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-7ab972f52bd122df41d9b65ba3a8cce96fd4f84ecdcaf8e50ec4956e95ba6930 | A sequence of positive integers is given by \( A_1 = 1 \) and \( A_n = \gcd(A_{n-1}, n) + 1 \) for \( n > 1 \). Compute \( A_{2000} \). | [
"3"
] | [
"A sequence of positive integers is given by \\( A_1 = 1 \\) and \\( A_n = \\gcd(A_{n-1}, n) + 1 \\) for \\( n > 1 \\). Compute \\( A_{2000} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8165506ae626da6e83370bfcaa703b4012357a9c0310d5bc9005b82ba0b39ab5 | A sequence of positive integers with \( n \) terms satisfies \( \sum_{i=1}^{n} a_i = 2007 \). Find the least positive integer \( n \) such that there exist some consecutive terms in the sequence with their sum equal to 30. | [
"1018"
] | [
"A sequence of positive integers with \\( n \\) terms satisfies \\( \\sum_{i=1}^{n} a_i = 2007 \\). Find the least positive integer \\( n \\) such that there exist some consecutive terms in the sequence with their sum equal to 30."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 6,
"tir": 2
} | 27 | 0 | 0 |
omr-ws-2c97c77c266577fe4bbe9e84d8ca4c060c6efbe516428f6850b9fd80a69d3974 | A sequence of positive numbers is defined by $(a_{n+1}+n)a_n=1$. How many rational elements exist in this sequence? | [
"0"
] | [
"A sequence of positive numbers is defined by $(a_{n+1}+n)a_n=1$. How many rational elements exist in this sequence?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-3811af97539bacb91da16838e0ebd7ddaaa1f25a3428a71f2d84f900bf79d5b5 | A sequence of positive real numbers is defined as follows: \[ Y(0) = 1 \] \[ Y(n+2) = 2 \cdot Y(n) - Y(n+1) \text{ for } n = 0, 1, 2, \ldots \] What is the value of \( Y(2007) \)? | [
"1"
] | [
"A sequence of positive real numbers is defined as follows:\n\\[ Y(0) = 1 \\]\n\\[ Y(n+2) = 2 \\cdot Y(n) - Y(n+1) \\text{ for } n = 0, 1, 2, \\ldots \\]\nWhat is the value of \\( Y(2007) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d83dd181a8d7742e78c5f44d7296e7fccaf4927f546ad48bde51fbfcca71cda | A sequence of real functions \( f_n \in C^2(\mathbb{R}) \) with \( \exists h \in C(\mathbb{R}), \forall n \in \mathbb{N}, f_n'' \leq h \), and the sequence simply converges to \( g \). If \( h(x) = 2 \) for all \( x \in \mathbb{R} \), find the value of \( g(5) - g(3) \) given that \( f_n(x) = \frac{1}{n} \sin(nx) + x^2... | [
"16"
] | [
"A sequence of real functions \\( f_n \\in C^2(\\mathbb{R}) \\) with \\( \\exists h \\in C(\\mathbb{R}), \\forall n \\in \\mathbb{N}, f_n'' \\leq h \\), and the sequence simply converges to \\( g \\). If \\( h(x) = 2 \\) for all \\( x \\in \\mathbb{R} \\), find the value of \\( g(5) - g(3) \\) given that \\( f_n(x)... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34738077adc830e90d6c2c53efad76fca1a0b5d936c687c59e5e72f6c1cdd974 | A sequence of real functions \( f_n \in C^3(\mathbb{R}) \) satisfies the following conditions: 1) \( \exists h \in C(\mathbb{R}), \forall n \in \mathbb{N}, f_n''' \leq h \), 2) \( \exists N > 0, \forall n \in \mathbb{N}, \max(|f_n'(0)|, |f_n''(0)|) \leq N \), 3) the sequence simply converges to \( g \). Let \( g(x) \) ... | [
"0"
] | [
"A sequence of real functions \\( f_n \\in C^3(\\mathbb{R}) \\) satisfies the following conditions:\n1) \\( \\exists h \\in C(\\mathbb{R}), \\forall n \\in \\mathbb{N}, f_n''' \\leq h \\),\n2) \\( \\exists N > 0, \\forall n \\in \\mathbb{N}, \\max(|f_n'(0)|, |f_n''(0)|) \\leq N \\),\n3) the sequence simply converge... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54d5829be9d3aef69dcc7f4790b82a80cb083397f42959632fa8d64259b28cad | A sequence of real numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = A$, $a_2 = B$, and $a_n = Ca_{n-1} + Da_{n-2}$ for all $n > 2$. Determine all possible conditions on the real constants $A, B, C$, and $D$ so that the sequence $\{a_n\}$ is a geometric sequence (geometric progression). | [
"B^2 = C A B + D A^2"
] | [
"A sequence of real numbers $a_1, a_2, a_3, \\ldots$ is defined by $a_1 = A$, $a_2 = B$, and $a_n = Ca_{n-1} + Da_{n-2}$ for all $n > 2$. Determine all possible conditions on the real constants $A, B, C$, and $D$ so that the sequence $\\{a_n\\}$ is a geometric sequence (geometric progression)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-41667f47d0c3fe1187a30bb5b3262129adc301e9a00f764153ed288e66df5104 | A sequence of real numbers $a_1, a_2, a_3, \ldots$ is defined by $a_1 = A$, $a_2 = B$, and $a_n = Ca_{n-1} + Da_{n-2}$ for all $n > 2$. Determine all possible conditions on the real constants $A, B, C$, and $D$ so that the sequence $\{a_n\}$ is an arithmetic sequence (arithmetic progression). | [
"(C = 2 \\text{ and } D = -1) \\text{ or } (A = B \\text{ and } (A = 0 \\text{ or } C + D = 1))"
] | [
"A sequence of real numbers $a_1, a_2, a_3, \\ldots$ is defined by $a_1 = A$, $a_2 = B$, and $a_n = Ca_{n-1} + Da_{n-2}$ for all $n > 2$. Determine all possible conditions on the real constants $A, B, C$, and $D$ so that the sequence $\\{a_n\\}$ is an arithmetic sequence (arithmetic progression)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eb67cd83c9be17d81cfb65f8bec40fc8d7c74b941a82e3083530d130b3c2c449 | A sequence of real numbers \( u_1, u_2, u_3, \dots \) is determined by \( u_1 \) and the recurrence relation for \( n \geq 1 \): \[ 4u_{n+1} = \sqrt[3]{64u_n + 15}. \] If the sequence converges, find the limit \( L \) of the sequence as \( n \to \infty \). Express \( L \) in simplest form. | [
"\\dfrac{1 + \\sqrt{61}}{8}"
] | [
"A sequence of real numbers \\( u_1, u_2, u_3, \\dots \\) is determined by \\( u_1 \\) and the recurrence relation for \\( n \\geq 1 \\):\n\\[ 4u_{n+1} = \\sqrt[3]{64u_n + 15}. \\]\nIf the sequence converges, find the limit \\( L \\) of the sequence as \\( n \\to \\infty \\). Express \\( L \\) in simplest form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fccc8dcee61da92d27b6a57deff62701f42f60f408739abf870d36f2ac101320 | A sequence of real numbers \( x_0, x_1, x_2, \ldots \) is defined as follows: \( x_0 = 1989 \) and for each \( n \geq 1 \), \[ x_n = - \frac{1989}{n} \sum_{k=0}^{n-1} x_k. \] Calculate the value of \( \sum_{n=0}^{1989} 2^n x_n. \) | [
"-1989"
] | [
"A sequence of real numbers \\( x_0, x_1, x_2, \\ldots \\) is defined as follows: \\( x_0 = 1989 \\) and for each \\( n \\geq 1 \\),\n\n\\[ x_n = - \\frac{1989}{n} \\sum_{k=0}^{n-1} x_k. \\]\n\nCalculate the value of \\( \\sum_{n=0}^{1989} 2^n x_n. \\)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8bcfce3ec91cb1c622716a0e3b8fb436d2d4f27b222e1f31a97223babbfe381 | A sequence of real numbers \( x_n \) is defined recursively as follows: \( x_0 \) and \( x_1 \) are arbitrary positive real numbers, and \( x_{n+2} = \frac{1 + x_{n+1}}{x_n} \) for \( n = 0, 1, 2, \ldots \). Find \( x_{1998} \). | [
"\\( \\frac{x_0 + x_1 + 1}{x_0 x_1} \\)"
] | [
"A sequence of real numbers \\( x_n \\) is defined recursively as follows: \\( x_0 \\) and \\( x_1 \\) are arbitrary positive real numbers, and \\( x_{n+2} = \\frac{1 + x_{n+1}}{x_n} \\) for \\( n = 0, 1, 2, \\ldots \\). Find \\( x_{1998} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-5a6e7b82e281c2a3ff45207fa96dbcbc918e9b357c8922c0bf2672b87fff1d73 | A sequence of real numbers \( x_n \) is defined recursively as follows: \( x_0 \) and \( x_1 \) are arbitrary positive real numbers, and \[ x_{n+2} = \frac{1 + x_{n+1}}{x_n}. \] Find \( x_{1998} \). | [
"\\( \\frac{x_0 + 1 + x_1}{x_0 x_1} \\)"
] | [
"A sequence of real numbers \\( x_n \\) is defined recursively as follows: \\( x_0 \\) and \\( x_1 \\) are arbitrary positive real numbers, and \n\\[ x_{n+2} = \\frac{1 + x_{n+1}}{x_n}. \\]\nFind \\( x_{1998} \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 13
} | 36 | 0 | 0 |
omr-ws-f6cc8e0e9d850d1b2e44d04c699e5d08a602f57a482f7432f26928ebd634ef2b | A sequence of real numbers \(\{a_i\}\) satisfies \[ n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + \ldots + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n \] for each integer \(n \ge 1\). Find the value of \(a_{2023}\). | [
"2022^2 \\cdot 2023^{2021}"
] | [
"A sequence of real numbers \\(\\{a_i\\}\\) satisfies\n\\[ n \\cdot a_1 + (n - 1) \\cdot a_2 + (n - 2) \\cdot a_3 + \\ldots + 2 \\cdot a_{n-1} + 1 \\cdot a_n = 2023^n \\]\nfor each integer \\(n \\ge 1\\). Find the value of \\(a_{2023}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-9b1077bfda65c1b7e98d7fde8f893f9a44a3e36fccda17f28eb0abdadc8b77a4 | A sequence of real numbers \(a_0, a_1, \ldots\) is said to be good if the following three conditions hold: (i) The value of \(a_0\) is a positive integer. (ii) For each non-negative integer \(i\), we have \(a_{i+1} = 2a_i + 1\) or \(a_{i+1} = \frac{a_i}{a_i + 2}\). (iii) There exists a positive integer \(k\) such that ... | [
"60"
] | [
"A sequence of real numbers \\(a_0, a_1, \\ldots\\) is said to be good if the following three conditions hold:\n(i) The value of \\(a_0\\) is a positive integer.\n(ii) For each non-negative integer \\(i\\), we have \\(a_{i+1} = 2a_i + 1\\) or \\(a_{i+1} = \\frac{a_i}{a_i + 2}\\).\n(iii) There exists a positive inte... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-55ee1edc9eeb6f9f354cc3926c2f5f12b2c735145a531a8bc3ddf4fac6d51d78 | A sequence of real numbers \(a_1, a_2, a_3, \ldots\) satisfies \(a_0 = 1\) and the recurrence relation \(a_{n+1} = 2016a_n + a_{n-1} - 1\). Evaluate \(a_{2016}\). | [
"1009"
] | [
"A sequence of real numbers \\(a_1, a_2, a_3, \\ldots\\) satisfies \\(a_0 = 1\\) and the recurrence relation \\(a_{n+1} = 2016a_n + a_{n-1} - 1\\). Evaluate \\(a_{2016}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7086d10532ec0abe5a744c78c53933e8c3a8002414530d1af494aad9772f1f13 | A sequence of real numbers \(a_1, a_2, a_3, \ldots\) satisfies \(a_1 = 1\), \(a_2 = 2\), and \(a_{n+2} = \frac{2}{a_{n+1}} + a_n\). Evaluate \(a_{2012}\). | [
"\\(\\frac{2012 \\cdot \\binom{2011}{1005}}{2^{2010}}\\)"
] | [
"A sequence of real numbers \\(a_1, a_2, a_3, \\ldots\\) satisfies \\(a_1 = 1\\), \\(a_2 = 2\\), and \\(a_{n+2} = \\frac{2}{a_{n+1}} + a_n\\). Evaluate \\(a_{2012}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b20b2140058a2e539d89c2d84c76c10fb4363681b2478f0b9871f1821805d8c7 | A sequence of real numbers has the first term \( a_0 = 1 \) and satisfies \( 2^{a_n} = 4^{a_{n-1}} - 1 \) for \( n \geq 1 \). Calculate the sum of all integer terms in the sequence. | [
"4"
] | [
"A sequence of real numbers has the first term \\( a_0 = 1 \\) and satisfies \\( 2^{a_n} = 4^{a_{n-1}} - 1 \\) for \\( n \\geq 1 \\). Calculate the sum of all integer terms in the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5229527ff10787f14336eb3ad9575bad323cbd216c4b620f69b9654adefacdb2 | A sequence of real numbers is defined as follows: $a_0=1$ and $a_{n+1}=\frac{a_n}{1+na_n}$ for $n=0, 1, 2, \ldots$. Find a general formula for $a_n$ that is non-recursive in terms of $n$. | [
"\\( a_n = \\frac{2}{n^2 + n + 2} \\)"
] | [
"A sequence of real numbers is defined as follows: $a_0=1$ and $a_{n+1}=\\frac{a_n}{1+na_n}$ for $n=0, 1, 2, \\ldots$. Find a general formula for $a_n$ that is non-recursive in terms of $n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11cf92c261332ebceb07ca2303027844c331b041a9d0e16865cd0191b3beeac3 | A sequence of real numbers is recursively defined by \( x_1 = 2014 \) and \[ x_{n+1} = \frac{(\sqrt{2}+1)x_n - 1}{(\sqrt{2}+1) + x_n}. \] Find the 2015-th term of this sequence. | [
"-\\dfrac{2015}{2013}"
] | [
"A sequence of real numbers is recursively defined by \\( x_1 = 2014 \\) and \n\\[ x_{n+1} = \\frac{(\\sqrt{2}+1)x_n - 1}{(\\sqrt{2}+1) + x_n}. \\]\nFind the 2015-th term of this sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0f84918acf6f74a38c7582030000ed01b1829d721eaa81574ba74d4c6d49fc64 | A sequence of rolls \(a_1, a_2, \ldots, a_n\) generated from a standard fair 6-sided die is called fluctuating if the \(n-1\) sets of two consecutive rolls, placed in order, alternately fluctuate between having a sum greater than or equal to 7 and having a sum less than or equal to 7. The probability that Jamie will pr... | [
"121"
] | [
"A sequence of rolls \\(a_1, a_2, \\ldots, a_n\\) generated from a standard fair 6-sided die is called fluctuating if the \\(n-1\\) sets of two consecutive rolls, placed in order, alternately fluctuate between having a sum greater than or equal to 7 and having a sum less than or equal to 7. The probability that Jam... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-110a25ef41fbe12802c8e4b266c887c928e2acf59e26dd7e7066f541e64522b0 | A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$. | [
"73"
] | [
"A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f01b49c0814e590d45c9f720e0fbc4edfcafdf7ba7ec836b7370066403043e8 | A sequence of tetrahedra \( T_n \) is defined such that \( T_{n+1} \) is formed by connecting the centers of the faces of \( T_n \). Let \( V_n \) be the volume of tetrahedron \( T_n \). Given that \( \frac{V_{2006}}{V_1} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, find the remainder when ... | [
"908"
] | [
"A sequence of tetrahedra \\( T_n \\) is defined such that \\( T_{n+1} \\) is formed by connecting the centers of the faces of \\( T_n \\). Let \\( V_n \\) be the volume of tetrahedron \\( T_n \\). Given that \\( \\frac{V_{2006}}{V_1} = \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are coprime positive integers, f... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-efc6c8262441a39b7d85c049f295a9525a957ea63c69603000657270df29d96f | A sequence of tetrahedra \( T_n \) is formed by connecting the centers of the faces of the previous tetrahedron \( T_{n-1} \). The volume of \( T_n \) is \( V_n \). Given \( \frac{V_{2006}}{V_1} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, find the remainder when \( m+n \) is divided by 100... | [
"908"
] | [
"A sequence of tetrahedra \\( T_n \\) is formed by connecting the centers of the faces of the previous tetrahedron \\( T_{n-1} \\). The volume of \\( T_n \\) is \\( V_n \\). Given \\( \\frac{V_{2006}}{V_1} = \\frac{m}{n} \\) where \\( m \\) and \\( n \\) are coprime positive integers, find the remainder when \\( m+... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cfefa60be8e3221515c5f675e850cdc386a96e82b125d62d639b3b669e772978 | A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression? | [
"1"
] | [
"A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e565fa18841a7536165d4c22e5452814932bf737e94af4f7d999969e41b6c782 | A sequence of vertices \( v_1, v_2, \ldots, v_k \) in a graph, where \( v_i = v_j \) only if \( i = j \) and \( k \) can be any positive integer, is called a cycle if \( v_1 \) is attached by an edge to \( v_2 \), \( v_2 \) to \( v_3 \), and so on to \( v_k \) connected to \( v_1 \). Rotations and reflections are disti... | [
"1001"
] | [
"A sequence of vertices \\( v_1, v_2, \\ldots, v_k \\) in a graph, where \\( v_i = v_j \\) only if \\( i = j \\) and \\( k \\) can be any positive integer, is called a cycle if \\( v_1 \\) is attached by an edge to \\( v_2 \\), \\( v_2 \\) to \\( v_3 \\), and so on to \\( v_k \\) connected to \\( v_1 \\). Rotations... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-204ef2b180f045d8a0feda0818912c1d39bcb40608fbb399c06a973ff3f3b93e | A sequence that contains the numbers $1, 2, 3, \ldots, n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3 \ldots w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $7 > 1$.... | [
"1024"
] | [
"A sequence that contains the numbers $1, 2, 3, \\ldots, n$ exactly once each is said to be a permutation of length $n$. A permutation $w_1w_2w_3 \\ldots w_n$ is said to be sad if there are indices $i < j < k$ such that $w_j > w_k$ and $w_j > w_i$. For example, the permutation $3142756$ is sad because $7 > 6$ and $... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-7abf6a839066266409e9dd6231a34bdbbb0d696c16c167fac220062cc00b4273 | A series is defined as follows: - \( A(1) = 1 \) - For \( n > 1 \), \( A(n) \) consists of \( f(m) \) numbers of \( f(m) \) followed by \( f(m) \) numbers of \( 0 \), where \( m \) is the number of digits in \( A(n-1) \). - Here, \( f(m) \) is the remainder when \( m \) is divided by 9. Find \( A(30) \). | [
"77777770000000"
] | [
"A series is defined as follows:\n- \\( A(1) = 1 \\)\n- For \\( n > 1 \\), \\( A(n) \\) consists of \\( f(m) \\) numbers of \\( f(m) \\) followed by \\( f(m) \\) numbers of \\( 0 \\), where \\( m \\) is the number of digits in \\( A(n-1) \\).\n- Here, \\( f(m) \\) is the remainder when \\( m \\) is divided by 9.\nF... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-1a79c745de7bb193467cec23008143b97f368d4ef8ecaba59ca86f9eff20768a | A series is defined as follows: - \( A(1) = 1 \) - \( A(n) = f(m) \) numbers of \( f(m) \) followed by \( f(m) \) numbers of 0, where \( m \) is the number of digits in \( A(n-1) \) - \( f(m) \) is the remainder when \( m \) is divided by 9. Find \( A(30) \). | [
"10"
] | [
"A series is defined as follows:\n- \\( A(1) = 1 \\)\n- \\( A(n) = f(m) \\) numbers of \\( f(m) \\) followed by \\( f(m) \\) numbers of 0, where \\( m \\) is the number of digits in \\( A(n-1) \\)\n- \\( f(m) \\) is the remainder when \\( m \\) is divided by 9.\nFind \\( A(30) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c7fb4ca6af7ba60f89a52052fcc4ef7a0272e21f75a06e345d8d352e490ec9d | A series is defined as follows: - \( A(1) = 1 \) - \( A(n) \) consists of \( f(m) \) numbers of \( f(m) \) followed by \( f(m) \) numbers of 0, where \( m \) is the number of digits in \( A(n-1) \) and \( f(m) \) is the remainder when \( m \) is divided by 9. Find \( A(30) \). | [
"77777770000000"
] | [
"A series is defined as follows:\n- \\( A(1) = 1 \\)\n- \\( A(n) \\) consists of \\( f(m) \\) numbers of \\( f(m) \\) followed by \\( f(m) \\) numbers of 0, where \\( m \\) is the number of digits in \\( A(n-1) \\) and \\( f(m) \\) is the remainder when \\( m \\) is divided by 9.\nFind \\( A(30) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60c11be73dc4c375c41f4d15ac5b05029c3cf987b1d1bf84e9070aaa58bf8787 | A series is defined as: $1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, \ldots$. The first term is $2^0$, the next two are $2^0, 2^1$, the next three are $2^0, 2^1, 2^2$, etc. Find the minimum integer $N$ so that the first $N$ terms of this sequence sum to an integer power of 2, where $N > 100$. | [
"470"
] | [
"A series is defined as: $1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, \\ldots$. The first term is $2^0$, the next two are $2^0, 2^1$, the next three are $2^0, 2^1, 2^2$, etc. Find the minimum integer $N$ so that the first $N$ terms of this sequence sum to an integer power of 2, where $N > 100$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cd16ff92a9a62324088a1e59b9377e9040db516201e51d0c884854b61342c96 | A series of $n$ light poles are arranged in a circle and numbered, clockwise, from $1$ to $n$. All the lights are initially on. You start at light pole #$1$ and begin walking clockwise around the circle of lights turning off every second lighted pole doing so until there is only one pole that remains lighted. Let $f(n)... | [
"1985"
] | [
"A series of $n$ light poles are arranged in a circle and numbered, clockwise, from $1$ to $n$. All the lights are initially on. You start at light pole #$1$ and begin walking clockwise around the circle of lights turning off every second lighted pole doing so until there is only one pole that remains lighted. Let ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f43882cb6d5afa45ff36a58a7b798f1428ea2ed25bccb14f41504bfac5e6827 | A series of $n$ light poles are arranged in a circle and numbered, clockwise, from $1$ to $n$. All the lights are initially on. You start at light pole #$1$ and begin walking clockwise around the circle of lights turning off every third lighted pole doing so until there is only one pole that remains lighted. Let $f(n)$... | [
"661"
] | [
"A series of $n$ light poles are arranged in a circle and numbered, clockwise, from $1$ to $n$. All the lights are initially on. You start at light pole #$1$ and begin walking clockwise around the circle of lights turning off every third lighted pole doing so until there is only one pole that remains lighted. Let $... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-da18e48700e0912a44a19f86cf024a8d5af0289d4df6cec8a18a722aba6da4a4 | A series of integers is defined by \(a_{n+1} = a_n^2 - a_n + 1\) with \(a_1 = 2\). Calculate the sum \(\sum_{n=1}^\infty \frac{1}{a_n}\). | [
"1"
] | [
"A series of integers is defined by \\(a_{n+1} = a_n^2 - a_n + 1\\) with \\(a_1 = 2\\). Calculate the sum \\(\\sum_{n=1}^\\infty \\frac{1}{a_n}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-521c9b2988ae9250dbab7f179249e636b8d3a7c213434b76e53935adf1b2285d | A service costs $2500 on average and yields a 5% profit. A fee of $25 is charged on every service. If a client increases their productivity by 20%, should the company accept the client's proposal to exclude the $25 fee from each service? | [
"Accept the client's proposal."
] | [
"A service costs $2500 on average and yields a 5% profit. A fee of $25 is charged on every service. If a client increases their productivity by 20%, should the company accept the client's proposal to exclude the $25 fee from each service?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-7d8e6b714a6eba23aab71ff7415576440b6193fad301d12f2d1ca8ac6ee46da7 | A service lift goes down a tower 320 m high. First its speed is 20 m/s then it switches over instantaneously to 50 m/s. Some time after the lift begins to move, a stone is thrown from the top of the tower which performs a free fall and reaches the ground at the same time as the lift. When falling, the stone was always ... | [
"2 s"
] | [
"A service lift goes down a tower 320 m high. First its speed is 20 m/s then it switches over instantaneously to 50 m/s. Some time after the lift begins to move, a stone is thrown from the top of the tower which performs a free fall and reaches the ground at the same time as the lift. When falling, the stone was al... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-558dbfd780562193e8fb0526841d4b914f1370f0b8713074d6178b09359fbd7e | A set $A$ of distinct positive integers is such that when any of its elements is removed, the remaining ones can be divided into two subsets with equal sums. What is the maximum possible cardinality of $A$? | [
"7"
] | [
"A set $A$ of distinct positive integers is such that when any of its elements is removed, the remaining ones can be divided into two subsets with equal sums. What is the maximum possible cardinality of $A$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-d0a03708d8c8e4a34bc0ce3f0d9515121763bb3c088447e8160edd599a78c503 | A set $D$ of $n$ straight lines has the property that every line intersects exactly 2011 other lines. Find all possible values of $n$. | [
"2012 or 4022"
] | [
"A set $D$ of $n$ straight lines has the property that every line intersects exactly 2011 other lines. Find all possible values of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-6b060790e694822faa4776e4e80e677de536dd6bdb057729840886139927fa26 | A set $S$ is called robust if for every pair of distinct elements $a, b \in S$, there exists an element $c \in S$ such that $c = a + b$. Let $T$ be the set of all linear combinations of the numbers $2^0, 2^1, 2^2, \ldots, 2^{10}$. Determine the number of elements in the intersection of $T$ and the set $\{1, 2, 3, \ldot... | [
"2047"
] | [
"A set $S$ is called robust if for every pair of distinct elements $a, b \\in S$, there exists an element $c \\in S$ such that $c = a + b$. Let $T$ be the set of all linear combinations of the numbers $2^0, 2^1, 2^2, \\ldots, 2^{10}$. Determine the number of elements in the intersection of $T$ and the set $\\{1, 2,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7462496d60b96d2302b80390b816cfef2b75ea5dd502fee4fdf413526e4a2d44 | A set $S$ is called set-friendly if there exists a bijection $f: S \to S$ such that for every $x \in S$, $f(f(x)) = x$ and $f(x) \neq x$. Let $S = \{1, 2, 3, 4, 5, 6, 7\}$. Find the number of distinct set-friendly bijections $f: S \to S$. | [
"0"
] | [
"A set $S$ is called set-friendly if there exists a bijection $f: S \\to S$ such that for every $x \\in S$, $f(f(x)) = x$ and $f(x) \\neq x$. Let $S = \\{1, 2, 3, 4, 5, 6, 7\\}$. Find the number of distinct set-friendly bijections $f: S \\to S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cc6ec1f2b50ebb13644dd014e17b2f1323a7d725d678a67285e58e2250e5007 | A set $S$ of integers has the following properties: - $1$ is a member of $S$. - $S$ has over $1000$ elements. - If $5n$ is a member of $S$, then $n$ is a member of $S$. - If $7n$ is a member of $S$, then $n$ and $3n$ are both members of $S$. - If $3n$ is not a member of $S$, then $3n+1$ is a member of $S$, unless this ... | [
"1093"
] | [
"A set $S$ of integers has the following properties:\n- $1$ is a member of $S$.\n- $S$ has over $1000$ elements.\n- If $5n$ is a member of $S$, then $n$ is a member of $S$.\n- If $7n$ is a member of $S$, then $n$ and $3n$ are both members of $S$.\n- If $3n$ is not a member of $S$, then $3n+1$ is a member of $S$, un... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-e55c6cd5f394865c2f82b16346f20551e3bcf438f6efe18e7c970f1c7bdaa2eb | A set $S$ of rational numbers is ordered in a tree-diagram such that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains all rationals $r$ with $0 < r < 1$? | [
"\\(\\frac{1}{2}\\)"
] | [
"A set $S$ of rational numbers is ordered in a tree-diagram such that each rational number $\\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\\frac{a}{a+b}$ and $\\frac{b}{a+b}$. How should the initial element be selected such that this tree contains all rationals $r$ with $0 < r ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cb8ef3048c572998cbd5af250730eb748fe5370bce465fddaa203c7494a7d01f | A set $X$ is called prime-difference free (pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number $n$ such that the following both hold: - There is a pdf set of size $n$ that is a subset of $\{1, \ldots, 2016\}$. - There is no pdf set of size $n + 1$ that is a subset of $\{1, \ldots, 2016\}$. | [
"504"
] | [
"A set $X$ is called prime-difference free (pdf) if for all $x, y \\in X$, $|x - y|$ is not prime. Find the number $n$ such that the following both hold:\n- There is a pdf set of size $n$ that is a subset of $\\{1, \\ldots, 2016\\}$.\n- There is no pdf set of size $n + 1$ that is a subset of $\\{1, \\ldots, 2016\\}... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-099256a95a4654518f481eb5482af7e047a16ebd84cfc96766a190199413aca7 | A set $\mathbb{A}$ of integers is called [i]respectable[/i] if for any $x, y \in \mathbb{A}$, the expression $x^2 + kxy + y^2$ is also in $\mathbb{A}$ for all $k \in \mathbb{Z}$. Find all pairs $(m,n)$ of nonzero integers such that if $m, n \in \mathbb{A}$ and $\mathbb{A}$ is respectable, then $\mathbb{A} = \mathbb{Z}$... | [
"$(m, n)$ such that $(m, n) = 1$"
] | [
"A set $\\mathbb{A}$ of integers is called [i]respectable[/i] if for any $x, y \\in \\mathbb{A}$, the expression $x^2 + kxy + y^2$ is also in $\\mathbb{A}$ for all $k \\in \\mathbb{Z}$. Find all pairs $(m,n)$ of nonzero integers such that if $m, n \\in \\mathbb{A}$ and $\\mathbb{A}$ is respectable, then $\\mathbb{A... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 13
} | 31 | 0 | 0 |
omr-ws-7cd3c539d33490a6862356ea2388e39af320ca11719c20c3129a8dd89cbfe0d4 | A set $\mathbb{S}=\{1,2,3,...,11,12\}$ is defined. Brian picks three numbers from $\mathbb{S}$ with replacement. What is the probability that the first number is less than or equal to the second number, and the second number is less than or equal to the third number? Express your answer as $m+n$, where $\frac{m}{n}$ is... | [
"2092"
] | [
"A set $\\mathbb{S}=\\{1,2,3,...,11,12\\}$ is defined. Brian picks three numbers from $\\mathbb{S}$ with replacement. What is the probability that the first number is less than or equal to the second number, and the second number is less than or equal to the third number? Express your answer as $m+n$, where $\\frac... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-819a872033ebcd4622e00bc1763277b8692ef0801088846e7ddb0c85b566d04e | A set $\mathcal U$ has $\frac{T}{10}$ elements. Compute the sum of $|S|$ over all subsets $S \subseteq \mathcal U$. (Note: For a finite set $\mathcal A$, the notation $|\mathcal A|$ represents the number of elements of $\mathcal A$.) | [
"24576"
] | [
"A set $\\mathcal U$ has $\\frac{T}{10}$ elements. Compute the sum of $|S|$ over all subsets $S \\subseteq \\mathcal U$. (Note: For a finite set $\\mathcal A$, the notation $|\\mathcal A|$ represents the number of elements of $\\mathcal A$.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c10315595dfd8bb05fdedd0197f0e92ba5283c257b2400271fc1b501f27ba753 | A set $\mathcal{M} \subset M_3(\mathbb{Z}_2)$ is such that $\det(A+B) \ne 0$ for every $A, B \in \mathcal{M}$ with $A \ne B$. Find the maximal possible number of matrices in the set $\mathcal{M}$. | [
"8"
] | [
"A set $\\mathcal{M} \\subset M_3(\\mathbb{Z}_2)$ is such that $\\det(A+B) \\ne 0$ for every $A, B \\in \\mathcal{M}$ with $A \\ne B$. Find the maximal possible number of matrices in the set $\\mathcal{M}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-dbcb1266ea1e5fcf2d4445d7af2b0542bcf900b04a59e2215951960fbfa6f2f6 | A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is t... | [
"030",
"30"
] | [
"A set $\\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\\mathcal{S}$ is an integer. Given that 1 belongs to $\\mathcal{S}$ and that 2002 is the largest element of $\\mathcal{S},... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 3
} | 33 | 0 | 0 |
omr-ws-8257d5de78921d4619d151c902001f9a9dccfc3514b79fdb5cf1de1f2890cff1 | A set $\mathcal{S}$ of distinct positive integers has the property that for every integer $x$ in $\mathcal{S}$, the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S}$, what is the gre... | [
"30"
] | [
"A set $\\mathcal{S}$ of distinct positive integers has the property that for every integer $x$ in $\\mathcal{S}$, the arithmetic mean of the set of values obtained by deleting $x$ from $\\mathcal{S}$ is an integer. Given that 1 belongs to $\\mathcal{S}$ and that 2002 is the largest element of $\\mathcal{S}$, what ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-35912dc4da3c6bae4675e10b45f60b22331826d24bacaed7b196da86ef1ff192 | A set $\ss$ of binary strings is defined inductively as follows: - The binary string $1$ is in $\ss$. - If $s_1, s_2, \ldots, s_n$ is in $\ss$ with $n$ odd, then both $s_1, s_2, \ldots, s_n, 0$ and $0, s_1, s_2, \ldots, s_n$ are in $\ss$. - If $s_1, s_2, \ldots, s_n$ is in $\ss$ with $n$ even, then both $s_1, s_2, \ldo... | [
"4"
] | [
"A set $\\ss$ of binary strings is defined inductively as follows:\n- The binary string $1$ is in $\\ss$.\n- If $s_1, s_2, \\ldots, s_n$ is in $\\ss$ with $n$ odd, then both $s_1, s_2, \\ldots, s_n, 0$ and $0, s_1, s_2, \\ldots, s_n$ are in $\\ss$.\n- If $s_1, s_2, \\ldots, s_n$ is in $\\ss$ with $n$ even, then bot... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-0dcd669eb850e4ec2d7ae9888c7c224c0f5131fc42c8d9e928b0c816cdf952be | A set $\{x_1, x_2, \ldots, x_n\}$ is called "godly" if it satisfies the property that $$\prod_{i=1}^{n} x_i = x_j^2$$ for some $j$ such that $1 \le j \le n$. How many such "godly" sets are there with the largest element as $2022$, where all elements are greater than 1? | [
"5"
] | [
"A set $\\{x_1, x_2, \\ldots, x_n\\}$ is called \"godly\" if it satisfies the property that $$\\prod_{i=1}^{n} x_i = x_j^2$$ for some $j$ such that $1 \\le j \\le n$. How many such \"godly\" sets are there with the largest element as $2022$, where all elements are greater than 1?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1da52924266b8c8fab15674ff82ebff3962eea532fd363c2bfe8ea40130f966 | A set \( A \) is endowed with a binary operation \( * \) satisfying the following four conditions: 1. If \( a, b, c \) are elements of \( A \), then \( a * (b * c) = (a * b) * c \), 2. If \( a, b, c \) are elements of \( A \) such that \( a * c = b * c \), then \( a = b \), 3. There exists an element \( e \) of \( A \)... | [
"3"
] | [
"A set \\( A \\) is endowed with a binary operation \\( * \\) satisfying the following four conditions:\n1. If \\( a, b, c \\) are elements of \\( A \\), then \\( a * (b * c) = (a * b) * c \\),\n2. If \\( a, b, c \\) are elements of \\( A \\) such that \\( a * c = b * c \\), then \\( a = b \\),\n3. There exists an ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1746ba2ac1dcdbe5cee1a765d8676f62665e6300786c29eaaf94a81829ed2f9 | A set \( A \) is formed by six different positive integers, where the largest of them is \( n \). All the subsets of \( A \) that have two elements, except for one subset, satisfy that its smallest element divides the largest element. Determine the smallest possible value of \( n \) and give as an answer the sum of its... | [
"6"
] | [
"A set \\( A \\) is formed by six different positive integers, where the largest of them is \\( n \\). All the subsets of \\( A \\) that have two elements, except for one subset, satisfy that its smallest element divides the largest element. Determine the smallest possible value of \\( n \\) and give as an answer t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-463de51b770ac225c19d9f704a1c503695c4076eeb8a377b849d00c7df1386f3 | A set \( A \) of positive integers is called uniform if, after any of its elements is removed, the remaining elements can be partitioned into two subsets with equal sums. Find the least positive integer \( n > 1 \) such that there exists a uniform set \( A \) with \( n \) elements. | [
"7"
] | [
"A set \\( A \\) of positive integers is called uniform if, after any of its elements is removed, the remaining elements can be partitioned into two subsets with equal sums. Find the least positive integer \\( n > 1 \\) such that there exists a uniform set \\( A \\) with \\( n \\) elements."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-11d574c10c2883ed1b4e5cad9149eadd5c5cddb608b9917f729a8ceceaf0156f | A set \( A_1, A_2, A_3, A_4 \) of 4 points in the plane is said to be an Athenian set if there is a point \( P \) of the plane satisfying: (*) \( P \) does not lie on any of the lines \( A_i A_j \) for \( 1 \leq i < j \leq 4 \); (**) the line joining \( P \) to the midpoint of the line \( A_i A_j \) is perpendicular to... | [
"The points \\( A_1, A_2, A_3, A_4 \\) form an orthocentric system."
] | [
"A set \\( A_1, A_2, A_3, A_4 \\) of 4 points in the plane is said to be an Athenian set if there is a point \\( P \\) of the plane satisfying:\n(*) \\( P \\) does not lie on any of the lines \\( A_i A_j \\) for \\( 1 \\leq i < j \\leq 4 \\);\n(**) the line joining \\( P \\) to the midpoint of the line \\( A_i A_j ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f02564e44bdd14c69e955934d898f2464161f9e871b5f96070fa67cf0dc55d99 | A set \( M \) consists of \( n \) elements. Find the greatest \( k \) for which there is a collection of \( k \) subsets of \( M \) such that for any subsets \( A_1, \ldots, A_j \) from the collection, there is an element belonging to an odd number of them. | [
"\\( n \\)",
"n"
] | [
"A set \\( M \\) consists of \\( n \\) elements. Find the greatest \\( k \\) for which there is a collection of \\( k \\) subsets of \\( M \\) such that for any subsets \\( A_1, \\ldots, A_j \\) from the collection, there is an element belonging to an odd number of them."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-231ba1009626bf7c89088a774663a58e6e39d65d6c7b8f0901e3efef72db0a58 | A set \( M \) of points in 3-dimensional space is called *interesting* if for any plane, there exist at least 100 points in \( M \) outside this plane. For which minimal \( d \) does any interesting set contain an interesting subset with at most \( d \) points? | [
"\\( 300 \\)"
] | [
"A set \\( M \\) of points in 3-dimensional space is called *interesting* if for any plane, there exist at least 100 points in \\( M \\) outside this plane. For which minimal \\( d \\) does any interesting set contain an interesting subset with at most \\( d \\) points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-3d72356df153e6b35d1a674202772fab97f6eb02e5e12d855c0ef2f4e8ac3b64 | A set \( M \) of real numbers is called *special* if it satisfies the following properties: - For each \( x, y \in M \) with \( x \ne y \), the numbers \( x + y \) and \( xy \) are not zero and exactly one of them is rational. - For each \( x \in M \), \( x^2 \) is irrational. Find the maximum number of elements in a *... | [
"4"
] | [
"A set \\( M \\) of real numbers is called *special* if it satisfies the following properties:\n- For each \\( x, y \\in M \\) with \\( x \\ne y \\), the numbers \\( x + y \\) and \\( xy \\) are not zero and exactly one of them is rational.\n- For each \\( x \\in M \\), \\( x^2 \\) is irrational.\n\nFind the maximu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-7313494ded9c89cec0acd1871cf96587eebbba3a613057bd71408c54c1015fee | A set \( M \in \mathbb{N} \) is called gappy if for every \( m \in M \), its successor \( (m+1) \not\in M \). The empty set is defined as gappy. Find an algorithm to determine all gappy subsets of \( M = \{1, 2, 3, \ldots, n\} \) and determine how many such subsets exist for each \( n \). | [
"\\( a_n = F_{n+2} \\) where \\( F_n \\) is the Fibonacci sequence with \\( F_0 = 0 \\) and \\( F_1 = 1 \\)."
] | [
"A set \\( M \\in \\mathbb{N} \\) is called gappy if for every \\( m \\in M \\), its successor \\( (m+1) \\not\\in M \\). The empty set is defined as gappy. Find an algorithm to determine all gappy subsets of \\( M = \\{1, 2, 3, \\ldots, n\\} \\) and determine how many such subsets exist for each \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-10a750010b2137fde4a1c5414c4533ea54f7ca0a6596e992b591b2922f81e86a | A set \( S \) consisting of 2019 different positive integers has the property that the product of every 100 elements of \( S \) is a divisor of the product of the remaining 1919 elements. What is the maximum number of prime numbers that \( S \) can contain? | [
"1820"
] | [
"A set \\( S \\) consisting of 2019 different positive integers has the property that the product of every 100 elements of \\( S \\) is a divisor of the product of the remaining 1919 elements. What is the maximum number of prime numbers that \\( S \\) can contain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-e197cf0ce9154f112432412e59443d3c3c9ec66c4f1e5f4d398ae702af246f8a | A set \( S \) consists of \( k \) sequences of \( 0, 1, 2 \) of length \( n \). For any two sequences \( (a_i), (b_i) \in S \), a new sequence \( (c_i) \) can be constructed such that \( c_i = \left\lfloor \frac{a_i + b_i + 1}{2} \right\rfloor \) and included in \( S \). After performing finitely many such operations, ... | [
"\\( n + 1 \\)"
] | [
"A set \\( S \\) consists of \\( k \\) sequences of \\( 0, 1, 2 \\) of length \\( n \\). For any two sequences \\( (a_i), (b_i) \\in S \\), a new sequence \\( (c_i) \\) can be constructed such that \\( c_i = \\left\\lfloor \\frac{a_i + b_i + 1}{2} \\right\\rfloor \\) and included in \\( S \\). After performing fini... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 6,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-2e8f8d47cf787a560c3e608c8b873f7f0f15762eb9c75ac49c71459ed82fd843 | A set \( S \) contains 372 integers from 1 to 1200. For every element \( a \in S \), the numbers \( a + 4 \), \( a + 5 \), and \( a + 9 \) do not belong to \( S \). Calculate the sum of all possible values of the largest element in \( S \) if \( 600 \) is included in \( S \). | [
"1200"
] | [
"A set \\( S \\) contains 372 integers from 1 to 1200. For every element \\( a \\in S \\), the numbers \\( a + 4 \\), \\( a + 5 \\), and \\( a + 9 \\) do not belong to \\( S \\). Calculate the sum of all possible values of the largest element in \\( S \\) if \\( 600 \\) is included in \\( S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-a44ce38dfb237c0ceaf22efa82284ec6a6b229aba8330e96004b60f9bc057d93 | A set \( S \) contains 372 integers from the set \(\{1, 2, \ldots, 1200\}\). For every element \( a \in S \), the numbers \( a+4 \), \( a+5 \), and \( a+9 \) do not belong to \( S \). Determine the number of possible values for the largest element in \( S \) if \( 600 \in S \). | [
"85"
] | [
"A set \\( S \\) contains 372 integers from the set \\(\\{1, 2, \\ldots, 1200\\}\\). For every element \\( a \\in S \\), the numbers \\( a+4 \\), \\( a+5 \\), and \\( a+9 \\) do not belong to \\( S \\). Determine the number of possible values for the largest element in \\( S \\) if \\( 600 \\in S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2fbd5e3b6326d778af270730d1057265366779f170b14e78be43797829dc1c9 | A set \( S \) has \( n \) integers. The maximum element is 2002, and the minimum element is 1. Given that when any element is removed from \( S \), the arithmetic mean of the remaining elements is an integer, find the maximum value of \( n \). | [
"30"
] | [
"A set \\( S \\) has \\( n \\) integers. The maximum element is 2002, and the minimum element is 1. Given that when any element is removed from \\( S \\), the arithmetic mean of the remaining elements is an integer, find the maximum value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-db99766b35b3e4571e99a6ef47305a532c2f71fd199b26d34bd281ad8598709f | A set \( S \) is called "good" if for any number \( x \) in \( S \), \( x \) does not divide the sum of the other numbers in \( S \). Find the largest subset of the set \([1, 2, \ldots, 63]\) that is good. | [
"32"
] | [
"A set \\( S \\) is called \"good\" if for any number \\( x \\) in \\( S \\), \\( x \\) does not divide the sum of the other numbers in \\( S \\). Find the largest subset of the set \\([1, 2, \\ldots, 63]\\) that is good."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-fd1b40c4ff0713bb27718d790ec77382425a537c61dbb141f00f38c85cfada2a | A set \( S \) is called *neighbouring* if it has the following two properties: a) \( S \) has exactly four elements. b) For every element \( x \) of \( S \), at least one of the numbers \( x - 1 \) or \( x + 1 \) belongs to \( S \). Find the number of all *neighbouring* subsets of the set \( \{1, 2, \ldots, n\} \). | [
"\\(\\binom{n-2}{2}\\)"
] | [
"A set \\( S \\) is called *neighbouring* if it has the following two properties:\na) \\( S \\) has exactly four elements.\nb) For every element \\( x \\) of \\( S \\), at least one of the numbers \\( x - 1 \\) or \\( x + 1 \\) belongs to \\( S \\).\n\nFind the number of all *neighbouring* subsets of the set \\( \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 12
} | 59 | 0 | 0 |
omr-ws-39f9f43eeaa9bd0649f30af5540d5f30ab1aa30f39fc29f90d5c51ebb033eb38 | A set \( S \) is constructed as follows. Initially, \( S = \{0, 10\} \). Repeatedly, if \( x \) is an integer root of some polynomial \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) for some \( n \geq 1 \), all of whose coefficients \( a_i \) are elements of \( S \), then \( x \) is added to \( S \). When no mor... | [
"9"
] | [
"A set \\( S \\) is constructed as follows. Initially, \\( S = \\{0, 10\\} \\). Repeatedly, if \\( x \\) is an integer root of some polynomial \\( a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0 \\) for some \\( n \\geq 1 \\), all of whose coefficients \\( a_i \\) are elements of \\( S \\), then \\( x \\) is adde... | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 14
} | 14 | 0 | 0 |
omr-ws-63ee080c7bc75649420e7bc3fa0ab3bee74951a5d3e7dbd477c72cfe30eca269 | A set \( S \) is initially defined as \( S = [0, 10] \). Repeatedly, if \( x \) is an integer root of some polynomial whose coefficients are elements of \( S \), then \( x \) is added to \( S \). This process continues until no more elements can be added. How many elements can \( S \) have when no more elements can be ... | [
"21"
] | [
"A set \\( S \\) is initially defined as \\( S = [0, 10] \\). Repeatedly, if \\( x \\) is an integer root of some polynomial whose coefficients are elements of \\( S \\), then \\( x \\) is added to \\( S \\). This process continues until no more elements can be added. How many elements can \\( S \\) have when no mo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-cba370c41e301e41f50a9f833b67a23d1eb757a9a055b490ab0beb153a5baf1a | A set \( S \) of distinct positive integers has the following property: for every integer \( x \) in \( S \), the arithmetic mean of the set of values obtained by deleting \( x \) from \( S \) is an integer. Given that \( 1 \) belongs to \( S \) and that \( 2002 \) is the largest element of \( S \), what is the greates... | [
"30"
] | [
"A set \\( S \\) of distinct positive integers has the following property: for every integer \\( x \\) in \\( S \\), the arithmetic mean of the set of values obtained by deleting \\( x \\) from \\( S \\) is an integer. Given that \\( 1 \\) belongs to \\( S \\) and that \\( 2002 \\) is the largest element of \\( S \... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 6
} | 15 | 0 | 0 |
omr-ws-7a8d80185130a5ae0dd08af67f73cfdc90400fca7eeffb200e83c92da945a619 | A set \( S \) of natural numbers is called "good" if for each element \( x \in S \), \( x \) does not divide the sum of the remaining numbers in \( S \). Find the maximal possible number of elements of a "good" set which is a subset of the set \( A = \{1, 2, 3, \ldots, 63\} \). | [
"61"
] | [
"A set \\( S \\) of natural numbers is called \"good\" if for each element \\( x \\in S \\), \\( x \\) does not divide the sum of the remaining numbers in \\( S \\). Find the maximal possible number of elements of a \"good\" set which is a subset of the set \\( A = \\{1, 2, 3, \\ldots, 63\\} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-ecfca6455eae61646c15772a4396ca22aaa20b9ee2a65d7dbf8bca52c271c274 | A set \( S \) of points in the \( xy \)-plane is symmetric about the origin, both coordinate axes, and the line \( y = x \). If \( (2, 3) \) is in \( S \), what is the smallest number of points in \( S \)? | [
"8"
] | [
"A set \\( S \\) of points in the \\( xy \\)-plane is symmetric about the origin, both coordinate axes, and the line \\( y = x \\). If \\( (2, 3) \\) is in \\( S \\), what is the smallest number of points in \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a16e1997f35feee8be3ded4a2c191b994c42ab832e06d2f595ab43ebc2c69617 | A set \( S \) of positive integers \( \leq 2004 \) satisfies the condition that no number in \( S \) is double in value of another number in \( S \). What is the largest possible number of integers in \( S \)? | [
"1003"
] | [
"A set \\( S \\) of positive integers \\( \\leq 2004 \\) satisfies the condition that no number in \\( S \\) is double in value of another number in \\( S \\). What is the largest possible number of integers in \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-492e1a9d0ea5db7756285d926c5cf31a5b49ee521f78b96231a790d957b8ac32 | A set \( S \) of positive integers all smaller than 2019 is such that, for any two pairs (not necessarily disjoint) of elements of \( S \), the non-negative differences between the two elements of each pair are different from each other. What is the maximum number of elements that \( S \) can have? | [
"63"
] | [
"A set \\( S \\) of positive integers all smaller than 2019 is such that, for any two pairs (not necessarily disjoint) of elements of \\( S \\), the non-negative differences between the two elements of each pair are different from each other. What is the maximum number of elements that \\( S \\) can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-e8dbe2fdf3619e794bb93be4b1e7c777d71a0bf3034fb2dc9a033673569c447c | A set \( S \) of positive integers is called good if for any \( x, y \in S \) either \( x = y \) or \( |x - y| \ge 3 \). How many subsets of \( \{1, 2, 3, \ldots, 13\} \) are good? Include the empty set in your count. | [
"189"
] | [
"A set \\( S \\) of positive integers is called good if for any \\( x, y \\in S \\) either \\( x = y \\) or \\( |x - y| \\ge 3 \\). How many subsets of \\( \\{1, 2, 3, \\ldots, 13\\} \\) are good? Include the empty set in your count."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2d63f452631fd6628ccbe638d782dc71ea5d8548464e30943601af5ee0089321 | A set \( S \) which contains some positive integers is called "Magical" if and only if it satisfies the property: for all \( i, j \in S \), then \( \frac{i+j}{\gcd(i,j)} \in S \). Find all "Magical" sets. | [
"\\( \\{1, 2\\} \\)"
] | [
"A set \\( S \\) which contains some positive integers is called \"Magical\" if and only if it satisfies the property: for all \\( i, j \\in S \\), then \\( \\frac{i+j}{\\gcd(i,j)} \\in S \\). Find all \"Magical\" sets."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d54b56ae54c35d12ce062284c6e04e16d7432c21c0ea5d1f2007475b8dd85ae6 | A set \( S \) with a finite number \( n \) of points in the plane, where \( n \geq 4 \) and at most \( n-2 \) of them are collinear, has the property that for any four distinct points \( A, B, C, D \), the intersection point of lines \( AB \) and \( CD \) (if the lines are different and not parallel) also belongs to \(... | [
"5"
] | [
"A set \\( S \\) with a finite number \\( n \\) of points in the plane, where \\( n \\geq 4 \\) and at most \\( n-2 \\) of them are collinear, has the property that for any four distinct points \\( A, B, C, D \\), the intersection point of lines \\( AB \\) and \\( CD \\) (if the lines are different and not parallel... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-30f5de8808f9ec2e2414197adb4a84c33666b183a288b78b43edbd24d8c53505 | A set \( S \subset \mathbb{N} \) is called a square set if for each \( x, y \in S \), the expression \( xy + 1 \) is a perfect square. Is \( S \) finite? | [
"No"
] | [
"A set \\( S \\subset \\mathbb{N} \\) is called a square set if for each \\( x, y \\in S \\), the expression \\( xy + 1 \\) is a perfect square. Is \\( S \\) finite?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a286b73e56ad81ec692e373da67a88430a9382ef775f3a8d48b0ddbbdadbb327 | A set \( X \) consisting of \( n \) positive integers is called \(\textit{good}\) if for any two different subsets \( A \) and \( B \) of \( X \), the number \( s(A) - s(B) \) is not divisible by \( 2^n \). (Here, \( s(A) \) denotes the sum of the elements of \( A \)). Given \( n \), find the number of good sets of siz... | [
"\\( 2^{\\frac{n(n-1)}{2}} \\)"
] | [
"A set \\( X \\) consisting of \\( n \\) positive integers is called \\(\\textit{good}\\) if for any two different subsets \\( A \\) and \\( B \\) of \\( X \\), the number \\( s(A) - s(B) \\) is not divisible by \\( 2^n \\). (Here, \\( s(A) \\) denotes the sum of the elements of \\( A \\)). Given \\( n \\), find th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 7,
"tir": 19
} | 43 | 0 | 0 |
omr-ws-0a8f7726859e93c4d47972abaaa5f4885a62fef734abc8f382a06cb113dc3c64 | A set \( X \) of positive integers is called nice if for each pair \( a, b \in X \), exactly one of the numbers \( a + b \) and \( |a - b| \) belongs to \( X \). Determine the number of nice sets containing the number 2008. | [
"8"
] | [
"A set \\( X \\) of positive integers is called nice if for each pair \\( a, b \\in X \\), exactly one of the numbers \\( a + b \\) and \\( |a - b| \\) belongs to \\( X \\). Determine the number of nice sets containing the number 2008."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-bc25d2c1c1d6534d567478350f4bafa4792a5a351e7cb4d279214a62e3f663fc | A set \( X \) of positive integers is said to be iberic if \( X \) is a subset of \(\{2, 3, \dots, 2018\}\), and whenever \( m, n \) are both in \( X \), \(\gcd(m, n)\) is also in \( X \). An iberic set is said to be olympic if it is not properly contained in any other iberic set. Find all olympic iberic sets that cont... | [
"\\(\\{ 3, 6, 9, \\ldots, 2016 \\}\\) and \\(\\{ 11, 22, 33, \\ldots, 2013 \\}\\)"
] | [
"A set \\( X \\) of positive integers is said to be iberic if \\( X \\) is a subset of \\(\\{2, 3, \\dots, 2018\\}\\), and whenever \\( m, n \\) are both in \\( X \\), \\(\\gcd(m, n)\\) is also in \\( X \\). An iberic set is said to be olympic if it is not properly contained in any other iberic set. Find all olympi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 15
} | 24 | 0 | 0 |
omr-ws-9a1e79748c6d62fb5a99fe6a10d1b3b525db110a1c85ea1878f57582848bcbfc | A set \((a_1, a_2, a_3, \ldots, a_n)\) with \(n\) numbers is called "good" if the following conditions are satisfied: 1. \(a_1 + a_2 + a_3 + \cdots + a_n = 2n\) 2. For any subset of \(\{a_1, a_2, \ldots, a_n\}\), the sum of the elements in the subset is not equal to \(n\). Find all "good" sets with \(n\) numbers. For e... | [
"\\((2, 2, 2, \\ldots, 2)\\) (with an odd number of twos) or \\((n+1, 1, 1, \\ldots, 1)\\) (with \\( n-1 \\) ones)"
] | [
"A set \\((a_1, a_2, a_3, \\ldots, a_n)\\) with \\(n\\) numbers is called \"good\" if the following conditions are satisfied:\n1. \\(a_1 + a_2 + a_3 + \\cdots + a_n = 2n\\)\n2. For any subset of \\(\\{a_1, a_2, \\ldots, a_n\\}\\), the sum of the elements in the subset is not equal to \\(n\\).\n\nFind all \"good\" s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 8,
"tir": 8
} | 35 | 0 | 0 |
omr-ws-60ca8ea67f89f86d48c21180c7306cdf4ba6704d2f3237efb27a8ebe7f56f19e | A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set? | [
"\\(-\\frac{1}{4}\\)"
] | [
"A set \\(A\\) of real numbers is framed when it is bounded and, for all \\(a, b \\in A\\), not necessarily distinct, \\((a-b)^{2} \\in A\\). What is the smallest real number that belongs to some framed set?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 6
} | 52 | 0 | 0 |
omr-ws-a609880fd2e05d1bba31eeb68d7c0140f2c18c94d4b3454aae474a4aefb648fb | A set \(P\) consists of 2005 distinct prime numbers. Let \(A\) be the set of all possible products of 1002 elements of \(P\), and \(B\) be the set of all products of 1003 elements of \(P\). Find a one-to-one correspondence \(f\) from \(A\) to \(B\) with the property that \(a\) divides \(f(a)\) for all \(a \in A\). | [
"\\( f(a) = a \\cdot p_j \\) where \\( p_j \\) is the prime in \\( P \\) that is not in \\( a \\)."
] | [
"A set \\(P\\) consists of 2005 distinct prime numbers. Let \\(A\\) be the set of all possible products of 1002 elements of \\(P\\), and \\(B\\) be the set of all products of 1003 elements of \\(P\\). Find a one-to-one correspondence \\(f\\) from \\(A\\) to \\(B\\) with the property that \\(a\\) divides \\(f(a)\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-0ea9573eaa51b181b26d058a846a4008efebf9e2c83731cca9db15d69d36438c | A set \(\mathcal{M} \subset M_3(\mathbb{Z}_2)\) is such that \(\det(A + B) \ne 0\) for every \(A, B \in \mathcal{M}\), \(A \ne B\). Find the maximal possible number of matrices in the set \(\mathcal{M}\). | [
"8"
] | [
"A set \\(\\mathcal{M} \\subset M_3(\\mathbb{Z}_2)\\) is such that \\(\\det(A + B) \\ne 0\\) for every \\(A, B \\in \\mathcal{M}\\), \\(A \\ne B\\). Find the maximal possible number of matrices in the set \\(\\mathcal{M}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 7
} | 31 | 0 | 0 |
omr-ws-9ad80ebef1b0ef1e3274153682380f316adb4e08d2abb7ee0fcb01815ab21473 | A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$. Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal total weights. Find all possible weights for the objects. | [
"w_1 = w_2 = \\cdots = w_{21}"
] | [
"A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$. Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal total weights. Find all possible weights for the objects."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-d61eb8e1f01a07d75f984a0c556517427f92bab4b7bb319b1273b226136a7277 | A set containing 7 elements is given. All its subsets containing 3 elements are colored with \( n \) colors such that every two disjoint subsets have different colors. Find the least value of \( n \) for which such a coloring is possible. | [
"3"
] | [
"A set containing 7 elements is given. All its subsets containing 3 elements are colored with \\( n \\) colors such that every two disjoint subsets have different colors. Find the least value of \\( n \\) for which such a coloring is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57365a71a245fa35b293bcc5b9ccba4de0ca8c328463c44bc83956418a5248ff | A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different sums that result is: 637, 669, 794, 915, 919, 951, 1040, 1072, 1197. What is the largest of the five integers in the set? | [
"659"
] | [
"A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different sums that result is: 637, 669, 794, 915, 919, 951, 1040, 1072, 1197. What is the largest of the five integers in the set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-7f3707eccff2702d8723241a95868b950966fcbd0b658588c927d0696bf8e819 | A set contains five integers. When distinct elements of this set are added together, two at a time, the resulting sums are: 637, 669, 794, 915, 919, 951, 1040, 1072, 1197. What is the largest of the five integers in the set? | [
"659"
] | [
"A set contains five integers. When distinct elements of this set are added together, two at a time, the resulting sums are: 637, 669, 794, 915, 919, 951, 1040, 1072, 1197. What is the largest of the five integers in the set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-547a76a79e647ffad598b76ae83e323bcb1b2345c6392e2422fb86823dd51dc9 | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$. | [
"791"
] | [
"A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$."
] | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 24
} | 79 | 0 | 0 |
omr-ws-9dce6d07362c13f0cc8d5ca8ffc64e844085c5d8ae281b136d388ffe210e1f87 | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and $y$. Find the greatest possible value of $x + y$. | [
"761"
] | [
"A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and $y$. Find the greatest possible value of $x + y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-85c05250d6e6a15443b6c99bc24209df887348f52515a42427737ebdddbee9aa | A set initially contains 2 distinct points. Every second, for each pair of points \(P, Q\) in the set, the reflections of \(P\) across \(Q\) and \(Q\) across \(P\) are added to the set. After 12 seconds, the set contains \(N\) more distinct points than it initially had. What is the largest prime factor of \(N\)? | [
"73"
] | [
"A set initially contains 2 distinct points. Every second, for each pair of points \\(P, Q\\) in the set, the reflections of \\(P\\) across \\(Q\\) and \\(Q\\) across \\(P\\) are added to the set. After 12 seconds, the set contains \\(N\\) more distinct points than it initially had. What is the largest prime factor... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 19
} | 42 | 0 | 0 |
omr-ws-f6b742b6ba0d40725fe38112de7460922387f04b2b56f1ab5990fc5c92b362c9 | A set is called a [i]square set[/i] if it contains exactly four positive integers \(a, b, x, y\) such that \[a^2 + b^2 = x^2 + y^2\] and \(x + y \neq a + b\). Determine the number of distinct square-sets with no element greater than 2007, and find the remainder when this number is divided by 1000. | [
"0"
] | [
"A set is called a [i]square set[/i] if it contains exactly four positive integers \\(a, b, x, y\\) such that\n\\[a^2 + b^2 = x^2 + y^2\\]\nand \\(x + y \\neq a + b\\). Determine the number of distinct square-sets with no element greater than 2007, and find the remainder when this number is divided by 1000."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-43170464cd34c918456464af6189e00501f33272d396e5991b54f5667eb60144 | A set of $ n(n+1)/2$ distinct values is randomly arranged in a triangle with $ n$ rows, where the first row contains one value, the second row two values, and so on. Let $ M_k$ denote the largest value in the $ k$th row. Find the probability that $ M_1 < M_2 < \dots < M_n$. | [
"\\(\\frac{2^n}{(n+1)!}\\)"
] | [
"A set of $ n(n+1)/2$ distinct values is randomly arranged in a triangle with $ n$ rows, where the first row contains one value, the second row two values, and so on. Let $ M_k$ denote the largest value in the $ k$th row. Find the probability that $ M_1 < M_2 < \\dots < M_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 14
} | 25 | 0 | 0 |
omr-ws-7b7507d06c814412897142ae993042f669c6d72575c68f1ce32d76d91a29a376 | A set of $16384$ numbers is given, where each number is chosen independently from the set $\{1, \frac{1}{2}, \frac{1}{2^2}, \ldots, \frac{1}{2^{13}}\}$. The probability of selecting $1$ is $\frac{1}{2^{13}}$, and for each natural number $k$ such that $1 \leq k \leq 13$, the probability of selecting $\frac{1}{2^k}$ is $... | [
"0.041"
] | [
"A set of $16384$ numbers is given, where each number is chosen independently from the set $\\{1, \\frac{1}{2}, \\frac{1}{2^2}, \\ldots, \\frac{1}{2^{13}}\\}$. The probability of selecting $1$ is $\\frac{1}{2^{13}}$, and for each natural number $k$ such that $1 \\leq k \\leq 13$, the probability of selecting $\\fra... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-2b84b28652a25c92e877c387d40b296d6cbb45334b0ba074561eb5cb86975b5a | A set of $1985$ points is distributed around the circumference of a circle, each marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. If the number of points mar... | [
"2"
] | [
"A set of $1985$ points is distributed around the circumference of a circle, each marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. If the number of point... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cc581b3f353f998e67f4abf141042039d62612095953c2ce8ec25d26cd0e97a | A set of $\frac{n(n+1)}{2}$ distinct numbers is arranged at random in a triangular array. If $M_k$ denotes the largest number in the $k$-th row from the top, find the probability that $M_1 < M_2 < \cdots < M_n$. | [
"\\(\\frac{2^n}{(n+1)!}\\)"
] | [
"A set of $\\frac{n(n+1)}{2}$ distinct numbers is arranged at random in a triangular array. If $M_k$ denotes the largest number in the $k$-th row from the top, find the probability that $M_1 < M_2 < \\cdots < M_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-7e87401e1760538e424b04fe9bd30e7342ab39f6add5b7c10da5ca76e558b4c7 | A set of $n \ge 5$ points on the plane, no three of which are collinear, is given. Hillary and Donald play a game where they alternately draw segments between pairs of points not yet joined. If after a player’s turn, each of the $n$ points is an endpoint of at least one drawn segment, then that player wins. If Hillary ... | [
"\\( n \\) is odd."
] | [
"A set of $n \\ge 5$ points on the plane, no three of which are collinear, is given. Hillary and Donald play a game where they alternately draw segments between pairs of points not yet joined. If after a player’s turn, each of the $n$ points is an endpoint of at least one drawn segment, then that player wins. If Hi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-981efeb044f4c1b83195ba6891a51e9c8c38ae82fc706e565a327f5b55aee1e6 | A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations: - change the state of lamp $1$, - change the state of lamp $2$, as long as lamp $1$ is on, - change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, \ldots, ... | [
"1023"
] | [
"A set of $n$ lights, numbered $1$ to $n$, are initially off. At every moment, it is possible to perform one of the following operations:\n- change the state of lamp $1$,\n- change the state of lamp $2$, as long as lamp $1$ is on,\n- change the state of lamp $k > 2$, as long as lamp $k - 1$ is on and all lamps $1, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f10df8365ab2f87b636d2e67b52f9cf730bb66504646b5927f096ee5d4a6ffa | A set of $n$ numbers $x_1, x_2, x_3, \ldots, x_n$ (where $n > 1$) has the property that the $k$-th number (that is, $x_k$) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript of $x_k$), and this is true for each $k = 1, 2, 3, \ldots, n$. Solve for these $n$ numbers. | [
"\\frac{n(n + 1)}{2(n - 1)} - k"
] | [
"A set of $n$ numbers $x_1, x_2, x_3, \\ldots, x_n$ (where $n > 1$) has the property that the $k$-th number (that is, $x_k$) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript of $x_k$), and this is true for each $k = 1, 2, 3, \\ldots, n$. Solve for these $n$ numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ccdd1d0530637f7283aa93b4766c9d6c48ed4adb567b4516c335648c3721697 | A set of $n$ people participate in an online video basketball tournament. Each person may be a member of $5$-player teams but no two teams may have exactly the same $5$ members. The average number of complete teams among all subsets of $9$ people is equal to the reciprocal of the average number of complete teams among ... | [
"595"
] | [
"A set of $n$ people participate in an online video basketball tournament. Each person may be a member of $5$-player teams but no two teams may have exactly the same $5$ members. The average number of complete teams among all subsets of $9$ people is equal to the reciprocal of the average number of complete teams a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-c60ee3cc865707c543597f6ac8957cb60ad05132b617b9b9d3f75bcad5bd6be3 | A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are amo... | [
"359"
] | [
"A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members ar... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
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