img src=\"//latex.artofproblemsolving.com/c/8/4/c847050bbc35c0d094dba30c243918f086abaad2.png\" class=\"latex\" alt=\"$=\\frac{\\sqrt{n^{2}+1}}{\\sqrt{n^{2}+2n+2}}* \\sqrt{2}* \\frac{(\\sqrt{n^{2}+2n+2})^{2}}{\\sqrt{(n+2)^{2}+1}\\sqrt{n^{2}+1}}$\" style=\"vertical-align: -20px\" width=\"369\" height=\"48\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/3/0/530d6f1a75b60c3e6d08f926b1e02d9c6b36d1f7.png\" class=\"latex\" alt=\"$=\\frac{(n+1)^{2}+1}{(n+1)^{2}+2(n+1)+1}*\\sqrt{2}$\" style=\"vertical-align: -17px\" width=\"248\" height=\"43\" ><br>\nand the result follows",
"post_id": 566165,
"post_number": 7,
"post_time_unix": 1152038598,
"post_time_utc": "2006-07-04 18:43:18 UTC",
"thanks_received": 2,
"user_id": 16759,
"username": "rem"
}
],
"source": null
}
|
Let \(a_n=\dfrac{n^{2}+1}{\sqrt{n^{4}+4}}\) for \(n=1,2,3,\dots\).
Define \(b_n=a_1a_2\cdots a_n\) for \(n\in\mathbb{N}\).
Prove that
\[
\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^{2}+1}}{\sqrt{n^{2}+2n+2}}.
\]
|
[
"/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity",
"/Mathematics/Algebra/Products/CumulativeProduct",
"/Mathematics/Algebra/Products/Product"
] |
Factor the denominator to rewrite a_n as a quotient of successive √(k^2+1) terms, making the product telescope.
|
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aops_999841
|
i did this problem differently and got a different answer. I set it up so that the y-component of the unknown vector was equal to 3500. This way, the upward y-component would equal the downward force of gravity and the truck would not move. Since the angle was 30 degrees, i made x the magnitude of the vector and made the equation
x*sin(30) = 3500
From here, I got that x= 7000 pounds
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "So, this question asks you to solve for the amount of force required to keep a 3500 lb car from rolling down an incline at 30 degrees. To do this, you must solve for the component of vector u along vector v. Therefore, you take the downward force of 3500 lb and multiply it by the cosine of (90-(degree of incline)), which gives you 3500*cos(60) = 1750 lb.",
"content_html": "So, this question asks you to solve for the amount of force required to keep a 3500 lb car from rolling down an incline at 30 degrees. To do this, you must solve for the component of vector u along vector v. Therefore, you take the downward force of 3500 lb and multiply it by the cosine of (90-(degree of incline)), which gives you 3500*cos(60) = 1750 lb.",
"post_id": 4422141,
"post_number": 1,
"post_time_unix": 1240189219,
"post_time_utc": "2009-04-20 01:00:19 UTC",
"thanks_received": 2,
"user_id": 46605,
"username": "gbeard"
},
{
"attachments": [],
"content_bbcode": "i did this problem differently and got a different answer. I set it up so that the y-component of the unknown vector was equal to 3500. This way, the upward y-component would equal the downward force of gravity and the truck would not move. Since the angle was 30 degrees, i made x the magnitude of the vector and made the equation\r\nx*sin(30) = 3500\r\nFrom here, I got that x= 7000 pounds",
"content_html": "i did this problem differently and got a different answer. I set it up so that the y-component of the unknown vector was equal to 3500. This way, the upward y-component would equal the downward force of gravity and the truck would not move. Since the angle was 30 degrees, i made x the magnitude of the vector and made the equation<br>\nx*sin(30) = 3500<br>\nFrom here, I got that x= 7000 pounds",
"post_id": 4422142,
"post_number": 2,
"post_time_unix": 1240190362,
"post_time_utc": "2009-04-20 01:19:22 UTC",
"thanks_received": 2,
"user_id": 47644,
"username": "LindsayA"
}
],
"source": null
}
|
A 3500 lb car rests on an incline of 30°. Find the amount of force required to keep the car from rolling down the incline by computing the component of the car’s weight acting parallel to the incline.
|
[
"/Mathematics/AppliedMathematics/Engineering"
] |
Resolve the weight into components and use the sine of the incline angle to find the parallel component.
|
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aops_99990
|
By Menelaos' theorem, we have $\frac{s}{1}\cdot \frac{t+1}{1}\cdot \frac{EG}{GB}=1\Longleftrightarrow EG: GB=1: st+s.$ Thus $\frac{\triangle{GAB}}{\triangle{ABC}}=\frac{AE}{AC}\cdot \frac{BG}{BE}=\frac{1}{t+1}\cdot \frac{st+s}{st+s+1}=\frac{s}{st+s+1}.$
Similarly we have $\frac{\triangle{HBC}}{\triangle{ABC}}=\frac{t}{tr+t+1},\ \frac{\triangle{ICA}}{\triangle{ABC}}=\frac{r}{rs+r+1}.$
$\therefore \triangle{GHI}=\triangle{ABC}-(\triangle{GAB}+\triangle{HBC}+\triangle{ICA})=.... .$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Can anyone show me a proof of the following theorem: Let $ABC$ be a triangle with area equals to 1. $D, E$, and $F$ are chosen from from the sides $BC, AC$, and $AB$ such that $AF/FB=r$, $BD/DC=s$, and $CE/EA=t$. Let $G=BE\\cap AD$, $H=BE\\cap CF$, and $I=AD\\cap CF$. Then the area of $GHI$ is \\[\\frac{(rst-1)^{2}}{(st+s+1)(tr+t+1)(rs+r+1)}.\\]",
"content_html": "Can anyone show me a proof of the following theorem: Let <img src=\"//latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png\" class=\"latex\" alt=\"$ABC$\" width=\"42\" height=\"13\" > be a triangle with area equals to 1. <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/8/7/0/87082784a051c7525966d9819cc379329a30e030.png\" class=\"latex\" alt=\"$D, E$\" style=\"vertical-align: -3px\" width=\"37\" height=\"16\" >,</span> and <img src=\"//latex.artofproblemsolving.com/a/0/5/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png\" class=\"latex\" alt=\"$F$\" width=\"14\" height=\"12\" > are chosen from from the sides <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/3/6/4/364e5edf8fda9e9697b5b06074688187f955452c.png\" class=\"latex\" alt=\"$BC, AC$\" style=\"vertical-align: -3px\" width=\"63\" height=\"16\" >,</span> and <img src=\"//latex.artofproblemsolving.com/5/7/e/57ee5125358c0606c9b588580ddfa66f83e607b7.png\" class=\"latex\" alt=\"$AB$\" width=\"27\" height=\"13\" > such that <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/4/0/5/405d32242c26c56ead559e2f990d0b88dc138ef1.png\" class=\"latex\" alt=\"$AF/FB=r$\" style=\"vertical-align: -4px\" width=\"97\" height=\"18\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/e/e/9/ee9ae64f496c8ac9367b3053339782fdf33e3fd0.png\" class=\"latex\" alt=\"$BD/DC=s$\" style=\"vertical-align: -4px\" width=\"101\" height=\"18\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/7/f/4/7f4fefd11b2bcd64ae9e826014ac79a639282fd7.png\" class=\"latex\" alt=\"$CE/EA=t$\" style=\"vertical-align: -4px\" width=\"96\" height=\"18\" >.</span> Let <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/2/3/0/23017966fff3e488e7f74f81acde69aa059ae269.png\" class=\"latex\" alt=\"$G=BE\\cap AD$\" style=\"vertical-align: 0px\" width=\"116\" height=\"13\" >,</span> <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/5/7/b/57b65046a43782b33fdfc88746c07140e8233530.png\" class=\"latex\" alt=\"$H=BE\\cap CF$\" style=\"vertical-align: 0px\" width=\"118\" height=\"13\" >,</span> and <span style=\"white-space:nowrap;\"><img src=\"//latex.artofproblemsolving.com/f/8/9/f89f6c50e83084c8fc80ddfba47c0b5da1aa7b1f.png\" class=\"latex\" alt=\"$I=AD\\cap CF$\" style=\"vertical-align: 0px\" width=\"111\" height=\"13\" >.</span> Then the area of <img src=\"//latex.artofproblemsolving.com/c/b/0/cb0d4bea7660be7d3d90c45a6c267589c5687ba4.png\" class=\"latex\" alt=\"$GHI$\" width=\"40\" height=\"12\" > is <img src=\"//latex.artofproblemsolving.com/9/1/5/9154b3c7a58b2e537f6995272620a885c2481d79.png\" class=\"latexcenter\" alt=\"\\[\\frac{(rst-1)^{2}}{(st+s+1)(tr+t+1)(rs+r+1)}.\\]\" width=\"280\" height=\"43\" >",
"post_id": 564590,
"post_number": 1,
"post_time_unix": 1151861727,
"post_time_utc": "2006-07-02 17:35:27 UTC",
"thanks_received": 2,
"user_id": 9472,
"username": "puuhikki"
},
{
"attachments": [],
"content_bbcode": "By Menelaos' theorem, we have $\\frac{s}{1}\\cdot \\frac{t+1}{1}\\cdot \\frac{EG}{GB}=1\\Longleftrightarrow EG: GB=1: st+s.$ Thus $\\frac{\\triangle{GAB}}{\\triangle{ABC}}=\\frac{AE}{AC}\\cdot \\frac{BG}{BE}=\\frac{1}{t+1}\\cdot \\frac{st+s}{st+s+1}=\\frac{s}{st+s+1}.$\r\n\r\nSimilarly we have $\\frac{\\triangle{HBC}}{\\triangle{ABC}}=\\frac{t}{tr+t+1},\\ \\frac{\\triangle{ICA}}{\\triangle{ABC}}=\\frac{r}{rs+r+1}.$\r\n\r\n$\\therefore \\triangle{GHI}=\\triangle{ABC}-(\\triangle{GAB}+\\triangle{HBC}+\\triangle{ICA})=.... .$",
"content_html": "By Menelaos' theorem, we have <img src=\"//latex.artofproblemsolving.com/2/2/3/223b7e5eb6177474f96251ce04ea8c404846bfe5.png\" class=\"latex\" alt=\"$\\frac{s}{1}\\cdot \\frac{t+1}{1}\\cdot \\frac{EG}{GB}=1\\Longleftrightarrow EG: GB=1: st+s.$\" style=\"vertical-align: -13px\" width=\"361\" height=\"38\" > Thus <img src=\"//latex.artofproblemsolving.com/d/a/c/dac7113c5811d0425ffddae292129d8502ffa467.png\" class=\"latex\" alt=\"$\\frac{\\triangle{GAB}}{\\triangle{ABC}}=\\frac{AE}{AC}\\cdot \\frac{BG}{BE}=\\frac{1}{t+1}\\cdot \\frac{st+s}{st+s+1}=\\frac{s}{st+s+1}.$\" style=\"vertical-align: -14px\" width=\"434\" height=\"40\" ><br>\n<br>\nSimilarly we have <img src=\"//latex.artofproblemsolving.com/f/3/b/f3beb9a4e1513f113d3d0ffb2636fbdd59d471cc.png\" class=\"latex\" alt=\"$\\frac{\\triangle{HBC}}{\\triangle{ABC}}=\\frac{t}{tr+t+1},\\ \\frac{\\triangle{ICA}}{\\triangle{ABC}}=\\frac{r}{rs+r+1}.$\" style=\"vertical-align: -14px\" width=\"357\" height=\"40\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/0/9/a/09a99fb9efbebd2eaf0e87e6e804515abc4b3ad6.png\" class=\"latex\" alt=\"$\\therefore \\triangle{GHI}=\\triangle{ABC}-(\\triangle{GAB}+\\triangle{HBC}+\\triangle{ICA})=.... .$\" style=\"vertical-align: -4px\" width=\"459\" height=\"18\" >",
"post_id": 564657,
"post_number": 2,
"post_time_unix": 1151869734,
"post_time_utc": "2006-07-02 19:48:54 UTC",
"thanks_received": 2,
"user_id": 3182,
"username": "Kunihiko_Chikaya"
},
{
"attachments": [],
"content_bbcode": "I never know that this can also be called a theorem......\r\n\r\nhttp://www.mathlinks.ro/Forum/viewtopic.php?t=40828&highlight=ratio+area",
"content_html": "I never know that this can also be called a theorem......<br>\n<br>\n<a target=\"_blank\" href=\"http://www.mathlinks.ro/Forum/viewtopic.php?t=40828&highlight=ratio+area\">http://www.mathlinks.ro/Forum/viewtopic.php?t=40828&highlight=ratio+area</a>",
"post_id": 564994,
"post_number": 3,
"post_time_unix": 1151905551,
"post_time_utc": "2006-07-03 05:45:51 UTC",
"thanks_received": 1,
"user_id": 6601,
"username": "shobber"
}
],
"source": null
}
|
Let \(ABC\) be a triangle with area \(1\). Points \(D\in BC\), \(E\in AC\), and \(F\in AB\) are chosen such that
\[
\frac{AF}{FB}=r,\qquad \frac{BD}{DC}=s,\qquad \frac{CE}{EA}=t.
\]
Let \(G=BE\cap AD\), \(H=BE\cap CF\), and \(I=AD\cap CF\). Prove that the area of triangle \(GHI\) is
\[
\frac{(rst-1)^{2}}{(st+s+1)(tr+t+1)(rs+r+1)}.
\]
|
[
"/Mathematics/Geometry/DivisionProblems/PlaneDivisionbyLines",
"/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry",
"/Mathematics/Geometry/GeneralGeometry/Geometry"
] |
Apply Menelaus to each cevian to find side ratios, convert them to area ratios, and subtract the three corner triangles from the whole.
|
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aops_999996
|
$ \ x\equal{}r*cos(\theta)$
$ \ y\equal{}r*sin(\theta)$
$ \ 4x\plus{}3y\equal{}12 \equal{}\equal{}> 4*(r*cos(\theta))\plus{}3*(r*sin(\theta))\equal{}12$
$ \ r*(4cos(\theta)\plus{}3sin(\theta))\equal{}12$
$ \ r\equal{}\frac{12}{4cos(\theta)\plus{}3sin(\theta)}$
| null |
{
"competition": null,
"dataset": "AOPS",
"posts": [
{
"attachments": [],
"content_bbcode": "Use conversion formulas from 6a to express 4x+3y=12 in polar coordinates.\r\n\r\nWhat conversion formulas is this referring to?",
"content_html": "Use conversion formulas from 6a to express 4x+3y=12 in polar coordinates.<br>\n<br>\nWhat conversion formulas is this referring to?",
"post_id": 4422365,
"post_number": 1,
"post_time_unix": 1243296424,
"post_time_utc": "2009-05-26 00:07:04 UTC",
"thanks_received": 2,
"user_id": 47708,
"username": "awoods"
},
{
"attachments": [],
"content_bbcode": "It is talking about the conversion formulas in the book.",
"content_html": "It is talking about the conversion formulas in the book.",
"post_id": 4422366,
"post_number": 2,
"post_time_unix": 1243298819,
"post_time_utc": "2009-05-26 00:46:59 UTC",
"thanks_received": 2,
"user_id": 48086,
"username": "kevkingsley"
},
{
"attachments": [],
"content_bbcode": "I was confused about this question too becuase there isnt a 6a on the worksheet, so I am assuming the question is talkin about the conversion formulas x=rcos(theta) and y=rsin(theta)",
"content_html": "I was confused about this question too becuase there isnt a 6a on the worksheet, so I am assuming the question is talkin about the conversion formulas x=rcos(theta) and y=rsin(theta)",
"post_id": 4422367,
"post_number": 3,
"post_time_unix": 1243305389,
"post_time_utc": "2009-05-26 02:36:29 UTC",
"thanks_received": 2,
"user_id": 47244,
"username": "ajiang"
},
{
"attachments": [],
"content_bbcode": "$ \\ x\\equal{}r*cos(\\theta)$\r\n$ \\ y\\equal{}r*sin(\\theta)$\r\n\r\n$ \\ 4x\\plus{}3y\\equal{}12 \\equal{}\\equal{}> 4*(r*cos(\\theta))\\plus{}3*(r*sin(\\theta))\\equal{}12$\r\n$ \\ r*(4cos(\\theta)\\plus{}3sin(\\theta))\\equal{}12$\r\n$ \\ r\\equal{}\\frac{12}{4cos(\\theta)\\plus{}3sin(\\theta)}$",
"content_html": "<img src=\"//latex.artofproblemsolving.com/6/7/a/67a9ee09ea8138af8e3bc61ab752e2b1fdb1b31a.png\" class=\"latex\" alt=\"$ \\ x=r*cos(\\theta)$\" style=\"vertical-align: -4px\" width=\"113\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/5/a/f/5af1ad4314e870576aa04b2e600188843e895a1f.png\" class=\"latex\" alt=\"$ \\ y=r*sin(\\theta)$\" style=\"vertical-align: -4px\" width=\"112\" height=\"18\" ><br>\n<br>\n<img src=\"//latex.artofproblemsolving.com/6/a/c/6ac6039efe36b3ff93c53b14d3b22345eb0e907b.png\" class=\"latex\" alt=\"$ \\ 4x+3y=12 ==> 4*(r*cos(\\theta))+3*(r*sin(\\theta))=12$\" style=\"vertical-align: -4px\" width=\"454\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/2/d/32d0eec0c9a684d85014eeb9dc349d425ba45cd0.png\" class=\"latex\" alt=\"$ \\ r*(4cos(\\theta)+3sin(\\theta))=12$\" style=\"vertical-align: -4px\" width=\"225\" height=\"18\" ><br>\n<img src=\"//latex.artofproblemsolving.com/3/d/2/3d2b69458a84fb508dcae17855fcfffeb9b67ec1.png\" class=\"latex\" alt=\"$ \\ r=\\frac{12}{4cos(\\theta)+3sin(\\theta)}$\" style=\"vertical-align: -17px\" width=\"178\" height=\"41\" >",
"post_id": 4422368,
"post_number": 4,
"post_time_unix": 1243307343,
"post_time_utc": "2009-05-26 03:09:03 UTC",
"thanks_received": 2,
"user_id": 47806,
"username": "jmarcinek"
}
],
"source": null
}
|
Use conversion formulas from (a) to express \(4x+3y=12\) in polar coordinates.
What conversion formulas is this referring to?
|
[
"/Mathematics/Geometry/CoordinateGeometry/Cartesian",
"/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/PolarCoordinateSystem",
"/Mathematics/Geometry/CoordinateGeometry/PolarCoordinates",
"/Mathematics/Geometry/CoordinateGeometry/PolarEquation",
"/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry",
"/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry"
] |
Replace x and y with r cosθ and r sinθ respectively, then solve the resulting equation for r.
|
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