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ours_565
The condition \(s(a+b) = s(a) + s(b) - 1\) implies that there is exactly one position where both \(a\) and \(b\) have a 1 in the same place value, and both have 0's in the next (left) bit. Both \(a\) and \(b\) must be represented with 6 bits (allowing leading 0's). The number of ways to achieve this is calculated as follows: - If the 1's are both in the 6th bit, there are \(3^5\) ways. - If the 1's are in any of the other bits, there are \(5 \cdot 3^4\) ways. Thus, the total number of ordered pairs \((a, b)\) is: \[ 3^5 + 5 \cdot 3^4 = 648 \] \(\boxed{648}\)
648
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSoln2.md" }
Let \( s(n) \) be the number of 1's in the binary representation of \( n \). Find the number of ordered pairs of integers \((a, b)\) with \(0 \leq a < 64, 0 \leq b < 64\) and \(s(a+b) = s(a) + s(b) - 1\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/IntegerFunctions" ]
The equality holds exactly when a and b share a single 1‑bit and have 0 in the next higher bit, so there is one overlap without a carry.
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ours_566
We want the number of odd coefficients in the generating function \(\left(x^{2}+x^{3}+x^{5}+x^{7}\right)^{4}\left(x^{1}+x^{2}+x^{3}+x^{4}\right)^{2}\). It is known that \( f(x)^{2^{k}}-f\left(x^{2^{k}}\right) \) has all even coefficients for any \( f \in \mathbb{Z}[x] \) and positive integer \( k \). Therefore, this is also the number of odd coefficients in \(\left(x^{8}+x^{12}+x^{20}+x^{28}\right)\left(x^{2}+x^{4}+x^{6}+x^{8}\right)\). The computation is as follows: \[ \left(x^{8}+x^{12}+x^{20}+x^{28}\right)\left(x^{2}+x^{4}+x^{6}+x^{8}\right) = x^{10}+x^{12}+2x^{14}+2x^{16}+x^{20}+x^{22}+x^{24}+x^{26}+x^{28}+x^{30}+x^{32}+x^{34}+x^{36} \] This expression has 12 odd coefficients. Thus, the number of \( n \) for which \( s_{n} \) is odd is \(\boxed{12}\).
12
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSoln2.md" }
Let \( s_{n} \) be the number of ordered solutions to \( a_{1}+a_{2}+a_{3}+a_{4}+b_{1}+b_{2}=n \), where \( a_{1}, a_{2}, a_{3}, \) and \( a_{4} \) are elements of the set \(\{2,3,5,7\}\) and \( b_{1} \) and \( b_{2} \) are elements of the set \(\{1,2,3,4\}\). Find the number of \( n \) for which \( s_{n} \) is odd.
[ "/Mathematics/Algebra/NumberTheory/GeneratingFunctions/GeneratingFunction", "/Mathematics/Algebra/NumberTheory/Parity/OddNumber", "/Mathematics/Algebra/Polynomials/Coefficient", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]
Reduce the generating function modulo 2 using the identity f(x)^{2^k}≡f(x^{2^k}) to count odd coefficients.
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ours_567
The condition $x \equiv 8 \pmod{34}$ can be rewritten as $x \equiv 110 \pmod{170}$ since $5 \mid x$. By testing values, we find that $x = 620$ is the smallest solution that satisfies both conditions. Therefore, the minimum possible value of $f(0)$ is 620. \(\boxed{620}\)
620
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
Find the smallest solution for $$ x \equiv 10,25,80,95 \quad(\bmod 105) \quad \text{and} \quad x \equiv 8 \quad(\bmod 34). $$
[ "/Mathematics/DiscreteMathematics/DivisionProblems/SubsetSumProblem", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/ChineseCongruence", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/LinearCongruenceEquation", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Use the Chinese Remainder Theorem to combine each of the four possible residues modulo 105 with the condition modulo 34 and select the smallest resulting solution.
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ours_568
Let \(O_{3}\) be the center of \(\Omega\). Note that \(P, O_{3}, O_{1}\) are collinear and \(P, Q, O_{2}\) are collinear. Since \(\triangle O_{3}PQ\) and \(\triangle O_{2}QF\) are isosceles, we have: \[ \angle O_{3}PF = \angle O_{3}PQ = \angle O_{3}QP = \angle O_{2}QF = \angle O_{2}FQ = \angle O_{2}FP \] Thus, \(\overline{O_{2}F} \parallel \overline{O_{3}P}\). Let \(\overline{O_{1}O_{2}}\) meet \(\overline{PF}\) at \(S\). Since \(\overline{O_{1}P} \parallel \overline{O_{2}F}\), \(S\) is the center of a negative homothety mapping \(\Gamma_{1}\) to \(\Gamma_{2}\). So \(S\) lies on \(\overline{PQ}\), and similarly \(S\) also lies on \(\overline{XY}\). This negative homothety maps \(P\) to the intersection past \(S\) of \(\overline{PS}\) with \(\Gamma_{2}\) and \(X\) to the intersection past \(S\) of \(\overline{XS}\) with \(\Gamma_{2}\). These intersections are \(R\) and \(Z\), respectively. Thus, this homothety maps \(\overline{PX}\) to \(\overline{RZ}\), and so \(\overline{PX} \parallel \overline{RZ}\). Therefore: \[ \frac{PX}{ZR} = \frac{O_{1}P}{O_{2}R} = \frac{13}{10} \] If lines \(PR\) and \(XZ\) meet at \(S\) inside quadrilateral \(PXRZ\), then lines \(PZ\) and \(RX\) meet at \(M\) outside this quadrilateral. Thus, \(\frac{MZ}{MP} = \frac{10}{13}\), and \(\frac{PM}{PZ} = \frac{13}{3}\). Since \(P\) is fixed, as \(\omega\) varies, \(Z\) can be any point on circle \(\Gamma_{2}\). Therefore, the locus of \(M\) is circle \(\Gamma_{2}\), dilated about \(P\) with scale factor \(\frac{13}{3}\). The area of this locus is: \[ \pi\left(\frac{13}{3} \cdot 10\right)^{2} = \left(\frac{16900}{9}\right) \pi \] Thus, the answer is \(16909\). \(\boxed{16909}\)
16909
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
Let \(\Gamma_{1}\) and \(\Gamma_{2}\) be circles in the plane with centers \(O_{1}\) and \(O_{2}\) and radii \(13\) and \(10\), respectively. Assume \(O_{1} O_{2}=2\). Fix a circle \(\Omega\) with radius \(2\), internally tangent to \(\Gamma_{1}\) at \(P\) and externally tangent to \(\Gamma_{2}\) at \(Q\). Let \(\omega\) be a second variable circle internally tangent to \(\Gamma_{1}\) at \(X\) and externally tangent to \(\Gamma_{2}\) at \(Y\). Line \(PQ\) meets \(\Gamma_{2}\) again at \(R\), line \(XY\) meets \(\Gamma_{2}\) again at \(Z\), and lines \(PZ\) and \(XR\) meet at \(M\). As \(\omega\) varies, the locus of point \(M\) encloses a region of area \(\frac{p}{q} \pi\), where \(p\) and \(q\) are relatively prime positive integers. Compute \(p+q\).
[ "/Mathematics/Geometry/Curves/AlgebraicCurves/AlgebraicCurve", "/Mathematics/Geometry/Curves/AlgebraicCurves/QuadraticCurve", "/Mathematics/Geometry/Curves/PlaneCurves/ConicSections", "/Mathematics/Geometry/Curves/PlaneCurves/GeneralPlaneCurves", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle-CircleTangents", "/Mathematics/Geometry/PlaneGeometry/Circles/TangentCircles", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Homothecy", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Homothetic", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/HomotheticCenter", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Scaling", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similar", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similarity", "/Mathematics/Geometry/Transformations/GeometricTransformations/CentralDilation", "/Mathematics/Geometry/Transformations/GeometricTransformations/Dilation" ]
Use the negative homothety that maps the larger circle to the smaller one to relate the tangency points and prove M is a fixed‑scale image of Γ₂.
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ours_569
The points on the planes in \(\mathcal{P}\) are of the form \((x, y, z) = \left(a_{1} e_{1}, a_{2} e_{2}, a_{3} e_{3}\right)\), where the \(a_{i}\) (axis intercepts) and \(e_{i}\) (weights) both sum to 1. By the Cauchy-Schwarz inequality, we have: \[ 1 = \left(a_{1} + a_{2} + a_{3}\right)\left(e_{1} + e_{2} + e_{3}\right) \geq \left(\sqrt{a_{1} e_{1}} + \sqrt{a_{2} e_{2}} + \sqrt{a_{3} e_{3}}\right)^{2} \] This implies: \[ \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \] Equality can be attained for any choice of \(x, y, z\) satisfying the equality by setting \(a_{1} = e_{1} = \sqrt{x}\) and so forth. Thus, we find \(S\) is the set of points with \(\sqrt{x} + \sqrt{y} + \sqrt{z} = 1\). Then by Hölder's inequality (or the weighted power mean), we have: \[ \left(\frac{1}{8} x^{2} + y^{2} + 8 z^{2}\right)\left(2 + 1 + \frac{1}{2}\right)^{3} \geq (\sqrt{x} + \sqrt{y} + \sqrt{z})^{4} = 1 \] Therefore: \[ \left(\frac{1}{4} x^{2} + 2 y^{2} + 16 z^{2}\right)^{-1} \leq \frac{1}{2}\left(2 + 1 + \frac{1}{2}\right)^{3} = \frac{343}{16} = 21 + \frac{7}{16} \] Since the values can be changed continuously, the largest possible integer value is \(21\). \(\boxed{21}\)
21
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
Let \(\mathcal{P}\) denote the set of planes in three-dimensional space with positive \(x, y\), and \(z\) intercepts summing to one. A point \((x, y, z)\) with \(\min \{x, y, z\}>0\) lies on exactly one plane in \(\mathcal{P}\). What is the maximum possible integer value of \(\left(\frac{1}{4} x^{2}+2 y^{2}+16 z^{2}\right)^{-1}\)?
[ "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/CoordinateGeometry/CoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/x-Intercept", "/Mathematics/Geometry/CoordinateGeometry/y-Intercept", "/Mathematics/Geometry/GeometricInequalities", "/Mathematics/Geometry/MultidimensionalGeometry", "/Mathematics/Geometry/SolidGeometry/GeneralSolidGeometry/SolidGeometry" ]
Transform the plane condition into √x+√y+√z=1 using Cauchy‑Schwarz, then bound the weighted quadratic form with Hölder’s inequality.
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ours_570
Let \(H_{n}=\frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{n}\). Note that $$ \binom{n+100}{n}=\left(\frac{n+100}{100}\right)\binom{n+99}{99} $$ and $$ \binom{n+100}{n}=\left(\frac{n+1}{100}\right)\binom{n+100}{99} $$ so $$ \frac{1}{\binom{n+99}{99}}-\frac{1}{\binom{n+100}{99}}=\frac{\frac{99}{100}}{\binom{n+100}{100}} $$ Thus, the sum is $$ \frac{100}{99}\left(H_{1}\left(\binom{100}{99}^{-1}-\binom{101}{99}^{-1}\right)+H_{2}\left(\binom{101}{99}^{-1}-\binom{102}{99}^{-1}\right)+\cdots\right) $$ which equals $$ \frac{100}{99}\left(\binom{100}{99}^{-1}+\frac{1}{2}\binom{101}{99}^{-1}+\frac{1}{3}\binom{102}{99}^{-1} \ldots\right) $$ Note that \(n\binom{99+n}{100}=100\binom{99+n}{100}\), so this becomes $$ \frac{1}{99}\left(\binom{100}{100}^{-1}+\binom{101}{100}^{-1}+\binom{102}{100}^{-1}+\cdots\right) $$ Using the first identity again, this telescopes and becomes $$ \frac{100}{99^{2}}\left(\binom{99}{99}^{-1}-\binom{100}{99}^{-1}+\binom{100}{99}^{-1}-\binom{101}{99}^{-1} \cdots\right)=\frac{100}{99^{2}}\left(\binom{99}{99}^{-1}\right)=\frac{100}{9801} $$ yielding a final answer of \(100+9801=9901\). Therefore, \(p+q = 9901\). \(\boxed{9901}\)
9901
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
If $$ \sum_{n=1}^{\infty} \frac{\frac{1}{1}+\frac{1}{2}+\cdots+\frac{1}{n}}{\binom{n+100}{100}}=\frac{p}{q} $$ for relatively prime positive integers \(p, q\), find \(p+q\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]
Rewrite each term using a binomial‑coefficient difference so the series telescopes.
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ours_571
Consider instead the pairs with both entries increased by one. Then the operation corresponds to starting with \((2,1)\) and repeatedly replacing \((m, n)\) by one of \((2m-n, m)\), \((2m+n, m)\), and \((m+2n, n)\). We can make a few simple observations about the resulting pairs: - The first entry always exceeds the second entry. - There is always exactly one even entry. - The entries are always relatively prime to each other. The crucial claim is that in fact this is a complete characterization for all achievable pairs. The proof is a simple strong induction on \(m+n\) where \((m, n)\) has the above properties. Indeed, one can actually show there is a unique construction of \((m, n)\) from \((2,1)\) using the above operations. Hence, the answer is the number of even integers relatively prime to 2015. Letting \(\varphi\) denote Euler's totient function, this is just \(\frac{1}{2} \varphi(2015) = \frac{1}{2} \cdot 4 \cdot 12 \cdot 30 = 720\). \(\boxed{720}\)
720
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
Qing initially writes the ordered pair \((1,0)\) on a blackboard. Each minute, if the pair \((a, b)\) is on the board, she erases it and replaces it with one of the pairs \((2a-b, a)\), \((2a+b+2, a)\), or \((a+2b+2, b)\). Eventually, the board reads \((2014, k)\) for some nonnegative integer \(k\). How many possible values of \(k\) are there?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Shift the pair by (+1,+1) to obtain simpler operations and use parity and gcd invariants to fully characterize reachable pairs.
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ours_572
Let \(g(n)\) be the number of ways for the frog to jump \(n\) times and land on a positive number that it has not landed on before. Let \(f(n, k)\) be the number of ways to do this such that its first \(k\) jumps are \(2\) to the right and its \(k+1\) 'th is not, for \(0 \leq k \leq n\). (We define \(f(n, n)=1\) for the one path that consists of \(n\) jumps \(2\) to the right.) Note that \[ g(n)=f(n, 0)+f(n, 1)+\cdots+f(n, n) \] for \(n \geq 0\). Note \(f(n, 0)=g(n-1)\) for \(n \geq 1\), since the frog's first jump must be \(1\) to the right, and starting from \(1\) with \(0\) already landed on is equivalent to starting from \(0\) again with one less step taken. Additionally, \(f(n, n-1)=2\) for \(n \geq 2\), since after moving \(n-1\) steps to the right, the frog can either move one to the left or one to the right. Let \(1_{a \geq b}\) be defined as taking the value \(1\) if \(a \geq b\) and \(0\) otherwise. Consider the value of \(f(j+k, k)\) for \(j \geq 2, k \geq 1\). After landing on the numbers \(2,4, \cdots 2k\), the possible remaining numbers to land on are \(1,3, \cdots 2k-1\), and \(2k+1,2k+2,2k+3, \cdots\). It is not hard to see that the frog can take \(4\) possible paths starting at \(2k\) and not moving to \(2k+2\) initially: - The frog can move to \(2k+1\) and remain to the right of \(2k\) for the next \(j-1\) steps. This is equivalent to starting at \(0\) and making \(j-1\) arbitrary legal steps, so the number of ways to do this is \(g(j-1)\). - The frog can move to \(2k-1\), and then continue to \(2k-3,2k-5\), etc. until it stops at a positive odd integer. In order for this to be possible, we must have \(j \leq k\), so the number of ways to do this is \(1_{j \geq k}\). - The frog can move to \(2k-1\), then hop to the right again to \(2k+1\), and remain to the right of \(2k\) for the next \(j-2\) steps. This is possible given \(k \geq 1, j \geq 2\), and similarly to the first case, the number of ways to do this is \(g(j-2)\). - The frog can move to \(2k+1\), then hop left to \(2k-1\) and continue moving along the path \(2k-1,2k-3, \cdots\). The number of ways to do this is \(1_{j \leq k+1}\). Thus if we substitute \(m=j+k\), we obtain \[ f(m, k)=g(m-k-1)+g(m-k-2)+1_{2k \geq m}+1_{2k+1 \geq m} \] for \(m \geq k+2\) and \(k \geq 1\). Substituting this into the definition of \(g(m)\) yields \[ \begin{aligned} g(m)= & \, g(m-1)+(g(m-2)+g(m-3))+\cdots+(g(1)+g(0))+2+1 \\ & +\sum_{m/2 \leq k \leq m-2} 1+\sum_{(m-1)/2 \leq k \leq m-2} 1 \\ = & \, g(m-1)+g(m-2)+2g(m-3)+\cdots+2g(1)+g(0)+3+\left\lfloor\frac{m}{2}\right\rfloor+\left\lceil\frac{m}{2}\right\rceil-2 \\ = & \, g(m-1)+g(m-2)+2g(m-3)+\cdots+2g(1)+g(0)+m+1 \end{aligned} \] for all \(m \geq 2\). This implies \[ g(m)=g(m-1)+2g(m-3)+1 \] for all \(m \geq 3\). Let \(h(n)\) be the probability that the frog makes \(n-1\) legal jumps and then must stop after its \(n\)th jump. The expected value of the number of jumps the frog makes is \[ E=\sum_{n \geq 0} n \cdot h(n) \] But note that \[ h(0)+h(1)+\cdots+h(k)+\frac{g(k)}{4^{k}}=1 \] and \[ \begin{aligned} 0 \cdot h(0)+\cdots+k \cdot h(k) = & \, (k)(h(0)+\cdots h(k))-(h(0))-(h(0)+h(1))-\cdots-(h(0)+\cdots+h(k-1)) \\ = & \, k\left(1-\frac{g(k)}{4^{k}}\right)-\left(1-\frac{g(0)}{4^{0}}\right)-\cdots-\left(1-\frac{g(k-1)}{4^{k-1}}\right) \\ = & \, \left(\sum_{0 \leq n<k} \frac{g(k)}{4^{k}}\right)-k \cdot \frac{g(k)}{4^{k}}. \end{aligned} \] But the last term approaches \(0\) since \(g(k) \leq 3^{k}\), so taking the limit to infinity yields \[ E=\sum_{n \geq 0} \frac{g(k)}{4^{k}} \] Thus \[ E-\frac{2E}{4}-\frac{E}{4^{3}}=\frac{1}{4^{0}}+\frac{1}{4^{2}}+\frac{1}{4^{3}}+\cdots \] and \[ \frac{31E}{64}=\frac{4}{3}-\frac{1}{4}=\frac{13}{12} \Rightarrow E=\frac{208}{93} \] This gives the answer of \(208+93=301\). \(\boxed{301}\)
301
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
A frog starts at \(0\) on a number line and plays a game. On each turn, the frog chooses at random to jump \(1\) or \(2\) integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is \(\frac{p}{q}\) for relatively prime positive integers \(p\) and \(q\), find \(p+q\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/RecurrenceEquations/LinearRecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceEquation", "/Mathematics/DiscreteMathematics/RecurrenceEquations/RecurrenceRelation", "/Mathematics/ProbabilityandStatistics/Probability", "/Mathematics/ProbabilityandStatistics/RandomWalks/RandomWalk", "/Mathematics/ProbabilityandStatistics/RandomWalks/RandomWalk1-Dimensional" ]
Count all legal jump sequences with a recurrence and express the expected number of jumps as the sum of those counts weighted by (1/4)^k.
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ours_573
Solution. Call a game position an \(A\)-position if it results in a win for the next player, a \(C\)-position if it results in a loss, and a \(B\)-position otherwise. For a nonnegative integer \(a=\overline{a_{n} a_{n-1} \cdots a_{1} a_{0}}\) written in base 2, define the ternation of \(a\), \(a_{T}\), to be \(\overline{a_{n} a_{n-1} \cdots a_{1} a_{0}}\), i.e., the number that results when the base 2 representation of \(a\) is interpreted as a base 3 integer. Define the trim-sum of two numbers \(a=\overline{a_{n} a_{n-1} \cdots a_{1} a_{0}}_{3}\) and \(b=\overline{b_{n} b_{n-1} \cdots b_{1} b_{0}}\) written in base 3 (possibly with leading zeroes), represented by \(a \boxplus b\), such that \[ a \boxplus b={\overline{c_{n}} c_{n-1} \cdots c_{1} c_{0}}_{3} \] where \(c_{i} \equiv a_{i}+b_{i}(\bmod 3)\) is \(0, 1\), or \(2\). The key is the following proposition. A position \(P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) in 3-Nim is a \(C\)-position if and only if \[ \left(x_{1}\right)_{T} \boxplus\left(x_{2}\right)_{T} \boxplus \cdots \boxplus\left(x_{n}\right)_{T}=0. \] In this case, we want \(101011_{3} \boxplus 1100011_{3} \boxplus x_{T} \boxplus y_{T}=0\), so we need \(x_{T} \boxplus y_{T}=2102011_{3}\). Since the base 3 digits in \(x_{T}\) and \(y_{T}\) are all zero or one, we see that \(x_{T}+y_{T}=2102011_{3}\). To maximize their product, we make them as close as possible. This is done by assigning the first occurrence of a 1 in the sum's base 3 representation to \(x\) and every other occurrence of a 1 to \(y\), yielding \(x=1101000_{2}=104\), \(y=1001011_{2}=75\). These multiply to give 7800. The proof of the key result used can actually generalize to more than 3 players, but for simplicity, we'll just include the 3-player version. We use strong induction on \(s=x_{1}+x_{2}+\cdots+x_{n}\). The theorem statement is trivial for \(s=0\) by the definition of a \(C\)-position. Assume for some \(k>0\) that the statement holds for all \(s<k\). Consider a position \(P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) with \(s=k\). Clearly, any move will decrease \(s\). Let \(s_{3}(P)=\left(x_{1}\right)_{T} \boxplus\left(x_{2}\right)_{T} \boxplus \cdots \boxplus\left(x_{n}\right)_{T}\). We will prove and use the following lemma. **Lemma 1.** Any position \(P\) with \(s_{3}(P) \neq 0\) can be moved to one with \(s_{3}(P)=0\) in at most two moves, while a position \(P\) with \(s_{3}(P)=0\) cannot be moved to another such position in at most two moves. **Proof.** For the first part of the lemma: Let \(s_{3}(P)=\overline{c_{d-1} c_{d-2} \cdots c_{1} c_{0}}\) in base 3 with \(c_{d-1}\) nonzero. If we can show that the lemma is true when the sizes of all piles have \(\leq d\) digits in base 2, then in any other case we can ignore all but the last \(d\) digits of each pile and perform the same operation on these digits to get the same result. Let the largest 3 piles in \(P\) have sizes \(a_{1}>a_{2}>a_{3}\). We are assuming that \(a_{1}\) has at most \(d\) digits in base 2. Let \(c_{d-1}=j\). We use strong induction on \(d\) to show that it is possible to perform the operation specified in the lemma with all of \(a_{1}, a_{2}, \cdots, a_{j}\) changed. For \(d=0\), this is trivial because all the nonempty piles are 1s, so we remove the \(c_{0}\) biggest piles and are done. Now assume it's true for all \(d<d_{0}\) for some \(d_{0}>0\), and consider \(d=d_{0}\). Let \(c_{d 0-1}=j_{0}\). Then replace \(a_{1}, a_{2}, \cdots, a_{j 0}\) with \(a^{*}=2^{d_{0}-1}-1\). This results in a position \(P^{\prime}\) with \(d=d^{\prime} \leq d_{0}-1\) (since \(c_{d_{0}-1}\) is now 0), so by induction the \(c_{d^{\prime}-1}^{\prime}\) piles whose last \(d^{\prime}\) digits form the largest base 2 integers possible can be used to do the operation demanded by the lemma. Since all digits of \(a^{*}\) in base 2 are 1, clearly \(a_{1}, a_{2}, \cdots, a_{j}\), which have all been replaced by \(a^{*}\), can be used from now on for any number of digits \(d<d_{0}\) until more than \(j_{0}\) digits are needed for some future digit, in which case we continue using them along with as many of the next largest integers as necessary. So the statement holds for \(d=d_{0}\) as well, and by strong induction holds for all \(d\). Now we prove the second part of the lemma. Assume the opposite, so there are piles \(a, b\) in \(P\) such that there exist nonnegative integers \(a^{\prime}<a, b^{\prime} \leq b\) with \(a_{T} \boxplus b_{T}=s_{3}(P) \boxplus a_{T}^{\prime} \boxplus b_{T}^{\prime}=a_{T}^{\prime} \boxplus b_{T}^{\prime}\). Since the digits of \(a_{T}, b_{T}, a_{T}^{\prime}, b_{T}^{\prime}\) in base 3 are 0s and 1s, summing them pairwise cannot produce "carry-overs", so \(a_{T} \boxplus b_{T}=a_{T}+b_{T}\) and \(a_{T}^{\prime} \boxplus b_{T}^{\prime}=a_{T}^{\prime}+b_{T}^{\prime}\). But then we get \(a_{T}+b_{T}=a_{T}^{\prime}+b_{T}^{\prime}\), which is impossible if \(a^{\prime}<a\) and \(b^{\prime} \leq b\) (since it is simple to see, e.g., by expanding out the integers' base representations into polynomials of the base, that \(a>a^{\prime}\) implies \(a_{T}>a_{T}^{\prime}\), etc.). This completes the proof of the lemma. Now we continue with the main proof. If \(s_{3}(P)\) is nonzero, then there exist two piles \(a, b\) that can be altered (with \(b\) possibly not changed) to produce a position \(P^{\prime}\) with \(s_{3}\left(P^{\prime}\right)=0\). Then the player (without loss of generality, player 1) who is faced with position \(P\) can alter \(a\). If this leaves a position \(P^{\prime}\) with \(s_{3}\left(P^{\prime}\right)=0\) already, then \(P^{\prime}\) is a \(C\) position by the inductive hypothesis, and \(P\) is an \(A\)-position. Otherwise, player 2 can alter \(b\) to leave a position \(P^{\prime}\) with \(s_{3}\left(P^{\prime}\right)=0\), which is a \(C\)-position for player 2, so player 1 would be in an \(A\)-position, implying \(P\) is a \(B\)-position. In either case, \(P\) is not a \(C\) position if \(s_{3}(P) \neq 0\). If instead \(s_{3}(P)=0\), notice that altering any pile will result in a position \(P^{\prime}\) with \(s_{3}\left(P^{\prime}\right) \neq 0\) (by, e.g., the lemma). \(P^{\prime}\) is then either an \(A\)-position or a \(B\)-position. However, by the lemma applied to \(P, P^{\prime}\) cannot be moved to a position \(P^{\prime \prime}\) with \(s_{3}\left(P^{\prime \prime}\right)=0\) in one move, and since by the inductive hypothesis such positions are the only \(C\)-positions, \(P^{\prime}\) cannot be an \(A\)-position, so it must be a \(B\)-position, and \(P\) is a \(C\)-position as wanted. So the theorem holds for \(s=k\) as well, and therefore for all \(s\). Thus, the maximum possible value of \(x y\) is \(\boxed{7800}\).
7800
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
In the game of Nim, players are given several piles of stones. On each turn, a player picks a nonempty pile and removes any positive integer number of stones from that pile. The player who removes the last stone wins, while the first player who cannot move loses. Alice, Bob, and Chebyshev play a 3-player version of Nim where each player wants to win but avoids losing at all costs (there is always a player who neither wins nor loses). Initially, the piles have sizes \(43, 99, x, y\), where \(x\) and \(y\) are positive integers. Assuming that the first player loses when all players play optimally, compute the maximum possible value of \(x y\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Games/GameTheory/CombinatorialGameTheory", "/Mathematics/RecreationalMathematics/Games/GameTheory/ImpartialGame", "/Mathematics/RecreationalMathematics/Games/GameTheory/Nim-Value", "/Mathematics/RecreationalMathematics/Games/GameTheory/PerfectInformation", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Translate each pile size to a base‑3 number by reading its binary digits (ternation) and require the digitwise mod‑3 sum of these numbers to be zero for a losing position.
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ours_574
Solution. Let \(H, G, O, H^{\prime}\) denote the orthocenter of \(\triangle ABC\), centroid of \(\triangle BCD\), circumcenter of \(\triangle CDA\), and orthocenter of \(\triangle DAB\), respectively. The standard perpendicularity criterion applied to \(PH\) and \(BC\) and to \(PG\) and \(BC\) gives \[ BH^{2} - CH^{2} = BP^{2} - CP^{2} = BG^{2} - CG^{2} \] (so \(\overline{BC} \perp \overline{GH}\)), and similarly for the other five pairs of sides between \(ABCD\) and \(GOH^{\prime}H\). We can cancel a whole bunch of terms in these expressions, using for example the identities \[ BH^{2} = 4R_{ABC}^{2} - AC^{2} \] and \[ BG^{2} = \frac{2}{9}(BC^{2} + CD^{2} + DB^{2}) - \frac{1}{3}CD^{2} \] Due to the circumcenter we get \[ GC^{2} - GD^{2} = HA^{2} - HC^{2} = H^{\prime}D^{2} - H^{\prime}A^{2} = 0 \] Hence we define \(s = BC = BD = BA\). The remaining equations give \[ \frac{CD^{2} - s^{2}}{3} = DA^{2} - s^{2} = AC^{2} - s^{2} \] Letting \(DA = AC = a\), we find \(3a^{2} - 2s^{2} = 9\). This is a Pell-type equation, and the only solutions small enough are \((a, s) = (3, 3), (27, 33), (267, 327), (2643, 3237)\). So, using the largest one, the answer is \(3 + 2 \cdot 2643 + 3 \cdot 3237 = 15000\). \(\boxed{15000}\)
15000
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
Let \(ABCD\) be a tetrahedron whose six side lengths are all integers, and let \(N\) denote the sum of these side lengths. There exists a point \(P\) inside \(ABCD\) such that the feet from \(P\) onto the faces of the tetrahedron are the orthocenter of \(\triangle ABC\), centroid of \(\triangle BCD\), circumcenter of \(\triangle CDA\), and orthocenter of \(\triangle DAB\). If \(CD=3\) and \(N<100,000\), determine the maximum possible value of \(N\).
[ "/Mathematics/Geometry/Distance/Point-PointDistance3-Dimensional", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes/Simplex", "/Mathematics/Geometry/SolidGeometry/Polyhedra/Tetrahedra", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/PellEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]
Translate the orthogonal foot conditions into equalities of squared distances, which force three edges to be equal and reduce the problem to solving a Pell‑type equation for the remaining edge lengths.
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ours_575
This problem involves generating functions. The main claim is that \[ \ln \sum_{n \geq 0}\left(T_{n} \frac{x^{n}}{n!}\right)=\frac{e^{x}-1}{2-e^{x}} \] From this, the answer is the solution to \(\frac{a-1}{2-a}=\frac{6}{29}\), which gives \(a=\frac{41}{35}\). Therefore, \(m+n=41+35=76\). To establish the claim, let \( S(n, k) \) denote the Stirling numbers of the second kind. **Lemma 1:** We have \[ \sum_{n \geq 0}\left(x^{n} \cdot \sum_{a_{1}+2 a_{2}+\ldots n a_{n}=n} \frac{1}{a_{1}!a_{2}!\ldots a_{n}!}\right)=e^{\frac{x}{1-x}} \] where the \( a_{i} \) in the second sum are nonnegative integers. **Proof:** Compute \[ \sum_{n \geq 0}\left(x^{n} \cdot \sum_{a_{1}+2 a_{2}+\ldots n a_{n}} \frac{1}{a_{1}!a_{2}!\ldots a_{n}!}\right)=\prod_{i \geq 1}\left(\sum_{j \geq 0} \frac{x^{i j}}{j!}\right)=\prod_{i \geq 1} e^{x^{i}}=e^{\frac{x}{1-x}} \] For simplicity later, let \[ A_{k}=\sum_{a_{1}+2 a_{2}+\ldots n a_{n}} \frac{1}{a_{1}!a_{2}!\ldots a_{n}!} \] Now, we directly count \( T_{n} \). We do casework on how many vertices we have not including the root. Say we have \( k \) vertices, and from under the root, we have \( i \) branches of length \( a_{i} \). Then clearly, \( a_{1}+2 a_{2}+\ldots+k a_{k}=k \). There are \( k!S(n, k) \) ways to assign the numbers \( 1 \) to \( n \) to \( k \) vertices such that each vertex has at least one number assigned to it. This overcounts slightly: we need to divide by \( a_{1}!a_{2}!\ldots a_{n}! \) to account for permutations of the branches of equal length. So \[ T_{n}=\sum_{k \geq 0} k!S(n, k) A_{k} \] To finish, \[ \sum_{n \geq 0} T_{n} \cdot \frac{x^{n}}{n!}=\sum_{n \geq 0} \sum_{k \geq 0} k!S(n, k) A_{k} \cdot \frac{x^{n}}{n!}=\sum_{k \geq 0} k!A_{k} \sum_{n \geq 0} S(n, k) \cdot \frac{x^{n}}{n!}=\sum_{k \geq 0} A_{k}\left(e^{x}-1\right)^{k}=e^{\frac{e^{x}-1}{2-e^{x}}} \] by Lemma 1. Here we used the fact that \(\sum_{n \geq 0} S(n, k) \cdot \frac{x^{n}}{n!}=\frac{\left(e^{x}-1\right)^{k}}{k!}\). Thus, the solution is \(\boxed{76}\).
76
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns-2.md" }
For a positive integer \( n \), an \( n \)-branch \( B \) is an ordered tuple \((S_{1}, S_{2}, \ldots, S_{m})\) of nonempty sets (where \( m \) is any positive integer) satisfying \( S_{1} \subset S_{2} \subset \ldots \subset S_{m} \subseteq \{1,2, \ldots, n\} \). An integer \( x \) is said to appear in \( B \) if it is an element of the last set \( S_{m} \). Define an \( n \)-plant to be an (unordered) set of \( n \)-branches \(\{B_{1}, B_{2}, \ldots, B_{k}\}\), and call it perfect if each of \( 1,2, \ldots, n \) appears in exactly one of its branches. Let \( T_{n} \) be the number of distinct perfect \( n \)-plants (where \( T_{0}=1 \) ), and suppose that for some positive real number \( x \) we have the convergence \[ \ln \left(\sum_{n \geq 0} T_{n} \cdot \frac{(\ln x)^{n}}{n!}\right)=\frac{6}{29} \] If \( x=\frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \), compute \( m+n \).
[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]
Model perfect plants via set partitions and use exponential generating functions with Stirling numbers to obtain a closed‑form expression for the logarithm of the EGF.
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ours_576
Note that the essay scores represent diminishing marginal returns - the number of additional points you gain with each additional half hour decreases. Therefore, your sum of scores is maximized if you distribute your time equally among all the essays. In the problem, there are eight half-hours to distribute among four essays, so one should spend an hour on each. This nets 75 points per essay and hence a final average of 75. \(\boxed{75}\)
75
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
In English class, you have discovered a mysterious phenomenon - if you spend \( n \) hours on an essay, your score on the essay will be \( 100\left(1-4^{-n}\right) \) points if \( 2n \) is an integer, and \( 0 \) otherwise. For example, if you spend 30 minutes on an essay you will get a score of 50, but if you spend 35 minutes on the essay you somehow do not earn any points. It is 4 AM, your English class starts at 8:05 AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?
[ "/Mathematics/AppliedMathematics/Optimization/IntegerProgramming", "/Mathematics/AppliedMathematics/Optimization/NumberPartitioningProblem", "/Mathematics/DiscreteMathematics/DivisionProblems/NumberPartitioningProblem" ]
Allocate the available half‑hour blocks equally to each essay because the score function is concave (diminishing marginal returns).
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ours_577
Observe that the base \(\overline{OO^{\prime}}\) is fixed at 2014, so we wish to maximize the height of the triangle. Since \( M \) lies on one of the circles, the maximum possible height from \( M \) to the line \(\overline{OO^{\prime}}\) is 1 (the radius of the circle). Therefore, the maximal area of triangle \( OMO^{\prime} \) is \(\frac{1}{2} \times 2014 \times 1 = 1007\). \(\boxed{1007}\)
1007
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Consider two circles of radius one, and let \( O \) and \( O^{\prime} \) denote their centers. Point \( M \) is selected on either circle. If \( OO^{\prime} = 2014 \), what is the largest possible area of triangle \( OMO^{\prime} \)?
[ "/Mathematics/Geometry/Distance/Point-LineDistance2-Dimensional", "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Radius", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties" ]
Maximize the triangle’s area by using the fixed base OO′ and noting the height from M cannot exceed the circle’s radius.
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ours_578
Let \( N=1007 \). Observe that \[ 2+4+6+\cdots+2N=2(1+2+\cdots+N)=N(N+1) \] while \[ 1+3+\cdots+(2N-1)=N^2 \] These can be proven using the standard formula for an arithmetic series. Hence, the fraction in question is \[ \frac{N+1}{N}-\frac{N}{N+1}=\frac{2N+1}{N(N+1)}=\frac{2015}{1007 \cdot 1008} \] Because \(\gcd(2015,1007)=\gcd(2015,1008)=1\), the answer is \( 2015 \). \(\boxed{2015}\)
2015
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Suppose that \( m \) and \( n \) are relatively prime positive integers with \( A=\frac{m}{n} \), where \[ A=\frac{2+4+6+\cdots+2014}{1+3+5+\cdots+2013}-\frac{1+3+5+\cdots+2013}{2+4+6+\cdots+2014} \] Find \( m \). In other words, find the numerator of \( A \) when \( A \) is written as a fraction in simplest form.
[ "/Mathematics/Algebra/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/Algebra/NumberTheory/Arithmetic/Fractions", "/Mathematics/Algebra/NumberTheory/Integers/EvenNumber", "/Mathematics/Algebra/NumberTheory/Integers/OddNumber", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/NumberTheory/RationalNumbers/Q", "/Mathematics/Algebra/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/Algebra/Sums/Accumulate", "/Mathematics/Algebra/Sums/CumulativeSum", "/Mathematics/Algebra/Sums/PartialSum", "/Mathematics/Algebra/Sums/RunningSum", "/Mathematics/Algebra/Sums/Sum", "/Mathematics/NumberTheory/Arithmetic/Fractions", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/RationalNumbers/RationalNumber" ]
Express the even and odd sums using arithmetic‑series formulas (N(N+1) and N²) to rewrite the expression as (N+1)/N – N/(N+1).
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ours_579
Note that the average of a set of consecutive integers is the median, which is simply half the sum of the smallest and largest numbers on the piece. We can sort the pieces in the order of their averages. The first piece, with the smallest average of 9.5, must contain the smallest numbers. The largest number on this piece is \(2 \times 9.5 - 1 = 18\). Therefore, the first piece contains the numbers \(1, 2, \ldots, 18\). The next piece, with an average of 19, contains just the number 19. The third piece, with an average of 128, contains the numbers from 20 to 236, since \(\frac{20 + 236}{2} = 128\). The fourth piece, with an average of 345, contains the numbers from 237 to 453. The last piece, with an average of 1234, contains the numbers from 454 to 2014. Thus, \(n = 2014\). \(\boxed{2014}\)
2014
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
The integers \(1, 2, \ldots, n\) are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are 1234, 345, 128, 19, and 9.5 in some order. Compute \(n\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Partitions", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Integers/ConsecutiveNumbers", "/Mathematics/NumberTheory/Numbers/LargeNumbers", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]
Recognize that the average of a block of consecutive integers equals its midpoint, allowing you to compute each block’s endpoints from the given averages.
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ours_580
The main idea is that the smallest grades which round to a \(90\) are precisely those of the form \[ 89 . \underbrace{4444 \ldots 44}_{n \text{ digits }} 5 . \] As \(n\) grows arbitrarily large, this limit approaches \(89+\frac{4}{9}=\frac{805}{9}\). So the answer is \(805+9=814\). Note that \(M\) itself is not a grade (since grades have finitely many digits after the decimal point), so the distinction between \(x \geq M\) and \(x>M\) is not relevant. \(\boxed{814}\)
814
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a grade is a nonnegative decimal number with finitely many digits after the decimal point. Given a grade with digits \(a_{1} a_{2} \ldots a_{m} . b_{1} b_{2} \ldots b_{n}\), Joe first rounds the number to the nearest \(10^{-n+1}\) th place. He then repeats the procedure on the new number, rounding to the nearest \(10^{-n+2}\) th, then rounding the result to the nearest \(10^{-n+3}\) th, and so on, until he obtains an integer. For example, he rounds the number \(2014.456\) via \(2014.456 \rightarrow 2014.46 \rightarrow 2014.5 \rightarrow 2015\). There exists a rational number \(M\) such that a grade \(x\) gets rounded to at least \(90\) if and only if \(x \geq M\). If \(M=\frac{p}{q}\) for relatively prime integers \(p\) and \(q\), compute \(p+q\).
[ "/Mathematics/Algebra/NumberTheory/RationalNumbers/Q", "/Mathematics/Algebra/NumberTheory/RationalNumbers/RationalNumber", "/Mathematics/Algebra/NumberTheory/Rounding/NearestIntegerFunction", "/Mathematics/Algebra/NumberTheory/Rounding/Nint", "/Mathematics/RecreationalMathematics/Puzzles" ]
Find the smallest numbers that round up to 90, which have the form 89.44…45, and take their limiting value as the number of 4s grows.
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ours_581
Because \( 31 \) is prime, we have \( L_{31} = 31 \times L_{30} \). Hence, it suffices to compute the remainder when \( 31 \times 62800 \) is divided by \( 10^{5} \), which is \( 46800 \). \(\boxed{46800}\)
46800
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( L_{n} \) be the least common multiple of the integers \( 1, 2, \ldots, n \). For example, \( L_{10} = 2,520 \) and \( L_{30} = 2,329,089,562,800 \). Find the remainder when \( L_{31} \) is divided by \( 100,000 \).
[ "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Use that a new prime factor multiplies the previous LCM, so L₃₁ = 31·L₃₀.
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ours_582
Let \( A_n \) denote the value when the last two digits of \( 17n \) are deleted. Notice that \( A_{n+1} - A_n \) is either \( 0 \) or \( 1 \) for every \( n \). Hence, the problem is asking for the number of \( n \) with \( 10 \leq n \leq 500 \) such that \( A_{n+1} - A_n = 1 \). As \( n \) ranges from \( 10 \) to \( 500 \), the smallest number is \( 17 \times 10 = 170 \) and the largest number is \( 17 \times 500 + 17 = 8517 \). In other words, \( A_{10} = 1 \) and \( A_{501} = 85 \). Hence, for \( 10 \leq n \leq 500 \), the value of \( A_n \) is increased exactly \( 84 \) times. Therefore, the answer is \(\boxed{84}\).
84
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
How many integers \( n \) with \( 10 \leq n \leq 500 \) have the property that the hundreds digit of \( 17n \) and \( 17n + 17 \) are different?
[ "/Mathematics/DiscreteMathematics/Combinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Arithmetic/MultiplicationandDivision", "/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" ]
The hundreds digit changes exactly when 17n crosses a multiple of 100, so count the multiples of 100 between 17·10 and 17·501.
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ours_583
Add eight times the first equation, four times the second, and two times the third to the fourth. We obtain: \[ 16 a_{1} + 16 a_{2} + 16 a_{3} + 16 a_{4} + 15 a_{5} = 30 + a_{4} + a_{3} + a_{2} + a_{1} \] Letting \( S \) be the desired sum, we derive \( 16S = 30 + S \), so \( S = 2 \). Thus, the sum \( a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \) is \(\boxed{2}\).
2
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \) be real numbers satisfying \[ \begin{aligned} 2 a_{1} + a_{2} + a_{3} + a_{4} + a_{5} &= 1 + \frac{1}{8} a_{4} \\ 2 a_{2} + a_{3} + a_{4} + a_{5} &= 2 + \frac{1}{4} a_{3} \\ 2 a_{3} + a_{4} + a_{5} &= 4 + \frac{1}{2} a_{2} \\ 2 a_{4} + a_{5} &= 6 + a_{1} \end{aligned} \] Compute \( a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \).
[ "/Mathematics/Algebra/AlgebraicEquations/LinearEquation", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/LinearAlgebra/LinearSystemsofEquations/LinearSystemofEquations", "/Mathematics/Algebra/Sums/CumulativeSum", "/Mathematics/Algebra/Sums/Sum" ]
Select multipliers for each equation so that the variables' coefficients align, then sum to obtain an equation in the total sum.
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ours_584
Let \(n=18\) and suppose \(1=r_{1} \leq r_{2} \leq \cdots \leq r_{n}\) is a set of ranks for the students. The main observation is that every such set of ranks gives rise to exactly one set of scores. Explicitly, we simply set the score of the \(n\)th student as \(1\), and then work backwards to construct the set of scores for the students. Because there are \(21 \geq n\) points available on the contest, this construction will always terminate successfully. Hence, the problem is equivalent to counting the number of tuples of ranks. Evidently \(r_{1}=1\). For each subsequent \(i\), either \(r_{i+1}=r_{i}\), or \(r_{i+1}\) is \(r_{i}\) plus the number of indices \(j\) with \(r_{i}=r_{j}\). In other words, we either stay the same or increase. This means that we have \(2\) choices at each step, and the answer is \(2^{n-1}=2^{17}=131072\). \(\boxed{131072}\)
131072
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Eighteen students participate in a team selection test with three problems, each worth up to seven points. All scores are nonnegative integers. After the competition, the results are posted in a table with three columns: the student's name, score, and rank (allowing ties), respectively. A student's rank is one greater than the number of students with strictly higher scores (for example, if seven students score \(0, 0, 7, 8, 8, 14, 21\), then their ranks would be \(6, 6, 5, 3, 3, 2, 1\) respectively). When the results are read, the rank column is accidentally read as the score column and vice versa. Coincidentally, the results still made sense! If the scores of the students were \(x_{1} \leq x_{2} \leq \cdots \leq x_{18}\), determine the number of possible values of the 18-tuple \((x_{1}, x_{2}, \ldots, x_{18})\). In other words, determine the number of possible multisets (sets with repetition) of scores.
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Map each possible rank sequence to a unique score multiset, noting that at each student you either keep the same rank or increase it by the size of the current tie group.
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ours_585
Let \(\omega\) denote the circumcircle of the 4000-gon and let \(O\) denote its center. One can verify that \(OXA_{2014}Y\) is a rectangle, hence \(1 = XY = OA_{2014}\) is a radius of \(\omega\). Consequently, the side length of the square is \(\sqrt{2}\) and the area is \((\sqrt{2})^{2} = 2\). \(\boxed{2}\)
2
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( A_{1} A_{2} \ldots A_{4000} \) be a regular \( 4000 \)-gon. Let \( X \) be the foot of the altitude from \( A_{1986} \) onto diagonal \( A_{1000} A_{3000} \), and let \( Y \) be the foot of the altitude from \( A_{2014} \) onto \( A_{2000} A_{4000} \). If \( XY = 1 \), what is the area of square \( A_{500} A_{1500} A_{2500} A_{3500} \)?
[ "/Mathematics/Geometry/Distance/Point-LineDistance2-Dimensional", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Square", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Squares/Square" ]
Observe that the center O together with the foot points X, Y and the vertex A_{2014} form a rectangle, so XY equals the circumradius.
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ours_586
Because \( AB = CD \), we can construct a point \( X' \) outside of \( ABCD \) such that \(\triangle AX'B \cong \triangle CXD\). Additionally, \[ \angle AXB + \angle AX'B = \angle AXB + \angle CXD = 180^\circ \] so \( AX'BX \) is cyclic, and it's sufficient to find the area of this quadrilateral. Let \( O \) be the center of its circumcircle. Note that we can rearrange the triangles \( OAX', OX'B, OBX, OXA \) while keeping each of the points equidistant from \( O \), so they are still cyclic and do not change the total area of \( AX'BX \). The side lengths of \( AX'BX \) are 10, 5, 11, 14, so we rearrange the lengths to be in the order 14, 5, 11, 10. But \( 14^2 + 5^2 = 11^2 + 10^2 = 221 \), so this new cyclic quadrilateral is composed of two right triangles with common hypotenuse \(\sqrt{221}\). Thus its total area is \[ \frac{1}{2} \cdot 14 \cdot 5 + \frac{1}{2} \cdot 11 \cdot 10 = 90 \] \(\boxed{90}\)
90
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( X \) be a point inside convex quadrilateral \( ABCD \) with \(\angle AXB + \angle CXD = 180^\circ\). If \( AX = 14 \), \( BX = 11 \), \( CX = 5 \), \( DX = 10 \), and \( AB = CD \), find the sum of the areas of \(\triangle AXB\) and \(\triangle CXD\).
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/CyclicQuadrangle", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/CyclicQuadrilateral", "/Mathematics/Geometry/PlaneGeometry/Triangles/SpecialTriangles", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Create a congruent triangle to form a cyclic quadrilateral whose side lengths satisfy a² + c² = b² + d², giving two right triangles and easy area sum.
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ours_587
By simple angle chasing, we find that \[ \angle A T C = \angle A T B + \angle B T C = \angle T E B + \angle D T E = \angle T C B. \] Thus, line \(A T\) is also a tangent to the circumcircle of triangle \(T E F\). Hence \[ A B \cdot A E = A T^2 = A C \cdot A D = 38 \cdot 53 = 2014. \] This implies \(A E = 1007\), whence \(D E = 1007 - 53 = 954\). \(\boxed{954}\)
954
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
The points \(A, B, C, D, E\) lie on a line \(\ell\) in this order. Suppose \(T\) is a point not on \(\ell\) such that \(\angle B T C = \angle D T E\), and \(\overline{A T}\) is tangent to the circumcircle of triangle \(B T E\). If \(A B = 2\), \(B C = 36\), and \(C D = 15\), compute \(D E\).
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/CirclePower", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles/CircularTriangle", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles", "/Mathematics/Geometry/Points/Point" ]
Use the equal-angle condition to show AT is tangent to both circles, then apply power of a point (AT² = AB·AE = AC·AD).
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ours_588
Let \( n = 10 \). By the Hockey Stick identity, we can factor the polynomial as follows: \[ p(x) = \left(x^{n} + x^{n-1} + x^{n-2} + \ldots + x + 1\right) \left(\sum_{i=0}^{n} \binom{n+i}{n} x^{n-i}\right) \] The polynomial \( x^{10} + x^{9} + \cdots + 1 \) is irreducible (it is the 11th cyclotomic polynomial), and this must be the unique factorization into two polynomials of degree \( n \). Therefore, the answer is: \[ g(2) = 1 + 2 + \cdots + 2^{10} = 2047 \] \(\boxed{2047}\)
2047
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Suppose that \( g \) and \( h \) are polynomials of degree 10 with integer coefficients such that \( g(2) < h(2) \) and \[ g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11} x^{k-1} \right) \] holds for all nonzero real numbers \( x \). Find \( g(2) \).
[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity", "/Mathematics/Algebra/Cyclotomy/CyclotomicFactorization", "/Mathematics/Algebra/Cyclotomy/CyclotomicPolynomial", "/Mathematics/Algebra/Polynomials/Binomial", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/IrreduciblePolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialFactorization", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial" ]
Factor the given polynomial using the Hockey‑Stick identity to expose the 11th cyclotomic polynomial as one degree‑10 factor.
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ours_589
Construct the points \( B' \) and \( C' \) on \(\overline{BC}\) such that \( CA = CB' \) and \( BA = BC' \). These are the reflections of \( B \) and \( C \) across lines \( CI \) and \( BI \), respectively. Thus, \( AI = B'I = C'I \), and hence \( B' \) and \( C' \) are precisely the points \( X \) and \( Y \). Therefore, the length \( XY \) is calculated as follows: \[ XY = AB + AC - BC = 1400 + 1800 - 2014 = 1186 \] Thus, the length \( XY \) is \(\boxed{1186}\).
1186
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( \triangle ABC \) be a triangle with incenter \( I \) and \( AB = 1400 \), \( AC = 1800 \), \( BC = 2014 \). The circle centered at \( I \) passing through \( A \) intersects line \( BC \) at two points \( X \) and \( Y \). Compute the length \( XY \).
[ "/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/Incenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles" ]
Reflect B and C across the internal angle bisectors to locate X and Y, turning the chord length into AB+AC-BC.
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ours_590
Let us ignore for now the indices modulo \(7\). The process described selects a 10-digit binary number between \(0\) and \(2^{10}-1=1023\) inclusive. We end at \(C_{0}\) if and only if the number is divisible by \(7\). There are \(147\) such multiples of \(7\), so the probability is \(\frac{147}{1024}\). Therefore, \(m = 147\). \(\boxed{147}\)
147
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
In Prime Land, there are seven major cities, labelled \(C_{0}, C_{1}, \ldots, C_{6}\). For convenience, we let \(C_{n+7}=C_{n}\) for each \(n=0,1, \ldots, 6\); i.e., we take the indices modulo \(7\). Al initially starts at city \(C_{0}\). Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city \(C_{k}\), he moves to city \(C_{2k}\); otherwise, he moves to city \(C_{2k+1}\). If the probability that Al is back at city \(C_{0}\) after \(10\) moves is \(\frac{m}{1024}\), find \(m\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/BinarySequences", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/ProbabilityandStatistics/MarkovProcesses/MarkoffChain", "/Mathematics/ProbabilityandStatistics/MarkovProcesses/MarkovChain", "/Mathematics/ProbabilityandStatistics/MarkovProcesses/MarkovProcess", "/Mathematics/ProbabilityandStatistics/Probability/CoinFlipping", "/Mathematics/ProbabilityandStatistics/Probability/CoinTossing", "/Mathematics/ProbabilityandStatistics/RandomWalks/AbsolutelyFair", "/Mathematics/ProbabilityandStatistics/RandomWalks/RandomWalk" ]
Interpret the coin‑flip sequence as a 10‑bit binary number and return to C₀ exactly when that number is divisible by 7.
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ours_591
Notice that counting such numbers is equivalent to counting multisets of primes such that one prime exceeds the product of the others. Let \( p \) be the largest prime, and express \( n = p \cdot x \). Note that \( 1 \leq x \leq p-1 \) and that any value of \( x \) is achievable in exactly one way (by unique prime factorization). Hence, for a fixed value of \( p \), there are exactly \( p-1 \) values of \( x \) possible. The answer is thus \[ \sum_{\substack{p \leq 30 \\ p \text{ prime }}} (p-1) = 129 \] \(\boxed{129}\)
129
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Say a positive integer \( n \) is radioactive if one of its prime factors is strictly greater than \(\sqrt{n}\). For example, \(2012=2^{2} \cdot 503\), \(2013=3 \cdot 11 \cdot 61\), and \(2014=2 \cdot 19 \cdot 53\) are all radioactive, but \(2015=5 \cdot 13 \cdot 31\) is not. How many radioactive numbers have all prime factors less than \(30\)?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/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-Plus", "/Mathematics/NumberTheory/Numbers", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization" ]
Fix the largest prime factor p; the rest of the number must be a product less than p, giving exactly p‑1 possible values.
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ours_592
Let \( O \) be the midpoint of \(\overline{A B}\), and let \( A B = 2 r \). The key observation is that \(\triangle L O Y \sim \triangle Y B K\). Since \(\angle Y Z A = \angle Y A Z = \angle L A Z\), we find \(\overline{L A}\) is a tangent to the semicircle. This follows because \[ \angle A O L = \frac{1}{2} \angle A O Y = \angle O B Y \Longrightarrow \overline{L O} \parallel \overline{Y B} \quad \text{and} \quad \angle A O Y = \frac{1}{2} \angle A O Z = \angle O B Z \Longrightarrow \overline{Y O} \parallel \overline{Z B} \] We set \( Y K = k h \) and \( K B = k r \), where \( A L = L Y = h \). First, we claim \(\frac{A K}{A X} = \frac{3}{2} k + 1\). By Ptolemy's Theorem on \( A B K L \), \[ B L \cdot A K = A L \cdot B K + A B \cdot L K = 2(k+1) r h + k h r = (3 k + 2) h r. \] But \(\frac{1}{2}[L A B] = B L \cdot A X = A L \cdot A B = h(2 r)\) and the conclusion follows. Next, we claim \( k = \frac{2 r^{2}}{r^{2} + h^{2}} \). This follows from \( L B^{2} = A L^{2} + A B^{2} = K L^{2} + K B^{2} \), whence \( h^{2} + 4 r^{2} = (k+1)^{2} h^{2} + k^{2} r^{2} \). Then, \( k(k+2) h^{2} = (4-k^{2}) r^{2} \). Cancelling a factor of \( 2+k \) gives \( k h^{2} = (2-k) r^{2} \), so \( k = \frac{2 r^{2}}{r^{2} + h^{2}} \). Thus, if we let \( x = \frac{h}{r} \), we get that \[ \begin{aligned} \frac{A K}{A X} + \left(\frac{A L}{A B}\right)^{2} & = \frac{3}{2} k + 1 + \left(\frac{h}{2 r}\right)^{2} \\ & = \frac{3}{1+x^{2}} + 1 + \frac{x^{2}}{4} \\ & = \frac{3}{4} + \frac{3}{1+x^{2}} + \frac{1+x^{2}}{4} \\ & \geq \frac{3}{4} + 2 \sqrt{\frac{3}{4}} \\ & = \frac{3}{4} + \sqrt{3} \end{aligned} \] and the answer is \(\boxed{343}\).
343
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( A X Y B Z \) be a convex pentagon inscribed in a circle with diameter \(\overline{A B}\). The tangent to the circle at \( Y \) intersects lines \( B X \) and \( B Z \) at \( L \) and \( K \), respectively. Suppose that \(\overline{A Y}\) bisects \(\angle L A Z\) and \( A Y = Y Z \). If the minimum possible value of \[ \frac{A K}{A X}+\left(\frac{A L}{A B}\right)^{2} \] can be written as \(\frac{m}{n}+\sqrt{k}\), where \( m, n \) and \( k \) are positive integers with \(\operatorname{gcd}(m, n)=1\), compute \( m+10 n+100 k \).
[ "/Mathematics/Geometry/Distance", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Diameter", "/Mathematics/Geometry/PlaneGeometry/Circles/Semicircle", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similar", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similarity", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Use triangle similarity to relate the unknown lengths, then apply Ptolemy’s theorem to obtain an algebraic expression that is minimized by AM‑GM.
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ours_593
This is equivalent to \((x+y+n)\left(x^{2}+y^{2}+n^{2}-x y-n(x+y)\right)=m+n^{3}\). Infinitely many solutions exist if and only if \(m=-n^{3}\). Given the range \(-2014 \leq m \leq 2014\), this condition allows for \(n\) values such that \(-2014 \leq -n^{3} \leq 2014\). Solving for \(n\), we find that \(-12 \leq n \leq 12\). Therefore, there are \(25\) possible integer values for \(n\), and for each \(n\), there is exactly one corresponding \(m\). Thus, there are \(\boxed{25}\) pairs \((m, n)\).
25
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Find the number of pairs \((m, n)\) of integers with \(-2014 \leq m, n \leq 2014\) such that \(x^{3}+y^{3}=m+3 n x y\) has infinitely many integer solutions \((x, y)\).
[ "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation3rdPowers", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/Z" ]
Factor the equation to isolate a linear factor whose vanishing forces the whole expression to be zero, yielding the condition m = -n³.
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ours_594
Solution. We can express \( n = 2u^{2} \), where \( u \) is a positive integer. Consider the function \[ g(n) = \frac{\tau(n)^{2}}{n} \] which is multiplicative. The exponent of 2 in \( n \) is odd, so we find that \( g(2) = g(8) = 2 \), but \( g(2^{j}) < 2 \) for \( j \geq 5 \). For any prime \( p \) other than 2, the exponent in \( n \) is even, and we can verify that \( g(p^{2k}) \leq 1 \), with equality only when \( p = 3 \) and \( k \in \{0, 1\} \). Therefore, \( g(n) \leq 2 \) for all \( n \), with equality at \( n = 2, 8, 18, 72 \). Thus, the sum of these integers is \( 2 + 8 + 18 + 72 = 100 \). \(\boxed{100}\)
100
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Find the sum of all positive integers \( n \) such that \(\tau(n)^{2} = 2n\), where \(\tau(n)\) is the number of positive integers dividing \( n \).
[ "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/DivisorFunction" ]
Exploit that τ(n)^2/n is multiplicative, bound each prime‑power factor, which forces n=2·u^2 with very limited prime exponents.
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ours_595
The key is the following lemma: if \( O \) lies inside triangle \( AEF \) and \( \angle EOF - \angle A = 90^\circ \), then \( \angle A = 45^\circ \). Indeed, the conditions imply that pentagon \( BEOF C \) is cyclic, so \( \angle BOC = 90^\circ \). Simple calculations give that \( \angle C = 70^\circ - \frac{1}{3} \angle A \). So we wish to minimize \( \angle A \). Note that if \( \angle A \leq 45^\circ \), then \( \angle BOC \leq 90^\circ \), which implies that \( O \) lies outside the circle with diameter \( \overline{BC} \) while inside \( \triangle ABC \), which causes the lemma to apply. In other words, \( \angle A \leq 45^\circ \Longrightarrow \angle A = 45^\circ \). So the minimum possible value of \( \angle C \) is \( 55^\circ \). This is achieved if \( (\angle A, \angle B, \angle C) = \left(45^\circ, 80^\circ, 55^\circ\right) \). Converting to radians, \( 55^\circ = \frac{11}{36} \pi \) and \( 11 + 36 = 47 \). \(\boxed{47}\)
47
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( \triangle ABC \) be an acute triangle with circumcenter \( O \), and select \( E \) on \( \overline{AC} \) and \( F \) on \( \overline{AB} \) so that \( \overline{BE} \perp \overline{AC} \) and \( \overline{CF} \perp \overline{AB} \). Suppose \( \angle EOF - \angle A = 90^\circ \) and \( \angle AOB - \angle B = 30^\circ \). If the maximum possible measure of \( \angle C \) is \(\frac{m}{n} \cdot 180^\circ\) for some positive integers \( m \) and \( n \) with \( m < n \) and \(\gcd(m, n) = 1\), compute \( m+n \).
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circumcenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/Geometry/Trigonometry/Angles/AcuteAngle", "/Mathematics/Geometry/Trigonometry/Angles/Angle" ]
Use the given angle condition to prove a cyclic configuration, which forces ∠A to be 45°.
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ours_597
Let \( p = 3, q = 5, \) and \( r = 7 \). Note that: \[ f(pq) f(qr) f(rp) = f(pq + qr + rp) f(0)^2 \pmod{pqr} \] This follows from the Chinese Remainder Theorem; we consider the equation modulo each of \( p, q, r \). For this particular problem, we see \( 2014 \equiv f(-34) \equiv f(15 + 21 + 34) \pmod{105} \) and \( f(15) f(21) f(35) \equiv 10 \pmod{105} \). Now, let \( x = f(0) \). Accordingly, we find: \[ x^2 \equiv 10 \cdot 2014^{-1} \equiv (-95) \cdot 19^{-1} \equiv -5 \pmod{105} \] The solutions are \( x \equiv 10, 25, 80, 95 \pmod{105} \) by the Chinese Remainder Theorem. The smallest non-negative solution is \( x = 10 \). Thus, the smallest possible value of \( f(0) \) is \(\boxed{10}\).
10
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring14Solns.md" }
Let \( f(x) \) be a polynomial with integer coefficients such that \( f(15) f(21) f(35) - 10 \) is divisible by 105. Given \( f(-34) = 2014 \) and \( f(0) \geq 0 \), find the smallest possible value of \( f(0) \).
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Apply the Chinese Remainder Theorem to turn the given divisibility into a congruence for f(0) and solve the resulting quadratic modulo 105.
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ours_598
Solution. All one-digit integers work, but for any integer \( n \) with more digits, the sum of the digits of \( n \) is strictly less than the value of \( n \) itself. Thus, the answer is the largest one-digit integer, which is 9. \(\boxed{9}\)
9
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
What is the largest positive integer which is equal to the sum of its digits?
[ "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/N", "/Mathematics/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus", "/Mathematics/NumberTheory/Numbers/SmallNumbers", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Observe that any integer with two or more digits is greater than the sum of its digits, so only one‑digit numbers can equal their digit sum.
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ours_600
It's clear that at least 12 hours must pass, since clock I accurately reflects the time while clock IV stands still at 7PM. But at twelve hours, clocks II and III also display the correct time of 7PM. Therefore, the answer is \(\boxed{12}\).
12
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
On a large wooden block there are four twelve-hour analog clocks of varying accuracy. At 7PM on April 3, 2015, they all correctly displayed the time. The first clock is accurate, the second clock is two times as fast as the first clock, the third clock is three times as fast as the first clock, and the last clock doesn't move at all. How many hours must elapse (from 7PM) before the times displayed on the clocks coincide again? (The clocks do not distinguish between AM and PM.)
[ "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Find the smallest time when each clock has completed an integer number of full 12‑hour cycles (the LCM of their periods).
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ours_601
Let \(f(x) = x^{2} - 4x + 100\). Note that \(f(1) = f(3)\) because \(x^{2} - 4x + 100 = (x-1)(x-3) + 97\). Therefore, the function \(f(x)\) takes on the same value for \(x = 1\) and \(x = 3\). To find the sum of all distinct values of \(f(x)\) for \(x\) from 1 to 100, we calculate: \[ \sum_{k=1}^{100} \left(k^{2} - 4k + 100\right) - f(1) \] This simplifies to: \[ \frac{100 \cdot 101 \cdot 201}{6} - 4 \cdot \frac{100 \cdot 101}{2} + 100 \cdot 100 - 97 = 328053 \] Thus, the sum of all distinct possible values is \(\boxed{328053}\).
328053
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Find the sum of all distinct possible values of \(x^{2}-4x+100\), where \(x\) is an integer between 1 and 100, inclusive.
[ "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialFunction", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/Algebra/Sums/PartialSum", "/Mathematics/Algebra/Sums/PowerSum", "/Mathematics/Algebra/Sums/Sum", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]
Notice the quadratic repeats a value (f(1)=f(3)) so subtract that duplicate when summing distinct outputs.
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ours_602
Let \( D' \) and \( E' \) be situated on \( AC \) and \( AB \) such that lines \( DD' \), \( EE' \), and \( BC \) are parallel. From the diagram, we see that \( XY \) is the average of \( DD' \) and \( EE' \). Therefore, the answer is \[ \frac{1}{2}(DD' + EE') = \frac{1}{2}(48 \cdot 2 + 52 \cdot 2) = 48 + 52 = 100 \] Thus, the length of \( XY \) is \(\boxed{100}\).
100
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( \triangle ABC \) be an isosceles triangle with \( \angle A = 90^\circ \). Points \( D \) and \( E \) are selected on sides \( AB \) and \( AC \), and points \( X \) and \( Y \) are the feet of the altitudes from \( D \) and \( E \) to side \( BC \). Given that \( AD = 48\sqrt{2} \) and \( AE = 52\sqrt{2} \), compute \( XY \).
[ "/Mathematics/Geometry/Distance", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Triangles/SpecialTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleLines", "/Mathematics/Geometry/PlaneGeometry/Triangles/TrianglePoints", "/Mathematics/Geometry/Points/PlanarDistance", "/Mathematics/Geometry/Points/Point", "/Mathematics/Geometry/Points/Point-PointDistance2-Dimensional" ]
Use lines through D and E parallel to BC so that XY equals the arithmetic mean of the parallel segments DD' and EE'
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ours_603
There are \(6 \cdot 5\) ways to select a rook for the pair of farthest edges, and \(6 \cdot 5\) ways for the remaining columns. Now it's straightforward to see that there are \(4!\) ways to finish from here, since the central \(6 \times 6\) square is essentially cut down to a \(4 \times 4\) subsquare. Hence the answer is \[ (6 \cdot 5)^{2} \cdot 4! = 21600 \] \(\boxed{21600}\)
21600
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
We delete the four corners of an \(8 \times 8\) chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Fix the rooks in the two outermost rows and columns first, then the problem reduces to a 4×4 permutation count for the remaining rooks.
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ours_604
The point is to count the powers of two. Let \( x_{1} = 10^{2015}, \ldots, x_{n} \) be the terms of the geometric progression, where \( x_n \) is odd. Call \( a_{k} \) the exponent of 2 in the prime factorization of \( x_{k} \). Then \( a_{1} = 2015 \), \( a_{n} = 0 \), and the \( a_{k} \) form an arithmetic progression. If \( d \) is the common difference, then we have \((n-1) d = 2015\). So the number of possible \( n \) corresponds to the factors of \( 2015 \), of which there are 8. \(\boxed{8}\)
8
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
A geometric progression of positive integers has \( n \) terms; the first term is \( 10^{2015} \) and the last term is an odd positive integer. How many possible values of \( n \) are there?
[ "/Mathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Divisors/DivisibilityRules", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/GreatestDividingExponent", "/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", "/Mathematics/NumberTheory/Sequences/CommonDifference", "/Mathematics/NumberTheory/Sequences/GeometricSequence" ]
Translate the odd final term condition into a linear drop of the exponent of 2, so (n‑1)·d equals the initial 2‑exponent.
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ours_605
Since \(10^{5}\) has exactly 10 prime factors, we find that \(\frac{x_{m+1}}{x_{m}} \in\{2,5\}\) for each \(m\). We must pick each of the two primes exactly 5 times. There are 10 positions, so the answer is \(\binom{10}{5}=252\). \(\boxed{252}\)
252
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Determine the number of sequences of positive integers \(1=x_{0}<x_{1}<\cdots<x_{10}=10^{5}\) with the property that for each \(m=0, \ldots, 9\) the number \(\frac{x_{m+1}}{x_{m}}\) is a prime number.
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/PrimeNumbers/PrimeArrangements", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberSequences", "/Mathematics/NumberTheory/Sequences/IncreasingSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]
Use the prime factorization of 10^5 to determine that the successive ratios must consist of exactly five 2's and five 5's.
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ours_606
We compute $$ \begin{aligned} 5 \sum_{k=1}^{99}\left(k^{2}+k\right)\left(k^{2}+k+1\right) & =\sum_{k=1}^{99}\left(5 k^{4}+10 k^{3}+10 k^{2}+5 k\right) \\ & =\sum_{k=1}^{99}\left((k+1)^{5}-k^{5}\right)-\sum_{k=1}^{99} 1 \\ & =100^{5}-1^{5}-99=9999999900 \end{aligned} $$ The sum of the decimal digits of \(9999999900\) is \(9 \cdot 8 = 72\). Thus, the answer is \(\boxed{72}\).
72
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Find the sum of the decimal digits of the number $$ 5 \sum_{k=1}^{99} k(k+1)\left(k^{2}+k+1\right) $$
[ "/Mathematics/Algebra/AlgebraicIdentities/AlgebraicIdentity", "/Mathematics/Algebra/AlgebraicIdentities/PolynomialIdentity", "/Mathematics/Algebra/Polynomials/Binomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialIdentity", "/Mathematics/Algebra/Sums/TelescopingSeries", "/Mathematics/Algebra/Sums/TelescopingSum", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]
Express the quartic sum as a telescoping difference of consecutive fifth powers.
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ours_607
Solution. Note that if a regular \( m \)-gon and a regular \( n \)-gon intersect at least one point, then they intersect at exactly \(\operatorname{gcd}(m, n)\) points. Now we claim that if two pairs of polygons intersect, then all three intersect at some unique point. Indeed, in this case, we can split up the initial circle evenly into 105 points, for which every 3 points lie on the 35-gon, every 5 points lie on the 21-gon, and every 7 points lie on the 15-gon, with variable starting positions. So one of these 105 points is a common vertex of all three polygons if it is the solution \( x \pmod{105} \) to some set of congruences \[ \begin{array}{ll} x \equiv a & \pmod{3} \\ x \equiv b & \pmod{5} \\ x \equiv c & \pmod{7} \end{array} \] By the Chinese Remainder Theorem, there is exactly one unique solution \( x \) to this set of congruences, so the claim is proven. Now we split the problem into several cases based on intersecting pairs. If no two polygons have a common vertex, there are \( 15 + 21 + 35 = 71 \) distinct vertices on the circle. If the 15-gon and 21-gon share 3 vertices but neither shares a vertex with the 35-gon, there are 68 distinct vertices. Similarly, in the cases for the other two pairs intersecting, we get \( 71 - 5 = 66 \) and \( 71 - 7 = 64 \) vertices. If all three polygons intersect, by the Principle of Inclusion-Exclusion, there are a total of \( 15 + 21 + 35 - 3 - 5 - 7 + 1 = 57 \) distinct vertices. Thus the final answer is \( 71 + 68 + 66 + 64 + 57 = 326 \). \(\boxed{326}\)
326
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Nicky has a circle. To make his circle look more interesting, he draws a regular 15-gon, 21-gon, and 35-gon such that all vertices of all three polygons lie on the circle. Let \( n \) be the number of distinct vertices on the circle. Find the sum of the possible values of \( n \).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Concyclic", "/Mathematics/Geometry/PlaneGeometry/Polygons/Pentadecagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularPolygon", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/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", "/Mathematics/NumberTheory/Numbers", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Model the polygon vertices as equally spaced points and use gcd for pairwise overlaps and the Chinese Remainder Theorem to determine if a triple overlap exists.
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ours_608
Clearly, any witness \( f \) must be surjective; otherwise, we can take \( Y \) to be the range of \( f \). Considering the cycle decomposition of \( f \), we find that \( f \) must consist of only a single cycle, or else we could take \( Y \) to be any such cycle. However, any such "single cycle" works. By a standard argument, there are \( 5! = 120 \) such witnesses. \(\boxed{120}\)
120
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( S \) be a set. We say \( S \) is \( D^{*} \)-finite if there exists a function \( f: S \rightarrow S \) such that for every nonempty proper subset \( Y \subsetneq S \), there exists a \( y \in Y \) such that \( f(y) \notin Y \). The function \( f \) is called a witness of \( S \). How many witnesses does \(\{0,1, \cdots, 5\}\) have?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]
The condition forces the function to be a single-cycle permutation (a cyclic bijection).
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ours_609
Since there are exactly \(7^{6}\) ways to obtain a nonzero total score, we discover everyone underneath the 117649th rank scored a zero. Hence this contestant's score must be \(1^{6} = 1\), the lowest possible nonzero score. \(\boxed{1}\)
1
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem, you can earn an integer score from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are \(8^{6} = 262144\) participants, and no two get the same score on every problem. Find the score of the participant whose rank was \(7^{6} = 117649\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Count the vectors with all scores non‑zero (7⁶) to see the rank where non‑zero products end, giving the lowest non‑zero product.
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ours_610
Suppose that the three side lengths are \(5a, 80b, 112c\), where \(a, b, c\) are positive integers. By the triangle inequality, we must have \[ 5a > |80b - 112c| = 16|5b - 7c| \] noting that the right-hand side cannot be \(0\), as the triangle is scalene. So \(5a \geq 16\), and thus \(5a \geq 20\). Clearly, \(80b\) and \(112c\) are at least \(20\), so the smallest side length is always at least \(20\). Conversely, we choose \(b, c\) such that \(|5b - 7c| = 1\) holds (which is possible since \(5\) and \(7\) are relatively prime): \(b = 3\) and \(c = 2\) suffice. Therefore, the triangle with side lengths \(20, 240\), and \(224\), which is a triangle, gives \(20\) as the minimum possible side length. \(\boxed{20}\)
20
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( ABC \) be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of \(5, 80\), and \(112\), respectively. What is the smallest possible side length that can appear in any stable triangle?
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/NumberTheory/DiophantineEquations/Diophantine", "/Mathematics/NumberTheory/DiophantineEquations/DiophantineEquation", "/Mathematics/NumberTheory/DiophantineEquations/IntegerTriangle", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z" ]
Use the triangle inequality to force 5a ≥ 16·|5b − 7c| and then pick b,c so that |5b − 7c|=1 using the coprimality of 5 and 7.
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ours_611
The main idea is to treat the disk as a single point, namely its center. If we consider an inner \(2013 \times 2013\) square, then we can treat the center of the ball as a particle being reflected by the "edges" of this inner square. With this insight, upon drawing out the various bounces, we see that the ball travels the diagonal of a \(3M \times 4M\) rectangle, where \(M = 2013\). This distance is \(5M = 10065\). Thus, the total distance the center of the disk traveled is \(\boxed{10065}\).
10065
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \(ABCD\) be a square with side length 2015. A disk with unit radius is packed neatly inside corner \(A\) (i.e., tangent to both \(\overline{AB}\) and \(\overline{AD}\)). Alice kicks the disk, which bounces off \(\overline{CD}, \overline{BC}, \overline{AB}, \overline{DA}, \overline{DC}\) in that order, before landing neatly into corner \(B\). What is the total distance the center of the disk traveled?
[ "/Mathematics/Geometry/Distance", "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Squares/PythagorassTheorem", "/Mathematics/Geometry/PlaneGeometry/Squares/Square", "/Mathematics/Geometry/PlaneGeometry/Squares/SquareInscribing", "/Mathematics/Geometry/Transformations/GeometricTransformations", "/Mathematics/Geometry/Transformations/Reflections/AlhazensBilliardProblem", "/Mathematics/Geometry/Transformations/Reflections/AlhazensProblem", "/Mathematics/Geometry/Transformations/Reflections/Billiards", "/Mathematics/Geometry/Transformations/Reflections/Reflection", "/Mathematics/Geometry/Transformations/Reflections/ReflectionProperty", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Unfold the reflections by treating the disk center as a point in a reduced square, turning the bounced path into a straight diagonal.
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ours_612
Consider a quadrilateral \(ABCD\) with side lengths \(AB=a, BC=b, CD=c, DA=d\), and \(\angle B=\angle D=90^{\circ}\) (which is possible by the first equation). By the second equation and the Law of Cosines, we have that \(\cos A=-\cos C=\frac{28}{53}\). Now we compute the area of \(ABCD\) in two ways: \[ [ABCD]=\frac{1}{2}(ab+cd)=\frac{1}{2}(bc+da) \sin A=\frac{45}{106}(bc+ad) \] Thus, \(\frac{ab+cd}{bc+ad}=\frac{45}{53}\). The maximum possible value \(M\) is \(\frac{45}{53}\), so \(m=45\) and \(n=53\). Therefore, \(100m+n = 100 \times 45 + 53 = 4553\). \(\boxed{4553}\)
4553
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \(a, b, c,\) and \(d\) be positive real numbers such that \[ a^{2}+b^{2}-c^{2}-d^{2}=0 \quad \text{and} \quad a^{2}-b^{2}-c^{2}+d^{2}=\frac{56}{53}(b c+a d) \] Let \(M\) be the maximum possible value of \(\frac{a b+c d}{b c+a d}\). If \(M\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers, find \(100 m+n\).
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/ConvexPolygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/PolygonArea", "/Mathematics/Geometry/PlaneGeometry/Polygons/ShoelaceFormula", "/Mathematics/Geometry/PlaneGeometry/Polygons/SimplePolygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/SurveyorsFormula", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Quadrangle", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Quadrilateral", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Cosine", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/LawofCosines", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/Sine" ]
Reinterpret the given equations as a quadrilateral with right angles and use two area expressions to relate the side ratios.
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ours_613
Consider the graph on \(5\) vertices with vertices numbered \(1\) to \(5\). For each swap \((i, j)\) draw an edge between \(i\) and \(j\). The key observation is that all other permutations are reachable from the starting one if and only if the graph is connected. Since we chose exactly \(4\) edges, it is connected if and only if it is a tree. So the problem amounts to counting the number of spanning trees on \(K_{5}\). A famous theorem of Cayley says that in general, the number of spanning trees of \(K_{n}\) is \(n^{n-2}=5^{3}=125\). \(\boxed{125}\)
125
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Joe is given a permutation \(p=\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of \((1,2,3,4,5)\). A swap is an ordered pair \((i, j)\) with \(1 \leq i<j \leq 5\), and this allows Joe to swap the positions \(i\) and \(j\) in the permutation. For example, if Joe starts with the permutation \((1,2,3,4,5)\), and uses the swaps \((1,2)\) and \((1,3)\), the permutation becomes \[ (1,2,3,4,5) \rightarrow(2,1,3,4,5) \rightarrow(3,1,2,4,5) \] Out of all \(\binom{5}{2}=10\) swaps, Joe chooses \(4\) of them to be in a set of swaps \(S\). Joe notices that from \(p\) he could reach any permutation of \((1,2,3,4,5)\) using only the swaps in \(S\). How many different sets are possible?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GraphTheory/Connectivity", "/Mathematics/DiscreteMathematics/GraphTheory/SimpleGraphs", "/Mathematics/DiscreteMathematics/GraphTheory/Trees" ]
Represent swaps as edges of a graph and use that reachability of all permutations is equivalent to the graph being connected, so with 4 edges it must be a spanning tree.
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ours_614
The key observation is that \(P\) is the radical center of \(\omega_{1}\), \(\omega_{2}\), and the circle of radius zero centered at \(M\). Indeed, \(PB\) is the radical axis of \(\omega_{1}\) and the circle at \(M\), since it passes through the point \(B\) and is perpendicular to the line \(O_{1}M\) through the centers. Similarly, \(PC\) is a radical axis. Consequently, by the characterization of the power of a point through distances, we have: \[ PO_{1}^{2} - 4^{2} = PO_{2}^{2} - 9^{2} \] Thus, \[ PO_{2}^{2} - PO_{1}^{2} = 9^{2} - 4^{2} = 81 - 16 = 65 \] \(\boxed{65}\)
65
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( A, B, M, C, D \) be distinct points on a line such that \( AB = BM = MC = CD = 6 \). Circles \(\omega_{1}\) and \(\omega_{2}\) with centers \(O_{1}\) and \(O_{2}\) and radii 4 and 9 are tangent to line \(AD\) at \(A\) and \(D\) respectively, such that \(O_{1}\) and \(O_{2}\) lie on the same side of line \(AD\). Let \(P\) be the point such that \(PB \perp O_{1}M\) and \(PC \perp O_{2}M\). Determine the value of \(PO_{2}^{2} - PO_{1}^{2}\).
[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/CirclePower", "/Mathematics/Geometry/Points/Point", "/Mathematics/Geometry/Points/Point-PointDistance2-Dimensional" ]
Identify P as the radical center of the two given circles and the point M, then equate its powers to obtain the difference of squared distances.
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ours_615
The answer is \(H_{10} = \frac{1}{1} + \cdots + \frac{1}{10}\). We show that in general, for \(n\) vertices (rather than 10), the answer is \(H_{n}\). We show that given any tree on \(n\) nodes, the expected value of \(S(v)\) increases by \(\frac{1}{n+1}\) when we add a new node. For a node \(v\), define \(d(v)\) as the number of nodes on the path from \(v\) to the root. By a simple double count, \[ \sum_{v} d(v) = \sum_{v} S(v) \] Clearly, when we randomly add a new child, we expect to increase \(\sum_{v} d(v)\) by \(\frac{1}{n} \cdot \sum_{v} d(v) + 1\). So our new average is \[ \frac{1}{n+1} \cdot \left(\sum d(v) + \frac{1}{n} \cdot \sum d(v) + 1\right) = \frac{1}{n} \cdot \sum d(v) + \frac{1}{n+1}. \] Therefore, our answer is \(H_{n}\), as claimed. It is now direct to compute \(H_{10} = \frac{7381}{2520}\), for an answer of \(9901\). \(\boxed{9901}\)
9901
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Alex starts with a rooted tree with one vertex (the root). For a vertex \(v\), let the size of the subtree of \(v\) be \(S(v)\). Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex \(v_1\)). The expected value of \(S(v_1)\) is of the form \(\frac{m}{n}\) for relatively prime positive integers \(m, n\). Find \(m+n\). Note: In a rooted tree, the subtree of \(v\) consists of its indirect or direct descendants (including \(v\) itself).
[ "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/DiscreteMathematics/GraphTheory/GeneralGraphTheory", "/Mathematics/DiscreteMathematics/GraphTheory/Trees", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilityMeasure", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace" ]
Relate total subtree sizes to total depths via double counting and observe each random addition raises the average by 1/(current size+1).
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ours_616
The main observation is that \( AD, BE, \) and \( CF \) concur. Since we have \( AP + PD \leq AD \) with equality if \( P \) lies on \( AD \), and similarly for the other cevians, we find that \[ AP + BP + CP + DP + EP + FP \leq AD + BE + CF \] and that equality occurs if \( P \) is the concurrency point. By the Pythagorean Theorem, we have \[ BE^2 = 80^2 + 45^2 \quad \text{and} \quad CF^2 = 60^2 + 20^2 \] Moreover, by the Law of Cosines on \(\triangle ADC\) (or by Stewart's Theorem or several other means), we find \[ AD^2 = 60^2 + 10^2 - 2 \cdot 60 \cdot 10 \cdot \frac{60}{100} = 60^2 + 10^2 - 720 \] Hence the quantity requested in the problem is \[ 10^2 + 20^2 + 45^2 + 2 \cdot 60^2 + 80^2 - 720 = 15405 \] Thus, the minimum possible value of \( AP + BP + CP + DP + EP + FP \) is \(\sqrt{100} + \sqrt{400} + \sqrt{2025}\), which corresponds to \( x = 100, y = 400, z = 2025 \). Therefore, \( x + y + z = 100 + 400 + 2025 = 2525 \). \(\boxed{2525}\)
2525
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( \triangle ABC \) be a triangle with \( AB = 80 \), \( BC = 100 \), \( AC = 60 \). Let \( D, E, F \) lie on \( BC, AC, AB \) such that \( CD = 10 \), \( AE = 45 \), \( BF = 60 \). Let \( P \) be a point in the plane of triangle \( ABC \). The minimum possible value of \( AP + BP + CP + DP + EP + FP \) can be expressed in the form \(\sqrt{x} + \sqrt{y} + \sqrt{z}\) for integers \( x, y, z \). Find \( x + y + z \).
[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/PlaneGeometry/Triangles/Cevians", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleLines", "/Mathematics/Geometry/Points/PlanarDistance", "/Mathematics/Geometry/Points/Point", "/Mathematics/Geometry/Points/Point-PointDistance2-Dimensional" ]
Place P at the common intersection of AD, BE, and CF so each pairwise sum AP+PD, BP+PE, CP+PF attains its minimum equal to the cevian length.
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ours_617
Note that 2011 is a prime. The following claim is the most important part of our solution: **Lemma:** Given the values \( P(0), P(1), \ldots, P(2010) \), there exists a unique polynomial of degree at most 2010 and coefficients in the set \(\{0,1, \ldots, 2010\}\) such that \( P(x) \equiv x \pmod{2011} \) for all \( x \). **Proof:** Work in \(\mathbb{F}_{2011}[x]\) throughout this proof. This is essentially polynomials with coefficients as integers \((\bmod 2011)\). First, I will prove uniqueness. If \( P(x), Q(x) \) are the same \((\bmod 2011)\) for all \( x \), then \( x^{2011}-x \mid P(x)-Q(x) \), contradicting the fact that \(\max (\deg P, \deg Q) \leq 2010\). Now I show existence. By the Lagrange Interpolation formula, \[ P(x)=\sum_{i=0}^{2010} P(i) \frac{\prod_{0 \leq j \leq 2010, j \neq i}(x-j)}{\prod_{0 \leq j \leq 2010, j \neq i}(i-j)}. \] Since every integer has a unique inverse \((\bmod 2011)\) and no denominators are a multiple of 2011, existence is shown. Now we will count the number of ways to assign the values 20, 15, 1234 to \( P(0), P(1), \ldots, P(2010) \) such that all 3 values are used at least once. This is essentially Stirling Numbers of the Second Kind and can be computed with the Principle of Inclusion-Exclusion. The answer is \( 3^{2011}-3 \cdot 2^{2011}+3 \cdot 1^{2011} \). Finally, since \( P(x) \) is degree 2015, write \( P(x)=(x^{2011}-x) \cdot Q(x)+R(x) \), where \(\deg R(x)< 2011\), \(\deg Q(x)=4\). So given the values of \( P(0), P(1), \ldots, P(2010) \), by the above claim, \( R(x) \) is determined uniquely. \( Q(x) \) can be chosen in \( 2010 \cdot 2011^{4} \) ways, just by choosing the coefficients independently. Therefore, our final answer is \( 2010 \cdot 2011^{4} \cdot(3^{2011}-3 \cdot 2^{2011}+3 \cdot 1^{2011}) \equiv 460 \pmod{1000} \). \(\boxed{460}\)
460
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Consider polynomials \( P \) of degree 2015, all of whose coefficients are in the set \(\{0,1, \ldots, 2010\}\). Call such a polynomial good if for every integer \( m \), one of the numbers \( P(m)-20, P(m)-15, P(m)-1234 \) is divisible by 2011, and there exist integers \( m_{20}, m_{15}, m_{1234} \) such that \( P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234 \) are all multiples of 2011. Let \( N \) be the number of good polynomials. Find the remainder when \( N \) is divided by 1000.
[ "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/FieldTheory/PrimeField", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/Congruent", "/Mathematics/Algebra/NumberTheory/Congruences/Mod", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/Congruences/Modulus", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialDegree", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]
Use that a polynomial of degree ≤2010 over ℤ₍₂₀₁₁₎ is uniquely determined by its values on the 2011 field elements, separating the low‑degree part from a free high‑degree factor.
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ours_618
First, note that the area condition implies that the side length of the pentagon is 1. Let the circumcircle of the pentagon be \(\omega\). Next, we show that \(C_{i} A_{i+2}=1\). Because \(C_{i} A_{i+2}^{2}=C_{i} A_{i} \cdot C_{i} A_{i+1}\), \(C_{i}\) is the intersection of the tangent from \(A_{i+2}\) to \(\omega\) with \(A_{i} A_{i+1}\). Therefore, \(\angle C_{i} A_{i+1} A_{i+2}=180^{\circ}-\angle A_{i} A_{i+1} A_{i+2}=72^{\circ}\). Also, \(\angle C_{i} A_{i+2} A_{i+1}=\angle A_{i+2} A_{i} A_{i+1}=36^{\circ}\) because of tangency. So \(\angle A_{i+1} C A_{i+2}=72^{\circ}=\angle C_{i} A_{i+1} A_{i+2} \Longrightarrow C A_{i+2}=A_{i+1} A_{i+2}=1\). This in fact means that \(B_{i} \equiv C_{i}\) because \(1^{2}=1\). Therefore, the pentagons are equal, so the ratio is 1. Our final answer would be \(\frac{1+0 \sqrt{5}}{1} \Longrightarrow 101\). \(\boxed{101}\)
101
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( A_{1} A_{2} A_{3} A_{4} A_{5} \) be a regular pentagon inscribed in a circle with area \(\frac{5+\sqrt{5}}{10} \pi\). For each \( i=1,2, \ldots, 5 \), points \( B_{i} \) and \( C_{i} \) lie on ray \(\overrightarrow{A_{i} A_{i+1}}\) such that \[ B_{i} A_{i} \cdot B_{i} A_{i+1}=B_{i} A_{i+2} \quad \text{and} \quad C_{i} A_{i} \cdot C_{i} A_{i+1}=C_{i} A_{i+2}^{2} \] where indices are taken modulo \(5\). The value of \(\frac{\left[B_{1} B_{2} B_{3} B_{4} B_{5}\right]}{\left[C_{1} C_{2} C_{3} C_{4} C_{5}\right]}\) (where \([\mathcal{P}]\) denotes the area of polygon \(\mathcal{P}\)) can be expressed as \(\frac{a+b \sqrt{5}}{c}\), where \(a, b\), and \(c\) are integers, and \(c>0\) is as small as possible. Find \(100 a+10 b+c\).
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/CirclePower", "/Mathematics/Geometry/PlaneGeometry/Circles/CircleTangentLine", "/Mathematics/Geometry/PlaneGeometry/Polygons/CyclicPentagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Pentagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularPentagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularPolygon" ]
Recognize the product condition as a power‑of‑a‑point/tangent relation to prove C_i lies on the tangent from A_{i+2}, giving C_iA_{i+2}=side length, so B_i=C_i.
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ours_619
The key observation is that by double-counting, the sum is equal to \[ \sum_{k \# \mid n} 1 = \sum_{1 \leq k \leq 2016} \left\lfloor \frac{N}{k \#} \right\rfloor \] where \( N = 2015 \# = 2016 \# \) (note \( N < 2017 \# \)). The quantity in the floor is an integer, and in fact, it's the product of the primes up to \( 2011 \) which are strictly greater than \( k \). Note that \( M = 3999991 = 2000^{2} - 3^{2} = 1997 \cdot 2003 \), both of which are primes. So in this sum, all terms \( k < 1997 \) vanish mod \( M \). The primes greater than \( 1997 \) are \( 1999, 2003, 2011, \) and \( 2017 \). - For \( 1997 \leq k \leq 1998 \), we have \( 1999 \cdot 2003 \cdot 2011 \), which contributes \( 2(2 \cdot 2003 \cdot 14) \equiv 56 \cdot 2003 \pmod{M} \). - For \( 1999 \leq k \leq 2002 \), we have \( 2003 \cdot 2011 \), which contributes \( 4(2003 \cdot 14) \equiv 56 \cdot 2003 \pmod{M} \). - For \( 2003 \leq k \leq 2010 \), we have \( 2011 \) which contributes \( 8 \cdot 2011 \). - For \( 2011 \leq k \leq 2016 \), we have \( 1 \) which contributes \( 6 \). Hence the answer is \( 112 \cdot 2003 + 8 \cdot 2011 + 6 = 224336 + 16088 + 6 = 240430 \). \(\boxed{240430}\)
240430
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
For a positive integer \( n \), let \( n \# \) denote the product of all primes less than or equal to \( n \) (or \( 1 \) if there are no such primes), and let \( F(n) \) denote the largest integer \( k \) for which \( k \# \) divides \( n \). Find the remainder when \( F(1) + F(2) + F(3) + \cdots + F(2015 \# - 1) + F(2015 \#) \) is divided by \( 3999991 \).
[ "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/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", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization", "/Mathematics/NumberTheory/PrimeNumbers/PrimeSumsandProducts" ]
Rewrite the total of F(n) as a double count → sum over k of ⌊N/(k#)⌋ and use the factorisation of the modulus to discard all but a few k terms.
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ours_620
Suppose \( U \) is a pseudo-ultrafilter. Note that inductively, the greatest common divisor of any \( n \) elements of \( U \) must also be in \( U \). Let \( g \) be the greatest common divisor of all elements of \( U \); then \( g \in U \) and \( g > 1 \). So all elements of \( U \) are multiples of \( g \), and by the first condition, all multiples of \( g \) are elements of \( U \). Thus we simply want to find all \( g \in X \) such that \[ \{h \in X \mid g \mid h\} \] is a pseudo-ultrafilter. This clearly satisfies the first two conditions, so it suffices to find \( g \) satisfying the third condition. If there are some integers \( a, b > 1 \) with \( g = ab \) and \(\operatorname{gcd}(a, b) = 1\), then we have a contradiction, as \(\operatorname{lcm}(a, b) = g\). Therefore, any valid \( g \) must satisfy \( g = p^k \) for some prime \( p \) and \( k \geq 1 \) (as \( g \) cannot be 1). Conversely, if \( g = p^k \), then \( p^k \nmid a \) and \( p^k \nmid b \) implies \( p^k \nmid \operatorname{lcm}(a, b) \). So it suffices to count the number of prime powers which are factors of \( 12! \), which is \( 10 + 5 + 2 + 1 + 1 = 19 \). \(\boxed{19}\)
19
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( N = 12! \) and denote by \( X \) the set of positive divisors of \( N \) other than \( 1 \). A pseudo-ultrafilter \( U \) is a nonempty subset of \( X \) such that for any \( a, b \in X \): - If \( a \) divides \( b \) and \( a \in U \), then \( b \in U \). - If \( a, b \in U \), then \(\operatorname{gcd}(a, b) \in U\). - If \( a, b \notin U \), then \(\operatorname{lcm}(a, b) \notin U\). How many such pseudo-ultrafilters are there?
[ "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/GCD-ClosedSet", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/Prime-RelatedNumbers", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization" ]
A pseudo-ultrafilter is exactly the set of all divisors that are multiples of a minimal element g, and g must be a prime power.
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ours_621
We solve the problem for the general case of $n$ balls and $n$ colors for $n \geq 2$. Let $C$ be the set of colors, so $|C|=10$. Let $B$ be the set of colors which are the color of some ball, so we want to find \(E\left[|B|^{2}\right]\). For a set \(T \subseteq C\), let \(f(T)\) be the probability that \(B=T\), and let \(g(T)\) be the probability that \(B \subseteq T\). So then \[ E\left[|B|^{2}\right]=\sum_{T \subseteq C} f(T)|T|^{2} \] and also \[ g(T)=\sum_{R \subseteq T} f(R) \] Clearly, \(g(T)=\frac{|T|^{n}}{n^{n}}\). By Mobius Inversion, \[ f(T)=\sum_{R \subseteq T}(-1)^{|T|-|R|} g(R) \] Thus, \[ \begin{aligned} E[|B|]^{2}=\sum_{T \subseteq C} \sum_{R \subseteq T}(-1)^{|T|-|R|} g(R)|T|^{2} & =\sum_{R \subseteq C} \sum_{R \subseteq T \subseteq C}(-1)^{|T|-|R|} g(R)|T|^{2} \\ & =\sum_{i=0}^{n} \sum_{j=i}^{n}\binom{n}{i}\binom{n-i}{j-i}(-1)^{j-i} \frac{i^{n}}{n^{n}} j^{2} \end{aligned} \] where we set \(i=|R|, j=|T|\), since there are \(\binom{n}{i}\) choices for \(R\) given fixed \(i\) and \(\binom{n-i}{j-i}\) choices for \(T\) given \(j\) and \(R \subseteq T\). If \(n-i>2\), \[ \sum_{j=i}^{n}\binom{n-i}{j-i}(-1)^{j-i} j^{2}=0 \] so our expression for \(E\left[|B|^{2}\right]\) simply reduces to \[ E\left[|B|^{2}\right]=\sum_{i=n-2}^{n} \sum_{j=i}^{n}\binom{n}{i}\binom{n-i}{j-i}(-1)^{j-i} \frac{i^{n}}{n^{n}} j^{2} \] Evaluating this sum yields \[ E\left[|B|^{2}\right]=\binom{n}{n-2} \frac{(n-2)^{n}}{n^{n}}\left(n^{2}-2(n-1)^{2}+(n-2)^{2}\right) \] \[ +\binom{n}{n-1} \frac{(n-1)^{n}}{n^{n}}\left((n-2)^{2}-(n-1)^{2}\right)+\binom{n}{n} \frac{n^{n}}{n^{n}}\left(n^{2}\right) \] which simplifies to \[ \frac{1}{n^{n}}\left(n(n-1)(n-2)^{n}+n(1-2 n)(n-1)^{n}+n^{n+2}\right) \] For \(n=10\), this is \[ 43.414772797 \] yielding a final answer of \(55\). \(\boxed{55}\)
55
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Suppose we have $10$ balls and $10$ colors. For each ball, we independently color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal.
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/ProbabilityandStatistics/Moments/Moment", "/Mathematics/ProbabilityandStatistics/Moments/RawMoment", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilityAxioms", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/Probability/SampleSpace" ]
Apply inclusion‑exclusion (Möbius inversion) to express the probability of exactly a given set of colors and sum over set sizes.
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ours_622
For ease of notation, let \( 0 = \varnothing, 1 = \{0\}, 2 = \{0,1\}, 3 = \{0,1,2\} \). Then: \[ V_{3} = \{0, 1, 2, \{1\}\} \] Let \( x \in V_{5} \), i.e., \( x \subseteq V_{4} \), be transitive. We now count the number of transitive sets in \( V_{4} \) by casework on \( y = x \cap V_{3} \). - If \( y = \varnothing \), this means \( x = \varnothing \), so one set here. - If \( y = \{0\} \), this means \( x = 1 \), so one set here. - If \( y = \{0,1\} \), this means \( x = 2 \), so one set here. - If \( y = \{0,1,2\} = 3 \), then there are four more elements of \( V_{4} \) we can add to \( x \) (namely the \( 2^{3} - 4 = 4 \) subsets of \( 3 \) not already in \( V_{2} \)), so \( 2^{4} = 16 \) cases here. - If \( y = \{0,1,\{1\}\} \), we again get \( 2^{4} = 16 \) cases. - If \( y = V_{3} \), then we can add any of the \( 2^{4} - 4 = 12 \) elements of \( V_{4} \setminus V_{3} \) we like, giving \( 2^{12} = 4096 \) cases. Tallying the total gives \( 4096 + 32 + 3 = 4131 \). \(\boxed{4131}\)
4131
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( V_{0} = \varnothing \) be the empty set and recursively define \( V_{n+1} \) to be the set of all \( 2^{|V_{n}|} \) subsets of \( V_{n} \) for each \( n=0,1, \ldots \). For example: \[ V_{2} = \{\varnothing, \{\varnothing\}\} \quad \text{and} \quad V_{3} = \{\varnothing, \{\varnothing\}, \{\{\varnothing\}\}, \{\varnothing, \{\varnothing\}\}\} \] A set \( x \in V_{5} \) is called transitive if each element of \( x \) is a subset of \( x \). How many such transitive sets are there?
[ "/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", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/AxiomaticSetTheory", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/AxiomofthePowerSet", "/Mathematics/FoundationsofMathematics/SetTheory/GeneralSetTheory/SetTheory", "/Mathematics/FoundationsofMathematics/SetTheory/SetProperties/Rank", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/Set", "/Mathematics/FoundationsofMathematics/SetTheory/Sets/Subset" ]
Use the transitivity condition to restrict the possible intersections with lower stages, then count independent choices of higher‑level subsets.
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ours_623
Let \(\alpha=\frac{3+\sqrt{5}}{2}\), so that \(L_{2n}=\alpha^{n}+\alpha^{-n}\). Note both \(359\) and \(179\) are prime. Since \[ \left(\frac{5}{359}\right)=1 \] (i.e. \(5\) is a quadratic residue modulo \(359\), as can be checked by quadratic reciprocity), we have that \(\alpha \in \mathbb{F}_{359}\). Finally, \[ \alpha=\left(\frac{1+\sqrt{5}}{2}\right)^{2} \] so \[ \alpha^{179}=1 \] By properties of Chebyshev polynomials, \(T_{n}\left(L_{2n}\right)=\alpha^{n^{2}}+\alpha^{-n^{2}}\). To finish, note that since \(179 \equiv 3 \pmod{4}\), \(-n^{2}\) is not a quadratic residue modulo \(179\). Therefore, \[ \sum_{n=0}^{178} T_{n}\left(L_{2n}\right)=\sum_{n=0}^{178}\left(\alpha^{n^{2}}+\alpha^{-n^{2}}\right)=2\left(\alpha^{0}+\alpha^{1}+\ldots+\alpha^{178}\right)=0 \] since we run through each residue and non-residue precisely twice. So in our desired sum almost everything goes to \(0\), except the last two terms, which are easily computed to be \(2+3=5\). \(\boxed{5}\)
5
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Consider a sequence \( T_{0}, T_{1}, \ldots \) of polynomials defined recursively by \( T_{0}(x)=2, T_{1}(x)=x \), and \( T_{n+2}(x)=x T_{n+1}(x)-T_{n}(x) \) for each nonnegative integer \( n \). Let \( L_{n} \) be the sequence of Lucas Numbers, defined by \( L_{0}=2, L_{1}=1 \), and \( L_{n+2}=L_{n}+L_{n+1} \) for every nonnegative integer \( n \). Find the remainder when \( T_{0}\left(L_{0}\right)+T_{1}\left(L_{2}\right)+T_{2}\left(L_{4}\right)+\cdots+T_{359}\left(L_{718}\right) \) is divided by \( 359 \).
[ "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Congruent", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticNonresidue", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticReciprocityTheorem", "/Mathematics/NumberTheory/ReciprocityTheorems/QuadraticResidue", "/Mathematics/NumberTheory/Sequences/LucasSequence", "/Mathematics/NumberTheory/Sequences/PolynomialSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]
Express each term as α^{n^2}+α^{-n^2} with α a 179‑th root of unity in ℤ₃₅₉, then the sum over n cancels by pairing quadratic residues and non‑residues.
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ours_624
To deal with mixtilinear incircles, we utilize the following key lemma: **Lemma 27.1**: For a triangle \(ABC\), suppose \(D\) lies on segment \(BC\) and a circle is tangent to segments \(AD\), \(BC\) at \(X\), \(Y\) and the circumcircle of \(ABC\) internally. Then the incenter \(I\) of \(ABC\) lies on line \(XY\). Now, let line \(CD\) intersect \(\Gamma\) at \(S, T\), and let \(U, V\) be the incenters of triangles \(AST, BST\), respectively. By Lemma 1, \(U\) lies on \(WX\) and \(V\) lies on \(YZ\). Let \(Q, R\) be the excenters of triangle \(EST\) across from \(T, S\) respectively. By an external analogue of Lemma 1, \(Q\) lies on \(WX\) as well, and \(R\) lies on \(YZ\). Thus \(P\) can be recharacterized as the intersection of \(QU\) and \(RV\). Let \(K\) be the midpoint of arc \(ST\) on \(\Gamma\) containing \(E\). Let \(\omega\) be a circle centered at \(K\) passing through \(S\) and \(T\). By a well-known result, \(Q, U, V, R\) also lie on \(\omega\). Of course, we also have that \(Q, R\) lie on \(KE\), the external bisector of \(\angle SET\), and similarly \(U\) lies on \(KA\), \(V\) lies on \(KB\). Now note that \[ \angle QRP = \angle KRV = 90^\circ - \frac{\angle RKB}{2} = 90^\circ - \frac{\angle EAB}{2} \] and similarly \(\angle RQP = 90^\circ - \frac{\angle EBA}{2}\). Thus \(PRQ \sim FGH\), where \(FGH\) is the excentral triangle of \(ABE\) (\(F, G, H\) are the excenters across \(E, A, B\) respectively). In particular, let \(M\) be the midpoint of arc \(AB\) on \(\Gamma\) containing \(E\). \(M\) is the second intersection of the nine-point center of \(FGH\) with \(GH\), so \(M\) is the midpoint of \(GH\). \(K\) is the center of \(\omega\) passing through \(Q, R\), so \(K\) is the midpoint of \(RQ\). Thus \(K, M\) are corresponding points on similar triangles \(FGH\) and \(PRQ\). In particular, \[ \angle FME = \angle FMH = \angle PKQ = \angle PME \] so \(M, P, F\) are collinear. The rest of the problem is standard: we can easily compute \(GH = 4576\), \(FH = 3146\), and \(FG = 3718\). Thus \[ FM = \sqrt{\frac{FG^2 + FH^2}{2} - \frac{GH^2}{4}} = 2574 \] and \[ FB = \frac{FH \cdot AB}{GH} = 374 \] so by Power of a Point \[ FP = \frac{FB \cdot FG / 2}{FM} = \frac{2431}{9} \] and the final answer is \(2431 + 9 = 2440\). \(\boxed{2440}\)
2440
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \(ABCD\) be a quadrilateral satisfying \(\angle BCD = \angle CDA\). Suppose rays \(AD\) and \(BC\) meet at \(E\), and let \(\Gamma\) be the circumcircle of \(ABE\). Let \(\Gamma_1\) be a circle tangent to ray \(CD\) past \(D\) at \(W\), segment \(AD\) at \(X\), and internally tangent to \(\Gamma\). Similarly, let \(\Gamma_2\) be a circle tangent to ray \(DC\) past \(C\) at \(Y\), segment \(BC\) at \(Z\), and internally tangent to \(\Gamma\). Let \(P\) be the intersection of \(WX\) and \(YZ\), and suppose \(P\) lies on \(\Gamma\). If \(F\) is the \(E\)-excenter of triangle \(ABE\), and \(AB = 544\), \(AE = 2197\), \(BE = 2299\), then find \(m+n\), where \(FP = \frac{m}{n}\) with \(m, n\) relatively prime positive integers.
[ "/Mathematics/Geometry/GeneralGeometry/Bisector", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/InversiveGeometry/MixtilinearIncirclesRadicalCircle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle-CircleTangents", "/Mathematics/Geometry/PlaneGeometry/Circles/CirclePower", "/Mathematics/Geometry/PlaneGeometry/Circles/Excenter", "/Mathematics/Geometry/PlaneGeometry/Circles/TangentCircles", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similar", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similarity", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles", "/Mathematics/Geometry/PlaneGeometry/Triangles/TrianglePoints" ]
Apply the mixtilinear incircle (and excenter) lemma to place P on the line through the E‑excenter F and the arc midpoint, yielding similar triangles that give FP.
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ours_625
We claim that the number of ordered pairs \((P(x), Q(x))\) for \(2^{n}\) is \(4 \cdot 3^{n-1}\) for \(n \geq 1\). The equation can be rewritten as: \[ (P(x)+i Q(x))(P(x)-i Q(x))=(x-1)^{2}(x+1)^{2} \prod_{k=0}^{n-2}\left(x^{2^{k}}+i\right)^{2}\left(x^{2^{k}}-i\right)^{2} \] This involves decomposing the right-hand side into the product of a polynomial and its conjugate. **Lemma:** The polynomial \(x^{2^{k}}-i\) is irreducible in \(\mathbb{Z}[i]\), meaning it cannot be decomposed over the Gaussian integers. **Proof 1:** Suppose \[ x^{2^{k}}-i=A(x) B(x) \] for some \(A, B \in \mathbb{Z}[i]\) of positive degree. Then \[ x^{2^{k}}+i=\overline{A(x) B \overline{(x)})} \] so \[ x^{2^{k+1}}+1=(A(x) A \overline{(x)})(B(x) B \overline{(x)}) \] However, \(x^{2^{k+1}}+1\) is irreducible, as it is a cyclotomic polynomial. Its irreducibility can also be shown by noting \[ (x+1)^{2^{k+1}}+1 \equiv x^{2^{k+1}}+2 \] where each coefficient is taken \((\bmod 2)\). By Eisenstein's Criterion, the shifted polynomial is irreducible, leading to a contradiction. **Proof 2:** Applying Eisenstein's Criterion to \(x^{2^{k}}-i\) itself, shift the polynomial by \(1\) to yield \[ (x+1)^{2^{k}}-i \] Each coefficient in this polynomial is divisible by \(2\) except for \(x^{2^{k}}\) and the constant term, \(1-i\). Note that \(1-i\) is irreducible in \(\mathbb{Z}[i]\) and \(1-i \mid 2\), so by Eisenstein on \(1-i\), this polynomial is irreducible. **Proof 3:** By Gauss' Lemma, it suffices to prove irreducibility in \(\mathbb{Q}[i]\). If \(z\) is a root of \(x^{2^{k}}-i\), then \(\mathbb{Q}[z][i]=\mathbb{Q}[z]\) since \(i\) is a power of \(z\). Also, \(z\) is a root of \(x^{2^{k+1}}+1\), which is irreducible in \(\mathbb{Q}[x]\). \[ 2[\mathbb{Q}[z][i]: \mathbb{Q}[i]]=[\mathbb{Q}[z][i]: \mathbb{Q}[i]][\mathbb{Q}[i]: \mathbb{Q}]=[\mathbb{Q}[z][i]: \mathbb{Q}]=2^{k+1} \] implying \([\mathbb{Q}[z][i]: \mathbb{Q}[i]]=2^{k}\), so \(x^{2^{k}}-i\) must be the minimal polynomial of \(z\). For each pair of conjugate polynomials in the RHS, one must become a factor of \(P+Q i\) and one must become a factor of \(P-Q i\). For each \(\left(x^{2^{k}}+i\right)^{2}\left(x^{2^{k}}-i\right)^{2}\), there are three choices: let \(\left(x^{2^{k}}+i\right)^{2}\) divide \(P+Q i\) and \(\left(x^{2^{k}}-i\right)^{2}\) divide \(P-Q i\), the reverse, or let \(\left(x^{2^{k}}+i\right)\left(x^{2^{k}}-i\right)\) divide both. We necessarily must have \((x-1)(x+1)\) dividing both, so up to units there are \(3^{n-1}\) choices for \(P+Q i\). Including the units \(1,-1, i,-i\) to multiply by, there are a total of \(4 \cdot 3^{n-1}\) choices. \(\boxed{708588}\)
708588
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Find the number of ordered pairs \((P(x), Q(x))\) of polynomials with integer coefficients such that \[ P(x)^{2}+Q(x)^{2}=\left(x^{4096}-1\right)^{2} \]
[ "/Mathematics/Algebra/FieldTheory/IrreduciblePolynomial", "/Mathematics/Algebra/NumberTheory/AlgebraicNumberTheory/AlgebraicInteger", "/Mathematics/Algebra/NumberTheory/AlgebraicNumberTheory/AlgebraicNumber", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/Z", "/Mathematics/Algebra/Polynomials/EisensteinsIrreducibilityCriterion", "/Mathematics/Algebra/Polynomials/Factorization", "/Mathematics/Algebra/Polynomials/IrreduciblePolynomial" ]
Factor the equation over Gaussian integers as (P+iQ)(P-iQ) and count assignments of each conjugate irreducible factor to the two factors.
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ours_626
The length of \( BC = 20x^{5/2} \) is extraneous and not used below. Define the \( A \)-mixtilinear incircle \(\omega_A\) to be the circle internally tangent to the circumcircle of \( \triangle ABC \) at a point \( T_A \) while simultaneously tangent to \(\overline{AB}\) and \(\overline{AC}\). Define \(\omega_B, \omega_C, T_B\), and \(T_C\) similarly. Let \(\omega\) and \(\Omega\) denote the incircle and circumcircle of \(\triangle ABC\). Let \( M_B \) and \( M_C \) be the midpoints of the arcs \(\widehat{AC}\) and \(\widehat{AB}\) of \(\Omega\) which do not contain the opposite vertices, and let the \( A \)-mixtilinear incircle be tangent to sides \( AB \) and \( AC \) at points \( A_1 \) and \( A_2 \). Also, observe that \( M \) is the midpoint of the arc \(\widehat{BC}\) of \(\Omega\) not containing \( A \), a well-known fact which can be established by angle chasing. First, we claim that points \(\angle AIA_1 = \angle AIA_2 = 90^\circ\). Indeed, note that the homothety taking \(\omega_A\) to \(\Omega\) sends \( A_1 \) to \( M_C \) and \( A_2 \) to \( M_B \). Therefore, applying Pascal's Theorem on hexagon \( ABT_BMT_CC \) (on \(\Omega\)) implies that \( A_1, A_2, I \) are collinear. The conclusion is immediate. By applying similar logic, we discover that \( B' \) and \( C' \) are the contact points of \(\omega_B\) and \(\omega_C\) on side \( BC \). Consequently, we discover that point \( M \) is the intersection of lines \( T_CC' \) and \( T_BB' \). Next, let \( X_{56} \) denote the center of the positive homothety taking the incircle to \(\Omega\). We claim that \( A, X_{56}, \) and \( T_A \) are collinear. Indeed, consider the composition of homotheties \(\omega \xrightarrow{A} \omega_A \xrightarrow{T_A} \Omega\). By definition, it has center \( X_{56} \). But the line \( AT_A \) is fixed by it, proving the claim. Also, note that \( D \) and \( M \) are images under the composed homothety. Hence, lines \( AT_A, BT_B, CT_C, DM \), and \( IO \) concur at \( X_{56} \). Next, by Pascal's Theorem on hexagon \( ABT_BMT_CC \) (on \(\Omega\)), we find that \( P, X_{56}, Q \) are collinear. It follows that \( X_{56} = S \). Now, let \( D' \) be the point diametrically opposite \( D \) on \(\omega\) and \( X \) the point diametrically opposite \( M \) on \(\Omega\). Of course, points \( S, D', X \) are collinear as the homothety maps \( X \) to \( D' \). We claim now that \( T_A, I, X \) are collinear. Indeed, note that rays \( TA \) and \( TI \) are the symmedian and median of \(\triangle T_AA_1A_2\), respectively; the result now follows since lines \( AX \) and \( M_BM_C \) are parallel. Denote by \( L \) the intersection of lines \( D'E \) and \( SI \). From the collinearity above, we find that under the homothety from \(\omega\) to \(\Omega\) at \( S \) maps the point \( L \) to \( I \). Moreover, we find that \( D'EDF \) is a rectangle inscribed in \(\omega\) by construction; thus \( LI = IK \). In summary, there is a homothety centered at \( S \) which sends: - \(\omega\) to \(\Omega\), - \( I \) to \( O \), - and \( L \) to \( I \). Let \( h \) be the homothety of this ratio. First, we have \[ h = \frac{SI}{SL} = \frac{SO}{SI} = \frac{IO}{LI} \] but also \[ IO^2 = R(R-2r) = h(h-2)r^2 \] In this way, we obtain that \[ IO^2 = LI^2 \cdot h^2 = h(h-2)r^2 \] so \[ LI^2 = \left(1-\frac{2}{h}\right) \cdot r^2 \] Substituting the known values \( LI = KI = 15x \) and \( SI = 20x + 15 \), and \( r^2 = DI^2 = 400x^3 \), we get \[ (15x)^2 = \left(1-2 \cdot \frac{5x+15}{20x+15}\right) \cdot 20^2 \cdot x^3 \] Simplifying, we get \[ \frac{9}{16x} = 1-2 \cdot \frac{x+3}{4x+3} = \frac{2x-3}{4x+3} \] id est, \[ 32x^2 - 84x - 27 = 0 \] Applying the quadratic formula and taking the positive root gives that \[ x = \frac{1}{64}(84+12\sqrt{73}) = \frac{3}{16}(7+\sqrt{73}) \] and the answer is \(3+16+7+73=99\). \(\boxed{99}\)
99
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( ABC \) be an acute scalene triangle with incenter \( I \), and let \( M \) be the circumcenter of triangle \( BIC \). Points \( D, B', \) and \( C' \) lie on side \( BC \) so that \(\angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^\circ\). Define \( P = \overline{AB} \cap \overline{MC'} \), \( Q = \overline{AC} \cap \overline{MB'} \), \( S = \overline{MD} \cap \overline{PQ} \), and \( K = \overline{SI} \cap \overline{DF} \), where segment \( EF \) is a diameter of the incircle selected so that \( S \) lies in the interior of segment \( AE \). It is known that \( KI = 15x \), \( SI = 20x + 15 \), \( BC = 20x^{5/2} \), and \( DI = 20x^{3/2} \), where \( x = \frac{a}{b}(n+\sqrt{p}) \) for some positive integers \( a, b, n, p \), with \( p \) prime and \(\operatorname{gcd}(a, b) = 1\). Compute \( a+b+n+p \).
[ "/Mathematics/Geometry/GeneralGeometry/Center", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/InversiveGeometry/MixtilinearIncirclesRadicalCircle", "/Mathematics/Geometry/PlaneGeometry/Arcs/Arc", "/Mathematics/Geometry/PlaneGeometry/Arcs/Mid-ArcPoints", "/Mathematics/Geometry/PlaneGeometry/Arcs/TriangleArcs", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Incenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles", "/Mathematics/Geometry/PlaneGeometry/Triangles/TrianglePoints" ]
Identify S as the homothety center sending the incircle to the circumcircle, then use the resulting length ratios to relate KI, SI, DI and solve for x.
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ours_627
Solution. Note that the series is conditionally convergent, so we introduce a regulator \( s \) in our sum: \[ f(s) = \sum_{n=1}^{\infty} \frac{d(n) + \sum_{m=1}^{\nu_{2}(n)}(m-3) d\left(\frac{n}{2^{m}}\right)}{n^{s}} \] We use the Riemann zeta function \(\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}}\). Then \[ \begin{aligned} f(s) & = \sum_{n=1}^{\infty} \frac{d(n)}{n^{s}} + \sum_{n=1}^{\infty} \frac{\sum_{m=1}^{\nu_{2}(n)}(m-3) d\left(\frac{n}{2^{m}}\right)}{n^{s}} \\ & = \zeta(s)^{2}\left(\frac{1}{2^{s}} + \frac{1}{4^{s}} + \frac{1}{8^{s}} + \cdots\right)^{2} \\ & = \zeta(s)^{2}\left(1 - \frac{1}{-1 + 2^{s}}\right)^{2} \\ & = \zeta(s)^{2}\left(\frac{-2 + 2^{s}}{-1 + 2^{s}}\right)^{2} \end{aligned} \] Now we take the limit of \( f(s) \) as \( s \rightarrow 1 \). The denominator \(-1 + 2^{s}\) vanishes, so we find \(\lim_{s \rightarrow 1} \zeta(s)^{2}\left(-2 + 2^{s}\right)^{2}\). Using the Laurent expansion of \(\zeta(s)\), we know: \[ \zeta(s) = \frac{1}{s-1} + c_{0} + c_{1}(s-1) + c_{2}(s-1)^{2} + \cdots \] So \[ \begin{aligned} \lim_{s \rightarrow 1} \zeta(s)^{2}\left(-2 + 2^{s}\right)^{2} & = \lim_{s \rightarrow 1}\left(\frac{1}{s-1} + c_{0} + c_{1}(s-1) + c_{2}(s-1)^{2} + \cdots\right)^{2}\left(-2 + 2^{s}\right)^{2} \\ & = \lim_{s \rightarrow 1} \frac{\left(-2 + 2^{s}\right)^{2}}{(s-1)^{2}} = \left(\lim_{s \rightarrow 1} \frac{-2 + 2^{s}}{s-1}\right)^{2} = \left(\lim_{s \rightarrow 1} \frac{2^{s} \ln 2}{1}\right)^{2} = (\ln 4)^{2} \end{aligned} \] Thus, \( S = (\ln 4)^{2} \), where \( m = 4 \) and \( n = 2 \). Therefore, \( 1000n + m = 1000 \times 2 + 4 = 2004 \). \(\boxed{2004}\)
2004
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring15Solns.md" }
Let \( S \) be the value of \[ \sum_{n=1}^{\infty} \frac{d(n)+\sum_{m=1}^{\nu_{2}(n)}(m-3) d\left(\frac{n}{2^{m}}\right)}{n} \] where \( d(n) \) is the number of divisors of \( n \) and \( \nu_{2}(n) \) is the exponent of \( 2 \) in the prime factorization of \( n \). If \( S \) can be expressed as \( (\ln m)^{n} \) for positive integers \( m \) and \( n \), find \( 1000n + m \).
[ "/Mathematics/CalculusandAnalysis/ComplexAnalysis/AnalyticContinuation", "/Mathematics/CalculusandAnalysis/Series/Convergence", "/Mathematics/CalculusandAnalysis/SpecialFunctions/Logarithms", "/Mathematics/CalculusandAnalysis/SpecialFunctions/RiemannZetaFunction", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/GeneralNumberTheory/AnalyticNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/DivisorFunction", "/Mathematics/NumberTheory/NumberTheoreticFunctions/GeneralNumberTheoreticFunctions", "/Mathematics/NumberTheory/NumberTheoreticFunctions/NumberTheoreticSums" ]
Rewrite the series as a Dirichlet series using ζ(s)^2 and a geometric sum in 2^{-s}, then evaluate its limit at s=1 via the pole of ζ(s).
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ours_628
First, we will determine the possible kinds of BQ subsets \( X \). Consider the convex hull of \( X \). It cannot have any obtuse angles, or else we take \( A, B, C \) with \(\angle ABC\) obtuse to get that the orthocenter of \( ABC \) is outside the convex hull, a contradiction. Thus, the convex hull has either 3 or 4 sides, since any convex polygon of more than 4 sides has an obtuse angle. If the convex hull of \( X \) has four sides, then it must be a rectangle. If the rectangle is \( ABCD \), we can't have a point \( P \) inside, or else the orthocenter of \( PAC \) is outside the rectangle, a contradiction. The rectangle contains \((0,0)\) as its bottom left vertex, so it must be of the form \((0,0), (2^x, 0), (0, 2^y), (2^x, 2^y)\) with \( 0 \leq x, y \leq 2015 \). This yields \( 2016^2 \) possibilities for \( X \). If the convex hull has three sides, it is a non-obtuse triangle \( ABC \). Then its orthocenter \( H \) is also in the set. No other points can be in the set, as any additional point \( P \) inside triangle \( BCH \) would lead to the orthocenter of \( PBC \) being outside the convex hull, a contradiction. Now we must count the number of triangles whose vertices are in the set and whose orthocenters are also in the set \( S \). We split into two cases: **Case 1.** The orthocenter is \((0,0)\). The other two vertices of the triangle are \((2^x, 0), (0, 2^y)\), which yields \( 2016^2 \) possibilities. **Case 2.** The orthocenter is not \((0,0)\). - **Case 2.1.** Suppose there is no point in \( X \) of the form \((a, 0)\) with \( a \neq 0 \). We need a horizontal segment somewhere in the convex hull, so the set \( X \) is \((0,0), (0, a), (b, a)\) for some \( a, b \). This yields \( 2016^2 \) more possibilities. - **Case 2.2.** There is another point in the set which is of the form \((a, 0)\). Let the third point of the triangle be \((b, c)\), and the orthocenter is \((b, \frac{b(a-b)}{c})\). Since the orthocenter is inside the triangle, \( b(a-b) \leq c^2 \). Since \( a, b, c, \frac{b(a-b)}{c} \) are powers of two, \( a-b \) is a power of two, so either \( a=b \) or \( a=2b \). - **Case 2.2.1.** \( a=b \). Then our set is \((0,0), (a, 0), (a, c)\) and the orthocenter is \((a, 0)\), which is already inside \( X \). So we need \( 1 \leq a \leq 2^{2015}, 1 \leq c \leq 2^{2015} \), yielding another \( 2016^2 \) possibilities. - **Case 2.2.2.** \( a=2b \). So our points are \((0,0), (2b, 0), (b, c), (b, \frac{b^2}{c})\). Let \( b=2^x \) with \( 0 \leq x \leq 2014 \) and \( c=2^y \). We know \( c \mid b^2, c \geq b \), which translates into \( x \leq y \leq \min(2x, 2015) \). Counting the number of pairs \((x, y)\) satisfying this is done by splitting the sum at \( x=1007 \): When \( x \leq 1007 \), we have \( x+1 \) choices for \( y \) and when \( 1008 \leq x \leq 2014 \), we have \( 2016-x \) choices for \( y \), yielding a total of \( (1+2+\ldots+1008)+(1008+1007+\ldots+2)=1008 \cdot 1009-1 \) pairs \((x, y)\). Thus, our final answer is the sum \( 4 \cdot 2016^2 + 1008 \cdot 1009 - 1 = 17274095 \). \(\boxed{17274095}\)
17274095
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Let \( S \) be the set of all \( 2017^2 \) lattice points \((x, y)\) with \( x, y \in \{0\} \cup \{2^0, 2^1, \cdots, 2^{2015}\} \). A subset \( X \subseteq S \) is called BQ if it has the following properties: (a) \( X \) contains at least three points, no three of which are collinear. (b) One of the points in \( X \) is \((0,0)\). (c) For any three distinct points \( A, B, C \in X \), the orthocenter of \(\triangle ABC\) is in \( X \). (d) The convex hull of \( X \) contains at least one horizontal line segment. Determine the number of BQ subsets of \( S \).
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Show that the convex hull cannot contain obtuse angles, so it must be a rectangle or a non‑obtuse triangle, which limits the set to just its vertices (and possibly the orthocenter).
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ours_629
This solution uses the language of posets, chains, and antichains. A poset is the structure defined in the problem. A chain is a subset of elements of the poset such that all pairs are comparable, and an antichain is a subset of elements of the poset such that no two are comparable. The problem asks to cover the poset with chains and antichains. Note that \(2016 = \frac{63 \cdot 64}{2}\). I claim that the answer is 63. The upper bound will be shown in the following lemma. **Lemma:** In a poset with \(\frac{k(k+1)}{2}\) elements, it can be covered using at most \(k\) chains or antichains. **Proof:** This can be done with Dilworth's Theorem, but here is a proof without it. For each \(v \in G\), label \(v\) with the longest path in the poset that starts at \(v\). Let \(f(v)\) denote this label. More explicitly, if we let \(<\) denote the binary comparator on \(G\), \(f(v)\) is the largest \(m\) such that there exists a sequence \(v = v_{1} < v_{2} < \cdots < v_{m}\), where \(v_{i} \in G\) for all \(i\). You can think of this as grouping the poset into "layers". If \(f(v) \geq k\) for some \(v\), then there exists a chain of length \(k\) starting at \(v\). Now delete this chain. Afterwards, we are left with a poset with \(\frac{(k-1)k}{2}\) elements, which can be covered in \(k-1\) chains or antichains by induction. So in this case, we use \(k\) in total. Otherwise, \(f(v) < k\) for all \(v\). This admits a decomposition into \(k-1\) antichains, where the \(i\)-th antichain is the set of all \(v\) such that \(f(v) = i\). To see that each of these sets is an antichain, assume that there exist \(u, v\) within the same set such that \(u < v\). But this means that \(f(u) \geq f(v) + 1\), a contradiction. So in this case, there exists a covering using only \(k-1\) antichains, as desired. Now we prove the lower bound by providing a construction. For the construction, let the elements be grouped into \(k\) groups \(G_{1}, G_{2}, \ldots, G_{k}\) such that \(|G_{i}| = i\) and for all \(i < j\), if \(v_{i} \in G_{i}, v_{j} \in G_{j}\), then \(v_{i} > v_{j}\). These are the only relations, which means that each \(G_{i}\) is an antichain. This has a total of \(\frac{k(k+1)}{2}\) elements. I claim that it needs at least \(k\) chains or antichains to cover it. We proceed by induction. If a chain is used, it might as well be of length \(k\), since deleting more vertices doesn't hurt us later on. After deleting this chain of length \(k\), we have reduced the poset to the case with \(\frac{(k-1)k}{2}\) elements of the same construction, which requires \(k-1\) more chains or antichains. If no chain is used, then obviously we need at least \(k\) antichains to cover everything. Therefore, we are done. \(\boxed{63}\)
63
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Bessie and her 2015 bovine buddies work at the Organic Milk Organization, for a total of 2016 workers. They have a hierarchy of bosses, where no cow is its own boss. For some pairs of employees \((A, B)\), \(B\) is the boss of \(A\). This relationship satisfies the condition: if \(B\) is the boss of \(A\) and \(C\) is the boss of \(B\), then \(C\) is also a boss of \(A\). Bessie hires an outside organizational company to partition the company into some number of groups. Every group is one of two forms: either no one in the group is the boss of another, or for every pair of cows in the group, one is the boss of the other. Let \(G\) be the minimum number of groups needed in such a partition. Find the maximum value of \(G\) over all possible company structures.
[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/Covers", "/Mathematics/DiscreteMathematics/Combinatorics/Partitions", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics" ]
Label each element by the length of its longest upward chain; this yields either a chain of size k or k antichain layers, bounding the cover number by k.
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ours_630
Let \( p = 2017 \). For an integer \( k \), let \( d(k, p-1) \) denote the largest divisor of \( p-1 \) that is relatively prime to \( k \). Using primitive roots, there are exactly \( d(k, p-1) + 1 \) \((p, k)\)-Hofstadterian residues, where the plus 1 accounts for including 0. We need to compute \( p-1 + \sum_{k=1}^{p-1} d(k, p-1) \). To compute this sum, consider the prime factors of \( p-1 \). If exactly the primes \( q_{a_{1}}, q_{a_{2}}, \ldots, q_{a_{j}} \) divide \(\gcd(k, p-1)\), then \((p-1) \prod_{i=1}^{j}\left(1-\frac{1}{q_{a_{i}}}\right)\) values of \( k \) satisfy this condition. For these \( k \), \( d(k, p-1) = \frac{\phi(p-1)}{\prod_{i=1}^{j}\left(q_{a_{i}}-1\right)} \). Thus, the sum can be rewritten as: \[ \sum_{q_{a_{1}}, q_{a_{2}}, \ldots, q_{a_{j}}} \frac{\phi(p-1)}{\prod_{i=1}^{j}\left(q_{a_{i}}-1\right)} \cdot \frac{p-1}{\prod q_{a_{i}}^{e_{a_{i}}}} = (p-1) \phi(p-1) \prod\left(1+\frac{1}{q_{i}^{e_{i}}\left(q_{i}-1\right)}\right) = 1296144 \] Thus, the final result is \(\boxed{1296144}\).
1296144
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Given a prime \( p \) and positive integer \( k \), an integer \( n \) with \( 0 \leq n < p \) is called a \((p, k)\)-Hofstadterian residue if there exists an infinite sequence of integers \( n_{0}, n_{1}, n_{2}, \ldots \) such that \( n_{0} \equiv n \) and \( n_{i+1}^{k} \equiv n_{i} \pmod{p} \) for all integers \( i \geq 0 \). If \( f(p, k) \) is the number of \((p, k)\)-Hofstadterian residues, then compute \(\sum_{k=1}^{2016} f(2017, k)\).
[ "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/PrimitiveRoot", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization", "/Mathematics/NumberTheory/PrimeNumbers/PrimeNumberProperties" ]
Apply primitive roots to relate Hofstadterian residues to the size of the subgroup of k‑th powers, i.e., the largest divisor of p‑1 coprime to k, then sum via the multiplicative structure of p‑1.
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ours_631
We can think of \(S\) as the set of all \(a+b \sqrt{2}\), where \(a, b\) are taken \((\bmod 2015)\). Then the sum and product work as expected: \[ \left(a_{1}+b_{1} \sqrt{2}\right)+\left(a_{2}+b_{2} \sqrt{2}\right)=\left(a_{1}+a_{2}\right)+\left(b_{1}+b_{2}\right) \sqrt{2} \] and \[ \left(a_{1}+b_{1} \sqrt{2}\right) \times\left(a_{2}+b_{2} \sqrt{2}\right)=\left(a_{1} a_{2}+2 b_{1} b_{2}\right)+\left(b_{1} a_{2}+a_{1} b_{2}\right) \sqrt{2} \] We want to find all functions \(f: S \rightarrow S\) that preserve both addition and multiplication. Suppose such a function \(f\) exists. For any \(a \in S\), we have \(f(n a)=n f(a)\) for any positive integer \(n\). Since all integers are taken \((\bmod 2015)\), we equivalently have \(f(n a)=n f(a)\) for any remainder \(n(\bmod 2015)\). Let \(f(1)=x\) and \(f(\sqrt{2})=y\). From \(f(n a)=n f(a)\) and preservation of addition, we get \[ f(a+b \sqrt{2})=a f(1)+b f(\sqrt{2})=a x+b y \] Additionally, note that \[ 2 x=2 f(1)=f(2)=f(\sqrt{2})^{2}=y^{2}, \] so \(x=\frac{y^{2}}{2}\). Also, \[ \frac{y^{3}}{2}=x y=f(1) f(\sqrt{2})=f(\sqrt{2})=y \] so \(y^{3}=2 y\). Thus any valid \(f\) must be in the form \[ f(a+b \sqrt{2})=a\left(\frac{y^{2}}{2}\right)+b y \] for some \(y \in S\) satisfying \(y^{3}=2 y\). We claim that this is also sufficient for \(f\) to preserve both addition and multiplication. It is clear that such an \(f\) preserves addition, and \[ \begin{aligned} f(a+b \sqrt{2}) f(c+d \sqrt{2}) & =\left(\frac{a y^{2}}{2}+b y\right)\left(\frac{c y^{2}}{2}+d y\right) \\ & =\frac{a c y^{4}}{4}+\frac{b c y^{3}}{2}+\frac{a d y^{3}}{2}+b d y^{2} \\ & =\frac{a c y^{2}}{2}+b c y+a d y+b d y^{2} \\ & =(a c+2 b d)\left(\frac{y^{2}}{2}\right)+(b d+a d)(y) \\ & =f((a c+2 b d)+(b d+a d) \sqrt{2}) \\ & =f((a+b \sqrt{2})(c+d \sqrt{2})) \end{aligned} \] so it preserves multiplication as well. Thus it suffices to find the number of solutions to \(y^{3}=2 y\) in \(S\). By the Chinese Remainder Theorem, the answer is the product of the number of solutions to \(y^{3}=2 y\) when the coefficients are taken \((\bmod 5),(\bmod 13)\), and \((\bmod 31)\). Let \(\mathbb{F}_{p}\) denote integers \((\bmod p)\), and let \(S_{p}\) be the set of all \(a+b \sqrt{2}\), where \(a, b \in \mathbb{F}_{p}\) for an odd prime \(p\). Suppose two elements of \(S_{p}\) multiply to \(0\); i.e. \[ 0=(a+b \sqrt{2})(c+d \sqrt{2})=(a c+2 b d)+(b c+a d) \sqrt{2} \] Then \(a c+2 b d \equiv 0(\bmod p)\) and \(b c+a d \equiv 0(\bmod p)\). Now if \(a=b=0\) or \(c=d=0\), we clearly get a solution. Assuming neither of these hold, now suppose \(a=0(\bmod p)\); then \(b d=0(\bmod p)\) and \(b c=0(\bmod p)\). This is only possible if either \(a=b=0\) or \(c=d=0\), so this is a contradiction. Similarly, we can assume that all of \(b, c, d\) are nonzero \((\bmod p)\). Then \[ c \equiv-\frac{2 b d}{a} \Longrightarrow \frac{2 b^{2} d}{a}=a d \] which is equivalent to \(\left(\frac{a}{b}\right)^{2}=2\). It's well-known that in \(\mathbb{F}_{p}\) for odd \(p\), the equation \(z^{2}=2\) has no solution if \(p \equiv 3,5(\bmod 8)\), and there exists \((\text{nonzero}) g \in \mathbb{F}_{p}\) such that \(z^{2}=2\) if and only \(z= \pm g\). So in \(S_{p}\), for \(p \equiv 3,5(\bmod 8),(a+b \sqrt{2})(c+d \sqrt{2})\) implies \(a+b \sqrt{2}=0\) or \(c+d \sqrt{2}=0\). If \((a+b \sqrt{2})(c+d \sqrt{2})=0\) in \(S_{p}\) for \(p \equiv 1,7(\bmod 8)\) (and neither \(a=b=0\) nor \(c=d=0\) holds), then \(a=g b\), which implies \(c=-g d\) from \(b c+a d=0\), or \(a=-g b\), which implies \(c=g d\). Thus \[ \{a+b \sqrt{2}, c+d \sqrt{2}\}=\{k(g-\sqrt{2}), \ell(g+\sqrt{2})\} \] for some \(k, \ell \in \mathbb{F}_{p}\). We finally now consider the number of solutions to \(y^{3}=2 y\) in \(S_{p}\) for \(p=5,13,31\). If \(p=5\) or \(p=13\), then \(p \equiv 3,5(\bmod 8)\), so \(m n=0\) implies \(m=0\) or \(n=0\) for \(m, n \in S_{p}\). Note then that if \(y^{3}=2 y\), then \(y(y-\sqrt{2})(y+\sqrt{2})=0\). Thus either \(y=0, y=\sqrt{2}\), or \(y=-\sqrt{2}\). So in both cases, there are \(3\) solutions to \(y^{3}=2 y\). If \(p=31\), then \(p \equiv 7(\bmod 8)\), so \(m n=0\) implies \(m=0, n=0\), or \(\{m, n\}=\{k(8-\sqrt{2}), \ell(8+\sqrt{2})\}\) (noting that \(8^{2} \equiv 2(\bmod 31)\)). So if \(y\left(y^{2}-2\right)=0\), either \(y=0, y^{2}=2\), or \(y=k(8 \pm \sqrt{2})\). If \(y=0\), of course we have \(1\) solution for \(y\). If \(y^{2}=2\), then either \(y= \pm \sqrt{2}\) or \(y-\sqrt{2}=k(8 \pm \sqrt{2})\). If the second case holds, then \(k(8 \pm \sqrt{2})+2 \sqrt{2}=\ell(8 \mp \sqrt{2})\), implying \(8 k=\mp 8(2 \pm k)\). Thus either \(8 k=16-8 k\) or \(8 k=-16-8 k\), yielding \(k=1\) and \(y-\sqrt{2}=8-\sqrt{2}\), or \(k=-1\) and \(y-\sqrt{2}=-8+\sqrt{2}\). These two cases give the solutions \(y=8\) and \(y=-8\) additionally, so there are \(4\) solutions to \(y^{2}=2\). If \(y=k(8 \pm \sqrt{2})\), then \(y^{2}-2=\ell(8 \mp \sqrt{2})\). Note that \[ y^{2}-2=-2+k^{2}(4 \pm 16 \sqrt{2}) \] so \[ \left(4 k^{2}-2\right)=\mp 8\left( \pm 16 k^{2}\right)=-4 k^{2} \] Implying \(4 k^{2}=1\) and so \(k=15\) or \(k=16\). So in this case we get the \(4\) solutions \(k=15(8+\sqrt{2})\), \(k=16(8+\sqrt{2}), k=15(8-\sqrt{2})\), and \(k=16(8-\sqrt{2})\). In total, the \(p=31\) case gives \(1+4+4=9\) solutions for \(y^{3}=2 y\) in \(S_{31}\). The other two primes, \(p=5\) and \(p=13\), each yield \(3\) solutions, so the number of solutions to \(y^{3}=y\) in \(S\) is \(3 \cdot 3 \cdot 9=81\) (by the Chinese Remainder Theorem). So our final answer is \(\boxed{81}\).
81
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Let \( S \) be the set of all pairs \((a, b)\) of integers satisfying \(0 \leq a, b \leq 2014\). For any pairs \(s_{1}=\left(a_{1}, b_{1}\right), s_{2}=\left(a_{2}, b_{2}\right) \in S\), define \[ s_{1}+s_{2}=\left(\left(a_{1}+a_{2}\right)_{2015},\left(b_{1}+b_{2}\right)_{2015}\right) \text{ and } s_{1} \times s_{2}=\left(\left(a_{1} a_{2}+2 b_{1} b_{2}\right)_{2015},\left(a_{1} b_{2}+a_{2} b_{1}\right)_{2015}\right) \] where \(n_{2015}\) denotes the remainder when an integer \(n\) is divided by \(2015\). Compute the number of functions \(f: S \rightarrow S\) satisfying \[ f\left(s_{1}+s_{2}\right)=f\left(s_{1}\right)+f\left(s_{2}\right) \text{ and } f\left(s_{1} \times s_{2}\right)=f\left(s_{1}\right) \times f\left(s_{2}\right) \] for all \(s_{1}, s_{2} \in S\).
[ "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/GeneralAlgebra/AbstractAlgebra", "/Mathematics/Algebra/GeneralAlgebra/Algebra", "/Mathematics/Algebra/GeneralAlgebra/Endomorphism", "/Mathematics/Algebra/NumberTheory/AlgebraicNumberTheory/AlgebraicNumber", "/Mathematics/Algebra/NumberTheory/Arithmetic/AdditionandSubtraction", "/Mathematics/Algebra/NumberTheory/Arithmetic/MultiplicationandDivision", "/Mathematics/Algebra/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/Algebra/NumberTheory/Congruences/Mod", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/Congruences/Residue", "/Mathematics/Algebra/NumberTheory/Congruences/ResidueClass", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/NumberTheory/Integers/Integer", "/Mathematics/Algebra/NumberTheory/Integers/NonnegativeInteger", "/Mathematics/Algebra/NumberTheory/Integers/PositiveInteger", "/Mathematics/Algebra/NumberTheory/Integers/RingofIntegers", "/Mathematics/Algebra/NumberTheory/Integers/Z", "/Mathematics/Algebra/NumberTheory/Integers/Z-Plus", "/Mathematics/Algebra/NumberTheory/Integers/Zero", "/Mathematics/Algebra/RingTheory/CommutativeRing", "/Mathematics/Algebra/RingTheory/Ring", "/Mathematics/Algebra/RingTheory/RingHomomorphism", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/AlgebraicNumberTheory/AlgebraicInteger", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/Congruences/ResidueClass", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Model S as the ring ℤ/2015ℤ[√2] and note any map preserving addition and multiplication is a ring homomorphism determined by y = f(√2) satisfying y³ = 2y.
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ours_632
Let \( \Omega \) be the circumcircle of \( \triangle ABC \), and \( A_1 \) is on \( \Omega \) such that \( AA_1 \parallel BC \). Let lines \( BB_1 \) and \( CC_1 \) intersect at \( P \), lines \( CC_1 \) and \( AA_1 \) intersect at \( Q \), and lines \( AA_1 \) and \( BB_1 \) intersect at \( R \). Note that \( A, G, \) and \( P \) are collinear. Also, note that \( H \) is the circumcenter and \( O \) is the nine-point center of \( \triangle PQR \), so \( L \) is the orthocenter of \( \triangle PQR \). Hence, \( L \) is the incenter of \( \triangle A_1B_1C_1 \). Let \( \omega_1 \) be the incircle of \( \triangle A_1B_1C_1 \), so \( \omega_1 \) has center \( L \) and let it touch side \( B_1C_1 \) at \( D \). Similarly, let \( \omega_2 \) be the \( A_1 \)-excircle of \( \triangle A_1B_1C_1 \), so \( \omega_2 \) has center \( P \) and let it touch side \( B_1C_1 \) at \( E \). Note that \( A \) is the midpoint of arc \( B_1A_1C_1 \), so let \( AA_1 \) (the external angle bisector of \( \angle B_1A_1C_1 \)) intersect \( B_1C_1 \) at \( K \). Let \( \mathcal{F} \) be the inversion centered at \( A_1 \) with radius \( \sqrt{A_1B_1 \cdot A_1C_1} \), and let \( \mathcal{G} \) be the reflection through the internal angle bisector of \( \angle B_1A_1C_1 \). Let \( \mathcal{H} = \mathcal{F} \circ \mathcal{G} \). Note that \( \angle LA_1K = \angle LDK = 90^\circ \), so \( A_1LDK \) is cyclic. Note that \( \mathcal{H}(\omega_1) \) is the \( A_1 \)-mixtilinear excircle of \( \triangle A_1B_1C_1 \). Hence, the line through \( \mathcal{H}(L) = P \) and \( \mathcal{H}(K) = A_1 \) intersects \( \Omega \) at \( \mathcal{H}(D) \), so then \( X \) is the tangency point of the \( A_1 \)-mixtilinear excircle of \( \triangle A_1B_1C_1 \) and \( \Omega \). Similarly, since \( \angle PA_1K = \angle PEK = 90^\circ \), \( A_1PEK \) is cyclic. Note that \( \mathcal{H}(\omega_2) \) is the \( A_1 \)-mixtilinear incircle of \( \triangle A_1B_1C_1 \), so the line through \( \mathcal{H}(P) = L \) and \( \mathcal{H}(K) = A_1 \) intersects \( \Omega \) at \( \mathcal{H}(E) \), so then \( Y \) is the tangency point of the \( A_1 \)-mixtilinear incircle of \( \triangle A_1B_1C_1 \) and \( \Omega \). Let the tangent of \( X \) to \( \Omega \) be \( \ell_1 \), and the tangent of \( Y \) to \( \Omega \) be \( \ell_2 \). Let \( \ell_1 \cap \ell_2 = T \). Note that \( \mathcal{H}(\ell_1) \) is the circle tangent to \( BC \) at \( D \) passing through \( A_1 \), and \( \mathcal{H}(\ell_2) \) is the circle tangent to \( BC \) at \( E \) passing through \( A_1 \). Let \( \mathcal{H}(T) = S \) be the second intersection of the two circles, so that \( A_1T \) is the radical axis of these two circles. But the midpoint \( M \) of side \( B_1C_1 \) is the midpoint of \( DE \), so \( A_1, S, M \) are collinear, hence \( A_1T \) is the \( A_1 \)-symmedian of triangle \( \triangle A_1B_1C_1 \). Let the tangents of \( B_1 \) and \( C_1 \) to \( \Omega \) be \( F \), so then note that \( Z \) lies on the polar of both \( T \) and \( F \) with respect to \( \Omega \). Hence, \( Z \) is the pole of line \( TF \), but \( T \) and \( F \) both lie on the \( A_1 \)-symmedian of \( \triangle A_1B_1C_1 \), so \( Z \) lies on the polar of \( A_1 \) and hence \( ZA_1 \) is tangent to \( \Omega \). Now, let the foot of the perpendicular from \( Z \) to line \( AA_1 \) be \( Z_1 \). Note that \( \angle ABA_1 = \angle B - \angle C = 15^\circ \) so \( AA_1 = \sqrt{6} - \sqrt{2} \), and \( \angle AA_1Z = 180^\circ - (\angle B - \angle C) = 165^\circ \) so \( \angle ZA_1Z_1 = 15^\circ \). Note that \( ZA_1 = 4 \), so \( A_1Z_1 = \sqrt{6} + \sqrt{2} \) and \( ZZ_1 = \sqrt{6} - \sqrt{2} \) so by the Pythagorean Theorem, \( AZ^2 = 32 - \sqrt{48} \), so our answer is \( 3248 \). \(\boxed{3248}\)
3248
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Let \( \triangle ABC \) be a triangle with circumradius \( 2 \) and \( \angle B - \angle C = 15^\circ \). Denote its circumcenter as \( O \), orthocenter as \( H \), and centroid as \( G \). Let the reflection of \( H \) over \( O \) be \( L \), and let lines \( AG \) and \( AL \) intersect the circumcircle again at \( X \) and \( Y \), respectively. Define \( B_1 \) and \( C_1 \) as the points on the circumcircle of \( \triangle ABC \) such that \( BB_1 \parallel AC \) and \( CC_1 \parallel AB \), and let lines \( XY \) and \( B_1C_1 \) intersect at \( Z \). Given that \( OZ = 2\sqrt{5} \), then \( AZ^2 \) can be expressed in the form \( m - \sqrt{n} \) for positive integers \( m \) and \( n \). Find \( 100m + n \).
[ "/Mathematics/Geometry/Centroids/GeometricCentroid", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/CircleInversion", "/Mathematics/Geometry/PlaneGeometry/Circles/Circumcenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCircles" ]
Apply an inversion‑reflection to turn the incircle/excircle of triangle A₁B₁C₁ into the A₁‑mixtilinear incircle and excircle, revealing X and Y as mixtilinear tangency points and allowing use of symmedian‑polar relations to locate Z and compute AZ.
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ours_634
Assume Yang has his sides parallel to the coordinate axes. Denote the reflection tricks as \(R_{1}, U_{1}, L_{1}, D_{1}\) depending on whether the reflection takes his body to the right, up, left, or down. Similarly define \(R_{2}, U_{2}, L_{2}, D_{2}\) for the rotations. First, we count the number of ways for Yang to arrive back at his original square after \(2016\) tricks, regardless of his orientation or direction. Assume his only moves are \(R, U, L, D\). Then it's clear that the number of ways is \(\sum_{k=0}^{1008}\binom{2016}{2k}\binom{2k}{k}\binom{2016-2k}{1008-k} = \binom{2016}{1008}\binom{1008}{k}\binom{1008}{1008-k}\). By Vandermonde's Identity, this is equal to \(\binom{2016}{1008}^{2}\). Multiplying by \(2^{2016}\) to take into account that each move may be a rotation or a reflection gives a total of \(\binom{2016}{1008}^{2} \cdot 2^{2016}\). Now, we consider the orientations and directions. Take the right side of Yang and denote it as his head. We give Yang two orientations: \(S\) for when his legs are in the same counterclockwise order as originally and \(O\) for when his legs are in the opposite counterclockwise order as originally. Let the subscripts \(0,1,2,3\) denote whether Yang's head is facing east, north, west, or south, respectively. Now, split the orientations/directions of Yang into two groups: Group \(A\) consists of \(S_{0}, S_{2}, O_{1}, O_{3}\) and Group \(B\) consists of \(S_{1}, S_{3}, O_{0}, O_{2}\). Note that each rotation keeps the same orientation but increases the index by \(1\), and each reflection flips the orientation but keeps the index the same modulo 2. Hence, each move switches the group the sequence is in, so Yang must be in Group \(A\) after \(2016\) tricks. Now, fix a sequence of moves that results in Yang at his original square. Note that a rotation with \(R\) and \(L\), and similarly for a reflection with \(R\) and \(L\), are indistinguishable when it comes to orientation and direction. Thus, we denote them both by \(H\). Similarly, we denote \(U\) and \(D\) by the single movement \(V\). We split into two cases: a case where there are two consecutive \(H\)'s or two consecutive \(V\)'s, or another case where the sequence is either \(H V H V \cdots H V\), or \(V H V H \cdots V H\). Case 1. Assume that there are two consecutive \(H\)'s. We will show that the sequence is equally likely to be any of the four orientations/directions in Group \(A\) after \(2016\) tricks. Let \(\mathcal{M}\) be the sequence before the two \(H\)'s. We perform a bijection by varying over all rotation/reflection possibilities of the two \(H\)'s. Let \(H_{1}\) be a horizontal reflection, and \(H_{2}\) be a horizontal rotation. \(H_{1} H_{1}\) gives the same orientation as \(\mathcal{M}\) and an index increase of \(2\), \(H_{2} H_{2}\) gives the same orientation as \(\mathcal{M}\) and an index increase of \(0\) or \(4\), and both of \(H_{1} H_{2}\) and \(H_{2} H_{1}\) flip the orientation and one gives an increase of \(1\) while the other gives an increase of \(3\), depending on the direction of \(\mathcal{M}\). Thus, Yang is equally likely to take any orientation/direction in the group he must be in after the two \(H\)'s, and no different choices for the rotation/reflection of the \(H\)'s will result in the same orientation/direction of Yang ever again. Hence, exactly one of the four choices for the rotation/reflection of the two \(H\)'s will allow Yang to go back to \(S_{0}\) after \(2016\) moves. Case 2. Assume the sequence is \(H V H V \cdots H V\). \(V H V H \cdots V H\) follows analogously. The number of ways this can result in Yang being in his original square is \(\binom{1008}{504}^{2} \cdot 2^{2016}\), since we must choose the \(H\) and \(V\)'s to be \(L\) and \(R\), or \(U\) and \(D\), respectively. Now, we show that after \(2n\) moves, Yang is equally likely to be in either \(S_{2n}\) or \(O_{2n+1}\), for \(n \geq 1\). Define \(V_{1}\) and \(V_{2}\) similarly to how \(H_{1}\) and \(H_{2}\) were defined in the previous case. After one move, note that both \(H_{1} V_{1}\) and \(H_{2} V_{2}\) give \(S_{2}\), and \(H_{1} V_{2}\) and \(H_{2} V_{1}\) both give \(O_{3}\). Now, assume that \(2n=k\) holds, we will show it for \(2n=k+2\). Assume that for each way Yang is in \(S_{2k}, H_{1} V_{1}\) and \(H_{2} V_{2}\) both give \(S_{2k+2}, H_{1} V_{2}\) and \(H_{2} V_{1}\) are both in \(O_{2k+3}\); and for each way Yang is in \(O_{2k+1}, H_{1} V_{1}\) and \(H_{2} V_{2}\) both give \(O_{2k+3}\) and \(H_{1} V_{2}\) and \(H_{2} V_{1}\) both give \(S_{2k+2}\). Hence, since \(2016 \equiv 0 \pmod{4}\), Yang has a \(\frac{1}{2}\) chance of being in his original position. Hence, our final answer is \(\binom{2016}{1008}^{2} \cdot 2^{2014} + \binom{1008}{504}^{2} \cdot 2^{2015}\) after doubling to account for \(V H V H \cdots V H\). Now, we wish to evaluate this modulo \(100000\), which by the Chinese Remainder Theorem we simply have to do for \(32\) and \(5^{5}=3125\). It's clear that our answer is divisible by \(32\). Note that to compute \(\binom{2016}{1008}\) and \(\binom{1008}{504}^{2} \pmod{3125}\), we only need to compute \(\binom{2016}{1008}\) and \(\binom{1008}{504} \pmod{125}\) since it's easy to check that \(v_{5}\binom{2016}{1008} = \binom{1008}{504} = 2\). Here, we use the fact that \(\binom{pa}{pb} \equiv \binom{a}{b} \pmod{p^{3}}\) for all integers \(a, b\) and \(p>3\). Using this fact, we get that \(\binom{2016}{1008} \equiv \binom{2000}{1000}\binom{16}{8} \equiv \binom{16}{8}^{2} \pmod{125}\). Then, \(\binom{2016}{1008}^{2} \equiv 5^{4} \cdot 2574^{4} \equiv 625 \pmod{3125}\). Similarly, \(\binom{1008}{504}^{2} \equiv 5^{4} \cdot 14^{4} \equiv 625 \pmod{3125}\). Also, note that \(2^{2014} \equiv 4 \pmod{5}\) and \(2^{2015} \equiv 3 \pmod{5}\), so \(\binom{2016}{1008}^{2} \cdot 2^{2014} + \binom{1008}{504}^{2} \cdot 2^{2015} \equiv 2500 + 1875 \equiv 1250 \pmod{3125}\). Then, since our answer is \(1250 \pmod{3125}\) and \(0 \pmod{32}\), our final answer modulo \(100000\) is \(\boxed{20000}\).
20000
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
Yang the Spinning Square Sheep is a square in the plane such that his four legs are his four vertices. Yang can do two different types of tricks: (a) Yang can choose one of his sides, then reflect himself over the side. (b) Yang can choose one of his legs, then rotate \(90^{\circ}\) counterclockwise around the leg. Yang notices that after \(2016\) tricks, each leg ends up in exactly the same place the leg started out in! Let there be \(N\) ways for Yang to perform his \(2016\) tricks. What is the remainder when \(N\) is divided by \(100000\)?
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/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", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Square", "/Mathematics/Geometry/PlaneGeometry/Squares/Square", "/Mathematics/Geometry/Symmetry/MirrorSymmetry", "/Mathematics/Geometry/Symmetry/RotationalSymmetry", "/Mathematics/Geometry/Symmetry/SymmetryOperation", "/Mathematics/Geometry/Transformations/GeometricTransformations", "/Mathematics/Geometry/Transformations/Reflections/MirrorSymmetry", "/Mathematics/Geometry/Transformations/Reflections/Reflection", "/Mathematics/Geometry/Transformations/Rotations/RigidMotion", "/Mathematics/Geometry/Transformations/Rotations/Rotation", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Translate each trick into a horizontal or vertical step, count balanced sequences with binomial coefficients, and use the orientation‑group switching symmetry to find the fraction that return to the original square.
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ours_635
Let \(O_1, O_2,\) and \(O_3\) be the circumcenters of \(AHO, BHO,\) and \(CHO\). Note that \(BE, CF\), and the line through \(A\) perpendicular to \(OH\) are mutually parallel and thus concur at a point at infinity perpendicular to \(OH\). Since \(AH\) and \(AO\) are isogonal with respect to \(\angle BAC\), \(AO_1, BO_2, CO_3\) concur at the isogonal conjugate of this point at infinity, which lies on the circumcircle \(\omega\). Let this point of concurrence be \(W\). **Lemma 1.** \(W\) lies on \(l'\). *Proof.* Let \(L\) be the midpoint of side \(BC\), and let \(G'\) be the reflection of the centroid \(G\) over \(M\). Let \(\mathcal{F}\) be the reflection over line \(BC\), \(\mathcal{G}\) be the reflection over the perpendicular bisector, and \(\mathcal{H}\) be the reflection over the point \(L\). Note that \(\mathcal{F}(l) = \mathcal{G}(\mathcal{H}(l))\). Now, note that \(\mathcal{H}(l)\) is a homothety of factor 2 from \(A\). Let \(U\) be the foot of the altitude from \(A\) to \(AOH\), and let \(V\) be where line \(AU\) intersects the circumcircle of \(ABC\) again. Note that \(\mathcal{H}(U) = V\) and \(\mathcal{G}(V) = W\), so \(W\) is on \(l'\). **Lemma 2.** Let \(AW\) intersect line \(OH\) at \(D\). Then \(D\) and \(K\) are isogonal conjugates. *Proof.* Let \(J\) be the intersection of \(l\) and line \(BC\). Note that \(AEWJP\) is a complete quadrilateral formed by the four lines \(AE, AW, JE\), and \(JW\). By the dual of Desargues' Involution Theorem, there must be an involution mapping \(BE \rightarrow BW, BA \rightarrow BJ\), and \(BP \rightarrow BD\). Since the reflection across the angle bisector of \(\angle ABC\) is an involution mapping \(BE \rightarrow BW\) and \(BA \rightarrow BJ\), it must be the same as the prior involution. Thus it also maps \(BP \rightarrow BD\), implying that \(BP\) and \(BD\) are isogonal with respect to \(\angle ABC\). Similarly, we can show that the lines \(CQ\) and \(CD\) are isogonal with respect to \(\angle BCA\), so \(D\) and \(K\) are isogonal conjugates with respect to triangle \(ABC\). **Lemma 3.** Let \(\mathcal{I}\) be the homothety of factor \(-\frac{1}{2}\) centered at \(G\). Then \(\mathcal{I}(D) = T\). *Proof.* We will show that \(N, S\), and \(\mathcal{I}(D)\) are collinear. Let the foot of \(K\) to sides \(BC\) and \(BA\) be \(K_1\) and \(K_3\), respectively. Let the reflection of \(K\) over \(K_1\) and \(K_3\) be \(K_A\) and \(K_C\), respectively. Note that \(AK_CBZ\) is a parallelogram because \(M\) is the midpoint of \(K_CZ\). Similarly, \(CK_ABX\) is a parallelogram. \(BK\) is the diameter of the circumcircle of \(BK_1KK_3\), so it follows that \(BD\) is perpendicular to \(K_1K_3\) from Lemma 2, and hence \(K_AK_C\). But since \(BK_A = BK_C\), \(BD\) is the perpendicular bisector of \(K_AK_C\). Now, note that \(\overrightarrow{SN} = \vec{N} - \vec{S} = \frac{\vec{A} + \vec{C}}{2} - \frac{\vec{X} + \vec{Z}}{2} = \frac{\vec{A} - \vec{Z}}{2} + \frac{\vec{C} - \vec{X}}{2} = \frac{\overrightarrow{ZA} + \overrightarrow{XC}}{2} = \frac{\overrightarrow{BK_C} + \overrightarrow{BK_A}}{2} = \overrightarrow{BM_B}\), where \(M_B\) is the midpoint of \(K_AK_C\), so hence \(BD \parallel SN\). But since \(\mathcal{I}(B) = N\), it follows that \(BD \parallel N\mathcal{I}(D)\) and hence \(N, S\), and \(\mathcal{I}(D)\) are collinear. Similarly, it follows that \(M, R\), and \(\mathcal{I}(D)\) are collinear, so \(\mathcal{I}(D) = T\), as desired. Now, note that the conditions give us that the circumradius \(R = 10\), \(\cos A = \frac{4}{5}\), and \(\cos(B-C) = \frac{1}{2}\). This gives \(AO = 10\), \(OH = 14\), and \(HA = 16\). Then, \(\sin DAH = \cos AOH = \frac{1}{7}\) and \(\sin DAO = \cos AHO = \frac{11}{14}\), so by the Ratio Lemma we get that \(HD: DO = 16: 55\). Then, since \(HG: GO = 2: 1\), we get that \(HT: TO = 63: 8\) from Lemma 3. Then, \(OT = 14 \times \frac{8}{71} = \frac{112}{71}\), so the answer is \(11271\). \(\boxed{11271}\)
11271
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns-2.md" }
In triangle \(ABC\), \(AB = 3\sqrt{30} - \sqrt{10}\), \(BC = 12\), and \(CA = 3\sqrt{30} + \sqrt{10}\). Let \(M\) be the midpoint of \(AB\) and \(N\) be the midpoint of \(AC\). Denote \(l\) as the line passing through the circumcenter \(O\) and orthocenter \(H\) of \(ABC\), and let \(E\) and \(F\) be the feet of the perpendiculars from \(B\) and \(C\) to \(l\), respectively. Let \(l'\) be the reflection of \(l\) in \(BC\) such that \(l'\) intersects lines \(AE\) and \(AF\) at \(P\) and \(Q\), respectively. Let lines \(BP\) and \(CQ\) intersect at \(K\). \(X, Y\), and \(Z\) are the reflections of \(K\) over the perpendicular bisectors of sides \(BC, CA\), and \(AB\), respectively, and \(R\) and \(S\) are the midpoints of \(XY\) and \(XZ\), respectively. If lines \(MR\) and \(NS\) intersect at \(T\), then the length of \(OT\) can be expressed in the form \(\frac{p}{q}\) for relatively prime positive integers \(p\) and \(q\). Find \(100p + q\).
[ "/Mathematics/Geometry/Centroids/GeometricCentroid", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circumcenter", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleLines", "/Mathematics/Geometry/PlaneGeometry/Triangles/TrianglePoints", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties" ]
Show that T is the image of D (the intersection of AO with OH) under the homothety of factor -½ centered at the centroid, then compute OT using known length ratios.
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ours_636
Since \( A_{1} \) is the answer to this problem, we know that \( A_{1} = 2 A_{1}\left(A_{1}+A_{2}+\cdots+A_{30}\right) \). This implies that either \( A_{1} = 0 \) or \( A_{1} + A_{2} + \cdots + A_{30} = \frac{1}{2} \). The latter is impossible because all answers are nonnegative integers. Therefore, \( A_{1} = 0 \). \(\boxed{0}\)
0
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \( A_{n} \) denote the answer to the \( n \)th problem on this contest (\( n=1, \ldots, 30 \)); in particular, the answer to this problem is \( A_{1} \). Compute \( 2 A_{1}\left(A_{1}+A_{2}+\cdots+A_{30}\right) \).
[ "/Mathematics/Algebra/AlgebraicEquations/LinearEquation", "/Mathematics/Algebra/Products", "/Mathematics/Algebra/Sums/Accumulate", "/Mathematics/Algebra/Sums/CumulativeSum", "/Mathematics/Algebra/Sums/PartialSum", "/Mathematics/Algebra/Sums/RunningSum", "/Mathematics/Algebra/Sums/Sum" ]
Equate the expression to A₁ itself and solve the resulting equation using the fact that answers are non‑negative integers.
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ours_637
Solution. We present three different solutions. Solution 1. Note that \(x+y+z\), \(x+2y+3z\), \(x+3y+5z\) form an arithmetic sequence, giving us the answer of 12. Solution 2. Subtracting the first equation from twice the second gives: \[ x+3y+5z = 2(x+2y+3z) - (x+y+z) = 2(16) - 20 = 12. \] Solution 3. Note that we are given three variables but only two equations, so assuming that the answer is constant, we can assume \(x=0\). Then, \(y+z=20\) and \(2y+3z=16\). Solving these gives \(y=44\) and \(z=-24\). Hence, \(3y+5z = 132 - 120 = 12\). Thus, the value of \(x+3y+5z\) is \(\boxed{12}\).
12
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \(x, y\), and \(z\) be real numbers such that \(x+y+z=20\) and \(x+2y+3z=16\). What is the value of \(x+3y+5z\)?
[ "/Mathematics/Algebra/AlgebraicEquations/LinearEquation", "/Mathematics/Algebra/AlgebraicEquations/SimultaneousEquations", "/Mathematics/Algebra/LinearAlgebra/LinearSystemsofEquations/LinearEquation", "/Mathematics/Algebra/LinearAlgebra/LinearSystemsofEquations/LinearSystemofEquations", "/Mathematics/Algebra/Sums/Sum" ]
Write the target expression as 2·(x+2y+3z) minus (x+y+z).
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ours_638
For every \$30, it's clear that our best option is to buy 40 pens through the latter option. After we do this 5 times, we are left with \$23, which we can use to either buy two packages of 12 pens or a package of 20 pens. The first option is better, giving \(5 \cdot 40 + 2 \cdot 12 = 224\) pens. \(\boxed{224}\)
224
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
A store offers packages of 12 pens for \$10 and packages of 20 pens for \$15. Using only these two types of packages of pens, find the greatest number of pens \$173 can buy at this store.
[ "/Mathematics/Algebra/RateProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/PackingProblems/Bin-PackingProblem" ]
Buy as many 20‑pen packages (the lower cost per pen) as possible, then spend the leftover money on 12‑pen packages.
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ours_639
Notice that it suffices to minimize the last term because its coefficient is as large as the sum of the other three. In other words, the slope of \( f(x) \) will be nonpositive when \( x + 10 < 0 \), and will be nonnegative when \( x + 10 > 0 \). Therefore, the minimum is achieved when \( x + 10 = 0 \), or \( x = -10 \). At this point, the minimum value is \( f(-10) = 9 + 3 \cdot 7 + 6 \cdot 4 = 54 \). \(\boxed{54}\)
54
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Given that \( x \) is a real number, find the minimum value of \( f(x) = |x+1| + 3|x+3| + 6|x+6| + 10|x+10| \).
[ "/Mathematics/Algebra/Sums/Sum" ]
When the largest coefficient equals the sum of the others, the minimum occurs at the breakpoint of that term.
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ours_640
Let \(\ell\) have a slope of \(-k\) for a positive real number \(k\). The line \(\ell\) intersects the \(x\)-axis at \(\left(20+\frac{16}{k}, 0\right)\) and the \(y\)-axis at \((0,16+20k)\). The area of the triangle formed by the \(x\)-axis, \(y\)-axis, and \(\ell\) is given by: \[ \text{Area} = \frac{1}{2} \left(20 + \frac{16}{k}\right)(16 + 20k) \] Simplifying, the area becomes: \[ \text{Area} = 320 + 200k + \frac{128}{k} \] To find the minimum area, we differentiate with respect to \(k\) and set the derivative to zero: \[ \frac{d}{dk} \left(320 + 200k + \frac{128}{k}\right) = 200 - \frac{128}{k^2} = 0 \] Solving for \(k\), we get: \[ 200k^2 = 128 \quad \Rightarrow \quad k^2 = \frac{128}{200} = \frac{16}{25} \quad \Rightarrow \quad k = \frac{4}{5} \] Substituting \(k = \frac{4}{5}\) back into the area expression: \[ \text{Area} = 320 + 200 \times \frac{4}{5} + \frac{128}{\frac{4}{5}} = 320 + 160 + 160 = 640 \] Thus, the minimum possible area of the triangle is \(\boxed{640}\).
640
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \(\ell\) be a line with negative slope passing through the point \((20,16)\). What is the minimum possible area of a triangle that is bounded by the \(x\)-axis, \(y\)-axis, and \(\ell\)?
[ "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Derivative", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/DerivativeTest", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/Differentiation", "/Mathematics/CalculusandAnalysis/Calculus/DifferentialCalculus/FirstDerivative", "/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/Extremum", "/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/GlobalExtremum", "/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/GlobalMinimum", "/Mathematics/CalculusandAnalysis/Calculus/MaximaandMinima/Minimum", "/Mathematics/Geometry/CoordinateGeometry/AnalyticGeometry", "/Mathematics/Geometry/CoordinateGeometry/Cartesian", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/CoordinateGeometry/CartesianPlane", "/Mathematics/Geometry/CoordinateGeometry/x-Axis", "/Mathematics/Geometry/CoordinateGeometry/x-Intercept", "/Mathematics/Geometry/CoordinateGeometry/y-Axis", "/Mathematics/Geometry/CoordinateGeometry/y-Intercept", "/Mathematics/Geometry/PlaneGeometry/Triangles" ]
Express the triangle area as a function of the line’s slope and then minimize that expression.
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ours_641
We will show that the answer is $7$. It's clear that each team with at least $16$ wins must have at most $3$ losses. Assume for the sake of contradiction that there are $8$ such teams with at most $3$ losses. Then, consider the $\binom{8}{2}=28$ games among these $8$ teams, which must consist of $28$ losses. By the Pigeonhole Principle, some team must have at least $\frac{28}{8}=3.5$ losses, which is a contradiction. Hence, our answer is at most $7$. For the construction, let the $7$ teams with at least $16$ wins be labeled $0,1, \cdots, 6$. Say that team $i$ beats team $j$ for $i \neq j$ if and only if $i-j(\bmod 7) \in\{1,2,3\}$. Then, each team will have $3$ wins and $3$ losses among these $7$ teams, and let these teams beat all of the $13$ other teams. In this scenario, it is clear that each team has $16$ wins. \(\boxed{7}\)
7
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
In a round-robin basketball tournament, each basketball team plays every other basketball team exactly once. If there are $20$ basketball teams, what is the greatest number of basketball teams that could have at least $16$ wins after the tournament is completed?
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]
Bound the number of high‑win teams by counting total losses among them and applying the pigeonhole principle.
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ours_642
Assume without loss of generality that \(a \leq b \leq c \leq d\). Clearly, \(d \leq 24\). If \(d = 24\), then \(a = b = c = 1\), giving the solution \((1, 1, 1, 24)\). If \(d < 24\), then \(23 \mid a! \cdot b! \cdot c! \cdot d!\), implying \(23 \mid d!\). Since 23 is prime, \(d = 23\). Thus, \(a! \cdot b! \cdot c! = 24\). Now, \(c \leq 4\). If \(c = 4\), then \(a = b = 1\), giving the solution \((1, 1, 4, 23)\). If \(c < 4\), then \(3 \mid a! \cdot b! \cdot c!\), implying \(c = 3\). Thus, \(a! \cdot b! = 4\), which means \(a = b = 2\). This gives the solution \((2, 2, 3, 23)\). The number of ordered quadruples is the sum of the permutations of \((1, 1, 1, 24)\), \((1, 1, 4, 23)\), and \((2, 2, 3, 23)\): \[ \frac{4!}{3!} + \frac{4!}{2!} + \frac{4!}{2!} = 4 + 12 + 12 = 28 \] \(\boxed{28}\)
28
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Compute the number of ordered quadruples of positive integers \((a, b, c, d)\) such that \[ a! \cdot b! \cdot c! \cdot d! = 24! \]
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations" ]
Use prime divisibility (especially the prime 23) to force the largest factorial to be 24! or 23! and then bound the remaining variables.
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ours_643
We present three solutions to this problem. **Solution 1:** Extend lines \( AB, CD, EF \) to intersect at \( AB \cap CD = G, CD \cap EF = H, EF \cap AB = I \). Then, \( GH = HI = IG = 9 \). Note that \( IX = 4, IZ = 5 \), and \(\angle XIZ = 60^\circ\), so by the Law of Cosines, \( XZ = \sqrt{21} \). Then, since \( XY = YZ = ZX \), the area of \( XYZ \) is \(\frac{21 \sqrt{3}}{4}\), so the answer is \( 2134 \). **Solution 2:** Notice that \(\frac{[XIZ]}{[GHI]} = \frac{4}{9} \cdot \frac{5}{9} = \frac{20}{81}\), so since \([XIZ] = [YGX] = [ZHY]\), we get that \(\frac{[XYZ]}{[GHI]} = \frac{21}{81}\). Because \([GHI] = \frac{81 \sqrt{3}}{4}\), it follows that \([XYZ] = \frac{21 \sqrt{3}}{4}\) and the answer is \( 2134 \). **Solution 3:** Let \( O \) be the center of equilateral triangle \( XYZ \), so it's the center of \( ABCDEF \) as well. Let the foot of \( O \) to \( AB \) be \( K \). Then \( XK = \frac{1}{2} \) and \( KO = \frac{3 \sqrt{3}}{2} \), so \( XO = \sqrt{7} \) giving \([XYZ] = \frac{21 \sqrt{3}}{4}\). Hence the answer is \( 2134 \). \(\boxed{2134}\)
2134
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \( ABCDEF \) be a regular hexagon of side length \( 3 \). Let \( X, Y, \) and \( Z \) be points on segments \( AB, CD, \) and \( EF \) such that \( AX = CY = EZ = 1 \). The area of triangle \( XYZ \) can be expressed in the form \(\frac{a \sqrt{b}}{c}\) where \( a, b, c \) are positive integers such that \( b \) is not divisible by the square of any prime and \(\gcd(a, c) = 1\). Find \( 100a + 10b + c \).
[ "/Mathematics/Geometry/Distance/Point-PointDistance2-Dimensional", "/Mathematics/Geometry/PlaneGeometry/Hexagons", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/Area", "/Mathematics/Geometry/PlaneGeometry/MiscellaneousPlaneGeometry/PlaneGeometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/Hexagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularHexagon", "/Mathematics/Geometry/PlaneGeometry/Polygons/RegularPolygon", "/Mathematics/Geometry/PlaneGeometry/Triangles/SpecialTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Extend the three alternating sides of the hexagon to form a large equilateral triangle and relate triangle XYZ to it via similarity or proportional angles
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ours_644
We evaluate the sum modulo 4 and modulo 25. First, consider modulo 4: - \( f(1) \equiv 1 \pmod{4} \) - \( f(2) \equiv 3 \pmod{4} \) - \( f(3) \equiv 3 \pmod{4} \) - \( f(4) \equiv 1 \pmod{4} \) These values repeat every four integers, so the sum \( f(1) + f(2) + f(3) + f(4) \equiv 1 + 3 + 3 + 1 \equiv 0 \pmod{4} \). Since there are 504 such complete cycles in the sum up to \( f(2016) \), the entire sum is \( 504 \times 0 \equiv 0 \pmod{4} \). Next, consider modulo 25: - Notice that \( f(n) \) is divisible by 25 for \( n \geq 8 \). - Therefore, we only need to consider \( f(1) + f(2) + \cdots + f(7) \pmod{25} \). Calculating these: - \( f(1) = 1 \equiv 1 \pmod{25} \) - \( f(2) = 1 \times 3 = 3 \equiv 3 \pmod{25} \) - \( f(3) = 1 \times 3 \times 5 = 15 \equiv 15 \pmod{25} \) - \( f(4) = 1 \times 3 \times 5 \times 7 = 105 \equiv 5 \pmod{25} \) - \( f(5) = 1 \times 3 \times 5 \times 7 \times 9 = 945 \equiv 20 \pmod{25} \) - \( f(6) = 1 \times 3 \times 5 \times 7 \times 9 \times 11 = 10395 \equiv 20 \pmod{25} \) - \( f(7) = 1 \times 3 \times 5 \times 7 \times 9 \times 11 \times 13 = 135135 \equiv 10 \pmod{25} \) Summing these: \[ f(1) + f(2) + \cdots + f(7) \equiv 1 + 3 + 15 + 5 + 20 + 20 + 10 \equiv 74 \equiv 24 \pmod{25} \] Combining the results using the Chinese Remainder Theorem: - The sum is \( 0 \pmod{4} \) and \( 24 \pmod{25} \). The number that satisfies these conditions is 24. Therefore, the remainder when \( f(1) + f(2) + \cdots + f(2016) \) is divided by 100 is \(\boxed{24}\).
24
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \( f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1) \). Compute the remainder when \( f(1) + f(2) + f(3) + \cdots + f(2016) \) is divided by 100.
[ "/Mathematics/DiscreteMathematics/DivisionProblems/SubsetSumProblem", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Congruences/Residue", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/NumberTheoreticSums", "/Mathematics/NumberTheory/Sequences/IntegerSequence", "/Mathematics/NumberTheory/Sequences/OddSequence", "/Mathematics/NumberTheory/Sequences/Sequence" ]
Break the sum modulo 4 and modulo 25, using periodicity for the former and the fact that f(n) is divisible by 25 for n≥8, then combine with CRT.
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ours_645
The problem is equivalent to the number of ways to choose 7 out of 9 squares in a 3 by 3 grid and label them from 1 to 7 inclusive such that the two blank squares don't lie on the same row or column. Once this configuration is fixed, we can map each column to a T-shirt and each row to a pair of pants (and therefore each square corresponds to a possible outfit), and the number in each square (or lack thereof) signifies which day to wear this outfit, if at all. There are \(\frac{3^{2} \cdot 2^{2}}{2} = 18\) ways to choose the two blanks and \(7! = 5040\) ways to label the 7 other squares, for a total of \(18 \cdot 5040 = 90720\) ways. \(\boxed{90720}\)
90720
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
[ "/Mathematics/DiscreteMathematics/Combinatorics/Configurations", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Map shirts and pants to a 3×3 grid and count placements of two empty cells that are not in the same row or column.
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ours_646
We need to solve the system of quadratic congruences \( x^2 \equiv 4 \pmod{16} \) and \( x^2 \equiv 16 \pmod{20} \). The first congruence is equivalent to \( 16 \mid (x-2)(x+2) \), while the second congruence implies \( x \equiv \pm 1 \pmod{5} \) and \( x \equiv 0 \pmod{2} \). These conditions give \( x \equiv 4, 6 \pmod{10} \). Now, we claim that \( x \equiv 2 \pmod{4} \). Assume not, then neither \( x-2 \) nor \( x+2 \) is divisible by \( 4 \), so \( (x-2)(x+2) \) is not divisible by \( 16 \), which is a contradiction. Thus, \( x \equiv 2 \pmod{4} \) and \( x \equiv 4, 6 \pmod{10} \), so \( x \equiv 6, 14 \pmod{20} \). It suffices to confirm that every number of this type works. Let \( x = 20k + 10 \pm 4 \), then \( x^2 = (20k + 10)^2 \pm 8(20k + 10) + 16 = 400k^2 + 400k + 116 \pm 160k \pm 80 \). Now, \( x^2 \equiv 4 \pmod{16} \), so \( x^2 - 20 \) is divisible by \( 16 \), and \( x^2 \equiv 16 \pmod{20} \), so \( x^2 - 16 \) is divisible by \( 20 \). Thus, every \( 20 \) integers, there will be \( 2 \) solutions, for a total of \( 4020 \div 20 \times 2 = 402 \) up to \( 4020 \). But between \( 4021 \) and \( 4032 \), there is only one: \( 4026 \), as \( 4034 \) is too large. Therefore, there are \( 403 \) such positive integers. \(\boxed{403}\)
403
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
For how many positive integers \( x \) less than \( 4032 \) is \( x^2 - 20 \) divisible by \( 16 \) and \( x^2 - 16 \) divisible by \( 20 \)?
[ "/Mathematics/Algebra/NumberTheory/Arithmetic", "/Mathematics/Algebra/NumberTheory/Congruences/Congruence", "/Mathematics/Algebra/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/Algebra/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/Algebra/NumberTheory/Congruences/QuadraticCongruenceEquation", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/DiscreteMathematics/DivisionProblems/SubsetSumProblem", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Arithmetic/GeneralArithmetic", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/QuadraticCongruenceEquation", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory" ]
Combine the two quadratic congruences to obtain a single modulus condition (x ≡ 6 or 14 mod 20).
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ours_647
Without loss of generality, let's consider the 9-cube as \([0,1]^{9}\) in the 9-dimensional Euclidean space. On each 5-dimensional face, there are \(9-5=4\) coordinates that are fixed, and each of them can be 0 or 1. Therefore, there are \(\binom{9}{4} \cdot 2^{4} = 2016\) five-dimensional faces in total. \(\boxed{2016}\)
2016
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
A 9-cube is a nine-dimensional hypercube (and hence has \(2^{9}\) vertices, for example). How many five-dimensional faces does it have? (An \(n\) dimensional hypercube is defined to have vertices at each of the points \((a_{1}, a_{2}, \cdots, a_{n})\) with \(a_{i} \in\{0,1\}\) for \(1 \leq i \leq n\).)
[ "/Mathematics/DiscreteMathematics/Combinatorics/BinomialCoefficients", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/CoordinateGeometry/Cartesian", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinateSystem", "/Mathematics/Geometry/CoordinateGeometry/CartesianCoordinates", "/Mathematics/Geometry/CoordinateGeometry/Coordinates", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes/Hypercube", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes/Polytope", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry/Hypercube" ]
Count faces by fixing the n‑k coordinates of the hypercube, then counting the ways to choose and assign those fixed coordinates.
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ours_649
First, assume that we have an arbitrary triangle with side lengths \( a, b, \) and \( c \) that satisfy this concurrency. Let \( D \) be the foot of the altitude from \( A \) to \( BC \). We will use a result known as Carnot's Theorem. **Lemma (Carnot's Theorem):** Let \( \triangle ABC \) be a triangle, and let \( D, E, F \) be points on \( BC, AC, AB \) respectively. If the perpendiculars through \( D, E, F \) to their respective sides concur, then: \[ BD^2 + CE^2 + AF^2 = CD^2 + AE^2 + BF^2 \] **Proof:** Suppose the perpendiculars concur at a point \( P \). Then: \[ PB^2 - PC^2 = (BD^2 + PD^2) - (CD^2 + PD^2) = BD^2 - CD^2 \] Similarly, \[ PC^2 - PA^2 = CE^2 - AE^2 \] and \[ PA^2 - PB^2 = AF^2 - BF^2 \] Summing these equations yields the desired result. Now, by Carnot's Theorem, we need: \[ BD^2 - DC^2 + \left(\frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + (s-a)^2 - (s-b)^2 = 0 \] Simplifying this gives: \[ BD^2 - DC^2 + c(b-a) = 0 \] By the Perpendicularity Lemma, \( BD^2 - DC^2 = BA^2 - AC^2 = c^2 - b^2 \), so we have: \[ c^2 + (b-a)c - b^2 = 0 \] With \( a = 20 \) and \( b = 16 \), we have: \[ c^2 - 4c - 256 = 0 \] Solving this quadratic equation, we find \( c = 2 \pm \sqrt{260} \). Since \( c \) must be positive, we have \( c = 2 + \sqrt{260} \). Therefore, \( 100m + n = 460 \). \(\boxed{460}\)
460
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \( \triangle ABC \) be a triangle with \( BC = 20 \) and \( CA = 16 \), and let \( I \) be its incenter. If the altitude from \( A \) to \( BC \), the perpendicular bisector of \( AC \), and the line through \( I \) perpendicular to \( AB \) intersect at a common point, then the length \( AB \) can be written as \( m+\sqrt{n} \) for positive integers \( m \) and \( n \). What is \( 100m+n \)?
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleCenters" ]
Use Carnot's theorem for the three perpendiculars to obtain an equation linking side lengths, then solve the resulting quadratic for AB.
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ours_650
Note that \(ab+bc+cd+da=(a+c)(b+d)=16\), which implies along with \((a+c)+(b+d)=20\) that \(a+c\) and \(b+d\) are roots of the equation \(x^2 - 20x + 16 = 0\). Solving the quadratic, we get \(a+c=10-2\sqrt{21}\) and \(b+d=10+2\sqrt{21}\). Now \[ abc+bcd+cda+dab=(a+c)bd+(b+d)ac \] so it suffices to maximize \(bd\) and \(ac\) subject to the constraints \(a+c=10-2\sqrt{21}\) and \(b+d=10+2\sqrt{21}\). By AM-GM, \(ac\) is maximized when \(a=c=5-\sqrt{21}\) and \(bd\) is maximized when \(b=d=5+\sqrt{21}\). Thus the answer is \[ (5+\sqrt{21})^2(10-2\sqrt{21})+(5-\sqrt{21})^2(10+2\sqrt{21})=80 \] \(\boxed{80}\)
80
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \(a, b, c, d\) be four real numbers such that \(a+b+c+d=20\) and \(ab+bc+cd+da=16\). Find the maximum possible value of \(abc+bcd+cda+dab\).
[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticEquation", "/Mathematics/Algebra/AlgebraicEquations/QuadraticFormula", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/QuadraticPolynomial", "/Mathematics/Algebra/Polynomials/VietasFormulas", "/Mathematics/Algebra/Products/Product", "/Mathematics/Algebra/Sums/Sum" ]
Rewrite ab+bc+cd+da as (a+c)(b+d) to determine a+c and b+d, then maximize ac and bd using AM‑GM
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ours_651
We want to use the dividers to split the permutation into sets of size \(3, 3, 2\) in some order. We consider all three possible uses of these dividers: \(U_1 = 3-3-2\), \(U_2 = 3-2-3\), and \(U_3 = 2-3-3\). In \(U_1\), \(p_1\) appears as the hundreds digit, and in \(U_2\), \(p_6\) appears as the hundreds digit. In \(U_3\), \(p_3\) appears as the hundreds digit, and \(p_4\) appears in \(U_1\). Note that \(2p_1 + 2p_6 + p_3 + p_4 \leq 2(8+7) + 6 + 5 = 41\). We wish to maximize \(m = \min(p_1 + p_4, p_1 + p_6, p_3 + p_6)\). We can achieve the maximum of \(m = 13\) by setting \(\{p_1, p_6\} = \{7, 8\}\), \(p_4 = 13 - p_1\), \(p_3 = 13 - p_6\), or \(\{p_1, p_6\} = \{6, 8\}\), \(p_4 = 13 - p_1\), \(p_3 = 13 - p_6\). For \(U_2\), \(m < p_1 + p_6\), so we do not worry about this case. Since \(p_3\) and \(p_6\) are units digits in \(U_1\), and \(p_1\) and \(p_4\) are tens digits in \(U_3\), choosing any of our possibilities for \((p_1, p_3, p_4, p_6)\) does not affect our minimum value for \(N\). Since \(p_7\) is a tens digit of both \(U_1\) and \(U_3\), and \(p_8\) is a units digit of both \(U_1\) and \(U_3\), we can set \(p_7 = 4\) and \(p_8 = 1\). Then we set \(\{p_2, p_5\} = \{2, 3\}\) in some order since \(p_2, p_5\) are both tens digits in \(U_1\) and both units digits in \(U_3\). Now, we get \(U_1 = 1404\) and \(U_3 = 1476\), so \(M = 1404\). \(\boxed{1404}\)
1404
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Jay is given a permutation \(\{p_1, p_2, \ldots, p_8\}\) of \(\{1, 2, \ldots, 8\}\). He may take two dividers and split the permutation into three non-empty sets, and he concatenates each set into a single integer. In other words, if Jay chooses \(a, b\) with \(1 \leq a < b < 8\), he will get the three integers \(\overline{p_1 p_2 \ldots p_a}\), \(\overline{p_{a+1} p_{a+2} \ldots p_b}\), and \(\overline{p_{b+1} p_{b+2} \cdots p_8}\). Jay then sums the three integers into a sum \(N = \overline{p_1 p_2 \ldots p_a} + \overline{p_{a+1} p_{a+2} \ldots p_b} + \overline{p_{b+1} p_{b+2} \cdots p_8}\). Find the smallest positive integer \(M\) such that no matter what permutation Jay is given, he may choose two dividers such that \(N \leq M\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialOptimization", "/Mathematics/DiscreteMathematics/Combinatorics/Permutations", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/RecreationalMathematics/Puzzles/Puzzle" ]
Analyze the three possible 3‑3‑2 block arrangements so that the largest digits must occupy high place values in at least one split, giving a uniform upper bound on the minimal sum.
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ours_652
Solution. Call a pair \((a, b)\) forcing if \(a, b \in S\) forces \(S=\mathbb{N}\). Lemma. \((a, b)\) is forcing if and only if \(a, b\) share no odd factor that is greater than \(1\). Proof. If \(a, b\) share an odd factor greater than \(1\), then neither operation can change this. Therefore, \(1 \notin S\), so \(S\) is not forced to be \(\mathbb{N}\). We proceed by induction on the pairs, where pairs are sorted by first coordinate, then by second coordinate. The base cases are clear, since if \(1 \in S\), then using condition (a) repeatedly shows that \(S=\mathbb{N}\). Assume that \(a<b\). If either is even, we can halve it and finish by induction. This doesn't change whether they have a common odd factor greater than \(1\). Now assume that \(a, b\) are both odd. Now because \(a, b \in S\), and \(a, b\) are both odd, \(\frac{a+b}{2}\) is in the set (use condition (a), then (b)). It's easy to check that \(a, \frac{a+b}{2}\) share no common odd factor if \(a, b\) do not. More explicitly, if an odd number \(p \mid \frac{a+b}{2}\) and \(p \mid a\), then \(p \mid a+b\) and therefore, \(p \mid (a+b)-a=b\). Now because \(\frac{a+b}{2}<b\), we can finish by induction. To finish, one could use the principle of inclusion and exclusion to eliminate pairs that share common odd prime factors. The odd primes less than \(50\) are \(3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\). Therefore, the final answer is \[ \begin{gathered} 50^{2}-\left\lfloor\frac{50}{3}\right\rfloor^{2}-\left\lfloor\frac{50}{5}\right\rfloor^{2}-\left\lfloor\frac{50}{7}\right\rfloor^{2}-\left\lfloor\frac{50}{11}\right\rfloor^{2}-\left\lfloor\frac{50}{13}\right\rfloor^{2}-\left\lfloor\frac{50}{17}\right\rfloor^{2}-\left\lfloor\frac{50}{19}\right\rfloor^{2}-\left\lfloor\frac{50}{23}\right\rfloor^{2}-\left\lfloor\frac{50}{29}\right\rfloor^{2}-\left\lfloor\frac{50}{31}\right\rfloor^{2}-\left\lfloor\frac{50}{37}\right\rfloor^{2} \\ -\left\lfloor\frac{50}{41}\right\rfloor^{2}-\left\lfloor\frac{50}{43}\right\rfloor^{2}-\left\lfloor\frac{50}{47}\right\rfloor^{2}+\left\lfloor\frac{50}{15}\right\rfloor^{2}+\left\lfloor\frac{50}{21}\right\rfloor^{2}+\left\lfloor\frac{50}{33}\right\rfloor^{2}+\left\lfloor\frac{50}{35}\right\rfloor^{2}+\left\lfloor\frac{50}{39}\right\rfloor^{2}=2068 \end{gathered} \] \(\boxed{2068}\)
2068
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
A set \( S \subseteq \mathbb{N} \) satisfies the following conditions: (a) If \( x, y \in S \) (not necessarily distinct), then \( x+y \in S \). (b) If \( x \) is an integer and \( 2x \in S \), then \( x \in S \). Find the number of pairs of integers \((a, b)\) with \(1 \leq a, b \leq 50\) such that if \(a, b \in S\) then \(S=\mathbb{N}\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/DivisionProblems", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory", "/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/Parity/EvenNumber", "/Mathematics/NumberTheory/Parity/OddNumber" ]
A pair (a,b) forces S=ℕ precisely when a and b share no odd divisor greater than 1.
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ours_653
Say \((1,1)\) is upper left, and \((100,200)\) is bottom right. Note that Kevin must hop in at least 200 squares to get from \((1,1)\) to \((100,200)\). He also must hop in each row and column at least once. Therefore, we can see that \[ M \geq \sum_{i=1}^{200} b_{i} + \sum_{i=1}^{100} a_{i} + 100 \min \left(a_{1}, a_{2}, \ldots, a_{100}\right) \] The extra term with the minimum at the end comes from the fact that the numbers written on the rows must be added at least 200 times in total, as Kevin hops on at least 200 squares. To see that this minimum is attainable, Kevin could just hop diagonally until he gets to the row with minimum sum, stay in that row for 100 steps while increasing his column number one by one, and then hop diagonally until the finish. By linearity of expectation, \[ \mathbb{E}[M] = \sum_{i=1}^{200} \mathbb{E}[b_{i}] + \sum_{i=1}^{100} \mathbb{E}[a_{i}] + 100 \cdot \mathbb{E}\left[\min \left(a_{1}, a_{2}, \ldots, a_{100}\right)\right] = 300 \cdot \frac{2017}{2} + 100 \mathbb{E}\left[\min \left(a_{1}, a_{2}, \ldots, a_{100}\right)\right] \] as \(\mathbb{E}[a_{i}] = \mathbb{E}[b_{i}] = \frac{2017}{2}\). To finish, we need to compute the expected value of the minimum. This is a classical problem with many ways to do it, one of which is a straightforward computation using the Hockey-Stick Identity. Instead, I will present an alternate proof, one that I like better. Say \(a_{1} < a_{2} < \cdots < a_{100}\). Consider the ranges of integers (some of which may be empty) \([1, a_{1} - 1], [a_{1} + 1, a_{2} - 1], \ldots, [a_{99} + 1, a_{100} - 1], [a_{100} + 1, 2016]\). There are 101 of these ranges, with 1916 total integers among them. We can compute the min by taking the first of these ranges and adding 1 to its length. By symmetry, the average length of the first range will be \(\frac{1916}{101}\), so the expected value of the minimum number is \(\frac{1916}{101} + 1 = \frac{2017}{101}\). Plugging this in, our final answer is \[ \frac{2017}{2} \cdot 300 + 100 \cdot \frac{2017}{101} = \frac{30759250}{101} \Longrightarrow p+q = 30759351 \] \(\boxed{30759351}\)
30759351
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Kevin is in kindergarten, so his teacher puts a \(100 \times 200\) addition table on the board during class. The teacher first randomly generates distinct positive integers \(a_{1}, a_{2}, \ldots, a_{100}\) in the range \([1,2016]\) corresponding to the rows, and then she randomly generates distinct positive integers \(b_{1}, b_{2}, \ldots, b_{200}\) in the range \([1,2016]\) corresponding to the columns. She then fills in the addition table by writing the number \(a_{i}+b_{j}\) in the square \((i, j)\) for each \(1 \leq i \leq 100, 1 \leq j \leq 200\). During recess, Kevin takes the addition table and draws it on the playground using chalk. Now he can play hopscotch on it! He wants to hop from \((1,1)\) to \((100,200)\). At each step, he can jump in one of 8 directions to a new square bordering the square he stands on a side or at a corner. Let \(M\) be the minimum possible sum of the numbers on the squares he jumps on during his path to \((100,200)\) (including both the starting and ending squares). The expected value of \(M\) can be expressed in the form \(\frac{p}{q}\) for relatively prime positive integers \(p, q\). Find \(p+q\).
[ "/Mathematics/DiscreteMathematics/Combinatorics/CombinatorialIdentities", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/ProbabilityandStatistics/Probability/ProbabilitySpace", "/Mathematics/ProbabilityandStatistics/RankStatistics/OrderStatistic" ]
Construct a path that uses the smallest row value for 100 steps, reducing the minimal sum to total of all a’s and b’s plus 100·min a, then compute its expectation via the expected minimum of a random sample.
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ours_654
I claim that \(f(n)=2^{n-s(n)}\), where \(s(n)\) is the sum of the digits of \(n\) in base 10. We proceed by induction on \(d\), the number of digits of \(n\) in base 10. For \(d=1\), we have \(f(\lfloor n / 10\rfloor)=f(0)=1\), and the \(512\) part becomes just \(1\) also, so \(f(n)=1=2^{n-n}\). Now, assume for some \(d=k \geq 1\), \(f(n)=2^{n-s(n)}\) for all \(n\) with \(k\) digits. The base case of \(k=1\) has just been proven. Then, let \(n=10a+b\) for a \(k\)-digit number \(a\) and a single-digit number \(b\). Then \(f(n)=512^{a} f(a)\). But \(a\) is a \(k\)-digit number, so \(f(a)=2^{a-s(a)}\). Thus, \(f(n)=2^{9a+a-s(a)}=2^{10a-s(a)}\). But note that \(s(a)=s(a)+s(b)-b\) as \(b=s(b)\), and \(s(a)+s(b)=s(10a)+s(b)=s(10a+b)=s(n)\), so \(f(n)=2^{10a+b-s(n)}=2^{n-s(n)}\). Thus, by induction on \(d\), we have \(f(n)=2^{n-s(n)}\) for all nonnegative integers \(n\). Then the condition about \(2500\) digits means that \(\left\lfloor\log_{16} f(n)\right\rfloor+1 \leq 2500\). This is equivalent to \(\left\lfloor\frac{n-s(n)}{4}\right\rfloor \leq 2499\), which in turn is equivalent to \(\frac{n-s(n)}{9}<1111+\frac{1}{9}\). But note that \(n-s(n)\) is always divisible by \(9\), so the LHS must be an integer, implying that \(\frac{n-s(n)}{9} \leq 1111\). Now, if we let \(n=\sum_{i=0}^{\infty} a_{i} 10^{i}\) for integers \(a_{i} \in[0,9]\), then \(\frac{n-s(n)}{9}=a_{1}+11a_{2}+111a_{3}+\ldots\). This shows that \(n-s(n)\) is always nonnegative. We can then easily determine that for every integer \(k \in[0,1111]\) except for \(1110\) and those in \([0,1109]\) that are \(10,21,32,43,54,65,76,87,98,109,110 \pmod{111}\), there exist exactly \(10\) solutions to \(\frac{n-s(n)}{9}=k\). This is a total of \(1001\) numbers, so \(10010\) solutions. \(\boxed{10010}\)
10010
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \(\mathbb{Z}_{\geq 0}\) denote the set of nonnegative integers. Define a function \(f: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{Z}\) with \(f(0)=1\) and \[ f(n)=512^{\lfloor n / 10\rfloor} f(\lfloor n / 10\rfloor) \] for all \(n \geq 1\). Determine the number of nonnegative integers \(n\) such that the hexadecimal (base 16) representation of \(f(n)\) contains no more than \(2500\) digits.
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/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/NonnegativeInteger", "/Mathematics/NumberTheory/Integers/RationalInteger", "/Mathematics/NumberTheory/Integers/WholeNumber", "/Mathematics/NumberTheory/Integers/Z", "/Mathematics/NumberTheory/Integers/Z-Plus" ]
Rewrite the recursion to express f(n) as 2^{n minus the sum of decimal digits of n}.
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ours_655
Notice that the function \( A \) is multiplicative, which means whenever \( m, n \) are relatively prime, we have \( A(m) A(n) = A(mn) \). We can focus on \( A\left(p^{t}\right) = \frac{\left(p^{t+1} - 1\right) / (p-1)}{t+1} \leq 42 \) for prime \( p \) and positive integer \( t \). We avoid prime factors other than \( 2, 3, 7 \) because any prime factor greater or equal to \( 11 \) in the numerator is too large, requiring high powers to cancel out in the denominator. For example, \( A\left(2^{10}\right) = \frac{2047}{11} > 42 \) and \( A\left(5^{4}\right) = \frac{781}{5} > 42 \). Thus, the candidate components are narrowed down to \( A\left(2^{t}\right) \) for \( t = 0, 1, 2, 5 \) and \( A(p) = \frac{p+1}{2} \) for primes \( p \) between \( 3 \) and \( 83 \) inclusive. Here is a complete list of all candidates: \[ \begin{align*} A(1) &= 1, & A(2) &= \frac{3}{2}, & A(4) &= \frac{7}{3}, & A(32) &= \frac{21}{2}, \\ A(3) &= 2, & A(5) &= 3, & A(7) &= 4, & A(11) &= 6, \\ A(13) &= 7, & A(17) &= 9, & A(23) &= 12, & A(31) &= 16, \\ A(41) &= 21, & A(47) &= 24, & A(53) &= 27, & A(71) &= 36, \\ A(83) &= 42. \end{align*} \] We now consider cases based on the maximum power of \( 2 \) dividing \( n \), denoted as \( 2^{t} \). - If \( t = 0 \), we look for distinct primes whose \( A \)-values multiply to \( 42 \): \( 83, 3 \cdot 41 = 123, 11 \cdot 13 = 143, 3 \cdot 5 \cdot 13 = 195 \). - If \( t = 1 \), the \( A \)-values multiply to \( \frac{42}{3/2} = 28 \): \( 2 \cdot 7 \cdot 13 = 182 \). - If \( t = 2 \), the \( A \)-values multiply to \( \frac{42}{7/3} = 18 \): \( 2^{2} \cdot 3 \cdot 17 = 204, 2^{2} \cdot 5 \cdot 11 = 220 \). - If \( t = 5 \), the \( A \)-values multiply to \( \frac{42}{21/2} = 4 \): \( 2^{5} \cdot 7 = 224 \). Summing up all cases, we find the solutions are \( 83, 123, 143, 182, 195, 204, 220, 224 \), and their sum is \( 1374 \). \(\boxed{1374}\)
1374
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Define \( A(n) \) as the average of all positive divisors of the positive integer \( n \). Find the sum of all solutions to \( A(n) = 42 \).
[ "/Mathematics/NumberTheory/Divisors/Coprime", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/Divisors/RelativelyPrime", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/NumberTheoreticFunctions/DivisorFunction" ]
Exploit that the divisor‑average function is multiplicative, so factor 42 into feasible A(p^k) values from prime powers and combine them.
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ours_656
Let the argument of \( z \), with \(|z|=1\), be \(\theta\). Note that \( f(z)=z^{4}+z^{3}+r z^{2}+z+1 \) is in the direction of argument \( 2 \theta \) with a signed magnitude of \( r+2 \cos \theta+2 \cos (2 \theta)=4 \cos ^{2} \theta+2 \cos \theta-2+r=4\left(\cos \theta+\frac{1}{4}\right)^{2}-\frac{9}{4}+r \). For \( r>\frac{9}{4} \), this signed magnitude is always positive. Then, \( f\left(z_{1}\right)=f\left(z_{2}\right) \) can only happen for \( z_{1} \neq z_{2} \) if their arguments \(\theta_{1}, \theta_{2}\) satisfy \( 2 \theta_{1}=2 \theta_{2} \Longrightarrow \theta_{1}=\theta_{2}+\pi \). But then their signed magnitudes can only be equal if \(\cos \theta_{1}=\cos \theta_{2}\), which implies \(\left\{z_{1}, z_{2}\right\}=\{i,-i\}\). Hence \( f\left(z_{1}\right) \neq f\left(z_{2}\right) \) for all distinct \( z_{1}, z_{2} \). For \( r<-4 \), the signed magnitude is always negative since \(\left|\cos \theta+\frac{1}{4}\right| \leq \frac{5}{4}\) over all \(\theta\). This implies that \( f\left(z_{1}\right) \neq f\left(z_{2}\right) \). However, if \(-4 \leq r \leq \frac{9}{4}\), then there exists a unique value for \(\cos \theta\) with \(-\frac{1}{4} \leq \cos \theta \leq 1\) such that the signed magnitude is equal to \(0\). For all \(\cos \theta \neq 1\), there are two values of \(\theta\) giving the desired value of \(\cos \theta\), showing there exist distinct \( z_{1}, z_{2} \) giving \( f\left(z_{1}\right)=f\left(z_{2}\right)=0 \). However, \(\cos \theta=1\) only when \(\theta=0\), meaning for \( r=-4 \) the signed magnitude is always negative except for \(\theta=0\), when the signed magnitude is exactly \(0\). Hence \( r=-4 \) is also repetitive. Thus, \( a=-4 \) and \( b=\frac{9}{4} \), so \(|a|+|b| = 4 + \frac{9}{4} = \frac{25}{4}\). Therefore, \( p = 25 \) and \( q = 4 \), giving \( 100p + q = 2504 \). \(\boxed{2504}\)
2504
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Say a real number \( r \) is repetitive if there exist two distinct complex numbers \( z_{1}, z_{2} \) with \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(\left\{z_{1}, z_{2}\right\} \neq\{-i, i\}\) such that \[ z_{1}\left(z_{1}^{3}+z_{1}^{2}+r z_{1}+1\right)=z_{2}\left(z_{2}^{3}+z_{2}^{2}+r z_{2}+1\right) \] There exist real numbers \( a, b \) such that a real number \( r \) is repetitive if and only if \( a<r \leq b \). If the value of \(|a|+|b|\) can be expressed in the form \(\frac{p}{q}\) for relatively prime positive integers \( p \) and \( q \), find \( 100 p+q \).
[ "/Mathematics/Algebra/AlgebraicEquations/AlgebraicEquation", "/Mathematics/Algebra/AlgebraicEquations/QuarticEquation", "/Mathematics/Algebra/Polynomials/ComplexPolynomial", "/Mathematics/Algebra/Polynomials/FundamentalTheoremofAlgebra", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PolynomialEquation", "/Mathematics/Algebra/Polynomials/PolynomialRoots", "/Mathematics/Algebra/Polynomials/UnivariatePolynomial" ]
Express the polynomial on |z|=1 using cos θ to obtain a simple quadratic whose sign determines the r‑range for duplicate values.
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ours_657
Let the pedal triangle of \( P \) with respect to \( \triangle ABC \) be \( \triangle DEF \) such that \( D \) is on \( BC \), \( E \) is on \( CA \), and \( F \) is on \( AB \). Note that \(\angle P Y X = \angle B Y X = \angle B A X = \angle F A P = \angle F E P\). Similarly, \(\angle P Y Z = \angle D E P\), so \(\angle X Y Z = \angle D E F\). Similarly, \(\angle Y Z X = \angle E F D\), so \(\triangle DEF \sim \triangle XYZ\). Then by the Law of Sines on triangles \( \triangle DEP \) and \( \triangle FEP \), \[ \begin{aligned} \frac{YX}{YZ} & = \frac{ED}{EF} \\ & = \frac{\left(\frac{EP \sin EPD}{\sin EDP}\right)}{\left(\frac{EP \sin EPF}{\sin EFP}\right)} \\ & = \frac{\sin EPD}{\sin EPF} \cdot \frac{\sin EFP}{\sin EDP} \\ & = \frac{\sin C}{\sin A} \cdot \frac{\sin EAP}{\sin ECP} \\ & = \frac{BA}{BC} \cdot \frac{PC}{PA}. \end{aligned} \] Symmetry shows that \( YZ: ZX: XY = PA \cdot BC: PB \cdot CA: PC \cdot AB = 7: 12: 15 \). Note that \(\cos BAC = \frac{5^2 + 8^2 - 7^2}{2 \cdot 5 \cdot 8} = \frac{1}{2}\), so \(\angle BAC = 60^\circ\), and the circumradius \( R \) of \(\omega\) is \(\frac{BC}{2 \sin A} = \frac{7}{\sqrt{3}}\). By Heron's formula, a triangle with side lengths \( 7, 12, 15 \) has area \(\sqrt{17 \cdot 10 \cdot 5 \cdot 2} = 10 \sqrt{17}\) and circumradius \(\frac{7 \cdot 12 \cdot 15}{4 \cdot 10 \sqrt{17}} = \frac{63}{2 \sqrt{17}}\). Since \( \triangle XYZ \) also has circumcircle \(\omega\), we can scale the \( 7-12-15 \) triangle to find the area of \( \triangle XYZ \) is \( 10 \sqrt{17} \cdot \left(\frac{\frac{7}{\sqrt{3}}}{\frac{63}{2 \sqrt{17}}}\right)^2 = \frac{680 \sqrt{17}}{243} \), so the answer is \( 940 \). \(\boxed{940}\)
940
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring16Solns.md" }
Let \( \triangle ABC \) be a triangle with \( AB = 5 \), \( BC = 7 \), \( CA = 8 \), and circumcircle \(\omega\). Let \( P \) be a point inside \( \triangle ABC \) such that \( PA: PB: PC = 2: 3: 6 \). Let rays \(\overrightarrow{AP}\), \(\overrightarrow{BP}\), and \(\overrightarrow{CP}\) intersect \(\omega\) again at \( X, Y, \) and \( Z \), respectively. The area of \( \triangle XYZ \) can be expressed in the form \(\frac{p \sqrt{q}}{r}\) where \( p \) and \( r \) are relatively prime positive integers and \( q \) is a positive integer not divisible by the square of any prime. What is \( p+q+r \)?
[ "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/GeneralGeometry/MensurationFormula", "/Mathematics/Geometry/PlaneGeometry/Circles/Circle", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles/AssociatedTriangles", "/Mathematics/Geometry/PlaneGeometry/Triangles/Cevians", "/Mathematics/Geometry/PlaneGeometry/Triangles/TriangleProperties", "/Mathematics/Geometry/Trigonometry/Angles/Angle", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry" ]
Establish that XYZ is similar to the pedal triangle of P, yielding side ratios proportional to PA·BC, PB·CA, and PC·AB.
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ours_658
First, we work in \(\mathbb{F}_{2}\). We claim that \(x^{5} + x^{4} + x^{2} + x + 1\) is irreducible. If it were reducible, it would have to be divisible by an irreducible polynomial of degree \(1\) or \(2\). Thus, we only have to consider divisibility by \(x\), \(x+1\), and \(x^{2} + x + 1\). But \(x^{5} + x^{4} + x^{2} + x + 1 = x(x+1)^{2}(x^{2} + x + 1) + 1\), so it is not divisible by any of those. Now consider a root \(z\) of \(x^{5} + x^{4} + x^{2} + x + 1\). Since it is irreducible, \(z\) is an element of \(\mathbb{F}_{32}\). The order of a nonzero element of \(\mathbb{F}_{32}\) divides \(31\), so it is either \(1\) or \(31\). Since the only element with order \(1\) is \(1\), \(z\) must have order \(31\), making it a primitive root in \(\mathbb{F}_{32}\). Hence, \(1, z, z^{2}, \ldots, z^{30}\) are the nonzero elements of \(\mathbb{F}_{32}\). Furthermore, \(x^{5} + x^{4} + x^{2} + x + 1 \mid x^{a} + x^{b} + x^{c} + x^{d} + x^{e}\) if and only if \(z^{a} + z^{b} + z^{c} + z^{d} + z^{e} = 0\). We need to find the number of ways \(5\) distinct elements of \(\mathbb{F}_{32}\) can add to \(0\). Suppose they are ordered and divide by \(5!\) at the end. We can choose \(4\) random distinct elements, and the fifth will be uniquely determined (it is their sum). This results in \(31 \cdot 30 \cdot 29 \cdot 28\) ways. The only issue is that the fifth element might be the same as one of the first four or \(0\). To resolve this, count the number of ways \(3\) distinct elements of \(\mathbb{F}_{32}\) can add to \(0\). By choosing \(2\) random distinct elements and taking their sum as the third, there are \(31 \cdot 30\) ways. It is not possible for their sum to be equal to one of them, because that implies the other is \(0\). The number of overcounted quintuples is \(31 \cdot 30 \cdot 4 \cdot 28\), since there are \(4\) ways to choose three of the first four entries to sum to \(0\) and \(28\) ways to choose the element for the remaining entry and the last entry. Also, the number of ways for four elements to sum to \(0\) is \(31 \cdot 30 \cdot 29 - 31 \cdot 30\). Hence, our answer is: \[ \frac{31 \cdot 30 \cdot 29 \cdot 28 - 31 \cdot 30 \cdot 4 \cdot 28 - 31 \cdot 30 \cdot 29 + 31 \cdot 30}{5!} = \frac{31 \cdot 30 \cdot 24 \cdot 28}{120} = 5208. \] \(\boxed{5208}\)
5208
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Determine the number of ordered quintuples \((a, b, c, d, e)\) of integers with \(0 \leq a < b < c < d < e \leq 30\) for which there exist polynomials \(Q(x)\) and \(R(x)\) with integer coefficients such that \[ x^{a} + x^{b} + x^{c} + x^{d} + x^{e} = Q(x)\left(x^{5} + x^{4} + x^{2} + x + 1\right) + 2R(x). \]
[ "/Mathematics/Algebra/FieldTheory/ExtensionField", "/Mathematics/Algebra/FieldTheory/FiniteField", "/Mathematics/Algebra/FieldTheory/IrreduciblePolynomial", "/Mathematics/Algebra/FieldTheory/PerfectField", "/Mathematics/Algebra/FieldTheory/PrimitiveElement", "/Mathematics/Algebra/Polynomials/IrreduciblePolynomial", "/Mathematics/Algebra/Polynomials/Polynomial", "/Mathematics/Algebra/Polynomials/PrimitivePolynomial", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/FiniteGroups/FiniteGroup", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics" ]
Convert the divisibility by the irreducible polynomial into the requirement that five distinct field elements sum to zero in the multiplicative group of \(\mathbb{F}_{32}\).
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ours_659
Let \( m = 2017 \). It is clear that \( A_{n} > 0 \) if and only if \( (m, n) = 1 \) and \( n \geq 2 \). Now, fix \( n \) that works. An integer \( k \) is not in \( S_{n} \) if and only if we can express \( k = x m + y n \) with \( x, y \in \mathbb{N}_{0} \) and \( x \leq n-1 \). Assume the contrary, that \( k = x_{0} m + y_{0} n \) with \( x_{0} \geq n \). Let \( z = \left\lfloor \frac{x_{0}}{n} \right\rfloor \). Then \( k = (x_{0} - n z) m + (y_{0} + m z) n \), which is a contradiction since \( x_{0} - n z \in \mathbb{N}_{0} \). The generating function for the set of \( k \) with \( k = x m + y n \) for \( x, y \in \mathbb{N}_{0} \) and \( x \leq n-1 \) is: \[ \frac{1-x^{m n}}{1-x^{m}} \cdot \frac{1}{1-x^{n}} \] Thus, the generating function for the integers in \( S_{n} \) is: \[ \frac{1}{1-x} - \frac{1-x^{m n}}{(1-x^{m})(1-x^{n})} \] Let \( F(x) = \frac{1}{1-x} - \frac{1-x^{m n}}{(1-x^{m})(1-x^{n})} \). Then \(|S_{n}| = \lim_{x \rightarrow 1} F(x)\) and the sum of the elements of \( S_{n} \) is \(\lim_{x \rightarrow 1} F'(x)\). Applying L'Hopital's rule twice, we find: \[ \lim_{x \rightarrow 1} F(x) = \frac{(m-1)(n-1)}{2} \] Thus, the size of \( S_{n} \) is \(|S_{n}| = \frac{(m-1)(n-1)}{2}\). The sum of the elements of \( S_{n} \) is \(\frac{(m-1)(n-1)(2 m n - m - n - 1)}{12}\), so the average of the elements of \( S_{n} \) is \(\frac{2 m n - m - n - 1}{6}\). Thus, if \( y = \frac{1}{2} \), \[ \sum_{n=1}^{\infty} \frac{A_{n}}{2^{n}} = -\frac{m-2}{12} + \frac{2 m - 1}{3} - \frac{m+1}{6} - \frac{m(2 m - 1) 2^{m}}{6(2^{m} - 1)^{2}} + \frac{m+1}{6(2^{m} - 1)} \] Simplifying, \[ -\frac{m-2}{12} + \frac{2 m - 1}{3} - \frac{m+1}{6} = \frac{5 m - 4}{12} = 840 + \frac{1}{12} \] Thus, \[ \left\lfloor\sum_{n=1}^{\infty} \frac{A_{n}}{2^{n}}\right\rfloor = 840 \] \(\boxed{840}\)
840
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
For any positive integer \( n \), let \( S_{n} \) denote the set of positive integers which cannot be written in the form \( a n + 2017 b \) for nonnegative integers \( a \) and \( b \). Let \( A_{n} \) denote the average of the elements of \( S_{n} \) if the cardinality of \( S_{n} \) is positive and finite, and \( 0 \) otherwise. Compute \[ \left\lfloor\sum_{n=1}^{\infty} \frac{A_{n}}{2^{n}}\right\rfloor \]
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/AdditiveNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/GeneratingFunctions/GeneratingFunction", "/Mathematics/NumberTheory/Integers/Integer", "/Mathematics/NumberTheory/Integers/PositiveInteger" ]
Apply the Frobenius coin problem formulas (via generating functions) to count and sum the non‑representable integers for coprime m and n.
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ours_660
Note that it does not matter that we are looking at a unit ball since all simple hyperplanes pass through the origin. We can replace it with \([-1,1]^{4}\) or simply its surface and still have the same answer. This hypercube has 3-dimensional faces that are separated by simple hyperplanes of the form \(x_{i} - x_{j} = 0\) or \(x_{i} + x_{j} = 0\) where \(i \neq j\), and it suffices to consider one of the faces, say \(x_{4} = 1\). Within this face, the simple hyperplanes \(x_{1} = 0, x_{2} = 0, x_{3} = 0\) separate the face into \(2^{3} = 8\) identical unit cubes, and it suffices to consider one of them, say \(0 \leq x_{1}, x_{2}, x_{3} \leq 1\). The simple hyperplanes that cut through this cube are: \[ x_{1} = x_{2}, \quad x_{2} = x_{3}, \quad x_{3} = x_{1}, \quad x_{1} + x_{2} = 1, \quad x_{1} + x_{3} = 1, \quad x_{2} + x_{3} = 1, \quad x_{1} + x_{2} + x_{3} = 1 \] \[ x_{1} + x_{2} - x_{3} = 0, \quad x_{1} + x_{2} - x_{3} = 1, \quad x_{1} - x_{2} + x_{3} = 0, \quad x_{1} - x_{2} + x_{3} = 1, \quad -x_{1} + x_{2} + x_{3} = 0, \quad -x_{1} + x_{2} + x_{3} = 1 \] (Note that the constant 1 is allowed since \(x_{4} = 1\) and one can set \(k_{4} = -1\).) In order to visualize the arrangement of these cuts, one can consider their intersections with \(x_{3} = t\) for different values of \(t\) and draw them inside the unit square \(0 \leq x_{1}, x_{2} \leq 1\) as moving/stationary lines parametrized by \(t\). The following six diagrams show the situation when \(0 < t < \frac{1}{4}, \frac{1}{4} < t < \frac{1}{3}, \frac{1}{3} < t < \frac{1}{2}, \frac{1}{2} < t < \frac{2}{3}, \frac{2}{3} < t < \frac{3}{4}, \frac{3}{4} < t < 1\) respectively: (Note that the asymmetry on the top-right corner comes from the absence of the plane \(x_{1} + x_{2} + x_{3} = 2\).) Whenever three (or more) lines pass through a common point or when two parallel lines coincide, a new region is created after they separate. These regions, along with the original ones, are shaded in gray in the above diagrams. One can count that there are \(42 + 4 + 8 + 22 + 6 + 2 = 84\) regions in total in this unit cube. Therefore, in total there are \(8 \cdot 8 \cdot 84 = 5376\) regions in the original unit ball. \(\boxed{5376}\)
5376
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
A simple hyperplane in \(\mathbb{R}^{4}\) has the form \[ k_{1} x_{1} + k_{2} x_{2} + k_{3} x_{3} + k_{4} x_{4} = 0 \] for some integers \(k_{1}, k_{2}, k_{3}, k_{4} \in \{-1, 0, 1\}\) that are not all zero. Find the number of regions that the set of all simple hyperplanes divide the unit ball \(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} \leq 1\) into.
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/Geometry/CombinatorialGeometry", "/Mathematics/Geometry/ComputationalGeometry/GeneralComputationalGeometry/ComputationalGeometry", "/Mathematics/Geometry/MultidimensionalGeometry/Four-DimensionalGeometry/4-DGeometry", "/Mathematics/Geometry/MultidimensionalGeometry/Four-DimensionalGeometry/4-DimensionalGeometry", "/Mathematics/Geometry/MultidimensionalGeometry/Polytopes", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry/Hypercube", "/Mathematics/Geometry/MultidimensionalGeometry/n-DimensionalGeometry/Hyperplane" ]
Use the symmetry of the origin‑passing hyperplanes to reduce the 4‑D region count to counting regions inside a single unit cube (or sub‑cube) via planar slices.
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ours_661
Let \( S = PQ \cap BC \) and \( S' \in BC \) such that \( BQ, CP, AS' \) are concurrent. Let \( X = (APQ) \cap (ABC) \), \( X' = (SS'X) \cap (ABC) \), and \( A' \) be the antipode of \( A \) on \( (ABC) \). It's clear that \( X \) is the center of a spiral similarity sending \( PQ \) to \( BC \). By Ceva's Theorem on \( BQ, CP, AS' \) and \( AP = AQ \), we know that \( r = \frac{XP}{XQ} = \frac{BP}{CQ} = \frac{BS'}{CS'} \), so that \( XS' \) bisects \(\angle BXC\), hence \( X \in S'M \). Similarly, by Menelaus, \( SB:SC = -r \), implying \( X \in SN \). Since \( X = (AI) \cap (AA') \), we deduce that \( X \in IA' \). Now let \( T' = XI \cap PQ \) so that \(\angle PXQ\) is bisected by \( XT \), implying that \(\frac{PT}{TQ} = r\). Then the spiral similarity sending \( PQ \) to \( BC \) sends \( T' \) to \( S' \), so we deduce \( XSS'T' \) is cyclic. But \( MX \perp NX \Longrightarrow SX \perp S'X\), so that \( T' \) is the projection of \( S' \) onto \( PQ \). Now I claim that \( T = T' \). Since \((S, S'; B, C)\) is harmonic and \( S'T \perp ST \), we deduce \( S'T, ST \) bisect \(\angle BTC\). It suffices to show that \( T'H, T'I \), or equivalently \( T'H, T'A' \), are isogonal in \(\angle BT'C\), as this would imply that \( H', T', I \) are collinear. To do this, we note that \( BPT' \sim CQT' \) by AA similarity, so that \(\angle T'BA = \angle T'CA\). Then \(\angle HBA = \angle HCA\) implies \(\angle HBT' = \angle HCT'\). If we let \( BH \cap CT' = B', CH \cap BT' = C' \), then \( BCB'C' \) is cyclic, hence \( T'B':T'C' = TB:TC \). Remark that \(\angle BT'C\) and \(\angle BAC\) have parallel angle bisectors. Then since \( B'H' \) belongs to the direction \(\perp_{AC}\), which is isogonal in \(\angle BTC\) to the direction of \( BA'\), which is \(\perp_{AB}\), and similarly for \( C'H', CA'\), we can deduce that figures \( T'B'HC', T'BA'C \) are inversely similar in \(\angle BT'C\), hence \( T'H, T'A' \) are isogonal as desired. Now we have established that \( T = T' \). Note that \((SS')\) must be an Apollonius circle in \(\triangle XBC\), hence \( XBX'C \) is harmonic so \( X'B:X'C = r\), so the Angle Bisector Theorem yields that \( X' \in SM, S'N \). Since \((S, S'; B, C)\) is harmonic, we know \((SS'), (ABC)\) are orthogonal. Let the S-Apollonius circle of \( SMN \) meet \((SS')\) at a point \( T'' \neq S \). Since \((SS')\) and the S-Apollonius circle of \( SMN \) are both orthogonal to \((ABC)\), we know the inversion about \((ABC)\) swaps \( S, T'' \), so it follows that \( T'' = T \). Then \(\frac{TM}{TN} = \frac{SM}{SN} = \frac{\sin SNM}{\sin SMN} = \frac{XM}{X'N}\). We can verify that \( XM = \frac{MB^2}{S'M}, NX' = \frac{NB^2}{NS'}\) so the ratio becomes \(\frac{NS'}{S'M} \frac{MB^2}{NB^2}\). Now we perform computations. \(\cos A = \frac{5}{13}\) by the Law of Cosines, \(\sin 0.5A = \frac{2}{\sqrt{13}}, \cos 0.5A = \frac{3}{\sqrt{13}}\) by half-angle, \([ABC] = 84\) by Heron's Formula, \( R = \frac{65}{8} \) by \([ABC] = \frac{abc}{4R}, BM = 2R \sin 0.5A = \frac{65}{2\sqrt{13}}, BN = 2R \cos 0.5A = \frac{65 \cdot 3}{4\sqrt{13}}\) by the Law of Sines. Next, we can compute \( AM = \frac{117}{2\sqrt{13}} \) by Ptolemy on \( ABMC \). Then since \( APMQ \) is a rhombus, we know \( AP = 0.5AM \frac{1}{\cos 0.5A} = \frac{39}{4} \), so that \( BP = \frac{13}{4}, CQ = \frac{17}{4}, r = \frac{13}{17} \). Next, by this ratio, we know \( BS' = \frac{13}{2}, CS' = \frac{17}{2} \). By Stewart's Theorem on \( BMC \), we know \( BM^2 = MS'^2 + BS' \cdot CS' \Longrightarrow MS' = \sqrt{26} \). Similarly, we find \( NS' = \frac{\sqrt{157 \cdot 13}}{4} \). Then \(\frac{NS'}{S'M} \frac{MB^2}{NB^2} = \frac{\sqrt{314}}{18}\), so the answer is \( 31418 \). \(\boxed{31418}\)
31418
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Let \( \triangle ABC \) be a triangle with \( AB = 13 \), \( BC = 15 \), \( AC = 14 \), circumcenter \( O \), and orthocenter \( H \). Let \( M, N \) be the midpoints of the minor and major arcs \( BC \) on the circumcircle of \( \triangle ABC \). Suppose \( P \in AB, Q \in AC \) satisfy that \( P, O, Q \) are collinear and \( PQ \parallel AN \), and point \( I \) satisfies \( IP \perp AB, IQ \perp AC \). Let \( H' \) be the reflection of \( H \) over line \( PQ \), and suppose \( H'I \) meets \( PQ \) at a point \( T \). If \(\frac{MT}{NT}\) can be written in the form \(\frac{\sqrt{m}}{n}\) for positive integers \( m, n \) where \( m \) is not divisible by the square of any prime, then find \( 100m + n \).
[ "/Mathematics/Geometry/GeneralGeometry/Bisector", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/InversiveGeometry/Inversion", "/Mathematics/Geometry/PlaneGeometry/Arcs/MajorArc", "/Mathematics/Geometry/PlaneGeometry/Arcs/Mid-ArcPoints", "/Mathematics/Geometry/PlaneGeometry/Arcs/MinorArc", "/Mathematics/Geometry/PlaneGeometry/Circles/ApolloniusCircle", "/Mathematics/Geometry/PlaneGeometry/Circles/Circumcenter", "/Mathematics/Geometry/PlaneGeometry/Circles/OrthogonalCircles", "/Mathematics/Geometry/PlaneGeometry/CircularTriangles/CircularTriangle", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/ProjectiveGeometry", "/Mathematics/Geometry/Trigonometry/GeneralTrigonometry/Trigonometry", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/LawofCosines", "/Mathematics/Geometry/Trigonometry/TrigonometricFunctions/LawofSines" ]
Identify the spiral similarity sending PQ to BC and use its harmonic and Apollonius circle properties to equate MT/NT with SM/SN.
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ours_662
First, note that if we send \( f(x) \rightarrow f(x)+8 \), the equation will still be true. Then, without loss of generality, assume that the image of \( f \) is contained within \(\{0, \ldots, 7\}\), and we'll multiply by \(2^{16}\) at the end to compensate. Let \( P(x, y) \) denote the statement \( f(x)^{2}+f(y)^{2}+f(x+y)^{2} \equiv 1+2 f(x) f(y) f(x+y) \pmod{16} \). Note that \( P(x, x) \) implies \( 2 f(x)^{2}+f(2x)^{2} \equiv 1+2 f(x)^{2} f(2x) \pmod{16} \), which leads to \( f(2x) \equiv 1 \pmod{2} \). So \( f \) maps evens to odds. Furthermore, for \( x, y \) odd, \( P(x, y) \) implies \( f(x)^{2}+f(y)^{2} \equiv 2 f(x) f(y) \pmod{4} \), leading to \((f(x)-f(y))^{2} \equiv 0 \pmod{4}\). So either \( f \) sends odds to odds or odds to evens. **Case 1:** \( f \) sends odds to evens. Then for \( x, y \) odd, \( P(x, y) \) implies \( f(x)^{2}+f(y)^{2} \equiv 0 \pmod{8} \). Then \( f(x) \equiv f(y) \pmod{4} \). - **Subcase 1:** \( f(\text{odd}) \equiv 0 \pmod{4} \). Then \( P(x, y) \) for \( x, y \) odd gives \( f(x+y)^{2} \equiv 1 \pmod{16} \), so \( f(\text{even}) \in \{1,5\} \). Now, \( P(a, b) \) for \( a, b \) even gives \( f(a) f(b) f(a+b) \equiv 1 \pmod{8} \). That is, an even number of \( f(a), f(b), f(a+b) \) are \( 5 \). In particular, \( f(2a)=1 \). If \( a, b \equiv 2 \pmod{4} \), \( P(a, b) \) implies \( f(a)=f(b) \). This value can be either \( 1 \) or \( 5 \), and by the above all equations will work out. Then there are \( 2 \cdot 2^{8} \) possibilities in this case; \( 2 \) for all the \( 2 \pmod{4} \) numbers, and \( 2 \) for each odd, which can be \( 0 \) or \( 4 \). - **Subcase 2:** \( f(\text{odd}) \equiv 2 \pmod{4} \). Take \( x, y \) odd. Note that \( f(x)^{2} \equiv 4 \pmod{16} \). \( P(x, y) \) implies \( 8+f(x+y)^{2} \equiv 9 \pmod{16} \). Then \( f(\text{even}) \in \{1,7\} \). Now for \( a, b \) even, \( 3 \equiv f(a)^{2}+f(b)^{2}+f(a+b)^{2} \equiv 1+2 f(a) f(b) f(a+b) \). Thus \( f(a) f(b) f(a+b) \equiv 1 \pmod{8} \). We finish as in subcase 1, and get that there are \( 2^{8} \cdot 2 \) possibilities in this subcase; \( 2 \) for assigning the evens, and \( 2 \) for each odd, which can be any of \(\{2,6\}\). **Case 2:** \( f \) sends odds to odds. Note that \( P(x, y) \Longleftrightarrow 3 \equiv 1+2 f(x) f(y) f(x+y) \pmod{8} \). Thus \( f(x) f(y) f(x+y) \equiv 1 \pmod{4} \). In particular, \( f(2x) \equiv 1 \pmod{4} \). Going back to the equation, if \( x, y \) are of the same parity, \( f(x) \equiv f(y) \pmod{4} \). Now, let \( f(1)=a \). Then for any \( n, m \), \[ \begin{aligned} & P(n, m) \Longleftrightarrow f(n)^{2}+f(m)^{2}+f(n+m)^{2} \\ & \equiv 1+2 f(n) f(m) f(n+m) \pmod{16} \\ & \Longleftrightarrow f(n)^{2}+(f(m)+4)^{2}+f(n+m)^{2} \equiv 1+2 f(n)(f(m)+4) f(n+m) \pmod{16} \end{aligned} \] This tells us that increasing or decreasing one of the values of \( f \) by \( 4 \) will not make a difference. We can then assume that \( f(\text{odd})=a \), and multiply by \( 2^{7} \) at the end. Furthermore, since \( f(2x) \equiv 1 \pmod{4} \), assume \( f(\text{even})=1 \), and multiply by \( 2^{8} \) at the end. It's easy to verify that any solution of this form works. There are \( 2^{17} \) solutions in this case; \( 2^{8} \) for the even numbers and \( 4 \cdot 2^{7} \) for the odds. The answer is \( 2^{16} \cdot (2^{9}+2^{9}+2^{17}) = 8657043456 \equiv 793 \pmod{2017} \). \(\boxed{793}\)
793
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Let \( N \) be the number of functions \( f: \mathbb{Z} / 16 \mathbb{Z} \rightarrow \mathbb{Z} / 16 \mathbb{Z} \) such that for all \( a, b \in \mathbb{Z} / 16 \mathbb{Z} \): \[ f(a)^{2}+f(b)^{2}+f(a+b)^{2} \equiv 1+2 f(a) f(b) f(a+b) \pmod{16} \] Find the remainder when \( N \) is divided by \( 2017 \).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/FiniteMathematics", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/CongruenceEquation", "/Mathematics/NumberTheory/Congruences/FunctionalCongruence", "/Mathematics/NumberTheory/Congruences/Mod", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/Modulus", "/Mathematics/NumberTheory/Parity/EvenNumber", "/Mathematics/NumberTheory/Parity/OddNumber" ]
Use parity reductions (mod 2, 4, 8) to force f(even) odd and restrict f(odd) to either all odd or all even, then count the resulting configurations.
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ours_663
It's well known that \[ \frac{1}{\prod_{i=0}^{\infty}\left(1-x^{2i+1}\right)} = \prod_{i=1}^{\infty}\left(1+x^{i}\right). \] For \( k \geq 1 \), \(\left[x^{k}\right] \prod_{i=1}^{\infty}\left(1+x^{i}\right)\) is the number of partitions of \( k \) into distinct positive integers. By splitting into the number of parts in each partition, we have: \[ \prod_{i=1}^{\infty}\left(1+x^{i}\right) = 1 + \sum_{i=1}^{\infty} \frac{x^{i(i+1)/2}}{(1-x)(1-x^{2})\cdots(1-x^{i})}. \] For \( x = \frac{1}{10} \), this becomes: \[ \prod_{i=1}^{\infty}\left(1+\frac{1}{10^{i}}\right) = 1 + \left(\frac{1}{9} + \frac{1}{9 \cdot 99}\right) + \left(\frac{1}{9 \cdot 99 \cdot 999} + \frac{1}{9 \cdot 99 \cdot 999 \cdot 9999} + \cdots\right) = 1 + \frac{100}{891} + \epsilon \] where \(\frac{1}{890109} < \epsilon < \frac{\frac{1}{890109}}{1-\frac{1}{9999}} < \frac{1}{890019}\). Let \( F_{n} \) be the set of positive irreducible fractions less than 1 with denominator less than or equal to \( n \). We need to find a lower bound \(\frac{m}{n}\) of \( N = \frac{100}{891} + \epsilon \) satisfying \( m+2n \leq 10000 \) and \( m \geq \frac{100}{891}n \). We approximate \( N \) by a fraction in \( F_{4735} \). By the theory of Farey Sequences, the smallest element \(\frac{a}{b}\) in \( F_{891} \) larger than \(\frac{100}{891}\) satisfies \( 891a - 100b = 1 \) and \( b \leq 891 \). Solving gives \( a = 11 \) and \( b = 98 \), and \(\frac{11}{98} - \frac{100}{891} = \frac{1}{98 \cdot 891} = \frac{1}{87318} > \epsilon\). The fraction with the smallest denominator between \(\frac{100}{891}\) and \(\frac{11}{98}\) is \(\frac{111}{989}\), and \(\frac{111}{989} - \frac{100}{891} = \frac{1}{989 \cdot 891} = \frac{1}{881199} > \epsilon\). The largest fraction in \( F_{4734} \) less than \(\frac{111}{989}\) is \(\frac{100 + 3 \cdot 111}{891 + 3 \cdot 989} = \frac{433}{3858}\). Note that \(\frac{433}{3858} - \frac{100}{891} = \frac{1}{1145826} < \epsilon\). Hence, the desired fraction in \( S \) is \(\frac{3858}{433 + 3858} = \frac{3858}{4291}\). Therefore, the answer is \( 1000 \times 3858 + 4291 = 3862291 \). \(\boxed{3862291}\)
3862291
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Let \( S \) denote the set of fractions \(\frac{m}{n}\) for relatively prime positive integers \( m \) and \( n \) with \( m+n \leq 10000 \). The least fraction in \( S \) that is strictly greater than \[ \prod_{i=0}^{\infty}\left(1-\frac{1}{10^{2i+1}}\right) \] can be expressed in the form \(\frac{p}{q}\), where \( p \) and \( q \) are relatively prime positive integers. Find \( 1000p+q \).
[ "/Mathematics/DiscreteMathematics/Combinatorics/Partitions", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/ConcreteMathematics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/HigherArithmetic", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/GeneratingFunctions/GeneratingFunction", "/Mathematics/NumberTheory/RationalApproximation", "/Mathematics/NumberTheory/RationalNumbers/RationalNumber" ]
Use the Farey‐sequence neighbor property (ad − bc = 1) to locate the smallest fraction with bounded denominator that exceeds the target product.
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ours_664
Consider \(\Psi\), the inversion at \( A \) with power \( AH \cdot AD \). I claim that \(\Psi(G) = S\), \(\Psi(M) = Q\). This is straightforward to verify, noting that \(\Psi(BC) = (AH)\). Then \(\Psi(L) = AR \cap \Psi(H) \Psi(M) = AR \cap DQ\). Call this point \( L_a \). Since \(\Psi(N) \) is the reflection of \( A \) over \( BC \), call it \( N_a \), \(\Psi(J) = N_a L_a \cap EF\). **Lemma:** \(\Psi(J)\) lies on \( AM \). This implies that \( J = AM \cap \Omega \). **Proof:** First, note that \( B, H, Q, C, N_a \) are cyclic on the reflection \(\gamma\) of \(\Omega\) over \( BC \). Then, \(-1 = (D, S; B, C) \stackrel{H}{=} (N_a, Q; B, C)_{\gamma} = (A, Q^*; B, C)_{\Omega}\). Here \( Q^* \) is the reflection of \( Q \) over \( BC \), which lies on the \( A \)-symmedian. In particular, \( A, R, Q' \) are collinear. Let \( I = AR \cap BC \cap N_aQ \). Then, working over \(\mathbb{RP}^2\), \[ (A, Q; \Psi(J), M) \stackrel{N_a}{=} (D, I; N_aL_a \cap BC, M) \stackrel{L_a}{=} (D, A; N_a, ML_a \cap AD) \stackrel{M}{=} (I, A; Q', L_a) \stackrel{Q}{=} (N_a, A; \infty_{AD}, D) = -1. \] Therefore, \(\Psi(J)\) lies on the polar of \( M \) with respect to circle \((AEHF)\), which is just line \( EF \), as desired. This completes the lemma. As a corollary, note that \( J \) is the reflection of \( Q \) over \( M \). **Lemma:** \( O \) is the reflection of \( R \) over \( M \). **Proof:** By repeated power of a point, \( MN \cdot MO = AM \cdot MJ = MB \cdot MC = ME^2 = MR \cdot MN \). **Lemma:** \( J, M, O, S' \) are cyclic on the circle of diameter \( MS' \). **Proof:** By the previous two lemmas, it suffices to show that \( S, R, Q, M, G \) are cyclic on the circle of diameter \( (SM) \). This follows from inversion with respect to the circle of diameter \( (BC) \), which sends \( S \rightarrow D, R \rightarrow N, Q \rightarrow A, G \rightarrow H \). **Lemma:** \( AG, AU \) are isogonal in \(\angle BAC\). **Proof:** Let \( AU \) intersect \((JMO)\) again at \( U_1 \), and let \( M_1 \) be the antipode of \( M \) on \((AHM)\). Note that \( M', U, S' \) are collinear. Then \[ \angle S'MU_1 = \angle S'UU_1 = \angle AUM' = \angle AMM' = \angle DMH = \angle SMG. \] Therefore, since \( G \) lies on \((SM)\), we have \( U_1 = G' \), the reflection of \( G \) over the perpendicular bisector of \( BC \). Since \( GG' \parallel BC \), and \( G, G' \in \Omega \), it follows that lines \( AG, AG' = AU \) are isogonal in \(\angle BAC\), and in particular \( BG = CG' \). **Lemma:** \( BG = CV \). **Proof:** Consider the circle at \( C \) with radius \( BG \), call it \(\omega\). By the previous lemma \( U_1 = G' \) lies on this circle. Clearly so does the reflection \( G'' \) of \( G' \) over \( AC \). \( G'' \) lies on \((AHC)\) since \( G' \) lies on \((ABC)\). Let \( P \) be the second intersection of \(\omega, (AHC)\). By construction \(\angle PAC = \angle CAG'' = \angle CAG'\), so \( P \in AG' \). Therefore, \( P = V \), and \( V \in \omega \), as desired. Since \( SBG \) and \( SAC \) are similar by antiparallels, \( BG = \frac{BS \cdot AC}{AS} \). Let \( AD = x, BD = y, CD = z \). We will find \( BS, AC \), and \( AS \) in terms of \( x, y, \) and \( z \). Note that \( DM \cdot DS = DB \cdot DC \), so \( DS = \frac{2yz}{z-y} \). Then \( BS = DS - y = \frac{y^2 + yz}{z-y} \), and \( AS^2 = DS^2 + AD^2 = x^2 + \frac{4y^2z^2}{(y-z)^2} = \frac{x^2(y-z)^2 + 4y^2z^2}{(y-z)^2} \). We also clearly have \( AC^2 = x^2 + z^2 \). Thus, \( BG^2 = \frac{(y^2 + yz)^2(x^2 + z^2)}{x^2(y-z)^2 + 4y^2z^2} \). Now it suffices to compute \( x, y, z \). We have \( y + z = 5 \) and \( y^2 - z^2 = -2 \), so \( y - z = -\frac{2}{5} \). This means that \( y = \frac{23}{10} \) and \( z = \frac{27}{10} \). The Pythagorean Theorem gives \( x^2 = 24 - \frac{529}{100} = \frac{1871}{100} \). Now, we just have to plug everything in for the answer, which is \(\frac{23^2 \cdot 50^2 \cdot 26}{1871 \cdot 4^2 + 4 \cdot 23^2 \cdot 27^2} = \frac{23^2 \cdot 25^2 \cdot 26}{1871 \cdot 4 + 23^2 \cdot 27^2} = \frac{23^2 \cdot 25^2 \cdot 26}{393125} = \frac{23^2 \cdot 26}{629} = \frac{13754}{629}\). Therefore, the answer is \( \boxed{1376029} \).
1376029
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Let \( \triangle ABC \) be a triangle with \( AB = 2\sqrt{6} \), \( BC = 5 \), \( CA = \sqrt{26} \), midpoint \( M \) of \( BC \), circumcircle \(\Omega\), and orthocenter \( H \). Let \( BH \) intersect \( AC \) at \( E \) and \( CH \) intersect \( AB \) at \( F \). Let \( R \) be the midpoint of \( EF \) and let \( N \) be the midpoint of \( AH \). Let \( AR \) intersect the circumcircle of \( AHM \) again at \( L \). Let the circumcircle of \( ANL \) intersect \(\Omega\) and the circumcircle of \( BNC \) at \( J \) and \( O \), respectively. Let circles \( AHM \) and \( JMO \) intersect again at \( U \), and let \( AU \) intersect the circumcircle of \( AHC \) again at \( V \neq A \). The square of the length of \( CV \) can be expressed in the form \(\frac{m}{n}\) for relatively prime positive integers \( m \) and \( n \). Find \( 100m + n \).
[ "/Mathematics/Geometry/InversiveGeometry/CirclePower", "/Mathematics/Geometry/InversiveGeometry/Inversion", "/Mathematics/Geometry/InversiveGeometry/Polar", "/Mathematics/Geometry/PlaneGeometry/Circles", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similar", "/Mathematics/Geometry/PlaneGeometry/GeometricSimilarity/Similarity", "/Mathematics/Geometry/PlaneGeometry/Triangles", "/Mathematics/Geometry/ProjectiveGeometry/GeneralProjectiveGeometry/ProjectiveGeometry", "/Mathematics/Geometry/Transformations/GeometricTransformations/Polarity", "/Mathematics/Geometry/Transformations/Reflections/InversionOperation", "/Mathematics/Geometry/Transformations/Reflections/Reflection" ]
Apply an inversion at A (power = AH·AD) to expose reflection symmetries and isogonal relations, turning the tangled configuration into simple reflections over M.
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ours_665
Notice that \(\frac{\left(p^{p}-1\right)\left(p^{p-1}-1\right)}{p-1} = \left(p^{p-1}-1\right)\left(\frac{p^{p}-1}{p-1}\right)\), and these two terms are relatively prime. Let \( B, C \) be two rows of the matrix. An \( n \)-determinant remains constant when we transform them to \((B+C, C)\) and \((B+2C, C)\), and so on. Thus \((B+(p-1)C, C)\) is equivalent, so \((B-C, C)\) is as well, and then \((B-C, B)\) is equivalent, so \((-C, B)\) is as well after adding \( p-1 \) times. Thus, we can "swap rows" to some extent. If \( xy \equiv 1 \pmod{p} \), then we can send \((B, C)\) to \((B+xC, C)\) to \((B+xC, C-y(B+xC)) = (B+xC, -yB)\) to \((B+xC+x(-yB), -yB) = (xC, -yB)\) to \((yB, xC)\). Thus, we can "scale rows" to some extent. We can perform steps equivalent to row reduction and reduce any matrix to a near-reduced row echelon form with the same \( n \)-determinant value. If the rows are linearly dependent, we will obtain an all-zero row, and using it to scale other rows, we can reduce fully to reduced row echelon form. Otherwise, we can reduce the matrix to the identity matrix, except the upper left element is the nonzero determinant of the original matrix, since the operations we perform preserve the determinant. Since the reduced row echelon form is unique, any matrix reduces to precisely one of these end matrices. Thus, the number of functions as desired is simply \( p^{S_{n}} \), where \( S_{n} \) is the number of such final matrices. Simple counting shows that \( S_{n} = (p-1) + \sum_{k=0}^{n-1} p^{-\binom{k}{2}} s_{k} \), counting the nonzero determinant matrices with \( p-1 \) and counting the other rref forms with the latter sum, where \( s_{k} = \sum_{a_{1}, \ldots, a_{k} \in \{0,1, \ldots, n-1\}} p^{a_{1}+\cdots+a_{k}} \) where we sum over distinct \( a_{i} \). \((s_{0} = 1\) since we must count the all-zero matrix.) Then \( p^{S_{n}} \pmod{p^{p-1}-1} \) reduces to \( p^{2^{n}-1} \pmod{p^{p-1}-1} \) while \( p^{S_{n}} \pmod{\frac{p^{p}-1}{p-1}} \) reduces to \( p^{n-1} \pmod{\frac{p^{p}-1}{p-1}} \). The value of \( n \pmod{p} \) determines the latter value, and all \( p \) possibilities are distinct, while the value of \( 2^{n} \pmod{p-1} \) determines the former. There are \( p \) possibilities for the first. For the second, let \( v = v_{2}(p-1) \) and \( d = \operatorname{ord}_{\frac{p-1}{2v}}(2) \). Then \( n = 1, \ldots, v-1 \) give \( v-1 \) distinct values while \( n = v, v+1, \ldots, v+d-1 \) give \( d \) values which repeat forever. This gives a total of \( pd + (v-1) \) residues for the answer by the Chinese Remainder Theorem, since \(\gcd(d, p) = 1\). For \( p = 2017 \), we have \( v = 5, d = 6 \). Thus, we find \( 2017 \cdot 6 + 4 = 12106 \). \(\boxed{12106}\)
12106
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns-2.md" }
Let \( p = 2017 \) be a prime. Given a positive integer \( n \), let \( T \) be the set of all \( n \times n \) matrices with entries in \( \mathbb{Z} / p \mathbb{Z} \). A function \( f: T \rightarrow \mathbb{Z} / p \mathbb{Z} \) is called an \( n \)-determinant if for every pair \( 1 \leq i, j \leq n \) with \( i \neq j \), \[ f(A) = f\left(A^{\prime}\right) \] where \( A^{\prime} \) is the matrix obtained by adding the \( j \)th row to the \( i \)th row. Let \( a_{n} \) be the number of \( n \)-determinants. Over all \( n \geq 1 \), how many distinct remainders of \( a_{n} \) are possible when divided by \(\frac{\left(p^{p}-1\right)\left(p^{p-1}-1\right)}{p-1}\)?
[ "/Mathematics/Algebra/LinearAlgebra/Determinants", "/Mathematics/Algebra/LinearAlgebra/GeneralLinearAlgebra", "/Mathematics/Algebra/LinearAlgebra/Matrices", "/Mathematics/Algebra/NumberTheory/Congruences", "/Mathematics/Algebra/NumberTheory/GeneralNumberTheory", "/Mathematics/Algebra/NumberTheory/PrimeNumbers", "/Mathematics/DiscreteMathematics/Combinatorics/Enumeration", "/Mathematics/DiscreteMathematics/Combinatorics/GeneralCombinatorics", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Congruences/ChineseRemainderTheorem", "/Mathematics/NumberTheory/Congruences/Congruence", "/Mathematics/NumberTheory/Congruences/ModularArithmetic", "/Mathematics/NumberTheory/Congruences/ModuloOrder", "/Mathematics/NumberTheory/Congruences/MultiplicativeOrder", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers" ]
Use the allowed row‑addition operations to reduce every matrix uniquely to a reduced row‑echelon form, turning the problem into counting canonical forms.
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ours_666
Solution. The smallest positive integer is \(1\). The number \(1\) is relatively prime to every positive integer. Therefore, the answer is \(\boxed{1}\).
1
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns.md" }
Find the smallest positive integer that is relatively prime to each of \(2, 20, 204\), and \(2048\).
[ "/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/Integers/N", "/Mathematics/NumberTheory/Integers/PositiveInteger", "/Mathematics/NumberTheory/Integers/Z-Plus" ]
Realize that 1 is relatively prime to every positive integer, making it the smallest possible candidate.
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ours_667
If \( n \) is a prime number or 1, then it is certainly bad since it has no more than one factor greater than 1. If \( n = p^2 \) for some prime \( p \), then the only way to express it as a product of integers greater than 1 is \( p \cdot p \), but the two numbers are not distinct. For all other cases of \( n \), setting one factor to be the smallest prime divisor of \( n \) always works. Therefore, since there are 26 non-composites less than 100 (including 1) and 4 prime squares \((2^2, 3^2, 5^2, 7^2)\), there are \( 26 + 4 = 30 \) bad positive integers less than 100. \(\boxed{30}\)
30
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns.md" }
A positive integer \( n \) is called bad if it cannot be expressed as the product of two distinct positive integers greater than 1. Find the number of bad positive integers less than 100.
[ "/Mathematics/DiscreteMathematics/DivisionProblems/NumberPartitioningProblem", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMath", "/Mathematics/DiscreteMathematics/GeneralDiscreteMathematics/DiscreteMathematics", "/Mathematics/NumberTheory/Divisors/Divides", "/Mathematics/NumberTheory/Divisors/Divisible", "/Mathematics/NumberTheory/Divisors/Divisor", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryMethods", "/Mathematics/NumberTheory/GeneralNumberTheory/ElementaryNumberTheory", "/Mathematics/NumberTheory/GeneralNumberTheory/NumberTheory", "/Mathematics/NumberTheory/PrimeNumbers/PrimeFactorization" ]
Recognize that only 1, primes, and squares of primes lack two distinct factors greater than 1.
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ours_668
Suppose that \(AR\) and \(BS\), \(BS\) and \(CP\), \(CP\) and \(DQ\), \(DQ\) and \(AR\) intersect at \(W, X, Y, Z\) respectively. Then the quadrilateral \(WXYZ\) is a rhombus, where \(XZ = AP = \frac{AB}{3} = 2 = PQ\). This also implies that triangles \(PQY\), \(XZY\), \(ZXW\), and \(RSW\) are all congruent, and thus \(WY = \frac{BC}{2} = 8\). The area of the union is therefore \(2 \cdot 16 \cdot 2 - \frac{2 \cdot 8}{2} = 56\). \(\boxed{56}\)
56
{ "competition": "us_comps", "dataset": "Ours", "posts": null, "source": "OMOSpring17Solns.md" }
In rectangle \(ABCD\), \(AB = 6\) and \(BC = 16\). Points \(P, Q\) are chosen on the interior of side \(AB\) such that \(AP = PQ = QB\), and points \(R, S\) are chosen on the interior of side \(CD\) such that \(CR = RS = SD\). Find the area of the region formed by the union of parallelograms \(APCR\) and \(QBSD\).
[ "/Mathematics/Geometry/Dissection/DissectionPuzzles", "/Mathematics/Geometry/GeneralGeometry/EuclideanGeometry", "/Mathematics/Geometry/GeneralGeometry/Geometry", "/Mathematics/Geometry/PlaneGeometry/Polygons/ConvexPolygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/Polygon", "/Mathematics/Geometry/PlaneGeometry/Polygons/PolygonArea", "/Mathematics/Geometry/PlaneGeometry/Quadrangles", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Parallelogram", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rectangle", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rhomb", "/Mathematics/Geometry/PlaneGeometry/Quadrilaterals/Rhombus", "/Mathematics/Geometry/PlaneGeometry/Rectangles/Rectangle" ]
Show that the intersecting lines form a rhombus with side equal to one‑third of AB and other diagonal half of BC, then compute area via subtraction.
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