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-3c74fd256701d270320e2287dc47fd6b856aee3f90bc65e8db4c8bc9570272aa | A sequence of \( n \) positive integers is called "full" if for each \( k > 1 \), \( k \) only occurs if \( k-1 \) occurs before the last occurrence of \( k \). How many full sequences are there for each \( n \)? | [
"\\( n! \\)"
] | [
"A sequence of \\( n \\) positive integers is called \"full\" if for each \\( k > 1 \\), \\( k \\) only occurs if \\( k-1 \\) occurs before the last occurrence of \\( k \\). How many full sequences are there for each \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect",
"tir"
] | {
"genselect": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-b24735966dcf1aa9a83db81d5fb21a7ff0edf5af31eb537ab1cd8033a74e10c7 | A sequence of bit-strings is constructed as follows: Let $s_1=001$ and $s_n$ is formed by appending the last half of $s_{n-1}$ to the end of $s_{n-1}$ (if $s_{n-1}$ has an odd length, divide the length by $2$ and round up). The first few elements of the sequence are: $s_1=001, s_2=00101, s_3=00101101, s_4=001011011101$... | [
"15"
] | [
"A sequence of bit-strings is constructed as follows: Let $s_1=001$ and $s_n$ is formed by appending the last half of $s_{n-1}$ to the end of $s_{n-1}$ (if $s_{n-1}$ has an odd length, divide the length by $2$ and round up). The first few elements of the sequence are: $s_1=001, s_2=00101, s_3=00101101, s_4=00101101... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf473a280c304eec49038b3bd7daf6fc187f035b4061f0825007a6ee5f1ef8a4 | A sequence of cards is initially ordered as A, B, C, D, E, F, G, H. After two rearrangements using the same algorithm, the order becomes F, A, E, G, D, H, C, B. Determine the order of the cards after the first rearrangement. | [
"D, H, C, B, F, A, E, G"
] | [
"A sequence of cards is initially ordered as A, B, C, D, E, F, G, H. After two rearrangements using the same algorithm, the order becomes F, A, E, G, D, H, C, B. Determine the order of the cards after the first rearrangement."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-016f706d376dedf98b4759d20cb2e41bf1d1aaad9e02d2109190b6fa6e8838a2 | A sequence of circles is constructed in the plane as follows: The first circle has its center at $(0,0)$ and radius $1$. For all $n \ge 2$, the $n$-th circle has a radius equal to one-third of the radius of the $(n-1)$-th circle and passes through its center. Determine the locus of the point that belongs to all these c... | [
"\\((0,0)\\)"
] | [
"A sequence of circles is constructed in the plane as follows: The first circle has its center at $(0,0)$ and radius $1$. For all $n \\ge 2$, the $n$-th circle has a radius equal to one-third of the radius of the $(n-1)$-th circle and passes through its center. Determine the locus of the point that belongs to all t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fea92086664305c3f4f05f6c14d26f38bf2a6a422de7f5e473f3f2a9cf2a1a10 | A sequence of complex numbers \( z_0, z_1, z_2, \ldots \) is defined by the rule \[ z_{n+1} = \frac{i z_n}{\overline{z_n}} \] where \( \overline{z_n} \) is the complex conjugate of \( z_n \) and \( i^2 = -1 \). Suppose that \( |z_0| = 1 \) and \( z_{2005} = 1 \). How many possible values are there for \( z_0 \)? | [
"\\( 2^{2005} \\)"
] | [
"A sequence of complex numbers \\( z_0, z_1, z_2, \\ldots \\) is defined by the rule\n\\[ z_{n+1} = \\frac{i z_n}{\\overline{z_n}} \\]\nwhere \\( \\overline{z_n} \\) is the complex conjugate of \\( z_n \\) and \\( i^2 = -1 \\). Suppose that \\( |z_0| = 1 \\) and \\( z_{2005} = 1 \\). How many possible values are th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4be87804cd2d6f3482ff10c7474a5d4023c6b749ade8401a9df8da992de08dda | A sequence of complex numbers \( z_{0}, z_{1}, z_{2}, \ldots \) is defined by the rule \[ z_{n+1} = \frac{iz_{n}}{\overline{z_{n}}}, \] where \( \overline{z_{n}} \) is the complex conjugate of \( z_{n} \) and \( i^{2} = -1 \). Suppose that \( |z_{0}| = 1 \) and \( z_{2005} = 1 \). How many possible values are there for... | [
"\\( 2^{2005} \\)"
] | [
"A sequence of complex numbers \\( z_{0}, z_{1}, z_{2}, \\ldots \\) is defined by the rule\n\n\\[ z_{n+1} = \\frac{iz_{n}}{\\overline{z_{n}}}, \\]\n\nwhere \\( \\overline{z_{n}} \\) is the complex conjugate of \\( z_{n} \\) and \\( i^{2} = -1 \\). Suppose that \\( |z_{0}| = 1 \\) and \\( z_{2005} = 1 \\). How many ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-15d2bd2610f3c65e02a07125b2f2dffc6276e2283730b1506695cdc861805352 | A sequence of complex numbers, $\{z_n\},$ is defined as: \[z_1=2+7i,~z_{n+1}=\left(i-\frac{2}{\sqrt{4n+1}}\right) z_n.\] Define two sequences of real numbers, $\{a_n\},~\{b_n\},$ as $z_n=a_n+b_n i.$ Find the value of \[\lim_{n\to\infty} \frac{1}{n^2}\left(\sum_{k=1}^{n}{a_k}^2+\sum_{k=1}^{n}{b_k}^2\right).\] | [
"\\(\\frac{106}{5}\\)"
] | [
"A sequence of complex numbers, $\\{z_n\\},$ is defined as:\n\\[z_1=2+7i,~z_{n+1}=\\left(i-\\frac{2}{\\sqrt{4n+1}}\\right) z_n.\\]\nDefine two sequences of real numbers, $\\{a_n\\},~\\{b_n\\},$ as $z_n=a_n+b_n i.$ Find the value of\n\\[\\lim_{n\\to\\infty} \\frac{1}{n^2}\\left(\\sum_{k=1}^{n}{a_k}^2+\\sum_{k=1}^{n}... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-e4a2e3e84acdeb0c845a8f438ac37ffe1ef16e311eed7e8ea5c147726d6bebac | A sequence of convex polygons $(P_n)$ for $n \geq 0$ is defined inductively as follows: $P_0$ is an equilateral triangle with side length $1$. Once $P_n$ is determined, its sides are trisected, and the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Express $\lim_{n\to\infty}[P_n]$ in th... | [
"$\\frac{\\sqrt{27}}{8}$"
] | [
"A sequence of convex polygons $(P_n)$ for $n \\geq 0$ is defined inductively as follows: $P_0$ is an equilateral triangle with side length $1$. Once $P_n$ is determined, its sides are trisected, and the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$. Express $\\lim_{n\\to\\infty}[P_... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-824551d410c46cac9419f7a7960c05203170891b5d135cefbf07071ed44ebaa5 | A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ such that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has a diameter of $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$. | [
"3995"
] | [
"A sequence of distinct circles $\\omega_1, \\omega_2, \\cdots$ is inscribed in the parabola $y=x^2$ such that $\\omega_n$ and $\\omega_{n+1}$ are tangent for all $n$. If $\\omega_1$ has a diameter of $1$ and touches the parabola at $(0,0)$, find the diameter of $\\omega_{1998}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 3
} | 48 | 0 | 0 |
omr-ws-6ae664793edc22386946d249f10ea0f68a0262fd835c9baa996dbde84a06fc38 | A sequence of finite sets of positive integers is defined as follows: \( S_0 = \{m\} \), where \( m > 1 \). Then given \( S_n \), you derive \( S_{n+1} \) by taking \( k^2 \) and \( k+1 \) for each element \( k \) of \( S_n \). Let \( f(n) \) be the number of distinct elements in \( S_n \). Calculate \( f(5) \). | [
"32"
] | [
"A sequence of finite sets of positive integers is defined as follows: \\( S_0 = \\{m\\} \\), where \\( m > 1 \\). Then given \\( S_n \\), you derive \\( S_{n+1} \\) by taking \\( k^2 \\) and \\( k+1 \\) for each element \\( k \\) of \\( S_n \\). Let \\( f(n) \\) be the number of distinct elements in \\( S_n \\). C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0adcab06b4eddbdec18f937e3134699c7d6310664c3d874c514172b32525ce64 | A sequence of fractions begins with \( x_1 = \frac{1}{2} \) and \( x_2 = \frac{1}{3} \). Thereafter, each succeeding fraction is the product of the last two fractions divided by the sum of the last two fractions. Find \( x_{20} \). | [
"\\(\\frac{1}{17711}\\)"
] | [
"A sequence of fractions begins with \\( x_1 = \\frac{1}{2} \\) and \\( x_2 = \\frac{1}{3} \\). Thereafter, each succeeding fraction is the product of the last two fractions divided by the sum of the last two fractions. Find \\( x_{20} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1dbc216c3a339b1e0d84dba8a63bb81d5d88da2b8aec8bd7cd586e68fd30d133 | A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows: \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*} (Recall that $\sqrt {\makebox[5mm]{}}$ is understood to represent the positive square root.) For ea... | [
"x = 4"
] | [
"A sequence of functions $\\, \\{f_n(x) \\} \\,$ is defined recursively as follows: \\begin{align*} f_1(x) &= \\sqrt {x^2 + 48}, \\quad \\text{and} \\\\ f_{n + 1}(x) &= \\sqrt {x^2 + 6f_n(x)} \\quad \\text{for } n \\geq 1. \\end{align*} (Recall that $\\sqrt {\\makebox[5mm]{}}$ is understood to represent the posi... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc8fcdb763e06a1e54e54a53ac81a9fbb18ac71ae8bdf7f69dc397a3a6c2fc1b | A sequence of functions $\{f_n(x)\}$ is defined recursively as follows: \[ f_1(x) = \sqrt{x^2 + 48}, \] \[ f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \] For each positive integer $n$, find all real solutions of the equation $f_n(x) = 2x$. | [
"\\( x = 4 \\)"
] | [
"A sequence of functions $\\{f_n(x)\\}$ is defined recursively as follows:\n\\[ f_1(x) = \\sqrt{x^2 + 48}, \\]\n\\[ f_{n+1}(x) = \\sqrt{x^2 + 6f_n(x)} \\quad \\text{for } n \\geq 1. \\]\nFor each positive integer $n$, find all real solutions of the equation $f_n(x) = 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f6b3dd031eb2b8cdc28259836888cc0080ba9cf7fe289dada6b5c70f94bcf202 | A sequence of integer numbers \( f_i \) is defined as follows: \( f_0 = 0 \), \( f_1 = 1 \), \( f_n = f_{n-1} + f_{n-2} \) for \( n \geq 2 \). Find all polynomials \( W \) with integer coefficients satisfying the condition that for every \( n = 0, 1, 2, \ldots \), there exists an integer number \( k \) such that \( W(k... | [
"\\pm x + c"
] | [
"A sequence of integer numbers \\( f_i \\) is defined as follows: \\( f_0 = 0 \\), \\( f_1 = 1 \\), \\( f_n = f_{n-1} + f_{n-2} \\) for \\( n \\geq 2 \\). Find all polynomials \\( W \\) with integer coefficients satisfying the condition that for every \\( n = 0, 1, 2, \\ldots \\), there exists an integer number \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e48a897868a08043bf7873bca9a647c4ae8c7755082eea1bdab13c25af8b8ecc | A sequence of integers $\{x_n\}_{n=1}^{\infty}$ is defined as follows: $x_1=0$ and $|x_{n+1}|=|x_n+k|$ for all positive integers $n$ and a given positive integer $k$. Let $A_s$ be the average of the first $s$ terms of the sequence. Find the smallest integer $M$ such that $A_s > \frac{-k}{2} + \frac{1}{M}$ for any posit... | [
""
] | [
"A sequence of integers $\\{x_n\\}_{n=1}^{\\infty}$ is defined as follows: $x_1=0$ and $|x_{n+1}|=|x_n+k|$ for all positive integers $n$ and a given positive integer $k$. Let $A_s$ be the average of the first $s$ terms of the sequence. Find the smallest integer $M$ such that $A_s > \\frac{-k}{2} + \\frac{1}{M}$ for... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-99ffb3adf8d83c5690c6dd7193e8845d2d0125540fdd52d998e694f4b3e06d62 | A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | [
"986"
] | [
"A sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6511f0a70387979cedd54d3d37fd85e40ae427ba4f78220be0bc46eafe79f795 | A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3.$ What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985,$ and the sum of the first $1985$ terms is $1492$? | [
"986"
] | [
"A sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3.$ What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985,$ and the sum of the first $1985$ terms is $1492$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3672b559c182c91dc7f78dd203a1d4197be29a76c35b927550ebd036843918f | A sequence of integers \( a_1, a_2, a_3, \ldots \) is defined such that \( a_1 = 2 \), \( a_2 = 3 \), and for all \( n \ge 2 \), \( a_{n+1} = 2a_{n-1} \) or \( 3a_n - 2a_{n-1} \). Let \( S \) be the sum of all distinct integers between 1600 and 2000 that can be an element of the sequence. Calculate \( S \). | [
"0"
] | [
"A sequence of integers \\( a_1, a_2, a_3, \\ldots \\) is defined such that \\( a_1 = 2 \\), \\( a_2 = 3 \\), and for all \\( n \\ge 2 \\), \\( a_{n+1} = 2a_{n-1} \\) or \\( 3a_n - 2a_{n-1} \\). Let \\( S \\) be the sum of all distinct integers between 1600 and 2000 that can be an element of the sequence. Calculate... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0350e1afd0bc6c4fe416e47b27d7886737a4c8d3cfc93356e9134f09edd73d6 | A sequence of integers \( a_1, a_2, a_3, a_4, a_5, \ldots, a_n \) satisfies \( a_{n+2} = a_{n+1} - a_n \) for all \( n \in \mathbb{N} \). Suppose the sum of the first 999 terms is 1003 and the sum of the first 1003 terms is -999. What is the sum of the first 2002 terms? | [
"1007"
] | [
"A sequence of integers \\( a_1, a_2, a_3, a_4, a_5, \\ldots, a_n \\) satisfies \\( a_{n+2} = a_{n+1} - a_n \\) for all \\( n \\in \\mathbb{N} \\). Suppose the sum of the first 999 terms is 1003 and the sum of the first 1003 terms is -999. What is the sum of the first 2002 terms?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c76dc51c907defccea306cca4dc02e562946e458b071834a5eea1e76af9d0982 | A sequence of integers \( a_{1}, a_{2}, a_{3}, \ldots \) is defined as follows: \( a_{1} = 1 \) and for \( n \geq 1 \), \( a_{n+1} \) is the smallest integer greater than \( a_{n} \) such that \( a_{i} + a_{j} \neq 3a_{k} \) for any \( i, j, \) and \( k \) in \( \{1, 2, 3, \ldots, n+1\} \), not necessarily distinct. De... | [
"4494"
] | [
"A sequence of integers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) is defined as follows: \\( a_{1} = 1 \\) and for \\( n \\geq 1 \\), \\( a_{n+1} \\) is the smallest integer greater than \\( a_{n} \\) such that \\( a_{i} + a_{j} \\neq 3a_{k} \\) for any \\( i, j, \\) and \\( k \\) in \\( \\{1, 2, 3, \\ldots, n+1\\} \\),... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-7ad5fc0b1b4542227adc28e42e32e9e42f81968ecba00de3cbcd2de58bd11c61 | A sequence of integers \( b_1, b_2, b_3, \ldots \) is "best" because for each \( m \) and \( n \), the following statements hold true: - If \( m \) divides \( n \) and \( m < n \), then \( b_m \) divides \( b_n \) with \( b_m < b_n \). In this best sequence, compute the least possible value of \( b_{5488} \). | [
"128"
] | [
"A sequence of integers \\( b_1, b_2, b_3, \\ldots \\) is \"best\" because for each \\( m \\) and \\( n \\), the following statements hold true:\n\n- If \\( m \\) divides \\( n \\) and \\( m < n \\), then \\( b_m \\) divides \\( b_n \\) with \\( b_m < b_n \\).\n\nIn this best sequence, compute the least possible va... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-a29eca5c269e138795881371cebff30c903a967f3c776463621cadc3d84e44a9 | A sequence of integers \( u_1, u_2, u_3, \ldots \) satisfies: \[ u_{n+1} - u_n = \begin{cases*} 1, &\text{if } n \text{ is odd} \\ 2, &\text{if } n \text{ is even}. \end{cases*} \] If \( u_1 + u_2 + \cdots + u_{20} = 360 \), then \( u_1 = \) ... | [
"1"
] | [
"A sequence of integers \\( u_1, u_2, u_3, \\ldots \\) satisfies:\n\\[\nu_{n+1} - u_n = \\begin{cases*} 1, &\\text{if } n \\text{ is odd} \\\\\n2, &\\text{if } n \\text{ is even}.\n\\end{cases*}\n\\]\nIf \\( u_1 + u_2 + \\cdots + u_{20} = 360 \\), then \\( u_1 = \\) ..."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-503f064a306732b678dc73f43b0e3d441bb04aecd62af9097072485d9aee490a | A sequence of integers \(a_0, a_1, \dots, a_n, \dots\) is defined by the following rules: \(a_0 = 0\), \(a_1 = 1\), \(a_{n+1} > a_n\) for each \(n \in \mathbb{N}\), and \(a_{n+1}\) is the minimum number such that no three numbers among \(a_0, a_1, \dots, a_{n+1}\) form an arithmetic progression. Calculate the value of ... | [
"59049"
] | [
"A sequence of integers \\(a_0, a_1, \\dots, a_n, \\dots\\) is defined by the following rules: \\(a_0 = 0\\), \\(a_1 = 1\\), \\(a_{n+1} > a_n\\) for each \\(n \\in \\mathbb{N}\\), and \\(a_{n+1}\\) is the minimum number such that no three numbers among \\(a_0, a_1, \\dots, a_{n+1}\\) form an arithmetic progression.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ea49f7d5c2b1e4241ec2e055669ecf47d3e122b51630f45afb93a686d3cb410 | A sequence of integers \(a_1, a_2, \cdots, a_n\) satisfies the relation: \[ 4a_{n+1}^2 - 4a_n a_{n+1} + a_n^2 - 1 = 0 \] for all positive integers \(n\). What are the possible values of \(a_1\)? | [
"All odd integers."
] | [
"A sequence of integers \\(a_1, a_2, \\cdots, a_n\\) satisfies the relation:\n\\[ 4a_{n+1}^2 - 4a_n a_{n+1} + a_n^2 - 1 = 0 \\]\nfor all positive integers \\(n\\). What are the possible values of \\(a_1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 25
} | 36 | 0 | 0 |
omr-ws-fdad12d199ac80018ee4f1ac7c9b4df5c12fad3ccdeda792c8cc8c2febb5b0c2 | A sequence of integers \(a_1, a_2, \ldots, a_n\) is called "nice" if for all \(i = 1, 2, \ldots, n-1\), the condition \(a_{i+1} + a_i = (a_{i+1} - a_i)^2\) holds. Find the number of nice sequences with \(n\) elements in terms of \(n\). | [
"\\(\\binom{n-1}{\\left\\lfloor \\frac{n-1}{2} \\right\\rfloor}\\)"
] | [
"A sequence of integers \\(a_1, a_2, \\ldots, a_n\\) is called \"nice\" if for all \\(i = 1, 2, \\ldots, n-1\\), the condition \\(a_{i+1} + a_i = (a_{i+1} - a_i)^2\\) holds. Find the number of nice sequences with \\(n\\) elements in terms of \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb14202cf4703316b7cfe323465dd71555b08545d80d9072a8ff7d00c644799d | A sequence of integers \(a_1, a_2, \ldots, a_{2010}\) satisfies the conditions: - \(a_2 = 1001\) - \(a_{n+2} = a_{n+1} - a_n\) for \(1 \leq n \leq 2008\) What is the value of \(a_1 + a_2 + a_3 + \cdots + a_{2010}\)? | [
"0"
] | [
"A sequence of integers \\(a_1, a_2, \\ldots, a_{2010}\\) satisfies the conditions:\n- \\(a_2 = 1001\\)\n- \\(a_{n+2} = a_{n+1} - a_n\\) for \\(1 \\leq n \\leq 2008\\)\n\nWhat is the value of \\(a_1 + a_2 + a_3 + \\cdots + a_{2010}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6a29d54e380aa35d3cf3b3fab6e813301a7b81dbd100bcdcb1388a7933b8ac7 | A sequence of integers \(a_1, a_2, \ldots\) is defined by \(a_1 = 1\), \(a_2 = 2\), and for \(n \geq 1\), \[ a_{n+2} = \begin{cases} 5a_{n+1} - 3a_{n}, & \text{if } a_n \cdot a_{n+1} \text{ is even}, \\ a_{n+1} - a_{n}, & \text{if } a_n \cdot a_{n+1} \text{ is odd}. \end{cases} \] Let \(P\) be the number of positive te... | [
"7"
] | [
"A sequence of integers \\(a_1, a_2, \\ldots\\) is defined by \\(a_1 = 1\\), \\(a_2 = 2\\), and for \\(n \\geq 1\\),\n\\[ a_{n+2} = \\begin{cases} \n5a_{n+1} - 3a_{n}, & \\text{if } a_n \\cdot a_{n+1} \\text{ is even}, \\\\\na_{n+1} - a_{n}, & \\text{if } a_n \\cdot a_{n+1} \\text{ is odd}.\n\\end{cases} \\]\nLet \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-09c9bd465d16c0cf15e220d0d235f2a1416324ebe86152ffea27b7b65a6cad01 | A sequence of integers \(a_1, a_2, \ldots\) is defined by \(a_1 = 1\), \(a_2 = 2\), and for \(n \geq 1\), \[ a_{n+2} = \begin{cases} 5a_{n+1} - 3a_{n}, & \text{if } a_n \cdot a_{n+1} \text{ is even}, \\ a_{n+1} - a_{n}, & \text{if } a_n \cdot a_{n+1} \text{ is odd}. \end{cases} \] Let \(S\) be the sum of all distinct p... | [
"7"
] | [
"A sequence of integers \\(a_1, a_2, \\ldots\\) is defined by \\(a_1 = 1\\), \\(a_2 = 2\\), and for \\(n \\geq 1\\),\n\\[ a_{n+2} = \\begin{cases} \n5a_{n+1} - 3a_{n}, & \\text{if } a_n \\cdot a_{n+1} \\text{ is even}, \\\\\na_{n+1} - a_{n}, & \\text{if } a_n \\cdot a_{n+1} \\text{ is odd}.\n\\end{cases} \\]\nLet \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d045d89bc2c4f2e95638a3e5eb4927ab7879792fecd6ded7d691ac38a570de72 | A sequence of integers \(a_1, a_2, a_3, \ldots\) is defined such that \(a_n = a_{n-1} - a_{n-2}\) for each \(n \ge 3\). What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | [
"986"
] | [
"A sequence of integers \\(a_1, a_2, a_3, \\ldots\\) is defined such that \\(a_n = a_{n-1} - a_{n-2}\\) for each \\(n \\ge 3\\). What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-2c1c03c32ebb607a113f6a3ce641f62f7e47e8112786cc46156aad2d8a756a0a | A sequence of integers \(a_1, a_2, a_3, \ldots\) is defined such that \(a_n = a_{n-1} - a_{n-2}\) for each \(n \geq 3\). Given that the sum of the first 1492 terms is 1985 and the sum of the first 1985 terms is 1492, what is the sum of the first 2001 terms of this sequence? | [
"986"
] | [
"A sequence of integers \\(a_1, a_2, a_3, \\ldots\\) is defined such that \\(a_n = a_{n-1} - a_{n-2}\\) for each \\(n \\geq 3\\). Given that the sum of the first 1492 terms is 1985 and the sum of the first 1985 terms is 1492, what is the sum of the first 2001 terms of this sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-769851c9f715a631e70dcefa2f34818dc85ef7a045295e79129f4e7cee707218 | A sequence of integers \(a_1, a_2, a_3, \ldots\) is given by the conditions \(a_1 = 1\), \(a_2 = 12\), \(a_3 = 20\), and \(a_{n+3} = 2a_{n+2} + 2a_{n+1} - a_n\) for every \(n \geq 1\). Let \(S_n = 1 + 4a_n a_{n+1}\). Calculate the value of \(S_5\). | [
"255025"
] | [
"A sequence of integers \\(a_1, a_2, a_3, \\ldots\\) is given by the conditions \\(a_1 = 1\\), \\(a_2 = 12\\), \\(a_3 = 20\\), and \\(a_{n+3} = 2a_{n+2} + 2a_{n+1} - a_n\\) for every \\(n \\geq 1\\). Let \\(S_n = 1 + 4a_n a_{n+1}\\). Calculate the value of \\(S_5\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92c4b0f66ad0d4bc8ab2bee85b7caa13323cd4510a6003a747c7947f22e58e61 | A sequence of integers \(a_1, a_2, a_3, \ldots\) satisfies the relation: \[ 4a_{n+1}^2 - 4a_n a_{n+1} + a_n^2 - 1 = 0 \] for all positive integers \(n\). What are the possible values of \(a_1\)? | [
"All odd integers."
] | [
"A sequence of integers \\(a_1, a_2, a_3, \\ldots\\) satisfies the relation:\n\\[ 4a_{n+1}^2 - 4a_n a_{n+1} + a_n^2 - 1 = 0 \\]\nfor all positive integers \\(n\\). What are the possible values of \\(a_1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 19
} | 29 | 0 | 0 |
omr-ws-f283a5ed5a9d2021283ab848fa53ac6154a46aa83e3975ef309f7e2b3410a902 | A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and \[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$ | [
"-1941"
] | [
"A sequence of integers is defined as follows: $a_i = i$ for $1 \\le i \\le 5,$ and\n\\[a_i = a_1 a_2 \\dotsm a_{i - 1} - 1\\]for $i > 5.$ Evaluate $a_1 a_2 \\dotsm a_{2011} - \\sum_{i = 1}^{2011} a_i^2.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-432ad0cdf82f21b43c3aa476f5bafce364cdafba559c4f42a95bcb03a6e146cc | A sequence of integers is defined by \( a_{0} = 0 \), \( a_{1} = 2 \), \( a_{2} = 9 \), and for \( n > 2 \), \( a_{n} = a_{n-1} + a_{n-2} - a_{n-3} \). Give a closed form expression for \( a_{n} \). | [
"\\( a_{n} = 2n + 5 \\left\\lfloor \\frac{n}{2} \\right\\rfloor \\)"
] | [
"A sequence of integers is defined by \\( a_{0} = 0 \\), \\( a_{1} = 2 \\), \\( a_{2} = 9 \\), and for \\( n > 2 \\), \\( a_{n} = a_{n-1} + a_{n-2} - a_{n-3} \\). Give a closed form expression for \\( a_{n} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-e798aea717be891848839fd2c40e0d620981ef400237f34291ac7e2dfecbcfb2 | A sequence of integers is written in blocks such that the $n$-th block consists of a single "1" followed by $n$ "2"s and then $1 + 2 + 3 + \ldots + n$ "3"s. The sequence begins as follows: $1, 2, 3, 1, 2, 2, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, \ldots$. Find the sum of all the numbers in the sequence when exactly ... | [
"6573"
] | [
"A sequence of integers is written in blocks such that the $n$-th block consists of a single \"1\" followed by $n$ \"2\"s and then $1 + 2 + 3 + \\ldots + n$ \"3\"s. The sequence begins as follows: $1, 2, 3, 1, 2, 2, 3, 3, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, \\ldots$. Find the sum of all the numbers in the sequence ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38248441c1f1a8c6262fcba7bf9aa6e6bd0cdbbdd26df4acf1b0bd0bcf5e8dbd | A sequence of letters $ASSARA...$ is formed by adding one of the three letters $A$, $R$, or $S$ such that no two consecutive sevens of letters are the same. At some point, it becomes impossible to add a new letter according to these rules. What letter could be written last? | [
"A, R, or S"
] | [
"A sequence of letters $ASSARA...$ is formed by adding one of the three letters $A$, $R$, or $S$ such that no two consecutive sevens of letters are the same. At some point, it becomes impossible to add a new letter according to these rules. What letter could be written last?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a1baa602e647c6daffc56eb120839181c116e99b7cddc6311314e304601141f | A sequence of moves is performed on the string of three letters $abc,$ where a move consists of inserting an instance of the three consecutive letters $abc$ between any two consecutive letters of the current string. After $n$ moves, how many distinct possible strings can be achieved? | [
"\\(\\binom{3n+1}{n} \\cdot \\frac{1}{n+1}\\)"
] | [
"A sequence of moves is performed on the string of three letters $abc,$ where a move consists of inserting an instance of the three consecutive letters $abc$ between any two consecutive letters of the current string. After $n$ moves, how many distinct possible strings can be achieved?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a27b4ae50eec0282f157e80d208263079bb5baeaa1812599b7650a3bdf1dbf07 | A sequence of moves is performed starting on the three-letter string "$BAD$". A move consists of inserting an additional instance of the three-letter string "$BAD$" between any two consecutive letters of the current string, to achieve an elongated string. After $n$ moves, how many distinct possible strings of length $3... | [
"\\prod_{k=1}^{n} (3k - 1)"
] | [
"A sequence of moves is performed starting on the three-letter string \"$BAD$\". A move consists of inserting an additional instance of the three-letter string \"$BAD$\" between any two consecutive letters of the current string, to achieve an elongated string. After $n$ moves, how many distinct possible strings of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-6cbe83fc7b291077a5a0f1ae484fd81d466bbc3c50eca9b3d5fc0b3dab9165d5 | A sequence of natural numbers $a_0,a_1,a_2,a_3,\ldots $ is formed according to the following rule: $$a_0a_1a_2=|a_0-a_1|, a_3=|a_1-a_2|,\ldots $$ (generally, $a_n=|a_{n-2}-a_{n-1}|$ for all $n\ge 2$). The elements of the sequence are computed until the first zero has been obtained. It is known that each of the numbers ... | [
"2952 (prove that the greatest number of members in a sequence satisfying the conditions of the problem which starts with the greatest number $a_1=n$ is equal to $[(3n + 1)/2]$)"
] | [
"A sequence of natural numbers $a_0,a_1,a_2,a_3,\\ldots $ is formed according to the following rule: $$a_0a_1a_2=|a_0-a_1|, a_3=|a_1-a_2|,\\ldots $$ (generally, $a_n=|a_{n-2}-a_{n-1}|$ for all $n\\ge 2$). The elements of the sequence are computed until the first zero has been obtained. It is known that each of the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-191549383b6060868f3f564db529a7443f53cc29352f5b9977fad1d3b6c701ef | A sequence of natural numbers \( c_1, c_2, \dots \) is called PERFECT if every natural number \( m \) with \( 1 \le m \le c_1 + \dots + c_n \) can be represented as \( m = \frac{c_1}{a_1} + \frac{c_2}{a_2} + \dots + \frac{c_n}{a_n} \). Given \( n \), find the maximum possible value of \( c_n \) in a perfect sequence \(... | [
"2^{n-1}"
] | [
"A sequence of natural numbers \\( c_1, c_2, \\dots \\) is called PERFECT if every natural number \\( m \\) with \\( 1 \\le m \\le c_1 + \\dots + c_n \\) can be represented as \\( m = \\frac{c_1}{a_1} + \\frac{c_2}{a_2} + \\dots + \\frac{c_n}{a_n} \\). Given \\( n \\), find the maximum possible value of \\( c_n \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-244dba8d3f5b83e2f3806fcfea2cbbaf04feb7b150f8040c18f27f98172ef958 | A sequence of natural numbers \( c_1, c_2, \dots \) is called perfect if every natural number \( m \) with \( 1 \le m \le c_1 + \dots + c_n \) can be represented as \( m = \frac{c_1}{a_1} + \frac{c_2}{a_2} + \dots + \frac{c_n}{a_n} \). Given \( n \), find the maximum possible value of \( c_n \) in a perfect sequence \(... | [
"\\( 4 \\times 3^{n-2} \\) for \\( n \\geq 2 \\) and \\( 2 \\) for \\( n = 1 \\)."
] | [
"A sequence of natural numbers \\( c_1, c_2, \\dots \\) is called perfect if every natural number \\( m \\) with \\( 1 \\le m \\le c_1 + \\dots + c_n \\) can be represented as \\( m = \\frac{c_1}{a_1} + \\frac{c_2}{a_2} + \\dots + \\frac{c_n}{a_n} \\). Given \\( n \\), find the maximum possible value of \\( c_n \\)... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ad2d096334475c675a982d0c248935e29289b6571479c95e927f3d004fc8b2a | A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? | [
"996506"
] | [
"A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 15,
"tir": 18
} | 60 | 0 | 0 |
omr-ws-c6028e7e657cd1a8e3c06722f3413362d613a1375416584d7e22f970a04b0c87 | A sequence of natural numbers is defined as follows: the first two numbers are 25 and 126. Each subsequent number is obtained by calculating the sum of the last two numbers, finding the greatest odd divisor of this sum, and then adding one to this greatest odd divisor. The sequence has 2015 numbers. Find the last numbe... | [
"4"
] | [
"A sequence of natural numbers is defined as follows: the first two numbers are 25 and 126. Each subsequent number is obtained by calculating the sum of the last two numbers, finding the greatest odd divisor of this sum, and then adding one to this greatest odd divisor. The sequence has 2015 numbers. Find the last ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-6bdfb869860e24311e919a8600ed6301bb30ae17fca37f2477295f6402bca092 | A sequence of nonzero complex numbers \(a_1, a_2, \dots, a_{2020}\) satisfies \(a_3 = a_2^2 + 2a_1a_2\) and \[ \frac{a_{n+2}}{a_{n+1}} - \frac{a_{n+1}}{a_{n}} = a_n + a_{n+1}, \] for all \(2 \leq n \leq 2018\). Given \(a_2 - a_{2020} = 2025\), how many integers \(0 \leq a_1 \leq 2020\) are there such that \(a_1 + a_2 +... | [
"0"
] | [
"A sequence of nonzero complex numbers \\(a_1, a_2, \\dots, a_{2020}\\) satisfies \\(a_3 = a_2^2 + 2a_1a_2\\) and \n\\[\n\\frac{a_{n+2}}{a_{n+1}} - \\frac{a_{n+1}}{a_{n}} = a_n + a_{n+1},\n\\]\nfor all \\(2 \\leq n \\leq 2018\\). Given \\(a_2 - a_{2020} = 2025\\), how many integers \\(0 \\leq a_1 \\leq 2020\\) are ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-ae8feeac3975b326ce7fef6bf9c7384c908d864c3ef2cf6b330543835e2026c4 | A sequence of numbers $a_1$, $a_2$, ..., $a_m$ is a geometric sequence modulo $n$ of length $m$ if for each index $i$ with $1 \le i \le m$ we have $a_i \in \{0, 1, 2, ..., n - 1\}$ and there is some integer $k$ such that $n$ divides $(a_{j+1} - ka_j)$ for $j = 1$, $2$, ..., $m-1$. How many geometric sequences modulo $1... | [
"129"
] | [
"A sequence of numbers $a_1$, $a_2$, ..., $a_m$ is a geometric sequence modulo $n$ of length $m$ if for each index $i$ with $1 \\le i \\le m$ we have $a_i \\in \\{0, 1, 2, ..., n - 1\\}$ and there is some integer $k$ such that $n$ divides $(a_{j+1} - ka_j)$ for $j = 1$, $2$, ..., $m-1$. How many geometric sequences... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-3c6ad06b60f429233dfc61beee5ffb4634726c7a1c9af6a8ace741255f758094 | A sequence of numbers $x_{1},x_{2},x_{3},\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$. | [
"173"
] | [
"A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-2db9efbf79d2ed0bf6355dcad5b70113282561f78f9a350e22e91593ef285d61 | A sequence of numbers \( a_1, a_2, a_3, \ldots \) satisfies \( a_1 = \frac{1}{2} \) and \( a_1 + a_2 + \cdots + a_n = n^2 a_n \) for \( n \geq 1 \). Find \( a_n \) in terms of \( n \). | [
"\\( \\frac{1}{n(n+1)} \\)"
] | [
"A sequence of numbers \\( a_1, a_2, a_3, \\ldots \\) satisfies \\( a_1 = \\frac{1}{2} \\) and \\( a_1 + a_2 + \\cdots + a_n = n^2 a_n \\) for \\( n \\geq 1 \\). Find \\( a_n \\) in terms of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-bcc2b2e9b5f30aea8a994168871a715e6dc09e7a7a97d6b97dab830b6afe5236 | A sequence of numbers \( a_1, a_2, a_3, \ldots \) satisfies the following conditions: (i) \( a_1 = \frac{1}{2} \) (ii) \( a_1 + a_2 + \cdots + a_n = n^2 a_n \) for \( n \geq 1 \) Determine the value of \( a_n \) for \( n \geq 1 \). | [
"\\( \\frac{1}{n(n+1)} \\)"
] | [
"A sequence of numbers \\( a_1, a_2, a_3, \\ldots \\) satisfies the following conditions:\n(i) \\( a_1 = \\frac{1}{2} \\)\n(ii) \\( a_1 + a_2 + \\cdots + a_n = n^2 a_n \\) for \\( n \\geq 1 \\)\n\nDetermine the value of \\( a_n \\) for \\( n \\geq 1 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-ef8e74aa624f058a74234e3a1dff5d0ff27f3f66ac62eb60547c35e38d4cf74b | A sequence of numbers \( x_1, x_2, x_3, \ldots, x_{100} \) has the property that, for every integer \( k \) between 1 and 100, inclusive, the number \( x_k \) is \( k \) less than the sum of the other 99 numbers. Given that \( x_{50} = \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, fi... | [
"173"
] | [
"A sequence of numbers \\( x_1, x_2, x_3, \\ldots, x_{100} \\) has the property that, for every integer \\( k \\) between 1 and 100, inclusive, the number \\( x_k \\) is \\( k \\) less than the sum of the other 99 numbers. Given that \\( x_{50} = \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-58fef5edaac068646835fb5fd83c0bd993a5d5e18d138e30591e05f98d3916a9 | A sequence of numbers \(a_1, a_2, \ldots\) is defined by the following rule: if the last digit of \(a_n\) is \(k\), then \(a_{n+1} = \frac{10^k - 1}{9} + \frac{a_n - k}{10}\). Given that \(a_1 = 20155102\), calculate the sum of the digits of \(a_{2015}\). | [
"16"
] | [
"A sequence of numbers \\(a_1, a_2, \\ldots\\) is defined by the following rule: if the last digit of \\(a_n\\) is \\(k\\), then \\(a_{n+1} = \\frac{10^k - 1}{9} + \\frac{a_n - k}{10}\\). Given that \\(a_1 = 20155102\\), calculate the sum of the digits of \\(a_{2015}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c2877b7504f97d95af30d4885f95d2268773d3af951419bbd1d8782846750dc | A sequence of numbers contains $2$ $a$'s, $2$ $b$'s, $2$ $c$'s, $2$ $d$'s, $2$ $e$'s, and $2$ $f$'s. This sequence is called abundantly even if between any two of the same letters $k$, there are two different letters $m$ between them. An example is $a b b a c d d e e c f f$. How many abundantly even sequences are there... | [
"518400"
] | [
"A sequence of numbers contains $2$ $a$'s, $2$ $b$'s, $2$ $c$'s, $2$ $d$'s, $2$ $e$'s, and $2$ $f$'s. This sequence is called abundantly even if between any two of the same letters $k$, there are two different letters $m$ between them. An example is $a b b a c d d e e c f f$. How many abundantly even sequences are ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-fa82b431dcaab7cbab68eba7cd059759dcd461092ee353de4ffa66103946c98a | A sequence of numbers is defined as $s_0 = 1$, $s_{2018} = 300$, and $s_n + 2s_{n-2} = 2s_{n-1}$. Find $\sum^{2019}_{i=2} s_i$. | [
"598"
] | [
"A sequence of numbers is defined as $s_0 = 1$, $s_{2018} = 300$, and $s_n + 2s_{n-2} = 2s_{n-1}$. Find $\\sum^{2019}_{i=2} s_i$."
] | [
"aops_c4_high_school_math"
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"cot",
"tir"
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"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-fe4059093b12abe4b3235289b361b9e3c58e2d5a0402276f32eb1def337add8e | A sequence of numbers is defined as follows: $a_1=1$, $a_2=1$, and $a_n=\frac{3}{4} * a_{n-2} + \frac{2}{3} * a_{n-1}$ for all integers $n \ge 3$. Compute the sum $S = a_1 + a_2 + a_3 + a_4 + \ldots$. | [
"\\(+\\infty\\)"
] | [
"A sequence of numbers is defined as follows: $a_1=1$, $a_2=1$, and $a_n=\\frac{3}{4} * a_{n-2} + \\frac{2}{3} * a_{n-1}$ for all integers $n \\ge 3$. Compute the sum $S = a_1 + a_2 + a_3 + a_4 + \\ldots$."
] | [
"aops_c6_high_school_olympiads"
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"cot",
"genselect",
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"cot": 6,
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"tir": 19
} | 33 | 0 | 0 |
omr-ws-769a150d62a25ea3dc00b6ef39cf7539890c035c8bc504d284c710a3d9e4f254 | A sequence of numbers is defined as follows: $x_{n}=\frac{4}{7}$ if $n=0$, and $x_{n}=7x_{(n-1)}-3$ for all positive integers $n$. It is given that $x_{n}=\frac{k^{n-1}+p}{w}$ for all non-negative integers $n$, where $k$, $p$, and $w$ are positive integers. Find the value of $k+p+w$. | [
"10"
] | [
"A sequence of numbers is defined as follows: $x_{n}=\\frac{4}{7}$ if $n=0$, and $x_{n}=7x_{(n-1)}-3$ for all positive integers $n$. It is given that $x_{n}=\\frac{k^{n-1}+p}{w}$ for all non-negative integers $n$, where $k$, $p$, and $w$ are positive integers. Find the value of $k+p+w$."
] | [
"aops_c4_high_school_math"
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"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5865d5dc62a0ccce1b6cb3a212fdd441396c9aa2c5ed1fe80796109ebe67215c | A sequence of numbers is defined by $D_0=0$, $D_1=0$, $D_2=1$, and $D_n=D_{n-1}+D_{n-3}$ for $n \ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023})$, where $E$ denotes even and $O$ denotes odd? | [
"(E, O, E)"
] | [
"A sequence of numbers is defined by $D_0=0$, $D_1=0$, $D_2=1$, and $D_n=D_{n-1}+D_{n-3}$ for $n \\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023})$, where $E$ denotes even and $O$ denotes odd?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-c307fc0a31b649886c2034d8df1f596a4c0f111983163a02f0c1ae5b1fdf499e | A sequence of numbers is defined by $D_0=0, D_1=0, D_2=1$, and $D_n=D_{n-1}+D_{n-3}$ for $n\ge3$. What are the parities (evenness or oddness) of the triple of numbers $D_{2021},D_{2022},D_{2023}$ where $E$ denotes even and $O$ denotes odd? | [
"\\(E, O, E\\)"
] | [
"A sequence of numbers is defined by $D_0=0, D_1=0, D_2=1$, and $D_n=D_{n-1}+D_{n-3}$ for $n\\ge3$. What are the parities (evenness or oddness) of the triple of numbers $D_{2021},D_{2022},D_{2023}$ where $E$ denotes even and $O$ denotes odd?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
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"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-4ce0aef69c8a27ec8448180f1df3d56b1467582f1a0fcdd2fbaf4426ff8c90a7 | A sequence of numbers is defined by \( a_1 = 19 \), \( a_2 = 77 \), and \( a_n = \frac{1 - a_{n-1}}{a_{n-2}} \) for \( n \geq 3 \). Determine the value of \( a_{2007} \). | [
"77"
] | [
"A sequence of numbers is defined by \\( a_1 = 19 \\), \\( a_2 = 77 \\), and \\( a_n = \\frac{1 - a_{n-1}}{a_{n-2}} \\) for \\( n \\geq 3 \\). Determine the value of \\( a_{2007} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2ac5f549d190f2da02ba8648fbd481f471e0bb9a8632930788b81f20e3d3866 | A sequence of numbers is defined by \(D_0 = 0\), \(D_1 = 1\), \(D_2 = 1\) and \(D_n = D_{n-1} + D_{n-3}\) for \(n \ge 3\). What are the parities (evenness or oddness) of the triple of numbers \((D_{2021}, D_{2022}, D_{2023})\), where \(E\) denotes even and \(O\) denotes odd? | [
"(E, O, E)"
] | [
"A sequence of numbers is defined by \\(D_0 = 0\\), \\(D_1 = 1\\), \\(D_2 = 1\\) and \\(D_n = D_{n-1} + D_{n-3}\\) for \\(n \\ge 3\\). What are the parities (evenness or oddness) of the triple of numbers \\((D_{2021}, D_{2022}, D_{2023})\\), where \\(E\\) denotes even and \\(O\\) denotes odd?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 16
} | 18 | 0 | 0 |
omr-ws-e57a57f311a87e6669996cdb1897fb95e4e56124883eef3b8812919de1c7a289 | A sequence of numbers is defined by the relation $a_n = -a_{n-1} + 6a_{n-2}$ with initial conditions $a_1 = 2$ and $a_2 = 1$. Find the value of $a_{100} + 3a_{99}$. | [
"\\( 7 \\cdot 4^{49} \\)"
] | [
"A sequence of numbers is defined by the relation $a_n = -a_{n-1} + 6a_{n-2}$ with initial conditions $a_1 = 2$ and $a_2 = 1$. Find the value of $a_{100} + 3a_{99}$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-b78859c8e31913ea8a54dab53ba0c169cd9a9156d99ab0180d38df38d9c47a42 | A sequence of numbers is defined by the relation \( a_n = -a_{n-1} + 6a_{n-2} \) with initial conditions \( a_1 = 2 \) and \( a_2 = 1 \). Find \( a_{100} + 3a_{99} \). | [
"\\( 7 \\cdot 2^{98} \\)"
] | [
"A sequence of numbers is defined by the relation \\( a_n = -a_{n-1} + 6a_{n-2} \\) with initial conditions \\( a_1 = 2 \\) and \\( a_2 = 1 \\). Find \\( a_{100} + 3a_{99} \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7e08324eeea5371eb9fee977ad8a267e8e29127eae1be744b92a9ba547636d1b | A sequence of numbers is defined recursively by \(a_1 = 1\), \(a_2 = 1\), and for \(k > 2\), \(a_k = \frac{1}{4} a_{k-2} + \frac{1}{3} a_{k-1}\). Evaluate the infinite sum \(a_1 + a_2 + a_3 + \cdots\). | [
"4"
] | [
"A sequence of numbers is defined recursively by \\(a_1 = 1\\), \\(a_2 = 1\\), and for \\(k > 2\\), \\(a_k = \\frac{1}{4} a_{k-2} + \\frac{1}{3} a_{k-1}\\). Evaluate the infinite sum \\(a_1 + a_2 + a_3 + \\cdots\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-92549cd18c18e52b01f168ed7fac611a5a79c32b5e39614d13580c55bf91ef1d | A sequence of numbers is defined recursively by \(a_1 = 1\), \(a_2 = \frac{3}{7}\), and \(a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\) for all \(n \geq 3\). Then \(a_{2019}\) can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. What is \(p + q\)? | [
"8078"
] | [
"A sequence of numbers is defined recursively by \\(a_1 = 1\\), \\(a_2 = \\frac{3}{7}\\), and \\(a_n = \\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\) for all \\(n \\geq 3\\). Then \\(a_{2019}\\) can be written as \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. What is \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3bc7c435d5f35180f852329967ab1ee3cc5beb87b224504881fa9f0526b191df | A sequence of numbers is defined recursively by \(a_1 = 1\), \(a_2 = \frac{3}{7}\), and \[a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\] for all \(n \ge 3\). Then \(a_{2019}\) can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. What is \(p + q\)? | [
"8078"
] | [
"A sequence of numbers is defined recursively by \\(a_1 = 1\\), \\(a_2 = \\frac{3}{7}\\), and \\[a_n = \\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\] for all \\(n \\ge 3\\). Then \\(a_{2019}\\) can be written as \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. What is \\(... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22ee1861ec7a7cbc8293e5ac99b13cf2dfe39fd69922a3301586ead76ceb38fc | A sequence of numbers is defined recursively by \(a_1 = 1\), \(a_2 = \frac{3}{7}\), and \[a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\] for all \(n \geq 3\). Then \(a_{2019}\) can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. What is \(p + q\)? | [
"8078"
] | [
"A sequence of numbers is defined recursively by \\(a_1 = 1\\), \\(a_2 = \\frac{3}{7}\\), and \n\\[a_n = \\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\]\nfor all \\(n \\geq 3\\). Then \\(a_{2019}\\) can be written as \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are relatively prime positive integers. What is... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-13fcb5689ea4bb97bafe089bdd18145e627ed89af9361c3bc76adf5ce7a9ff07 | A sequence of numbers is formed, each of which is equal to the product of 3 consecutive numbers, as follows: $1 \times 2 \times 3$, $2 \times 3 \times 4$, $3 \times 4 \times 5$, ..., $98 \times 99 \times 100$. Juan adds the inverses of all these numbers and obtains a rational number, whose simplified expression is $p /... | [
"24749"
] | [
"A sequence of numbers is formed, each of which is equal to the product of 3 consecutive numbers, as follows: $1 \\times 2 \\times 3$, $2 \\times 3 \\times 4$, $3 \\times 4 \\times 5$, ..., $98 \\times 99 \\times 100$. Juan adds the inverses of all these numbers and obtains a rational number, whose simplified expre... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-5b6e9c3ae49f3774b1d2680ab293e146ce61ac1a9751fcb7ebb54d2de8ce7299 | A sequence of numbers is generated as follows: Let \( a_1 = 1 \) and for \( n \ge 1 \), choose \( a_{n+1} \) at random from the positive integers from 1 to \( 10 - a_n \), inclusive. On average, what fraction of the terms in the sequence will be equal to 1? | [
"\\(\\frac{1}{5}\\)"
] | [
"A sequence of numbers is generated as follows: Let \\( a_1 = 1 \\) and for \\( n \\ge 1 \\), choose \\( a_{n+1} \\) at random from the positive integers from 1 to \\( 10 - a_n \\), inclusive. On average, what fraction of the terms in the sequence will be equal to 1?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 19
} | 32 | 0 | 0 |
omr-ws-dc765a57c94381a98dd11b185a99227b9763d539be35b9344da4e3c2888d7f86 | A sequence of numbers is generated by the following rule: the first number is given by Camilo; thereafter, the program calculates the integer division of the last number generated by $18$, obtaining a quotient and a remainder. The sum of that quotient plus that remainder is the next number generated. Determine the numb... | [
"17"
] | [
"A sequence of numbers is generated by the following rule: the first number is given by Camilo; thereafter, the program calculates the integer division of the last number generated by $18$, obtaining a quotient and a remainder. The sum of that quotient plus that remainder is the next number generated. Determine the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-112f7c341abe1eca6e1a9d579017f7181e5e4dfdbb18e5f5bb725082538b4c65 | A sequence of numbers starts with \(2017^{2017}\) and each subsequent number is obtained by subtracting the sum of the digits of the current number from the current number. What is the last number in the sequence that is not 0? | [
"9"
] | [
"A sequence of numbers starts with \\(2017^{2017}\\) and each subsequent number is obtained by subtracting the sum of the digits of the current number from the current number. What is the last number in the sequence that is not 0?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-199e77fcaf239a901cca0b46ebf2adc9fdfc2496e19c834bb43299d4bfe2ff9c | A sequence of numbers, $x_1, x_2, \ldots, x_{100}$, has the property that for every integer $k$, $1 \le k \le 100$, $x_k$ is $k$ less than the sum of the other numbers. Let $x_{50} = \frac{m}{n}$. Find $m+n$. | [
"199"
] | [
"A sequence of numbers, $x_1, x_2, \\ldots, x_{100}$, has the property that for every integer $k$, $1 \\le k \\le 100$, $x_k$ is $k$ less than the sum of the other numbers. Let $x_{50} = \\frac{m}{n}$. Find $m+n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5615931fef077a08f728cfc85b45ffba196f95c733ec668094d3fbdd8e5ec530 | A sequence of points \((a_n, b_n)\) in the coordinate plane satisfies \((a_{n+1}, b_{n+1}) = (\sqrt{3}a_n - b_n, \sqrt{3}b_n + a_n)\) for \(n = 1, 2, 3, \ldots\). Given that \((a_{100}, b_{100}) = (2, 4)\), what is \(a_1 + b_1\)? | [
"\\(\\frac{1}{2^{98}}\\)"
] | [
"A sequence of points \\((a_n, b_n)\\) in the coordinate plane satisfies \\((a_{n+1}, b_{n+1}) = (\\sqrt{3}a_n - b_n, \\sqrt{3}b_n + a_n)\\) for \\(n = 1, 2, 3, \\ldots\\). Given that \\((a_{100}, b_{100}) = (2, 4)\\), what is \\(a_1 + b_1\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 14
} | 61 | 0 | 0 |
omr-ws-4a143ad7122e56b710ec07243ef77aa1a16102fed146d7547174510ae56b3efd | A sequence of points \((a_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3}), \ldots\) in the coordinate plane satisfies \((a_{n+1}, b_{n+1}) = (\sqrt{3}a_{n} - b_{n}, \sqrt{3}b_{n} + a_{n})\) for all positive integers \(n\). Given that \((a_{100}, b_{100}) = (2, 4)\), what is \(a_{1} + b_{1}\)? | [
"\\(\\frac{1}{2^{98}}\\)"
] | [
"A sequence of points \\((a_{1}, b_{1}), (a_{2}, b_{2}), (a_{3}, b_{3}), \\ldots\\) in the coordinate plane satisfies \\((a_{n+1}, b_{n+1}) = (\\sqrt{3}a_{n} - b_{n}, \\sqrt{3}b_{n} + a_{n})\\) for all positive integers \\(n\\). Given that \\((a_{100}, b_{100}) = (2, 4)\\), what is \\(a_{1} + b_{1}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-20da5f85f1b729ebbffdbd5e315a2b16049ba0257db76d4b52ea1d31c680b113 | A sequence of polyhedra \( P_i \) is defined recursively as follows: \( P_0 \) is a regular tetrahedron with volume 1. To obtain \( P_{i+1} \), replace the midpoint triangle of every face of \( P_i \) by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of \( P_3 \) is \(\frac... | [
"101"
] | [
"A sequence of polyhedra \\( P_i \\) is defined recursively as follows: \\( P_0 \\) is a regular tetrahedron with volume 1. To obtain \\( P_{i+1} \\), replace the midpoint triangle of every face of \\( P_i \\) by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of \\( P_3... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1e8b44e2bddc7aa3e945d0c940aadf182a134fa4bd51520f0330bdfe9c9a3d03 | A sequence of polynomials $Q_0(x), Q_1(x), \ldots$ is defined by $Q_0(x) = 1$, $Q_1(x) = x$, and $Q_{i+1}(x) = xQ_i(x) + iQ_{i-1}(x)$ for $i \geq 1$. Let $n$ be an odd whole number. Calculate the remainder when $Q_n(x) - x^n$ is divided by $n$ for $n = 5$. | [
"0"
] | [
"A sequence of polynomials $Q_0(x), Q_1(x), \\ldots$ is defined by $Q_0(x) = 1$, $Q_1(x) = x$, and $Q_{i+1}(x) = xQ_i(x) + iQ_{i-1}(x)$ for $i \\geq 1$. Let $n$ be an odd whole number. Calculate the remainder when $Q_n(x) - x^n$ is divided by $n$ for $n = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c85ce553806344263684bbdb8f22fce06d4ad21edc8963c0d8e0e392498704ab | A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and \[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \quad k=1,2,\ldots\] Calculate the value of $f_{10}(10)$. | [
"1024"
] | [
"A sequence of polynomials $\\{f_n\\}_{n=0}^{\\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and\n\\[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \\quad k=1,2,\\ldots\\]\nCalculate the value of $f_{10}(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27175747e161a0c87fe0aa9c7aa86ac69a29d5ee4a7197b86f159dc473663873 | A sequence of polynomials \( Q_0(x), Q_1(x), \ldots \) is defined by \( Q_0(x) = 1 \), \( Q_1(x) = x \), and \( Q_{i+1}(x) = xQ_i(x) + iQ_{i-1}(x) \) for \( i \geq 1 \). For an odd whole number \( n \geq 1 \), calculate the coefficient of \( x^{n-1} \) in the polynomial \( Q_n(x) - x^n \). | [
"0"
] | [
"A sequence of polynomials \\( Q_0(x), Q_1(x), \\ldots \\) is defined by \\( Q_0(x) = 1 \\), \\( Q_1(x) = x \\), and \\( Q_{i+1}(x) = xQ_i(x) + iQ_{i-1}(x) \\) for \\( i \\geq 1 \\). For an odd whole number \\( n \\geq 1 \\), calculate the coefficient of \\( x^{n-1} \\) in the polynomial \\( Q_n(x) - x^n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14bcec000fe948e8b85df26837405f32fd1dd29e43d1f0ef426650d39bfcd385 | A sequence of polynomials \( f_n(x) \) is defined as follows: \[ f_0(x) = 2, \] \[ f_1(x) = x, \] \[ f_n(x) = x f_{n-1}(x) - f_{n-2}(x) \quad \text{for} \quad n \geq 2. \] Let \( x_n \) be the maximum real root of the equation \( f_n(x) = 0 \) for \( |x| \leq 2 \). Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_{x... | [
"2\\pi - 4"
] | [
"A sequence of polynomials \\( f_n(x) \\) is defined as follows:\n\\[ f_0(x) = 2, \\]\n\\[ f_1(x) = x, \\]\n\\[ f_n(x) = x f_{n-1}(x) - f_{n-2}(x) \\quad \\text{for} \\quad n \\geq 2. \\]\n\nLet \\( x_n \\) be the maximum real root of the equation \\( f_n(x) = 0 \\) for \\( |x| \\leq 2 \\).\n\nEvaluate the limit:\n... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65c1e4a18cf96c0e712cc636adff4432fac39e647e2a0d3498fc1f3ca1bb796f | A sequence of polynomials \( f_n(x) \) is defined recursively by \((k+1)f_{k+1}(x) - (x+1)f_k(x) + (x-k)f_{k-1}(x) = 0\) for natural number \( k \) with initial conditions \( f_0(x) = 1 \) and \( f_1(x) = 1 + x \). Calculate the value of \( f_{10}(2) \). | [
"4"
] | [
"A sequence of polynomials \\( f_n(x) \\) is defined recursively by \\((k+1)f_{k+1}(x) - (x+1)f_k(x) + (x-k)f_{k-1}(x) = 0\\) for natural number \\( k \\) with initial conditions \\( f_0(x) = 1 \\) and \\( f_1(x) = 1 + x \\). Calculate the value of \\( f_{10}(2) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-622731d16646fbacf4d0d9e3676bfb8dd05933e8d849226a4d11099124259f3f | A sequence of polynomials \( f_n(x) \) is defined recursively by \[ (k+1) f_{k+1}(x) - (x+1)f_k(x) + (x-k)f_{k-1}(x) = 0 \] for \( k \in \mathbb{N} \) with initial conditions \( f_0(x) = 1 \) and \( f_1(x) = 1 + x \). Calculate the value of \( f_5(5) \). | [
"32"
] | [
"A sequence of polynomials \\( f_n(x) \\) is defined recursively by\n\\[ (k+1) f_{k+1}(x) - (x+1)f_k(x) + (x-k)f_{k-1}(x) = 0 \\]\nfor \\( k \\in \\mathbb{N} \\) with initial conditions \\( f_0(x) = 1 \\) and \\( f_1(x) = 1 + x \\). Calculate the value of \\( f_5(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bc604fad678af27de2c009e929d8bb3a36079b3e552f234f08051c8e4282992 | A sequence of positive integers $(a_n)_{n \geq 1}$ is defined by the relation $$a_n = \varphi(a_{n-1}) + \varphi(a_{n-2}),$$ where $\varphi(x)$ is the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Additionally, there exists an $n < 20$ such that $\varphi(a_{n}) \geq \varphi(a_{... | [
"896"
] | [
"A sequence of positive integers $(a_n)_{n \\geq 1}$ is defined by the relation $$a_n = \\varphi(a_{n-1}) + \\varphi(a_{n-2}),$$ where $\\varphi(x)$ is the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Additionally, there exists an $n < 20$ such that $\\varphi(a_{n}) \\geq ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb9a23d1bd419de9b2e49c178094ba19e0d31ba592d6b95ba19b1a9f21cdc9c8 | A sequence of positive integers $\{a_n\}$ is defined by $a_1 = 1$, $a_2 = 3$, and $a_n = (n+1)a_{n-1} - na_{n-2}$ for $n \geq 3$. Find all positive integers $n$ such that $11 \mid a_n$. | [
"\\( n = 4, 8, 10, 11, \\) and all \\( n > 9 \\)"
] | [
"A sequence of positive integers $\\{a_n\\}$ is defined by $a_1 = 1$, $a_2 = 3$, and $a_n = (n+1)a_{n-1} - na_{n-2}$ for $n \\geq 3$. Find all positive integers $n$ such that $11 \\mid a_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 23
} | 34 | 0 | 0 |
omr-ws-ded0debca2540701766bcc28bc09dbe5616191194784dfacf5949cbe3f64d18a | A sequence of positive integers $a_1, a_2, a_3, \ldots$ satisfies $$a_{n+1} = n \left \lfloor \frac{a_n}{n} \right \rfloor + 1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_1$ take? | [
"274"
] | [
"A sequence of positive integers $a_1, a_2, a_3, \\ldots$ satisfies $$a_{n+1} = n \\left \\lfloor \\frac{a_n}{n} \\right \\rfloor + 1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_1$ take?"
] | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-0e7265d6e72c6f6f659f2e138aeeacd2ddb8908be2e234175f3fda0f73256134 | A sequence of positive integers $a_n$ begins with $a_1=1$ and $a_2=b$ for positive integers $a$ and $b$. Subsequent terms in the sequence satisfy the following two rules for all positive integers $n$: $$a_{2n+1}=a_{2n}a_{2n-1},~a_{2n+2}=a_{2n+1}+4.$$ Exactly $m$ of the numbers $a_1$, $a_2$, $a_3$, $\ldots$, $a_{2022}$ ... | [
"1012"
] | [
"A sequence of positive integers $a_n$ begins with $a_1=1$ and $a_2=b$ for positive integers $a$ and $b$. Subsequent terms in the sequence satisfy the following two rules for all positive integers $n$:\n$$a_{2n+1}=a_{2n}a_{2n-1},~a_{2n+2}=a_{2n+1}+4.$$\nExactly $m$ of the numbers $a_1$, $a_2$, $a_3$, $\\ldots$, $a_... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 8,
"tir": 8
} | 35 | 0 | 0 |
omr-ws-fd40c861d2eecc11d3b85fe016c6bf555f79fca2ffa9d2c911c7e704d32e882f | A sequence of positive integers $a_n$ is defined by $a_1 = x$ and $a_n = \frac{k}{a^2_{n-1}}$ for $n \ge 2$. The sequence terminates once the next term would not be a positive integer. Find the sum of all positive integers $k$ less than or equal to 1000 such that $a_n$ is an infinite sequence for some value of $x$. | [
"3025"
] | [
"A sequence of positive integers $a_n$ is defined by $a_1 = x$ and $a_n = \\frac{k}{a^2_{n-1}}$ for $n \\ge 2$. The sequence terminates once the next term would not be a positive integer. Find the sum of all positive integers $k$ less than or equal to 1000 such that $a_n$ is an infinite sequence for some value of $... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e7acc2f0f1380af09ebaa8fbbc493d48b62e25e0ef584376135c645e3ed69b3 | A sequence of positive integers $a_{n}$ begins with $a_{1}=a$ and $a_{2}=b$ for positive integers $a$ and $b.$ Subsequent terms in the sequence satisfy the following two rules for all positive integers $n:$ $$a_{2n+1}=a_{2n}a_{2n-1},$$ $$a_{2n+2}=a_{2n+1}+4.$$ Exactly $m$ of the numbers $a_{1}, a_{2}, \ldots, a_{2022}$... | [
"1012"
] | [
"A sequence of positive integers $a_{n}$ begins with $a_{1}=a$ and $a_{2}=b$ for positive integers $a$ and $b.$ Subsequent terms in the sequence satisfy the following two rules for all positive integers $n:$\n$$a_{2n+1}=a_{2n}a_{2n-1},$$\n$$a_{2n+2}=a_{2n+1}+4.$$\nExactly $m$ of the numbers $a_{1}, a_{2}, \\ldots, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba06e5c34a7b6d1e72fc6dc8f3244db739f7bc2b8813c79441b311e1eab238b3 | A sequence of positive integers \( x_1 < x_2 < \dots < x_n \) is called an exquisite \( n \)-array if for all \( 1 \le i \le n \), \( x_i \) divides \( \prod_{j \neq i} x_j + 1 \). Find all exquisite \( n \)-arrays. | [
"\\{2, 3, 7, 43, 1807, \\ldots\\}"
] | [
"A sequence of positive integers \\( x_1 < x_2 < \\dots < x_n \\) is called an exquisite \\( n \\)-array if for all \\( 1 \\le i \\le n \\), \\( x_i \\) divides \\( \\prod_{j \\neq i} x_j + 1 \\). Find all exquisite \\( n \\)-arrays."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-99398acd119485790fea8c7b080b2afc46b5ce7d04988565d565154c9b6b5e8b | A sequence of positive integers \((a_n)_{n\ge 1}\) contains each positive integer exactly once. Assume that if \(m \neq n\), then \(\frac{1}{1998} < \frac{|a_m - a_n|}{|m - n|} < 1998\). Calculate the maximum possible value of \(|a_n - n|\) for all \(n\). | [
"1997"
] | [
"A sequence of positive integers \\((a_n)_{n\\ge 1}\\) contains each positive integer exactly once. Assume that if \\(m \\neq n\\), then \\(\\frac{1}{1998} < \\frac{|a_m - a_n|}{|m - n|} < 1998\\). Calculate the maximum possible value of \\(|a_n - n|\\) for all \\(n\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-873347e00edf19cb4700ff6a7a51f7915f121e65b52e178dd477952f46b25a03 | A sequence of positive integers \(a_0, a_1, a_2, \ldots, a_{2005}\) has \(a_0 = 2^{2006}\) and satisfies the inequality \[ \frac{a_{n-1}}{2} - 1 \le a_n \le \frac{a_{n-1}}{2}. \] If there are \(N\) possible such sequences, find the sum of the digits of \(N\). | [
"8"
] | [
"A sequence of positive integers \\(a_0, a_1, a_2, \\ldots, a_{2005}\\) has \\(a_0 = 2^{2006}\\) and satisfies the inequality\n\\[\n\\frac{a_{n-1}}{2} - 1 \\le a_n \\le \\frac{a_{n-1}}{2}.\n\\]\nIf there are \\(N\\) possible such sequences, find the sum of the digits of \\(N\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-b97c94f547da29048310f90ed88dcb7c54ebf5a5373e878dee3add988913b75d | A sequence of positive integers \(a_1, a_2, \ldots, a_{2014}\) is called [b][i]Cool[/i][/b] if it satisfies the following conditions: 1. \(a_i \leq 2014\) for every \(i\), 2. \(a_i \neq a_j\) for every \(i \neq j\), 3. \(a_i + i \leq a_j + j\) for every \(i < j\). Determine the number of [i][b]Cool[/b][/i] sequences. | [
"\\(2^{2013}\\)"
] | [
"A sequence of positive integers \\(a_1, a_2, \\ldots, a_{2014}\\) is called [b][i]Cool[/i][/b] if it satisfies the following conditions:\n1. \\(a_i \\leq 2014\\) for every \\(i\\),\n2. \\(a_i \\neq a_j\\) for every \\(i \\neq j\\),\n3. \\(a_i + i \\leq a_j + j\\) for every \\(i < j\\).\n\nDetermine the number of [... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 7,
"tir": 12
} | 55 | 0 | 0 |
omr-ws-4b2d938ea59e7ee9d1c334fc5f5d128e943d3b77960e130b7b23c1a4a1a4b399 | A sequence of positive integers \(a_1, a_2, \ldots\) is such that for each \(m\) and \(n\), if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_m\) is a divisor of \(a_n\) and \(a_m < a_n\). Find the least possible value of \(a_{2000}\). | [
"128"
] | [
"A sequence of positive integers \\(a_1, a_2, \\ldots\\) is such that for each \\(m\\) and \\(n\\), if \\(m\\) is a divisor of \\(n\\) and \\(m < n\\), then \\(a_m\\) is a divisor of \\(a_n\\) and \\(a_m < a_n\\). Find the least possible value of \\(a_{2000}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-959aefa7e33d75d7fdf2e9c94559ec92666cf7c96913f96dc96d4bd7136023a4 | A sequence of positive integers \(a_1, a_2, \ldots\) satisfies the inequality \[ a_n > \frac{a_{n+1} + a_{n+2} + \ldots + a_{2n}}{n+2016} \] for all positive integers \(n\). Determine the smallest possible value of \(C\) such that \(a_n < C\) for all positive integers \(n\), given that \(a_1 = 2017\). | [
"2017^2"
] | [
"A sequence of positive integers \\(a_1, a_2, \\ldots\\) satisfies the inequality\n\\[ a_n > \\frac{a_{n+1} + a_{n+2} + \\ldots + a_{2n}}{n+2016} \\]\nfor all positive integers \\(n\\). Determine the smallest possible value of \\(C\\) such that \\(a_n < C\\) for all positive integers \\(n\\), given that \\(a_1 = 20... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-93c722cf86b0591189327bfee95d928834cfc65cdb5440a338e8f0edf2d2d93a | A sequence of positive integers \(a_n\) begins with \(a_1 = a\) and \(a_2 = b\) for positive integers \(a\) and \(b\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \(n\): \[a_{2n+1} = a_{2n}a_{2n-1}, \quad a_{2n+2} = a_{2n+1} + 4.\] Exactly \(m\) of the numbers \(a_1, a_2, ... | [
"1012"
] | [
"A sequence of positive integers \\(a_n\\) begins with \\(a_1 = a\\) and \\(a_2 = b\\) for positive integers \\(a\\) and \\(b\\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \\(n\\):\n\\[a_{2n+1} = a_{2n}a_{2n-1}, \\quad a_{2n+2} = a_{2n+1} + 4.\\]\nExactly \\(m\\) of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ba5ad2ee30a455f9d3019968d048ca520b9279375cb4158a6948041d49fcd9c8 | A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is defined as follows: \(a_{1}=1\) and, for \(n \geq 1\), \(a_{n+1}\) is the smallest integer greater than \(a_{n}\) such that for any \(i, j, k\) (not necessarily distinct) in \(\{1, \ldots, n+1\}\) we have \(a_{i} + a_{j} \neq 3 a_{k}\). Determine \(a_{1999}\). | [
"4495"
] | [
"A sequence of positive integers \\(a_{1}, a_{2}, \\ldots\\) is defined as follows: \\(a_{1}=1\\) and, for \\(n \\geq 1\\), \\(a_{n+1}\\) is the smallest integer greater than \\(a_{n}\\) such that for any \\(i, j, k\\) (not necessarily distinct) in \\(\\{1, \\ldots, n+1\\}\\) we have \\(a_{i} + a_{j} \\neq 3 a_{k}\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 10
} | 18 | 0 | 0 |
omr-ws-5d2d8fb13b2da50e59d6d4fc8a23e8ee934bd9b48ed1d25c9255a373fd12b9a6 | A sequence of positive integers includes the number $68$ and has an arithmetic mean of $56$. When $68$ is removed, the arithmetic mean of the remaining sequence is $55$. What is the largest number that can occur in such a sequence? | [
"649"
] | [
"A sequence of positive integers includes the number $68$ and has an arithmetic mean of $56$. When $68$ is removed, the arithmetic mean of the remaining sequence is $55$. What is the largest number that can occur in such a sequence?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-923f239f928df9a7af9ad596501da1124ac9cb42cfe2d2a84dc0e2de84878433 | A sequence of positive integers is constructed as follows: the first term is $1$, the following two terms are $2$, $4$, the following three terms are $5$, $7$, $9$, the following four terms are $10$, $12$, $14$, $16$, etc. Find the $n$-th term of the sequence. | [
"\\( 2n - 2 \\left\\lfloor \\frac{\\sqrt{8n+1} - 1}{2} \\right\\rfloor \\)"
] | [
"A sequence of positive integers is constructed as follows: the first term is $1$, the following two terms are $2$, $4$, the following three terms are $5$, $7$, $9$, the following four terms are $10$, $12$, $14$, $16$, etc. Find the $n$-th term of the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 6,
"tir": 20
} | 62 | 0 | 0 |
omr-ws-a56056387d248caa6e9c67744be5616d07cf9215e2e9c1ad48958bfac3035110 | A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=(a_n)^2+1$ for all $n\geq0$. Find the greatest common divisor between $a_{999}$ and $a_{2004}$. | [
"677"
] | [
"A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=(a_n)^2+1$ for all $n\\geq0$. Find the greatest common divisor between $a_{999}$ and $a_{2004}$."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-854668f59983c7f27b9c07648df566bc3c6541ad7a07c8320796db3a94edb028 | A sequence of positive integers is defined by \( a_0 = 1 \) and \( a_{n+1} = a_n^2 + 1 \) for all \( n \geq 0 \). Find \( \gcd(a_{999}, a_{2004}) \). | [
"677"
] | [
"A sequence of positive integers is defined by \\( a_0 = 1 \\) and \\( a_{n+1} = a_n^2 + 1 \\) for all \\( n \\geq 0 \\). Find \\( \\gcd(a_{999}, a_{2004}) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-64a2c806fdfa383c70da84a806ddfb9711367fd6dd6003cb641d604839ed3826 | A sequence of positive integers is defined by \( a_0 = 1 \) and \( a_{n+1} = a_n^2 + 1 \) for each \( n \geq 0 \). Find \( \gcd(a_{999}, a_{2004}) \). | [
"677"
] | [
"A sequence of positive integers is defined by \\( a_0 = 1 \\) and \\( a_{n+1} = a_n^2 + 1 \\) for each \\( n \\geq 0 \\). Find \\( \\gcd(a_{999}, a_{2004}) \\)."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-9be2a0d39833015bc0372057adda1e924b96e9e1c903f41e600a4c9b3c713c8e | A sequence of positive integers is defined by \(a_0 = 1\) and \(a_{n+1} = (a_n)^2 + 1\). Find \(\gcd(a_{999}, a_{2004})\). | [
"677"
] | [
"A sequence of positive integers is defined by \\(a_0 = 1\\) and \\(a_{n+1} = (a_n)^2 + 1\\). Find \\(\\gcd(a_{999}, a_{2004})\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-b0f1caebd73b8b8f941c0b99f4f79fbaeabde40df269996724153f612817bce1 | A sequence of positive integers is defined by \(a_{0}=1\) and \(a_{n+1}=a_{n}^{2}+1\) for each \(n \geq 0\). Find \(\gcd(a_{999}, a_{2004})\). | [
"677"
] | [
"A sequence of positive integers is defined by \\(a_{0}=1\\) and \\(a_{n+1}=a_{n}^{2}+1\\) for each \\(n \\geq 0\\). Find \\(\\gcd(a_{999}, a_{2004})\\)."
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-9dad2aaba1d512c1ee89d00f36ea677676d1e8d5f702254328fe3d5973444ab3 | A sequence of positive integers is defined such that \( a_1 = k \) where \( k \) is a positive integer, and \( a_{200} = 30300 \). The sequence is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and in general, for all \( n \geq 1 \),... | [
"3"
] | [
"A sequence of positive integers is defined such that \\( a_1 = k \\) where \\( k \\) is a positive integer, and \\( a_{200} = 30300 \\). The sequence is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and in general, for all \\( ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
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