id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-6189ab7f2d4a8889ed166a4036beee917623937626765b10aac78081e3a2b0ed | $x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$? | [
"2"
] | [
"$x$ and $y$ are complex numbers such that $x^3 + y^3 = -16$ and $(x + y)^2 = xy$. What is the value of $|x + y|$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07d9052dd209fe6d5b30126fb77feea4e1bda26e865f0054201923f7a2790458 | $x$ is a real number with the property that $x+\tfrac1x = 3$. Let $S_m = x^m + \tfrac{1}{x^m}$. Determine the value of $S_7$. | [
"843"
] | [
"$x$ is a real number with the property that $x+\\tfrac1x = 3$. Let $S_m = x^m + \\tfrac{1}{x^m}$. Determine the value of $S_7$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d6b31e025e26a94977794d22f3a345b70e971acbd57c0bfe5ef4fdef6b0cfc4 | $x$ is directly proportional to $y$ and inversely proportional to $z$. If $x = \frac{1}{2}$ when $y = \frac{3}{4}$ and $z=\frac{2}{3}$, find $x$ when $y=\frac{7}{8}$ and $z=\frac{7}{9}$. | [
"\\dfrac{1}{2}"
] | [
"$x$ is directly proportional to $y$ and inversely proportional to $z$. If $x = \\frac{1}{2}$ when $y = \\frac{3}{4}$ and $z=\\frac{2}{3}$, find $x$ when $y=\\frac{7}{8}$ and $z=\\frac{7}{9}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a76c1e2498ca292d4adba3746ab8fea428342eba9af4d8bd30a7d2831f93473 | $x+\sqrt x=a$. | [
"$x=\\left(2a+1-\\sqrt {1+4a}\\right)/2$ for $a\\ge 0$; no solution for $a<0$"
] | [
"$x+\\sqrt x=a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c08d15afa52ae7551f2eac4a4e74947241af80d9212f2c9bf1c2c7e3f80112c | $x, y$, and $z$ are positive real numbers that satisfy the equations: \[ x + \frac{1}{y} = 4, \] \[ y + \frac{1}{z} = 1, \] \[ z + \frac{1}{x} = \frac{7}{3}. \] Compute $xyz$. | [
"1"
] | [
"$x, y$, and $z$ are positive real numbers that satisfy the equations:\n\\[ x + \\frac{1}{y} = 4, \\]\n\\[ y + \\frac{1}{z} = 1, \\]\n\\[ z + \\frac{1}{x} = \\frac{7}{3}. \\]\nCompute $xyz$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e36a5585bf5c2577dd247f139f1e19f67d40657fa4f518306a7ffca95b0e45ae | $x=\sin \pi \frac{x+1}{3}\sin \pi \frac{1-x}{3},\ 0\le x\le 1$. | [
"$x=1/2$. Reduce the equation to the form $4x-1=2\\cos (2\\pi x/3)$. Its root is $x=1/2$. There are no other roots because for $0\\le x<1/2$ we have $4x-1<2\\cos (2\\pi x/3)$ and foe $1/2<x\\le 1$ we have $4x-1>2\\cos (2\\pi x/3)$"
] | [
"$x=\\sin \\pi \\frac{x+1}{3}\\sin \\pi \\frac{1-x}{3},\\ 0\\le x\\le 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-341c080b5dfe8a25c77c893f25055f0493ef477a83d1a1997ec6dec951d78469 | $x\cdot 2^x=x(3-x)+2(2^x-1)$. | [
"$x_1=0$, $x_2=2$. Reduce the equation to the form \\ $(x-2)2^x=(x-2)(1-x)$"
] | [
"$x\\cdot 2^x=x(3-x)+2(2^x-1)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a86f44c5c77bca9686f032dfc44690e5ccc40dfdd9e4fe0ffc721bb6342e4a8 | $x^2+1=0$ and $\frac{x^2+1}{x}=0$. | [
"Yes"
] | [
"$x^2+1=0$ and $\\frac{x^2+1}{x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6dd2969482bc9d233481813817d4ca288a483db352b771da59d1d5345dcb0117 | $x^2+1=\sqrt x$ and $x^2+1+\sqrt {1-x}=\sqrt x+\sqrt {1-x}$. | [
"Yes"
] | [
"$x^2+1=\\sqrt x$ and $x^2+1+\\sqrt {1-x}=\\sqrt x+\\sqrt {1-x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35acc0a753575fd768309f0383287f6f9805e48e97751600af8cfa3fe844ab5a | $x^2+2x\sin xy+1=0$. | [
"$\\left(1;\\frac{\\pi }{2}+\\pi k\\right);\\ \\left(-1;\\frac{\\pi }{2}+\\pi n\\right)$"
] | [
"$x^2+2x\\sin xy+1=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ee6448a68ed56daf0ace698ef3628ccb3c6c7ed3a462a84ed3a831f6ff4c75a | $x^2+4x\cos xy+4=0$. | [
"$x_1=2$, $y_1=(\\pi /2)+k\\pi $; $x_2=-2$, $y_2=k\\pi $"
] | [
"$x^2+4x\\cos xy+4=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3299aaeae6c2bb4e476fef46a60600698d8d5a4c2c0269e4ddffc6c56985bb0d | $x^2-1=\sqrt x$ and $x^2-1+\sqrt {1-x}=\sqrt x+\sqrt {1-x}$. | [
"No"
] | [
"$x^2-1=\\sqrt x$ and $x^2-1+\\sqrt {1-x}=\\sqrt x+\\sqrt {1-x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d25994dbb4b1ce3a52e3139e4c8d0b7d63ee786296d98a483ac54fda310a4731 | $x^2=1+6\log _4 y$ \ $y^2=y\cdot 2^x+2^{2x+1}$. | [
"$x_1=4,\\ y_1=32$; $x_2=-1,\\ y_2=1$"
] | [
"$x^2=1+6\\log _4 y$ \\ $y^2=y\\cdot 2^x+2^{2x+1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce29061059a25e6e3af8fcb03faf9df0eb68bd292a4d241ea047a2eb0633ec11 | $x^3+x=0$ and $\frac{x^3+x}{x}=0$. | [
"No"
] | [
"$x^3+x=0$ and $\\frac{x^3+x}{x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1074d48aa020dcf3eefaf94b0ddd91a6e8b9c572d8a0865f9724152a036aa4e | $x^{\sqrt [4]{x}+\sqrt y}=y^{8/3}$ \ $y^{\sqrt [4]{x}+\sqrt y}=x^{2/3}$. | [
"$x_1=1,\\ y_1=1$; $x_2=16/81,\\ y_2=4/9$"
] | [
"$x^{\\sqrt [4]{x}+\\sqrt y}=y^{8/3}$ \\ $y^{\\sqrt [4]{x}+\\sqrt y}=x^{2/3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d53a9f9896ce360612f62379411968dcf62c324851f4c4079cf8c59a011d0b88 | $x^{y^2-15y+56}=1$ \ $y-x=5$. | [
"$x_1=1,\\ y_1=6$; $x_2=2,\\ y_2=7$; $x_3=3,\\ y_3=8$"
] | [
"$x^{y^2-15y+56}=1$ \\ $y-x=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7c3e9b0b6a5833f027d2839c13d5ed22fa07d34c8f2170c20494dc14dea731d | $x_1, x_2$, and $x_3$ are complex numbers such that \[ x_1 + x_2 + x_3 = 0, \] \[ x_1^2 + x_2^2 + x_3^2 = 16, \] \[ x_1^3 + x_2^3 + x_3^3 = -24. \] Let $\gamma = \min\left(\left|x_1\right|, \left|x_2\right|, \left|x_3\right|\right)$, where $|a + bi| = \sqrt{a^2 + b^2}$ and $i = \sqrt{-1}$. Determine the value of $\gamm... | [
"56"
] | [
"$x_1, x_2$, and $x_3$ are complex numbers such that\n\\[\nx_1 + x_2 + x_3 = 0,\n\\]\n\\[\nx_1^2 + x_2^2 + x_3^2 = 16,\n\\]\n\\[\nx_1^3 + x_2^3 + x_3^3 = -24.\n\\]\nLet $\\gamma = \\min\\left(\\left|x_1\\right|, \\left|x_2\\right|, \\left|x_3\\right|\\right)$, where $|a + bi| = \\sqrt{a^2 + b^2}$ and $i = \\sqrt{-1... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5556f9c07a2b31a0c3ad7a8c2339fe35541d85a655e690a22b12cc93f3924d19 | $z=|z|$. | [
"A positive semiaxis $Ox$ which includes the point $O$"
] | [
"$z=|z|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c17dd4479cb63426d544e3bb5037bd052649cc40686b02e9b1c44c1286b86113 | $|\cot x|=\cot x+\frac{1}{\sin x}$. | [
"$\\frac{2\\pi }{3}+2\\pi k$"
] | [
"$|\\cot x|=\\cot x+\\frac{1}{\\sin x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e867c8dcf8e9651f2d5edaf003aff9c47e9cc6e3a93ff2fe11dec37ff93e072 | $|z+1|=|z-1|$. | [
"The $Oy$ axis"
] | [
"$|z+1|=|z-1|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-149d90394e82ad2c35ce22d2824ec810be26515a78acd94e6629bc4866b8d0df | ( Dick Gibbs ) For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$. | [
" N = 2k^3 + 3k^2 + 3k "
] | [
"( Dick Gibbs ) For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$."
] | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-bd59e97dfe1d424b1584b36b45e31c25b8a796eb72bada629b0487b79111fdc7 | ( Ricky Liu ) Find all positive integers $n$ such that there are $k\ge 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \cdots + a_k = a_1\cdot a_2\cdots a_k = n$. | [
"n > 3 \\text{ and } n \\neq 5"
] | [
"( Ricky Liu ) Find all positive integers $n$ such that there are $k\\ge 2$ positive rational numbers $a_1, a_2, \\ldots, a_k$ satisfying $a_1 + a_2 + \\cdots + a_k = a_1\\cdot a_2\\cdots a_k = n$."
] | [
"MATH_training_set"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-1dd7bd44563f4edfa3776862cb208d05e860ba17840d5797ff056ae813063b81 | ( Ricky Liu ) For what values of $k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons? | [
"$k \\neq 1$"
] | [
"( Ricky Liu ) For what values of $k > 0$ is it possible to dissect a $1 \\times k$ rectangle into two similar, but incongruent, polygons?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-64b2fc625b2c2e5ff58ea321578995f3ab0ad35b29bb991770459d0a6c14d163 | (1) $\sin x<\frac{1}{5}$; (2) $\cos x\ge -0.7$; (3) $\tan x\le 5$; (4) $\cot x>-\frac{\sqrt 3}{4}$. | [
"(1) $\\pi -\\arcsin \\frac{1}{5}+2\\pi k<x<2\\pi +\\arcsin \\frac{1}{5}+2\\pi k$; (2) $-\\arccos (-0.7)+2\\pi k\\le x\\le \\arccos (-0.7)+2\\pi k$; (3) $-\\frac{\\pi }{2}+\\pi k<x\\le \\arctan 5+\\pi k$; (4) $\\pi k<x<{\\rm arccot}\\left(-\\frac{\\sqrt 3}{4}\\right)+\\pi k$"
] | [
"(1) $\\sin x<\\frac{1}{5}$; (2) $\\cos x\\ge -0.7$; (3) $\\tan x\\le 5$; (4) $\\cot x>-\\frac{\\sqrt 3}{4}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf723563f3b32d68b4f36f21d9c6cfe76eb2ce24c92bd396a0522f85a4395715 | (1) $\sin x>-\frac{1}{2}$; (2) $\cos x<\frac{\sqrt 3}{2}$; (3) $\tan x\ge -\frac{\sqrt 3}{3}$; (4) $\cot x\le -1$. | [
"(1) $-\\frac{\\pi }{6}+2\\pi k<x<\\frac{7\\pi }{6}+2\\pi k$; (2) $\\frac{\\pi }{6}+2\\pi k<x<\\frac{11\\pi }{6}+2\\pi k$; (3) $-\\frac{\\pi }{6}+\\pi k\\le x<\\frac{\\pi }{2}+\\pi k$; (4) $\\frac{3\\pi }{4}+\\pi k\\le x<\\pi +\\pi k$"
] | [
"(1) $\\sin x>-\\frac{1}{2}$; (2) $\\cos x<\\frac{\\sqrt 3}{2}$; (3) $\\tan x\\ge -\\frac{\\sqrt 3}{3}$; (4) $\\cot x\\le -1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-082e9f5d388df6c7b365a3cfa0d50571eed6d9090d7965bf79ef1c9c69c4d5d2 | (1) Determine whether the function \( y = \frac{\log x}{x} \) is increasing or decreasing for \( x > 0 \). (2) Find all pairs \( (a, b) \) such that \( a^b = b^a \) and \( a \neq b \). (3) Prove that 3 is the only positive rational number \( x \) such that \( 3^x = x^3 \). | [
"The function \\( y = \\frac{\\log x}{x} \\) is increasing for \\( x \\in (0, e) \\) and decreasing for \\( x \\in (e, \\infty) \\). There are no pairs \\( (a, b) \\) such that \\( a^b = b^a \\) and \\( a \\neq b \\). The only positive rational number \\( x \\) such that \\( 3^x = x^3 \\) is \\( x = 3 \\)."
] | [
"(1) Determine whether the function \\( y = \\frac{\\log x}{x} \\) is increasing or decreasing for \\( x > 0 \\).\n(2) Find all pairs \\( (a, b) \\) such that \\( a^b = b^a \\) and \\( a \\neq b \\).\n(3) Prove that 3 is the only positive rational number \\( x \\) such that \\( 3^x = x^3 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2e2ba8c281bc5220d571c997289eed036ec97c88389caf6a0b54d590a7ef84a | (1) Find the possible number of roots for the equation \( |x+1| + |x+2| + |x+3| = a \), where \( x \in \mathbb{R} \) and \( a \) is a parameter. (2) Let \( \{ a_1, a_2, \ldots, a_n \} \) be an arithmetic progression, \( n \in \mathbb{N} \), and satisfy the condition \( \sum_{i=1}^n |a_i| = \sum_{i=1}^n |a_i + 1| = \sum... | [
"39"
] | [
"(1) Find the possible number of roots for the equation \\( |x+1| + |x+2| + |x+3| = a \\), where \\( x \\in \\mathbb{R} \\) and \\( a \\) is a parameter.\n(2) Let \\( \\{ a_1, a_2, \\ldots, a_n \\} \\) be an arithmetic progression, \\( n \\in \\mathbb{N} \\), and satisfy the condition \\( \\sum_{i=1}^n |a_i| = \\su... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5be75187750b6888b93cf937e9668200ec5d2b12afa6f47ec41694a0d95dfbbe | (A) If \(\theta\) is any quadrant 2 angle, what quadrant(s) could \(2\theta\) lie in? (B) Given \(\sin \theta = \frac{8}{17}\) where \(\frac{\pi}{2} < \theta < \pi\), find: - \(\sin 2\theta\) - \(\cos 2\theta\) - The quadrant containing \(2\theta\) (C) Simplify the following: \((2 \sin x \cos x)^2 + (\cos^2 x - \sin^2 ... | [
"1"
] | [
"(A) If \\(\\theta\\) is any quadrant 2 angle, what quadrant(s) could \\(2\\theta\\) lie in?\n(B) Given \\(\\sin \\theta = \\frac{8}{17}\\) where \\(\\frac{\\pi}{2} < \\theta < \\pi\\), find:\n- \\(\\sin 2\\theta\\)\n- \\(\\cos 2\\theta\\)\n- The quadrant containing \\(2\\theta\\)\n(C) Simplify the following: \\((2... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86355a415323f33d9500676d2faa3b82ad0c6d6aef0e64b291734a5a2e17e4ea | (Estimation) We randomly generated 100 positive integers \( a_1, \ldots, a_{100} \) from 1 to 100, and for each of the \( \binom{100}{2} = 4950 \) pairs \( (i, j) \) with \( i < j \), computed \( |a_i^2 - a_j^2| \). We then ordered these quantities from least to greatest. Estimate the 2023rd smallest integer in this re... | [
"2023"
] | [
"(Estimation) We randomly generated 100 positive integers \\( a_1, \\ldots, a_{100} \\) from 1 to 100, and for each of the \\( \\binom{100}{2} = 4950 \\) pairs \\( (i, j) \\) with \\( i < j \\), computed \\( |a_i^2 - a_j^2| \\). We then ordered these quantities from least to greatest. Estimate the 2023rd smallest i... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c0818e965bc3de4b4c59a5bd00bbce5f4e71858bedbbeab6bc59f3b21804fb24 | (Estimation) What is the smallest positive integer value of $x$ such that $1000! < x^x$? If the correct answer is $x$ and you submit positive integer $y$, your answer will be $\max \left(0, 100 - 800\frac{|y - x|}{x}\right)\%$ of the maximum possible score. Otherwise, your score is $0$. | [
"874"
] | [
"(Estimation) What is the smallest positive integer value of $x$ such that $1000! < x^x$? If the correct answer is $x$ and you submit positive integer $y$, your answer will be $\\max \\left(0, 100 - 800\\frac{|y - x|}{x}\\right)\\%$ of the maximum possible score. Otherwise, your score is $0$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8733f8e0dbf03aec9e1da96b9fdf027d0d0f41af0d55c03c1ddb0e9fa1ecf75d | (a) A regular polyhedron A has 8 faces, with 4 edges meeting each vertex, and all its faces are equilateral triangles. How many vertices does it have? (b) A regular polyhedron B has 30 edges, and all its faces are regular pentagons. How many vertices does it have? How many faces? (c) A regular polyhedron C has 12 verti... | [
"20"
] | [
"(a) A regular polyhedron A has 8 faces, with 4 edges meeting each vertex, and all its faces are equilateral triangles. How many vertices does it have? (b) A regular polyhedron B has 30 edges, and all its faces are regular pentagons. How many vertices does it have? How many faces? (c) A regular polyhedron C has 12 ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5323728c22062ffae30d2c31433fcd36b549ad8b66578865a8c647be5e8b085a | (a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle \( \alpha \). Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections. | [
"97"
] | [
"(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle \\( \\alpha \\). Determine the smallest number of lines for which this is possible.\n(b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersectio... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-8717bdfd9a30d0e8e7259dff7f9775e3ac948978711f03d95d0be8f2ed1f1854 | (a) Compute the remainder of $2^{100} + 2^{99} + \cdots + 2^1 + 2^0$ when divided by $3$. | [
"1"
] | [
"(a) Compute the remainder of $2^{100} + 2^{99} + \\cdots + 2^1 + 2^0$ when divided by $3$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-562fc0b4db23cac427096f6f7a6f53462a9db94280c254b0a3d086d6f9b685bb | (a) Consider a finite collection of lines in a plane $P$. Let $C(r)$ denote a circle of radius $r$ centered at the origin. Determine the smallest integer $r$ such that $C(r)$ does not intersect any of the given lines. (b) Construct an infinite sequence of lines in $P$ such that every circle in $P$ intersects at least o... | [
"55"
] | [
"(a) Consider a finite collection of lines in a plane $P$. Let $C(r)$ denote a circle of radius $r$ centered at the origin. Determine the smallest integer $r$ such that $C(r)$ does not intersect any of the given lines.\n\n(b) Construct an infinite sequence of lines in $P$ such that every circle in $P$ intersects at... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70f731622a5d8e262cdb1d26158f76a14e25f65553f671a6db47fcf365040980 | (a) Consider the set of all triangles \(ABC\) which are inscribed in a circle with radius \(R\). When is \(AB^2 + BC^2 + CA^2\) maximum? Find this maximum. (b) Consider the set of all tetragons \(ABCD\) which are inscribed in a sphere with radius \(R\). When is the sum of squares of the six edges of \(ABCD\) maximum? F... | [
"\\(9R^2\\) for the triangle and \\(16R^2\\) for the tetragon."
] | [
"(a) Consider the set of all triangles \\(ABC\\) which are inscribed in a circle with radius \\(R\\). When is \\(AB^2 + BC^2 + CA^2\\) maximum? Find this maximum.\n\n(b) Consider the set of all tetragons \\(ABCD\\) which are inscribed in a sphere with radius \\(R\\). When is the sum of squares of the six edges of \... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-90f06db8abd6e9b87c2d591f2c87784ca705e43a0f8cc9835edff4f1fc620812 | (a) Determine the conditions that the coefficients of a fourth-degree polynomial \( P(x) \) must satisfy so that \( P(x) = P(1-x) \) for all \( x \in \mathbb{R} \). (b) Determine all the roots of the polynomial \( P(x) = 16x^4 - 32x^3 - 56x^2 + 72x + 77 \). | [
"\\(-1, \\frac{1}{2}, 2, \\frac{7}{4}\\)"
] | [
"(a) Determine the conditions that the coefficients of a fourth-degree polynomial \\( P(x) \\) must satisfy so that \\( P(x) = P(1-x) \\) for all \\( x \\in \\mathbb{R} \\).\n\n(b) Determine all the roots of the polynomial \\( P(x) = 16x^4 - 32x^3 - 56x^2 + 72x + 77 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ba81ede37b5f5ff7efd85535930978c9b5db9b5796e4d560ddd33ce23e06f4b | (a) Determine the number of ways the number $216$ can be written as a product of three positive integers. (Different orderings of the same factors, such as $1\times 2\times 108$ and $2\times 1\times 108$ are to be considered the same product). (b) Repeat (a) for the number $1000$. | [
"19"
] | [
"(a) Determine the number of ways the number $216$ can be written as a product of three positive integers. (Different orderings of the same factors, such as $1\\times 2\\times 108$ and $2\\times 1\\times 108$ are to be considered the same product).\n(b) Repeat (a) for the number $1000$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12eba2c0b12b08f11ca97062fd18e1485c57cdec9d29d7ec2dfaa81c0ff1535b | (a) Evaluate \( \sum_{k=1}^5 \frac{1}{k(k+1)(k+2)} \). (b) Find an integer \( m \) in terms of \( n \) such that \( \sum_{k=1}^n \frac{1}{k(k+1)(k+2)} = \frac{1}{4} - \frac{1}{m} \). | [
"\\(\\frac{5}{21}\\) and \\(m = 2n^2 + 6n + 4\\)."
] | [
"(a) Evaluate \\( \\sum_{k=1}^5 \\frac{1}{k(k+1)(k+2)} \\).\n(b) Find an integer \\( m \\) in terms of \\( n \\) such that \\( \\sum_{k=1}^n \\frac{1}{k(k+1)(k+2)} = \\frac{1}{4} - \\frac{1}{m} \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-d5a679b1b167f22aee942b4e4093470e42c4d9aeaf454d84b0b987b0d5eeddf2 | (a) Evaluate the integral \( \int_{S} (x^{2} + y^{2}) \, d\sigma \) where \( S \) is the sphere of radius 1 centered at \( (0,0,0) \) and \( \sigma \) is surface area. (b) Determine whether the series \( \sum_{n \in S} \frac{1}{n} \) converges, where \( S \) consists of those positive integers whose decimal expansion d... | [
"\\text{Converges}"
] | [
"(a) Evaluate the integral \\( \\int_{S} (x^{2} + y^{2}) \\, d\\sigma \\) where \\( S \\) is the sphere of radius 1 centered at \\( (0,0,0) \\) and \\( \\sigma \\) is surface area.\n(b) Determine whether the series \\( \\sum_{n \\in S} \\frac{1}{n} \\) converges, where \\( S \\) consists of those positive integers ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06be4ebdf988b550b05fb2b62f50b57f3bdbeba5fe2db2b1c7456c5cb57ce6e6 | (a) Express the area of a right triangle in terms of its circumradius and inradius. (b) Express the lengths of the three sides of a right triangle in terms of its circumradius and inradius. | [
"R + r - \\sqrt{R^2 - 2Rr - r^2}"
] | [
"(a) Express the area of a right triangle in terms of its circumradius and inradius.\n(b) Express the lengths of the three sides of a right triangle in terms of its circumradius and inradius."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-516e80a82fc581b763ef01146435a1a09cc06185f1d35cf0ca80a1c15e44eae0 | (a) Find a number \( R \) such that for any three points on or inside a square of side length one, at least two of the given points are at a distance at most \( R \). (b) Determine the smallest number \( R_0 \) such that, given any three points on or inside a square of side length one, at least two of the given points ... | [
"1"
] | [
"(a) Find a number \\( R \\) such that for any three points on or inside a square of side length one, at least two of the given points are at a distance at most \\( R \\).\n(b) Determine the smallest number \\( R_0 \\) such that, given any three points on or inside a square of side length one, at least two of the g... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-873447bdbc0a180e424888475f81a4fb1bb93fb4e5caf538af03e2e299f46cdc | (a) Find a set \(A\) of ten positive integers such that no six distinct elements of \(A\) have a sum which is divisible by 6. (b) Is it possible to find such a set if "ten" is replaced by "eleven"? | [
"No"
] | [
"(a) Find a set \\(A\\) of ten positive integers such that no six distinct elements of \\(A\\) have a sum which is divisible by 6.\n(b) Is it possible to find such a set if \"ten\" is replaced by \"eleven\"?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8d00df539b0f8c29f89a062f59ed9dec42d59ab991e527c28e31dc41114480e9 | (a) Find all possible values of \( y \), where \( y = x + \frac{1}{x} \) and \( x \) satisfies the equation \( x^4 + 2x^3 - 22x^2 + 2x + 1 = 0 \). (b) Solve the quartic polynomial \( x^4 + 2x^3 - 22x^2 + 2x + 1 = 0 \). | [
"\\( y = -6, 4 \\) and the solutions to the quartic polynomial are \\( x = -3 \\pm 2\\sqrt{2} \\) and \\( x = 2 \\pm \\sqrt{3} \\)."
] | [
"(a) Find all possible values of \\( y \\), where \\( y = x + \\frac{1}{x} \\) and \\( x \\) satisfies the equation \\( x^4 + 2x^3 - 22x^2 + 2x + 1 = 0 \\).\n(b) Solve the quartic polynomial \\( x^4 + 2x^3 - 22x^2 + 2x + 1 = 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-bc1184236837386ae4a61c4ad21efbb7be4fcac043498ec3417698cf14afc561 | (a) Find the highest power of 3 contained in $23!$. (b) Find the highest power of 3 contained in $2003!$. | [
"9"
] | [
"(a) Find the highest power of 3 contained in $23!$. (b) Find the highest power of 3 contained in $2003!$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dcf61718baed93c67094d551e57b1bae2b84a53d281afe9f5c4cf00a6457bd00 | (a) Find the length of the median $| AD |$ of the triangle $ABC$ knowing the lengths of the sides $|AC|=b$, $|AB|=c$ and the magnitude of the angle $A$. (b) Find the length of the bisector $| AE |$ of the triangle knowing the lengths of the sides $|AC|=b$, $|AB|=c$ and the magnitude of the angle $A$. | [
"(a) $m_a=\\frac{1}{2}\\sqrt {b^2+2bc\\cos A+c^2}$; (b) $l_a=\\frac{2bc\\cos (A/2)}{b+c}$"
] | [
"(a) Find the length of the median $| AD |$ of the triangle $ABC$ knowing the lengths of the sides $|AC|=b$, $|AB|=c$ and the magnitude of the angle $A$. (b) Find the length of the bisector $| AE |$ of the triangle knowing the lengths of the sides $|AC|=b$, $|AB|=c$ and the magnitude of the angle $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ef96d317b64da2334effd4558097f5ea93ee60fc85800d462ac874108c26293 | (a) Find the minimum number of edges in a graph $G$ with $n$ vertices $A_1,A_2,\ldots,A_n$ such that for each pair of non-adjacent vertices $A_i$ and $A_j$, there exists another vertex $A_k$ that is adjacent to both $A_i$ and $A_j$. (b) If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6$, find th... | [
"2"
] | [
"(a) Find the minimum number of edges in a graph $G$ with $n$ vertices $A_1,A_2,\\ldots,A_n$ such that for each pair of non-adjacent vertices $A_i$ and $A_j$, there exists another vertex $A_k$ that is adjacent to both $A_i$ and $A_j$.\n\n(b) If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6$... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-d2af35a5dcdcde61a595af9f617fd2e6bdb8276f3c0e20286d214c2befd67286 | (a) Find the power series about 0 for the function \( f(z) = e^{z^3} \), and give its radius of convergence. (b) Using the series obtained in part (a), compute \( f^{(20)}(0) \) and \( f^{(21)}(0) \). | [
"\\( f^{(20)}(0) = 0 \\) and \\( f^{(21)}(0) = \\frac{21!}{7!} \\)"
] | [
"(a) Find the power series about 0 for the function \\( f(z) = e^{z^3} \\), and give its radius of convergence.\n(b) Using the series obtained in part (a), compute \\( f^{(20)}(0) \\) and \\( f^{(21)}(0) \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-ca99db28091b401161606c6e5f6fa843d7076f9cf01385f77d51dd30d60c17d9 | (a) Fix \( 0 < p < 1 \). Given a fair coin, simulate a biased coin that flips head with probability \( p \). (b) Given a biased coin that flips heads with some probability \( 0 < p < 1 \), simulate a fair coin. You do not know \( p \). | [
"(a) Binary expand \\( p \\) and use the described method. (b) Flip a coin twice and use the described method."
] | [
"(a) Fix \\( 0 < p < 1 \\). Given a fair coin, simulate a biased coin that flips head with probability \\( p \\).\n(b) Given a biased coin that flips heads with some probability \\( 0 < p < 1 \\), simulate a fair coin. You do not know \\( p \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 19
} | 26 | 0 | 0 |
omr-ws-0a2fa1003f2c0655dcc00acc7302b1bbe94f05a048bc1fa9b11a0911abc7dc1c | (a) Given 6 points on a circle, how many chords are there having two of these 6 points as endpoints? What is the maximum possible number of intersections these chords can make in the interior of the circle? What is the maximum possible number of regions these chords can divide the interior of the circle? (b) Given 12 p... | [
"562"
] | [
"(a) Given 6 points on a circle, how many chords are there having two of these 6 points as endpoints? What is the maximum possible number of intersections these chords can make in the interior of the circle? What is the maximum possible number of regions these chords can divide the interior of the circle?\n(b) Give... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce6a5f433f25512c49a0e07e063057ed5279be0c132194edfda30764f808ff33 | (a) Given the quadrilateral $ABCD$ with $AB$ parallel to $CD$ and $P$ the midpoint of $DC$, suppose $AB = a$, $AD = BC = b$ and $x$ is the distance from $P$ to each of $A, B, C, D$. Find a formula for $x$ in terms of $a$ and $b$. (b) A hexagon is inscribed in a circle. If two opposite sides have length $2$ and the othe... | [
"\\( \\pi \\left(1 + \\frac{\\sqrt{3}}{2}\\right) \\)"
] | [
"(a) Given the quadrilateral $ABCD$ with $AB$ parallel to $CD$ and $P$ the midpoint of $DC$, suppose $AB = a$, $AD = BC = b$ and $x$ is the distance from $P$ to each of $A, B, C, D$. Find a formula for $x$ in terms of $a$ and $b$.\n(b) A hexagon is inscribed in a circle. If two opposite sides have length $2$ and th... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-2fd8a2420ca5596cdef6a439084aef968dada75bd79d14aea15a6ec2cad1f2d6 | (a) How many triangles are there with integer-length sides and perimeter 24? (b) How many triangles are there with integer-length sides and perimeter 36? (c) How many isosceles triangles are there with integer side lengths and perimeter 12*n, where n is an integer? (d) How many scalene triangles are there with integer ... | [
"3n^2 - 3n + 1"
] | [
"(a) How many triangles are there with integer-length sides and perimeter 24?\n(b) How many triangles are there with integer-length sides and perimeter 36?\n(c) How many isosceles triangles are there with integer side lengths and perimeter 12*n, where n is an integer?\n(d) How many scalene triangles are there with ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a3c1798f27d588419f3f268ec246a0d3c29e6b88963d06cf00135f050581e06 | (a) How many ways can the faces of a cube be painted by six different colors if no two faces are to be painted by the same color? (b) What if the cube in Part (a) is replaced by a box (i.e., a rectangular parallelepiped) none of whose faces is a square? | [
"720"
] | [
"(a) How many ways can the faces of a cube be painted by six different colors if no two faces are to be painted by the same color?\n(b) What if the cube in Part (a) is replaced by a box (i.e., a rectangular parallelepiped) none of whose faces is a square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e135f9b8a98767f82b54acc68bf67ff4e1207211daa46e1866858ff03c6db23 | (a) If \( a^b = b^a \) and \( a > b \) are positive integers, find \( (b-a)^{(a-b)} \). (b) Let TNYWR be \( k \). The area of the largest triangle that has vertices on a cube with side length \( k \) is \( a\sqrt{3} \). Find \( a \). (c) What is the largest prime factor of \( 3^{\text{TNYWR}} - 2^{\text{TNYWR}} \)? | [
"4, 8, 97"
] | [
"(a) If \\( a^b = b^a \\) and \\( a > b \\) are positive integers, find \\( (b-a)^{(a-b)} \\).\n(b) Let TNYWR be \\( k \\). The area of the largest triangle that has vertices on a cube with side length \\( k \\) is \\( a\\sqrt{3} \\). Find \\( a \\).\n(c) What is the largest prime factor of \\( 3^{\\text{TNYWR}} - ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-827507253cea3038a31da201aef779bad5e1e16a9351b6774c797fb55c1ddd4f | (a) If \((x+1)(x-1) = 80\), find \(x^2\). (b) If \((x+1)(x-1) = 22\), find \(x^2\). | [
"\\( x^2 = 81 \\) or \\( x^2 = 23 \\)"
] | [
"(a) If \\((x+1)(x-1) = 80\\), find \\(x^2\\). \n(b) If \\((x+1)(x-1) = 22\\), find \\(x^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-b10e4ef705428903412d8bd165502864005f6424422588ca8a7457752e7419da | (a) In how many ways can three pints of ice cream out of 12 different flavors be chosen if the order does not matter? (b) In how many ways can three pints of ice cream be chosen if the flavors don't need to be different and the order does not matter? | [
"(a) 220, (b) 364"
] | [
"(a) In how many ways can three pints of ice cream out of 12 different flavors be chosen if the order does not matter?\n(b) In how many ways can three pints of ice cream be chosen if the flavors don't need to be different and the order does not matter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 26,
"tir": 18
} | 44 | 0 | 0 |
omr-ws-d696fa6775a04e56163df101f2e2145fe9f69e94c4c4de15c192b8e4a994677f | (a) In the \( x - y \) plane, determine the length of the path \( P \to A \to C \to B \to P \) around the circle \( (x - 6)^2 + (y - 8)^2 = 25 \) from the point \( P(12, 16) \) to itself. (b) Determine the coordinates of the points \( A \) and \( B \). | [
"B \\left( \\frac{15 - 4\\sqrt{3}}{2}, \\frac{20 + 3\\sqrt{3}}{2} \\right)"
] | [
"(a) In the \\( x - y \\) plane, determine the length of the path \\( P \\to A \\to C \\to B \\to P \\) around the circle \\( (x - 6)^2 + (y - 8)^2 = 25 \\) from the point \\( P(12, 16) \\) to itself.\n(b) Determine the coordinates of the points \\( A \\) and \\( B \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f131bcd0c2454964606c360f9abb4477064b558dc50f22ba4a8132b2e6b04b7a | (a) Let $P$ be a point in the plane of the equilateral triangle $ABC$. The lengths of the perpendiculars from $P$ to the lines determined by $BC$, $CA$, and $AB$ are 1, 3, and 7 units, respectively. Find the length of a side of $\triangle ABC$. (b) Is the solution unique? If not, find all other solutions. | [
"\\dfrac{22\\sqrt{3}}{3}"
] | [
"(a) Let $P$ be a point in the plane of the equilateral triangle $ABC$. The lengths of the perpendiculars from $P$ to the lines determined by $BC$, $CA$, and $AB$ are 1, 3, and 7 units, respectively. Find the length of a side of $\\triangle ABC$. (b) Is the solution unique? If not, find all other solutions."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-09e5e1852ca959fa7461be82e9045b0b90fb2c0f15e44d34947c59d385a1a35b | (a) Let \(\mathbf{u}, \mathbf{v} \in \mathbb{R}^2\). Describe the vectors \(\mathbf{x} = s\mathbf{u} + t\mathbf{v}\), where \(s + t = 1\). (b) Let \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^3\). Describe the vectors \(\mathbf{x} = r\mathbf{u} + s\mathbf{v} + t\mathbf{w}\), where \(r + s + t = 1\). | [
"\\text{The plane through } \\mathbf{u}, \\mathbf{v}, \\text{ and } \\mathbf{w}"
] | [
"(a) Let \\(\\mathbf{u}, \\mathbf{v} \\in \\mathbb{R}^2\\). Describe the vectors \\(\\mathbf{x} = s\\mathbf{u} + t\\mathbf{v}\\), where \\(s + t = 1\\).\n(b) Let \\(\\mathbf{u}, \\mathbf{v}, \\mathbf{w} \\in \\mathbb{R}^3\\). Describe the vectors \\(\\mathbf{x} = r\\mathbf{u} + s\\mathbf{v} + t\\mathbf{w}\\), where... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d339c119f155d0b34f9bbb92ad1a43847853e45f21ed5fed46e5965b0980adbb | (a) Sketch the diagram of the function \( f \) if \[ f(x) = 4x(1 - |x|), \quad |x| \leq 1. \] (b) Does there exist a derivative of \( f \) at \( x = 0 \)? (c) Let \( g \) be a function such that \[ g(x) = \begin{cases} \frac{f(x)}{x} & \text{if } x \neq 0, \\ 4 & \text{if } x = 0. \end{cases} \] Is the function \( g \)... | [
"The derivative of \\( f \\) at \\( x = 0 \\) is 4, and \\( g \\) is continuous at \\( x = 0 \\)."
] | [
"(a) Sketch the diagram of the function \\( f \\) if\n\\[ f(x) = 4x(1 - |x|), \\quad |x| \\leq 1. \\]\n\n(b) Does there exist a derivative of \\( f \\) at \\( x = 0 \\)?\n\n(c) Let \\( g \\) be a function such that\n\\[ g(x) = \\begin{cases} \n\\frac{f(x)}{x} & \\text{if } x \\neq 0, \\\\\n4 & \\text{if } x = 0.\n\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 12
} | 22 | 0 | 0 |
omr-ws-ecd08490e0895724cf4ccd4161b1ced0fe81fbdcdcb08ef133ec901bf87cbdf6 | (a) Solve for $x$: $\log_x 8 - 3 = 0$. (b) Solve for $y$: $\log_y 9 - 2 \log_{3y} 9 = 0$. (c) Solve for $z$: $\log_z 4 - 2 \log_{4z} 4 - \log_{16z} 4 = 0$. | [
"\\( x = 2 \\), \\( y = 3 \\), \\( z = 4 \\)"
] | [
"(a) Solve for $x$: $\\log_x 8 - 3 = 0$. (b) Solve for $y$: $\\log_y 9 - 2 \\log_{3y} 9 = 0$. (c) Solve for $z$: $\\log_z 4 - 2 \\log_{4z} 4 - \\log_{16z} 4 = 0$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25a9d950a07183758d441f605f2121a7df11ee916bc52b9ac17ea46f8390a51c | (a) Suppose \( A \), \( B \), \( R \), and \( Q \) are four points in the plane such that the points \( R \) and \( Q \) lie on one side of the line through the points \( A \) and \( B \). Determine the position of the point \( P \) in the interior of the segment \( AB \) so that the sum of the lengths of the segments ... | [
"PQR"
] | [
"(a) Suppose \\( A \\), \\( B \\), \\( R \\), and \\( Q \\) are four points in the plane such that the points \\( R \\) and \\( Q \\) lie on one side of the line through the points \\( A \\) and \\( B \\). Determine the position of the point \\( P \\) in the interior of the segment \\( AB \\) so that the sum of the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-cd06a6cd69a20b0510c135fd7e69bc4a1a2a77aa2a3170d0b8c42802470f9b42 | (a) Suppose \( f(x) = 1 - \frac{|x|}{\pi} \) for \( x \in [-\pi, \pi] \). Compute the integrals \( a_{n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-2\pi i n x} \, dx \). (b) Suppose \( f(x) = (1 - \frac{|x|}{\pi})^{4} \). Determine \( \lim_{n \to \infty} n^{3} a_{n} \). | [
"0"
] | [
"(a) Suppose \\( f(x) = 1 - \\frac{|x|}{\\pi} \\) for \\( x \\in [-\\pi, \\pi] \\). Compute the integrals \\( a_{n} = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) e^{-2\\pi i n x} \\, dx \\).\n(b) Suppose \\( f(x) = (1 - \\frac{|x|}{\\pi})^{4} \\). Determine \\( \\lim_{n \\to \\infty} n^{3} a_{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-fb5931d279cc97d665d19cceacd602bb8eeba9dd1556c62c3cc50e505bc94610 | (a) Suppose there are $m$ Martians and $n$ Earthlings seated in a line such that no two Martians sit together. How many ways can the Earthlings and Martians be seated in a line? (b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table such that no two Martians sit together. How ma... | [
"(n-1)! \\cdot \\binom{n}{m} \\cdot m!"
] | [
"(a) Suppose there are $m$ Martians and $n$ Earthlings seated in a line such that no two Martians sit together. How many ways can the Earthlings and Martians be seated in a line?\n\n(b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table such that no two Martians sit together... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7f49fac2e00ad6515103c95960ff60b13ca15acb3f20819b73ea097c7c65c39 | (a) The consumer association estimates that 39% of households have 3 or more mobile phones. A mobile phone company suspects this percentage is not 39%. A random sample of 160 households found that 45 had 3 or more mobile phones. (i) Calculate a 95% confidence interval for the percentage of households with 3 or more mob... | [
"\\text{Reject } H_0"
] | [
"(a) The consumer association estimates that 39% of households have 3 or more mobile phones. A mobile phone company suspects this percentage is not 39%. A random sample of 160 households found that 45 had 3 or more mobile phones.\n(i) Calculate a 95% confidence interval for the percentage of households with 3 or mo... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-120bbae1bb0330f368a92a6432c2daa5e5e37eb739abc7f97863574a8b432e56 | (a) What is the magic sum for a 3-by-3 magic square? (b) What is the magic sum for an \(n\)-by-\(n\) magic square? (c) Place the numbers 1 through 9 into the nine cells of a 3-by-3 square to make it a magic square. | [
"15"
] | [
"(a) What is the magic sum for a 3-by-3 magic square?\n(b) What is the magic sum for an \\(n\\)-by-\\(n\\) magic square?\n(c) Place the numbers 1 through 9 into the nine cells of a 3-by-3 square to make it a magic square."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76ee1de509b1e3b5d4586a3459b2e6abe00e6d4e50567c75379e03a10d05e49b | (b) Compute the remainder of $100!$ when divided by $101$. | [
"100"
] | [
"(b) Compute the remainder of $100!$ when divided by $101$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-740906b0aa18aebc1514d5525e8b15dcc3c142020a982d28c23f3fdc324692b4 | (b) Find the maximum value of the expression \( x^2y + y^2x + z^2x - x^2z - y^2z - z^2y \) when \( 0 \leq x, y, z \leq 1 \). | [
"\\(\\frac{1}{4}\\)"
] | [
"(b) Find the maximum value of the expression \\( x^2y + y^2x + z^2x - x^2z - y^2z - z^2y \\) when \\( 0 \\leq x, y, z \\leq 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b849e4f476296c5755071783b49460d210fb1b95866a4851d07a8b30f0ee49c | (i) Express the area of the cross section $S(t)$ in terms of $t$ and $n$ when the solid $K_n$ (swept by a sphere of radius 1 centered at a point traveling along the sides of $P$) is cut by the plane $z=t$ where $-1 \leq t \leq 1$. (ii) Express the volume $V(n)$ of $K_n$ in terms of $n$. | [
"\\( S(t) = \\pi (1 - t^2) \\) and \\( V(n) = \\frac{4\\pi}{3} \\)"
] | [
"(i) Express the area of the cross section $S(t)$ in terms of $t$ and $n$ when the solid $K_n$ (swept by a sphere of radius 1 centered at a point traveling along the sides of $P$) is cut by the plane $z=t$ where $-1 \\leq t \\leq 1$.\n(ii) Express the volume $V(n)$ of $K_n$ in terms of $n$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a05facb7b4f6a7c680ef3c4dd9b12f637649cdbd906c1809014be92ec8ca0461 | (ii) Given a set $A = \{a_1, a_2, a_3, \ldots, a_n\}$, find the total number of binary operations on $A$ such that $a_i * a_j \neq a_i * a_k$ if $j \neq k$. | [
"\\((n!)^n\\)"
] | [
"(ii) Given a set $A = \\{a_1, a_2, a_3, \\ldots, a_n\\}$, find the total number of binary operations on $A$ such that $a_i * a_j \\neq a_i * a_k$ if $j \\neq k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a5493c5b73d52cf410f18a9a3e3669a41c5a1dc0316eb54afcf3cd92ee4e5868 | (iii) Given a set $A = \{a_1, a_2, a_3, \ldots, a_n\}$, find the total number of binary operations on $A$ such that $a_i * a_j < a_i * a_{j+1}$ for all $i, j$. | [
"1"
] | [
"(iii) Given a set $A = \\{a_1, a_2, a_3, \\ldots, a_n\\}$, find the total number of binary operations on $A$ such that $a_i * a_j < a_i * a_{j+1}$ for all $i, j$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4469cf46da3d1feb5a515bdac737a4495387978bbbb83e3932d4280b6f4686c5 | * Find all positive integers which are 4 or 9 times as small as their reversions. | [
"A number which is 5, 6, 8 or 7 times as small as its reversion must begin with the digit 1; a number which is twice or three times as small as its reversion can begin with the digits 1, 2, 3, 4 or 1, 2, 3 respectively"
] | [
"* Find all positive integers which are 4 or 9 times as small as their reversions."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13fd08631a14d206da19da53f29bad2cda6ba3e9319f956c0b39675b676cc521 | * Find all whole numbers which decrease an integral number of times when their second digits are deleted. | [
"The numbers whose all digits except the first two are zeros"
] | [
"* Find all whole numbers which decrease an integral number of times when their second digits are deleted."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42ac022226e912785470d5e6db566def8f5da3844363b64d0af39bd4c8c1f5d4 | * Find the integral solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ in $x, y, z$ (derive the general formula expressing all the solutions). | [
"Put $=x_1+n$, $y=y_1+n$"
] | [
"* Find the integral solutions of the equation $\\frac{1}{x}+\\frac{1}{y}=\\frac{1}{z}$ in $x, y, z$ (derive the general formula expressing all the solutions)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b067258e756dafe0cc7432a09813edd04e7059d388a50b546c5452f6d54ee63 | * Find the last two digits of the number $14^{(14^{14})}$. | [
"9; 2"
] | [
"* Find the last two digits of the number $14^{(14^{14})}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cafa03ccb835b7110aa2bbcec1445d73831b0469b4809f307ecee814d4d1f9d5 | * Find the value of the root $\sqrt {0.\underbrace{1111\ldots 111}_{100\ {\rm ones}}}$ to an accuracy of (1) 100; (2) 101; (3) 200 and (4) 300 decimal places after the decimal point. | [
"Prove that if $\\alpha <1-(0.1)^{100}$ then also $\\sqrt \\alpha <1-(0.1)^{100}$"
] | [
"* Find the value of the root $\\sqrt {0.\\underbrace{1111\\ldots 111}_{100\\ {\\rm ones}}}$ to an accuracy of (1) 100; (2) 101; (3) 200 and (4) 300 decimal places after the decimal point."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fccfa185e1efa8799b4b0e3900e69e7e8687ecee8f7c96540f68cc0ac5ed781e | * Given $n$ pairwise distinct numbers $a_1,a_2,a_3,\ldots ,a_n$, it is required to arrange them in a certain order as $a_{i_1},a_{i_2},a_{i_3},\ldots ,a_{i_n}$ to make the sum $$\Phi=(a_{i_1}-a_{i_2})^2+(a_{i_2}-a_{i_3})^2+\ldots + (a_{i_{n-1}}-a_{i_n})^2+(a_{i_n}-a_{i_1})^2$$ assume the least possible value. | [
"$a_1,a_2,a_4,a_3$"
] | [
"* Given $n$ pairwise distinct numbers $a_1,a_2,a_3,\\ldots ,a_n$, it is required to arrange them in a certain order as $a_{i_1},a_{i_2},a_{i_3},\\ldots ,a_{i_n}$ to make the sum $$\\Phi=(a_{i_1}-a_{i_2})^2+(a_{i_2}-a_{i_3})^2+\\ldots + (a_{i_{n-1}}-a_{i_n})^2+(a_{i_n}-a_{i_1})^2$$ assume the least possible value."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe16f9069aefc45dfdf56e149f13074f437dd0688f28dc60c1f326cf825b789a | 1) In a class with 17 boys and 20 girls, how many ways can 37 students be arranged in a column such that there are exactly 1 pair of boy-girl where the boy is standing in front of the girl? 2) A row of 19 rooms is numbered from 1 to 19. Initially, each room has 1 person. Every day, two random people are moved to the ne... | [
"\\text{No}"
] | [
"1) In a class with 17 boys and 20 girls, how many ways can 37 students be arranged in a column such that there are exactly 1 pair of boy-girl where the boy is standing in front of the girl?\n2) A row of 19 rooms is numbered from 1 to 19. Initially, each room has 1 person. Every day, two random people are moved to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-c7e23d0a723529a73507e93a63d0e4db443c59e37a5bb105c6333fa9830a75bb | 1. Investigate all the possible dollar values that can be obtained using a $3 coin and a $5 coin. Possible hint: 2 · 3 - 1 · 5 = 1 and 2 · 5 - 3 · 3 = 1. 2. Your local cookie store sells cookies in packs of 6, 10, and 15. Which numbers of cookies is it possible to order? 3. Develop a general strategy for answering such... | [
"\\text{Yes, } N = ab - a - b + 1 \\text{ for two coprime denominations}"
] | [
"1. Investigate all the possible dollar values that can be obtained using a $3 coin and a $5 coin. Possible hint: 2 · 3 - 1 · 5 = 1 and 2 · 5 - 3 · 3 = 1.\n2. Your local cookie store sells cookies in packs of 6, 10, and 15. Which numbers of cookies is it possible to order?\n3. Develop a general strategy for answeri... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1e91117ff12f13656e7cd7a376c8f5f105453eac35135b45190a68739484253 | 10 lines and 10 circles divide the plane into at most \( n \) disjoint regions. Compute \( n \). | [
"346"
] | [
"10 lines and 10 circles divide the plane into at most \\( n \\) disjoint regions. Compute \\( n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5db83f8f631d06a6cdb292c52f1138ad8538145001e5922faaa5ccb4b1963d79 | 10 people are assigned seats at random at a circular table with 10 chairs. Three of these people are members of the same family. What is the probability that no two members of this family are sitting next to each other? | [
"\\dfrac{5}{12}"
] | [
"10 people are assigned seats at random at a circular table with 10 chairs. Three of these people are members of the same family. What is the probability that no two members of this family are sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ee645a683d44b43b37be8cec2613ce99213e5fa3ba8e842353a4b2b33b5d7d24 | 10 people are seated around a circular table with chairs marked 0 through 9 in a clockwise manner. A ball starts at chair 0 and is passed clockwise. In the $n$-th step, the ball is passed $n^n$ chairs. Determine the chair number where the ball will be after 2020 steps. | [
"8"
] | [
"10 people are seated around a circular table with chairs marked 0 through 9 in a clockwise manner. A ball starts at chair 0 and is passed clockwise. In the $n$-th step, the ball is passed $n^n$ chairs. Determine the chair number where the ball will be after 2020 steps."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3353a701552d4fca87c0c05163570b528c0c47fe899441188a2b680972c2f1fd | 10 people are seated around a circular table with chairs marked 0 through 9 in a clockwise manner. A ball starts at chair 0 and is passed clockwise. In the $n$-th step, the ball is passed $n^n$ chairs. Determine the chair number where the ball will be after the 2017th step. | [
"7"
] | [
"10 people are seated around a circular table with chairs marked 0 through 9 in a clockwise manner. A ball starts at chair 0 and is passed clockwise. In the $n$-th step, the ball is passed $n^n$ chairs. Determine the chair number where the ball will be after the 2017th step."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe6811d3acf6192dd7c7f4011e3b1db7719b84ad4afe0305a439015ed51b9ab3 | 10 points lie on the circumference of a circle. How many distinct convex polygons can be formed by connecting some or all of the points? | [
"968"
] | [
"10 points lie on the circumference of a circle. How many distinct convex polygons can be formed by connecting some or all of the points?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8546be5f6c54fe6767ff49268995a4b46a5d4764ad648784cff6214d5f23f91 | 10 students form a line, each wearing a black or white hat. For any number of consecutive students, the absolute difference between the number of students wearing black hats and the number of students wearing white hats is less than 3. Find the number of valid sequences of hats. | [
"94"
] | [
"10 students form a line, each wearing a black or white hat. For any number of consecutive students, the absolute difference between the number of students wearing black hats and the number of students wearing white hats is less than 3. Find the number of valid sequences of hats."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 6
} | 9 | 0 | 0 |
omr-ws-395998c3b6eb5e47b6f37cb83875164cf0a33f5c92361f21169bbea847874dc8 | 100 blocks are selected from a crate containing 33 blocks of each of the following dimensions: \( 13 \times 17 \times 21 \), \( 13 \times 17 \times 37 \), \( 13 \times 21 \times 37 \), and \( 17 \times 21 \times 37 \). The chosen blocks are stacked on top of each other (one per cross section) forming a tower of height ... | [
"596"
] | [
"100 blocks are selected from a crate containing 33 blocks of each of the following dimensions: \\( 13 \\times 17 \\times 21 \\), \\( 13 \\times 17 \\times 37 \\), \\( 13 \\times 21 \\times 37 \\), and \\( 17 \\times 21 \\times 37 \\). The chosen blocks are stacked on top of each other (one per cross section) formi... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e689f43b8896b51fad60720fa7bfce3433a9b7e5445f4ddb821adb044df47439 | 100 cards are marked from 1 to 100 and 2 are drawn at random. Find the probability that the product of the two cards drawn is a prime number. | [
"$\\frac{2}{33}$"
] | [
"100 cards are marked from 1 to 100 and 2 are drawn at random. Find the probability that the product of the two cards drawn is a prime number."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f401e27fe7cf7b2925dfa6da1f1b8fedcdc6a8f6715f2c16af3e360f0ae5800d | 100 chicks are sitting in a circle. The first chick in the circle says the number 1, then the chick 2 seats away from the first chick says the number 2, then the chick 3 seats away from the chicken that said the number 2 says the number 3, and so on. The process will always go clockwise. Some of the chicks in the circl... | [
"21"
] | [
"100 chicks are sitting in a circle. The first chick in the circle says the number 1, then the chick 2 seats away from the first chick says the number 2, then the chick 3 seats away from the chicken that said the number 2 says the number 3, and so on. The process will always go clockwise. Some of the chicks in the ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-4fe3ed1e50e4865840c144de8b0d47fab5ba6715754c9a22dd7681c6ca26fe0e | 100 chicks sit in a circle. Each chick has a 50/50 chance of chirping to a chick next to it. What is the probability that half of the chicks are not chirped at? | [
"0"
] | [
"100 chicks sit in a circle. Each chick has a 50/50 chance of chirping to a chick next to it. What is the probability that half of the chicks are not chirped at?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-a6e8f11f18fa46f519e8805f0bfe192cca1a355d4072e2c7b45824624e53f831 | 100 fractions are written on a board, their numerators are numbers from 1 to 100 (each once) and denominators are also numbers from 1 to 100 (also each once). It appears that the sum of these fractions equals \(a/2\) for some odd \(a\). Find the minimum possible value of \(a\) such that it is possible to interchange nu... | [
"201"
] | [
"100 fractions are written on a board, their numerators are numbers from 1 to 100 (each once) and denominators are also numbers from 1 to 100 (also each once). It appears that the sum of these fractions equals \\(a/2\\) for some odd \\(a\\). Find the minimum possible value of \\(a\\) such that it is possible to int... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-d44e3aa3d73df89e4dfba8a0320bd5383f9eb99de0345b2c7bf3908c3e718e35 | 100 numbers are written in a circle. The sum of all numbers is equal to 100, and the sum of any 6 consecutive numbers (along the circle) is less than or equal to 6. One of the numbers is equal to 6. Restore all written numbers. | [
"The numbers are \\(6, -4, 6, -4, \\ldots\\). \n\nTo provide the final answer in the requested format:"
] | [
"100 numbers are written in a circle. The sum of all numbers is equal to 100, and the sum of any 6 consecutive numbers (along the circle) is less than or equal to 6. One of the numbers is equal to 6. Restore all written numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-9b04126ba57ac1b59eaa4e6b101694fb50214d9ecdb6229721b88a06600ee057 | 100 people are standing in a line, and each person has an arbitrarily large amount of red envelopes. Every second, the \(n\)th person gives the \(n+1\)th person \(n+1\) red envelopes or the \(n-1\)th person \(n-1\) red envelopes, both with equal probability. The two people on the end give the one person next to them th... | [
"50"
] | [
"100 people are standing in a line, and each person has an arbitrarily large amount of red envelopes. Every second, the \\(n\\)th person gives the \\(n+1\\)th person \\(n+1\\) red envelopes or the \\(n-1\\)th person \\(n-1\\) red envelopes, both with equal probability. The two people on the end give the one person ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bf20f4c704c919a3bfdfbf03ad4bd66d7f0e3d9d3b3d22940f46e4e925a100d | 100 people from 25 countries, four from each country, are seated on a circle. Find the minimum number of groups, $n$, into which they can be partitioned such that no two people from the same country or adjacent on the circle are in the same group. Calculate $n^2$. | [
"25"
] | [
"100 people from 25 countries, four from each country, are seated on a circle. Find the minimum number of groups, $n$, into which they can be partitioned such that no two people from the same country or adjacent on the circle are in the same group. Calculate $n^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-2bda4381c69c75926d44f520d6a0837352e26715dca9cd889c65876d05318e65 | 100 people from 25 nations (4 people from each nation) sit around a round table. Each nation is represented by the numbers 1 through 25, and each person from a nation is uniquely identified by a pair $(n, k)$ where $n$ is the nation number and $k \in \{1, 2, 3, 4\}$. Define a "valid grouping" as a division of these peo... | [
"6^{25}"
] | [
"100 people from 25 nations (4 people from each nation) sit around a round table. Each nation is represented by the numbers 1 through 25, and each person from a nation is uniquely identified by a pair $(n, k)$ where $n$ is the nation number and $k \\in \\{1, 2, 3, 4\\}$. Define a \"valid grouping\" as a division of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ca8d4750970ac7b5ac2dd2e0b87a6bc8f1f407af7ab14da11726642f756cf44 | 100 people from 50 countries, two from each country, are seated in a circle. Let $S$ be the number of ways to partition them into two groups such that no two countrymen and no three consecutive people on the circle are in the same group. Find the remainder when $S$ is divided by $1000$. | [
"624"
] | [
"100 people from 50 countries, two from each country, are seated in a circle. Let $S$ be the number of ways to partition them into two groups such that no two countrymen and no three consecutive people on the circle are in the same group. Find the remainder when $S$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-2485841830d7ca9ac97d4ec10e6eb3aa776e634185ea9256edc0165976d65590 | 100 people from 50 nations (2 people from each nation) sit around a round table. Find the number of ways to divide these people into 2 groups of 50 such that each group contains one person from each nation and no one sits between two people of the same group. | [
"2"
] | [
"100 people from 50 nations (2 people from each nation) sit around a round table. Find the number of ways to divide these people into 2 groups of 50 such that each group contains one person from each nation and no one sits between two people of the same group."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-80f802c8c641a0c7cc7b9fed2e4cc9f76c6aaffb9a5b3be2df8529a9c5f2a009 | 100 people went to a party. The entrance fee for males is $ \$5$, for women is $ \$1$, and for children is 5 cents. If $ \$100$ were collected at the party, how many of each went, given that at least one of each attended? | [
"19 men, 1 woman, and 80 children."
] | [
"100 people went to a party. The entrance fee for males is $ \\$5$, for women is $ \\$1$, and for children is 5 cents. If $ \\$100$ were collected at the party, how many of each went, given that at least one of each attended?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 29
} | 37 | 0 | 0 |
omr-ws-ca6b184caf68416d14f1e6779f27560a9f7dc55f0c1e005110cb08add0dcf72d | 100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.'' How many of those 100 people believe that f... | [
"50"
] | [
"100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''\n\nHow many of those 100 people belie... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d2c3ae71114423a01cd21a0c4bd8812a44e9672cad408b1968374bbd96f23a2 | 100 pictures of BDMO math campers were painted by Urmi. Exactly \( k \) colors were used in each picture. There is a common color in every 20 pictures, but there is no common color in all 100 pictures. Find the smallest possible value of \( k \). | [
"20"
] | [
"100 pictures of BDMO math campers were painted by Urmi. Exactly \\( k \\) colors were used in each picture. There is a common color in every 20 pictures, but there is no common color in all 100 pictures. Find the smallest possible value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2af224b2bd295a9a8b1951a2c61cabfd01266fc46bfea533b48d9b3d176e112c | 100 soldiers stand on the Deccan Plateau, and each soldier's field of vision is a 100-degree angle. Each soldier announces the number of other soldiers they see. What is the largest possible sum of these 100 numbers? | [
"9900"
] | [
"100 soldiers stand on the Deccan Plateau, and each soldier's field of vision is a 100-degree angle. Each soldier announces the number of other soldiers they see. What is the largest possible sum of these 100 numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-9d30eebcd13d3553c39d32b3dfafc6332a10f07dcd917f31d9ac77f72d7a7a9b | 100 students in the Arvine Unified School District are taking a quiz. Each student randomly submits a real number between 0 and 1. All students submit a different real number. A student's score is the minimum positive difference between his or her number and another student's number. Let $M$ be the maximum number of di... | [
"1"
] | [
"100 students in the Arvine Unified School District are taking a quiz. Each student randomly submits a real number between 0 and 1. All students submit a different real number. A student's score is the minimum positive difference between his or her number and another student's number. Let $M$ be the maximum number ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
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