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-5a4bdc22e37f93cbdf5f52b7fb8f519367c5a9ed4183b17d075bffc4ef3e0573 | What is the inverse of the function \( y = \frac{\pi}{2} + \cos^{-1} x \)? | [
"\\sin x"
] | [
"What is the inverse of the function \\( y = \\frac{\\pi}{2} + \\cos^{-1} x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7403ef3d22782363068f393540fe0106357e5e4be0e9166d78e1ff266c600830 | What is the inverse of the subtraction of two Vandermonde matrices \(X\) and \(Y\)? Specifically, find the inverse of the matrix \(X - Y\), where \[ X = \begin{bmatrix} x_{1} & x_{1}^{2} & \cdots & x_{1}^{k} \\ x_{2} & x_{2}^{2} & \cdots & x_{2}^{k} \\ x_{3} & x_{3}^{2} & \cdots & x_{3}^{k} \\ \vdots & \vdots & \ddots ... | [
"(X - Y)^{-1}"
] | [
"What is the inverse of the subtraction of two Vandermonde matrices \\(X\\) and \\(Y\\)? Specifically, find the inverse of the matrix \\(X - Y\\), where\n\\[ X = \\begin{bmatrix}\nx_{1} & x_{1}^{2} & \\cdots & x_{1}^{k} \\\\\nx_{2} & x_{2}^{2} & \\cdots & x_{2}^{k} \\\\\nx_{3} & x_{3}^{2} & \\cdots & x_{3}^{k} \\\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-9e2dae8509aca254797565cb4ab32de25667ba584be8d576de8196abf7954d20 | What is the joint characteristic function for a chi-square distribution on $k$ standard normal variables and the usual normal distribution with mean $0$ and variance $k$? | [
"(1 - 2i t_1)^{-k/2} \\exp\\left( -\\frac{k t_2^2}{2(1 - 2i t_1)} \\right)"
] | [
"What is the joint characteristic function for a chi-square distribution on $k$ standard normal variables and the usual normal distribution with mean $0$ and variance $k$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-4be005f9bb07c791b03d24b8ef4ee26728645e84cfc67eb28ca15071a7e9e581 | What is the joint distribution of several independent binomial variables (with the same parameter but different indices) given that their sum takes a specified value? | [
"\\text{Multivariate hypergeometric distribution with parameters } (n_1, n_2, \\ldots, n_m) \\text{ and } s"
] | [
"What is the joint distribution of several independent binomial variables (with the same parameter but different indices) given that their sum takes a specified value?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-84e3d284967e977ba0c9e3d9973f1166251db74d03e063bc3079ac6b44336941 | What is the kernel \( G(t - t^{\prime}) \) of the integral operator in the expression \[ x(t) = \int_{-\infty}^{\infty} G(t - t^{\prime})\; F(t^{\prime})\; dt^{\prime} \] for the solution of the Newton equation \( m\ddot{x} = F(t) \)? | [
"\\( G(t - t^{\\prime}) = \\begin{cases} 0 & \\text{if} \\ t < t^{\\prime} \\\\ \\frac{t - t^{\\prime}}{m} & \\text{if} \\ t > t^{\\prime} \\end{cases} \\)"
] | [
"What is the kernel \\( G(t - t^{\\prime}) \\) of the integral operator in the expression \n\\[ x(t) = \\int_{-\\infty}^{\\infty} G(t - t^{\\prime})\\; F(t^{\\prime})\\; dt^{\\prime} \\]\nfor the solution of the Newton equation \\( m\\ddot{x} = F(t) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-8561b48142f4c1d9222daa536e41705a5efc303b6fff98d1da654312ee506739 | What is the kernel of the homomorphism \( z \to |z| \) from the multiplicative group \( \mathbb{C}^* \) of nonzero complex numbers to itself? | [
"\\( S^1 \\)"
] | [
"What is the kernel of the homomorphism \\( z \\to |z| \\) from the multiplicative group \\( \\mathbb{C}^* \\) of nonzero complex numbers to itself?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-34dde6543ef7d240a840018ec2706a884303d34bbe22da71dddf243779ecaf75 | What is the largest $2$ -digit prime factor of the integer $n = {200\choose 100}$? | [
"061"
] | [
"What is the largest $2$ -digit prime factor of the integer $n = {200\\choose 100}$?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-cb80f8a05237889f57ebd91391f13dca9fcd7fca37f0b25a67e0b724d56f7cc1 | What is the largest $2$-digit prime factor of the integer $n = {200\choose 100}$? | [
"61"
] | [
"What is the largest $2$-digit prime factor of the integer $n = {200\\choose 100}$?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d982f3b12bbbb45cb1d011c5e49501e5488b728c3363a23d9246ff8feec557a1 | What is the largest $4$ digit integer congruent to $15 \pmod{22}?$ | [
"9981"
] | [
"What is the largest $4$ digit integer congruent to $15 \\pmod{22}?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf90467c31fdd5eacdacdba0f17a1ac4eb862d07bcf26980396e1983605522cd | What is the largest $k$ for which the inequality $$\left(a^2+k\right)\left(b^2+k\right)\left(c^2+k\right)\left(d^2+k\right)\leq (1+k)^4$$ holds for all $a, b, c, d \geq 0$ satisfying $a + b + c + d = 4$? | [
"\\dfrac{1}{3}"
] | [
"What is the largest $k$ for which the inequality\n$$\\left(a^2+k\\right)\\left(b^2+k\\right)\\left(c^2+k\\right)\\left(d^2+k\\right)\\leq (1+k)^4$$\nholds for all $a, b, c, d \\geq 0$ satisfying $a + b + c + d = 4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-0a63ca770bcfb276d2c361c83d946afa1ba523e21a3b13ba8c27da59f9b13413 | What is the largest $k$ such that for any partition of an infinite cell grid into dominoes, there exists a $100 \times 100$ square that contains at least $k$ whole dominoes? | [
"4800"
] | [
"What is the largest $k$ such that for any partition of an infinite cell grid into dominoes, there exists a $100 \\times 100$ square that contains at least $k$ whole dominoes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-c700fb1cea4e79de50756f08c77c2fcd9b6c924322399273c93f5d222596edba | What is the largest $k$ such that for positive reals $x, y, z$, the inequality $$\sum_{sym}x^6 + k\sum_{sym}x^3y^3 \ge (k+1)\sum_{sym}x^4y^2$$ holds true? | [
"2"
] | [
"What is the largest $k$ such that for positive reals $x, y, z$, the inequality $$\\sum_{sym}x^6 + k\\sum_{sym}x^3y^3 \\ge (k+1)\\sum_{sym}x^4y^2$$ holds true?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-4dee10a5f7493c4413ed6d9fbaeb4c0f0c162c2bbbc22a6296edc8c99e9e0264 | What is the largest $k$ such that in any triangle, there are two sides whose ratio is greater than $k$ but less than $1/k$? | [
"\\(\\frac{-1 + \\sqrt{5}}{2}\\)"
] | [
"What is the largest $k$ such that in any triangle, there are two sides whose ratio is greater than $k$ but less than $1/k$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 20
} | 39 | 0 | 0 |
omr-ws-fc12cc7d13c73008f23e038b8b801a643fa316de89d27ae7220bb7696c9f83e8 | What is the largest $k$ such that there is an element of $G$ of order $2^k$, where $G$ is the set of all invertible $4 \times 4$ matrices over the integers mod 2? | [
"4"
] | [
"What is the largest $k$ such that there is an element of $G$ of order $2^k$, where $G$ is the set of all invertible $4 \\times 4$ matrices over the integers mod 2?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cee4d278bf5d4dd661821949928716410e2535efef253cdbe5b02c5854db78f | What is the largest $n$ for which there exists a convex $n$-gon whose diagonal lengths are at most two different values? | [
"7"
] | [
"What is the largest $n$ for which there exists a convex $n$-gon whose diagonal lengths are at most two different values?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-dfa793c5f1b0102e7a5ef67e782b2eb4ef013a2e8132cfd1307c5ed2d56b04a0 | What is the largest $n$ such that the sum \[ \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + \cdots + \dfrac{1}{n} < 3 \]? | [
"11"
] | [
"What is the largest $n$ such that the sum \\[ \\dfrac{1}{1} + \\dfrac{1}{2} + \\dfrac{1}{3} + \\cdots + \\dfrac{1}{n} < 3 \\]?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59547687b620767f56bd9ccce4b376fd6616b5caea6e3054eb210c696f504f0f | What is the largest $n$ such that there exist $n$ seven-digit numbers that are successive members of a geometric progression? | [
"11"
] | [
"What is the largest $n$ such that there exist $n$ seven-digit numbers that are successive members of a geometric progression?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-f3b51478d520ceb4f7abf5df8bf5bb62360f3a5854fec5051914f1d8ff5f54b0 | What is the largest $n$ such that there exist natural numbers $a_1, a_2, \ldots, a_n$ with the property that all numbers $a_i^2 + a_j^2$ for $1 \le i < j \le n$ are perfect squares? | [
"3"
] | [
"What is the largest $n$ such that there exist natural numbers $a_1, a_2, \\ldots, a_n$ with the property that all numbers $a_i^2 + a_j^2$ for $1 \\le i < j \\le n$ are perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-fe85417f8f1d2364662088bd92c58ce31b7144f9563d60379d451c5a49701705 | What is the largest $n$ such that there exists an $n$-gon with two adjacent sides of length $1$ and all diagonals having integer lengths? | [
"4"
] | [
"What is the largest $n$ such that there exists an $n$-gon with two adjacent sides of length $1$ and all diagonals having integer lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-e0e1763e44415f6db7d8b2c71adf6be99a2dca26177bf89e44d62dc40112f3da | What is the largest (in area) triangle that can fit within the overlapping area of two circles, each with radius \( x \) units, where the center of one circle lies on the circumference of the other circle? | [
"\\(\\frac{\\sqrt{3}}{4} x^2\\)"
] | [
"What is the largest (in area) triangle that can fit within the overlapping area of two circles, each with radius \\( x \\) units, where the center of one circle lies on the circumference of the other circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 6
} | 16 | 0 | 0 |
omr-ws-be4d01f2038e22a9d8283d24cde61c29711e273f09f40d01dfa683a230a1ec66 | What is the largest (in terms of area) equiangular hexagon that can fit inside a unit cube? | [
"\\dfrac{3\\sqrt{3}}{4}"
] | [
"What is the largest (in terms of area) equiangular hexagon that can fit inside a unit cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-604db0353cddce69c47e02496ac554035624881d3422a65e6bac0782a700d355 | What is the largest (smallest) perimeter that a figure consisting of $10$ cells can have? | [
"22"
] | [
"What is the largest (smallest) perimeter that a figure consisting of $10$ cells can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-b742639d99245f1a315ed464b3ebcc5cce506da3b3c4216734b6558f82763f77 | What is the largest 2-digit prime factor of $ {200 \choose 100}$? | [
"97"
] | [
"What is the largest 2-digit prime factor of $ {200 \\choose 100}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af7dbf76592174b488434d013d2f65b89a6ee86cf74b78a98c5d177e2fd295d6 | What is the largest 2-digit prime factor of the binomial coefficient \(\binom{200}{100}\)? | [
"61"
] | [
"What is the largest 2-digit prime factor of the binomial coefficient \\(\\binom{200}{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d4c0136b9360eede60a6fe2b7323c4100d4931d1709acdca4744d4b5a34d5921 | What is the largest 2-digit prime factor of the integer $n = \binom{200}{100}$? | [
"61"
] | [
"What is the largest 2-digit prime factor of the integer $n = \\binom{200}{100}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-e002a27d3788f4ed5cffdc10abf8acde6d5f8f7e183c69fb8ce96514cc28685e | What is the largest 2-digit prime factor of the integer \( n = \binom{200}{100} \)? | [
"61"
] | [
"What is the largest 2-digit prime factor of the integer \\( n = \\binom{200}{100} \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 49
} | 80 | 0 | 0 |
omr-ws-c6320949d12442f24f0d4e793cdcbdf44d50777ba0d200b412125bb49f02f589 | What is the largest 2-digit prime factor of the integer \(n = \binom{200}{100}\)? | [
"61"
] | [
"What is the largest 2-digit prime factor of the integer \\(n = \\binom{200}{100}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-101350e479e5da2054a81d123f5d53a9f4f4d6c913ec7950ab259e9ca34d1f94 | What is the largest 2-digit prime factor of the integer \({200 \choose 100}\)? | [
"61"
] | [
"What is the largest 2-digit prime factor of the integer \\({200 \\choose 100}\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f14b6845fd2008cd08bf3605fcac2825c998bf22229e304d9299725d78d8d939 | What is the largest 3-digit integer \( k \) such that concatenating \( k \) and the 3-digit number \( k + 23 \) (in that order) gives a square number? | [
"938"
] | [
"What is the largest 3-digit integer \\( k \\) such that concatenating \\( k \\) and the 3-digit number \\( k + 23 \\) (in that order) gives a square number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd93965608b910908f17e39de0b38b7aef47054df2a4886200da5d451bb698e1 | What is the largest 3-digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$? | [
"598"
] | [
"What is the largest 3-digit number $\\overline{abc}$ such that $b \\cdot \\overline{ac} = c \\cdot \\overline{ab} + 50$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f090c66b15e01bbd7da719c831d93a793b69cb3123552fcd7e08788e0deb21c | What is the largest 3-digit number $\overline{abc}$ such that \[b \cdot \overline{ac} = c \cdot \overline{ab} + 50?\] | [
"598"
] | [
"What is the largest 3-digit number $\\overline{abc}$ such that\n\n\\[b \\cdot \\overline{ac} = c \\cdot \\overline{ab} + 50?\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5adee440a41eb3369b5bb961d7053cece5f6691b6aae2f1d3654491dd6dbb826 | What is the largest 3-digit palindrome that, when multiplied by 111, remains a palindrome? | [
"414"
] | [
"What is the largest 3-digit palindrome that, when multiplied by 111, remains a palindrome?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-09457a4c05d024b525710a22ae4aa502ca37952c3d82ff7ed09e50de65b08e5d | What is the largest 4-digit number that is both open (its one’s digit is a perfect square that is not 0) and cubedd (divisible by a perfect cube that is not 1)? | [
"9984"
] | [
"What is the largest 4-digit number that is both open (its one’s digit is a perfect square that is not 0) and cubedd (divisible by a perfect cube that is not 1)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a61e17566dd7a3393305b4de4177b0128678d47abe3be031f4b40635dd0fc71c | What is the largest 4-digit palindrome that can be expressed as the sum of two different 3-digit palindromes? | [
"1221"
] | [
"What is the largest 4-digit palindrome that can be expressed as the sum of two different 3-digit palindromes?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 7,
"tir": 22
} | 63 | 0 | 0 |
omr-ws-756d55050e8cb5980ba213b4d76a1dd799e16572e8a599f30ddaf88c016dd4e6 | What is the largest 5-digit number such that its square has 9 digits? | [
"31622"
] | [
"What is the largest 5-digit number such that its square has 9 digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-5481915b5d1a5a7de92cea1c1ce38c249a177827453f067490d124123c57b2d8 | What is the largest Pythagorean triple in which two numbers are consecutive and each number is less than 115? | [
"15, 112, 113"
] | [
"What is the largest Pythagorean triple in which two numbers are consecutive and each number is less than 115?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-56d542b56801f80d50a336a7991d2eecfee7c42ba01682c22c918bc84daf8964 | What is the largest \( n \) such that \( n! \) is a product of \( n - 3 \) consecutive positive integers? | [
"23"
] | [
"What is the largest \\( n \\) such that \\( n! \\) is a product of \\( n - 3 \\) consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-3163723f94715176659afc793ab94d9c5d44fa5c15cc63f67eda48d5875c1366 | What is the largest \(n\) for which you can create a sequence \(a_1, a_2, a_3, \ldots, a_n\) of natural numbers such that for all \(i < n - 2\), \({a_i}^2 + {a_{i+1}}^2 = {a_{i+2}}^2\)? | [
"4"
] | [
"What is the largest \\(n\\) for which you can create a sequence \\(a_1, a_2, a_3, \\ldots, a_n\\) of natural numbers such that for all \\(i < n - 2\\), \\({a_i}^2 + {a_{i+1}}^2 = {a_{i+2}}^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-58d44322d262b15da493fd35905a5cd0c1f23022f90b142f7dadf498a7aa78b3 | What is the largest angle for which the tangent of the angle can be approximated with an error less than 10%? | [
"0.54"
] | [
"What is the largest angle for which the tangent of the angle can be approximated with an error less than 10%?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-a16473e15e2302d5e13539910d1df6365aa9fe89bf4b64b6d58285094d3295cf | What is the largest area a heptagon can have if its vertices lie at the nodes of an integer lattice with a side of $1$ m, and the lengths of its sides are $1, 2, 3, 4, 5, 6,$ and $7$ meters (in some order)? | [
"36"
] | [
"What is the largest area a heptagon can have if its vertices lie at the nodes of an integer lattice with a side of $1$ m, and the lengths of its sides are $1, 2, 3, 4, 5, 6,$ and $7$ meters (in some order)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-03346f5d6fc278c58e825fc2d22103853544e096b987f252877c31b972315b7c | What is the largest area a triangle can have on the surface of a cube with an edge of length $1$? | [
"\\dfrac{\\sqrt{3}}{2}"
] | [
"What is the largest area a triangle can have on the surface of a cube with an edge of length $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-5ed1e243dc1915dd5b921df17e72751b61b1a1a05c40071d3e2eeb7547939763 | What is the largest area a triangle can have, whose sides \(a\), \(b\), and \(c\) satisfy the conditions \(0 < a \leq 1 \leq b \leq 2 \leq c \leq 3\)? | [
"1"
] | [
"What is the largest area a triangle can have, whose sides \\(a\\), \\(b\\), and \\(c\\) satisfy the conditions \\(0 < a \\leq 1 \\leq b \\leq 2 \\leq c \\leq 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a58b880d7cd249919f3ca187b982d9a3fa4dcfa8950edf97ead1ab609a6c7bab | What is the largest area of a regular hexagon that can be drawn inside an equilateral triangle with side length $3$? | [
"\\(\\frac{3\\sqrt3}{2}\\)"
] | [
"What is the largest area of a regular hexagon that can be drawn inside an equilateral triangle with side length $3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-8824f4cec14683d75768aba55fd5571b7a0204c818e0252bd924ba08cca59182 | What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$, and $c$ from a certain point, where $a$ is the distance to the vertex of the right angle? | [
"\\(\\frac{1}{2}bc\\)"
] | [
"What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$, and $c$ from a certain point, where $a$ is the distance to the vertex of the right angle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63fd731d7110a12cc1d654b600a7b146dd7e3be84183b19df74357199ac92641 | What is the largest area of a slice if it has a right angle when cutting a rectangular parallelepiped with edges $2, 4, 8$? | [
"\\(16\\sqrt{5}\\)"
] | [
"What is the largest area of a slice if it has a right angle when cutting a rectangular parallelepiped with edges $2, 4, 8$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 7,
"tir": 15
} | 48 | 0 | 0 |
omr-ws-0c4e286b36b76fd0156f9e0fbc2d5db1fee957902381f61ff88eeaf66e6628f1 | What is the largest area of a square that can be covered with two circles, each having a radius of 1/2? | [
"1"
] | [
"What is the largest area of a square that can be covered with two circles, each having a radius of 1/2?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-2a1ef175b3dcace755da0ab40607d8ab411d16b992c14757a516a124c5ecca3c | What is the largest area that a closed polyline of length $2004$ can cover when constructed on checkered paper along the lines of a rectangular grid with side $1$? | [
"251001"
] | [
"What is the largest area that a closed polyline of length $2004$ can cover when constructed on checkered paper along the lines of a rectangular grid with side $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-4819c207c724fa3239bdef35ccd63341eb48190e2ffaaf5807f3cac801061d59 | What is the largest area that a closed polyline of length $24$ can cover when constructed on checkered paper along the lines of a rectangular grid with side $1$? | [
"36"
] | [
"What is the largest area that a closed polyline of length $24$ can cover when constructed on checkered paper along the lines of a rectangular grid with side $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-97de2c82e7a6f53ecd6024ceced9e6384e394126a56f9d7e0c75702439dab913 | What is the largest area that can be covered by an $n$-sided cyclic polygon in a unit circle? | [
"\\( n \\sin\\left(\\frac{2\\pi}{n}\\right) \\)"
] | [
"What is the largest area that can be covered by an $n$-sided cyclic polygon in a unit circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-755f21a0d3fb58e55636bf4e6201bc969cdbac2ec256c5e4f732f6693eb22a16 | What is the largest ball that can be contained within a cube in $\mathbb{R}^m$? | [
"\\(\\frac{s}{2}\\)"
] | [
"What is the largest ball that can be contained within a cube in $\\mathbb{R}^m$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-fe444eba30f91b8607dac26ffe39c0305f02ef16f2e6f171d1c60adeacc7b6cb | What is the largest base-4 number that has four digits? Express your answer in base 10. | [
"255"
] | [
"What is the largest base-4 number that has four digits? Express your answer in base 10."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb4488c6577f1585dda20174a64e63492d7673aa0282b3fc73ea82ca7a93a8cc | What is the largest cardinality of a set of pairwise disjoint 'T' letters in the Euclidean plane, where the horizontal and vertical lines of each 'T' are of equal size? | [
"\\(\\aleph_0\\)"
] | [
"What is the largest cardinality of a set of pairwise disjoint 'T' letters in the Euclidean plane, where the horizontal and vertical lines of each 'T' are of equal size?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-c3c54f5a5d2a4187422b026835144749e81ba9adede93de9c443e46778ea7e6d | What is the largest coefficient that occurs when expanding the expression \((x_1 + x_2 + x_3 + x_4)^6\)? | [
"180"
] | [
"What is the largest coefficient that occurs when expanding the expression \\((x_1 + x_2 + x_3 + x_4)^6\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-b3f019f7437f0a11afca46519b030dd72530f7b733960188dd6f54ba6387a357 | What is the largest digit $N$ for which $2345N$ is divisible by 6? | [
"4"
] | [
"What is the largest digit $N$ for which $2345N$ is divisible by 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-211dffdc0d9ba36cfd7c3e8f1e552409dfe1fa8f320db0668b48877e58db1809 | What is the largest dimension of a subspace of $M_{31}(\mathbb{R})$ in which all matrices except the zero matrix are invertible? | [
"1"
] | [
"What is the largest dimension of a subspace of $M_{31}(\\mathbb{R})$ in which all matrices except the zero matrix are invertible?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-76f73cd61dbe80d02ff79a46eab40981a647833a2a83d77dd9cd34da4afc40e4 | What is the largest disk around $z_0=1$ on which the function $f(z) = e^z$ is one-to-one? | [
"\\( D(1, \\pi) \\) or radius \\( \\pi \\)."
] | [
"What is the largest disk around $z_0=1$ on which the function $f(z) = e^z$ is one-to-one?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-1184465e050360403aa905cdf51ef8ac3cd84eadd08c96b4bb5be29d3a90b712 | What is the largest divisor of 342 that is also a factor of 285? | [
"57"
] | [
"What is the largest divisor of 342 that is also a factor of 285?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74da695bcd76b8c30054aba06f3592071be6e97f1a178759d8908dd83cd54e7c | What is the largest equilateral triangle that can be fitted into a cone as a function of the height of the cone, given that the base of the cone is a circle with diameter \(d\)? Note: The problem specifies that if the height is zero, the largest equilateral triangle has sides of \(\frac{\sqrt{3} \times d}{2}\), and if ... | [
"\\( s(h) = \\begin{cases} \n\\frac{\\sqrt{3} \\times d}{2} & \\text{if } h = 0 \\\\\nd & \\text{if } h \\geq \\frac{\\sqrt{3} \\times d}{2}\n\\end{cases} \\)"
] | [
"What is the largest equilateral triangle that can be fitted into a cone as a function of the height of the cone, given that the base of the cone is a circle with diameter \\(d\\)? \n\nNote: The problem specifies that if the height is zero, the largest equilateral triangle has sides of \\(\\frac{\\sqrt{3} \\times d... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-ba56ff45f40446657bb2550bf55a97a07ab8a52ebe178a8e6503f62fec22664f | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | [
"38"
] | [
"What is the largest even integer that cannot be written as the sum of two odd composite numbers?"
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-5c6f2ae608f21c50abd117d289564fd018ecc51d7fd2cb99bf690d9b5b4259b7 | What is the largest even integer that cannot be written as the sum of two odd composite positive integers? | [
"38"
] | [
"What is the largest even integer that cannot be written as the sum of two odd composite positive integers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ebfde43f92c5c98526e1054c644f1724bd8883216a47a07610ff3341a17da80 | What is the largest factorial that ends in exactly 200 zeros? | [
"809"
] | [
"What is the largest factorial that ends in exactly 200 zeros?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-8f68e4299a25ffdb8e3e9c36ecf97f6275761308d5e26d4f74d20b7478586b65 | What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$? | [
"98752"
] | [
"What is the largest five-digit integer whose digits have a product equal to the product $(7)(6)(5)(4)(3)(2)(1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1340127f53a55891623bd1ea82314c76e35af8a7ade27cb749117ece5b01df5 | What is the largest four-digit negative integer congruent to $1 \pmod{23}?$ | [
"-1011"
] | [
"What is the largest four-digit negative integer congruent to $1 \\pmod{23}?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa17e525b5613b17a44a1919c99a96a77746c3b399b36e1b3331ec1826a9b231 | What is the largest four-digit number that is divisible by 4? | [
"9996"
] | [
"What is the largest four-digit number that is divisible by 4?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-940d579af5649ee2d4d67f17308cd7a2c0ace6f71d521c9f31f9fbbc811f5c9f | What is the largest four-digit number whose digits add up to 16? | [
"9700"
] | [
"What is the largest four-digit number whose digits add up to 16?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b9f57e007152a462e9dc14fd106f85a0cc850dcbd5aac5b70faf85c46808c72 | What is the largest group of people such that among any 3 of them, at least 2 are acquainted, and no four of them are mutually acquainted? | [
"8"
] | [
"What is the largest group of people such that among any 3 of them, at least 2 are acquainted, and no four of them are mutually acquainted?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c7338c5090557522692beb806dc6c53a0d62c527b39e9cdb9b9139e8c35d7d11 | What is the largest integer $ N$ for which $ n^{5} - 5n^{3} + 4n$ is divisible by $ N$ for every integer $ n$? | [
"120"
] | [
"What is the largest integer $ N$ for which $ n^{5} - 5n^{3} + 4n$ is divisible by $ N$ for every integer $ n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-0ef01dea8c2dd94faeafadb3a39a0d172e6dd2545383a1381d92f527df613704 | What is the largest integer $n$ for which $n^3+1631$ is divisible by $n+11$? | [
"289"
] | [
"What is the largest integer $n$ for which $n^3+1631$ is divisible by $n+11$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-7b5c7abe7583aa799fdee15652910161ed074a11bc282bd3c6e521d4604fa944 | What is the largest integer $n$ for which a circle with center $(\sqrt{2}, \sqrt{2})$ can pass through exactly $n$ points with integer coordinates? | [
"2"
] | [
"What is the largest integer $n$ for which a circle with center $(\\sqrt{2}, \\sqrt{2})$ can pass through exactly $n$ points with integer coordinates?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aa3e248dcc7ef63db687f6dd9cb2bd807f3419bf27798d7a166dc7a9609e4696 | What is the largest integer $n$ less than 999 such that $(n-1)^2$ divides $n^{2016}-1$? | [
"673"
] | [
"What is the largest integer $n$ less than 999 such that $(n-1)^2$ divides $n^{2016}-1$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-41f7ad59f750a6b9f857e2accf17e46d82ab65faaf6c017b23b26a776ed01781 | What is the largest integer $n$ less than or equal to $2000$ such that $3^7$ will divide $\binom{2n}{n}$? | [
"1943"
] | [
"What is the largest integer $n$ less than or equal to $2000$ such that $3^7$ will divide $\\binom{2n}{n}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-d01f9d91cc026c066f8acce0804bb4564b0d2a2c6fa9716ca2355389cf0400b3 | What is the largest integer $n$ such that $2^{869} > 10^n$? | [
"261"
] | [
"What is the largest integer $n$ such that $2^{869} > 10^n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-695ceca9e72b380f94c3d8bb581f9a413858e241fa6dc1589eba68b9989f47de | What is the largest integer $n$ such that $5^n$ divides $\frac{2006!}{(1003!)^2}$? | [
"2"
] | [
"What is the largest integer $n$ such that $5^n$ divides $\\frac{2006!}{(1003!)^2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-824685fd5a89dc75d00327804cc4fe01a68f76c79f81924603d37411527ccacb | What is the largest integer $n$ such that $\frac{n^2-2012}{n+7}$ is an integer? | [
"1956"
] | [
"What is the largest integer $n$ such that $\\frac{n^2-2012}{n+7}$ is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-aff186cb6f766957273998a6d85f8534033b702456777779615651fa6b51e811 | What is the largest integer $n$ such that $n - 2$ is a divisor of $n + 10$? | [
"14"
] | [
"What is the largest integer $n$ such that $n - 2$ is a divisor of $n + 10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-815af3be09b48f67abaf5618429fac3b638e82cb4de973e052b576ab08a5c86d | What is the largest integer $n$ such that at least one of the players in an $n$-in-a-row game on an infinite board has a winning strategy? | [
"5"
] | [
"What is the largest integer $n$ such that at least one of the players in an $n$-in-a-row game on an infinite board has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-20aeebe5a9b7a0f4acb94bc1f5dc4c9cd2f5a340c025e95ef27ce64df187d274 | What is the largest integer $n$ such that the number $4^{37} + 4^{1000} + 4^n$ is a perfect square? | [
"1962"
] | [
"What is the largest integer $n$ such that the number $4^{37} + 4^{1000} + 4^n$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-d54de7c5d2b1a858d1196cba2dafd4925ffaf2151a46cfb3c272fbd9f0b4b678 | What is the largest integer $x$ such that $\frac{x}{3}+\frac{4}{5} < \frac{5}{3}$? | [
"2"
] | [
"What is the largest integer $x$ such that $\\frac{x}{3}+\\frac{4}{5} < \\frac{5}{3}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b8ee6527601804217a586e5b762126a2ae96ce6cf4b97735ba09a5d4e5ea7cd | What is the largest integer N such that there is certainly some class in some public school with at least N students sharing the same birth month, given that class sizes range from 15 to 65 students? | [
"6"
] | [
"What is the largest integer N such that there is certainly some class in some public school with at least N students sharing the same birth month, given that class sizes range from 15 to 65 students?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-6829c46dfb6f37a2f7e93ed070b6959f07ed227f88a8da4a386ef0dc8b6f5741 | What is the largest integer \( a < 999 \) such that \( (a-1)^2 \) divides \( a^{2020} - 1 \)? | [
"506"
] | [
"What is the largest integer \\( a < 999 \\) such that \\( (a-1)^2 \\) divides \\( a^{2020} - 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-7039b4bf5b9acc946c89d0c3bb487a83ba527f90710a29f5440d6b0ed78ef664 | What is the largest integer \( a \) such that \( b^5 - 5b^3 + 4b \) is divisible by \( a \) for any integer \( b \)? | [
"120"
] | [
"What is the largest integer \\( a \\) such that \\( b^5 - 5b^3 + 4b \\) is divisible by \\( a \\) for any integer \\( b \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-83bda957e6d91a1bb8892cc550c2ab1582bf79bba71b77de7098ae7bd5aa93c1 | What is the largest integer \( k \) such that \( 5^k \) divides \( 3(10!) + 12(5!) + 4(7!) \)? | [
"3"
] | [
"What is the largest integer \\( k \\) such that \\( 5^k \\) divides \\( 3(10!) + 12(5!) + 4(7!) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b534f85b0d087f72a6a570338c7fa152e3399294b5b27719af82a62ed4144279 | What is the largest integer \( k \) such that \( \frac{3}{2} \cdot \frac{2}{1} \cdot \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \ldots \cdot \frac{k}{k+1} \geq \frac{1}{8} \)? | [
"23"
] | [
"What is the largest integer \\( k \\) such that \\( \\frac{3}{2} \\cdot \\frac{2}{1} \\cdot \\frac{1}{2} \\cdot \\frac{2}{3} \\cdot \\frac{3}{4} \\cdot \\ldots \\cdot \\frac{k}{k+1} \\geq \\frac{1}{8} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5c9aa957ccf38dadae37398854dc7e5a62f8230ab0a53f5cdb4938c4448571e | What is the largest integer \( k \) such that \( k^2 \) is a factor of \( 10! \)? | [
"720"
] | [
"What is the largest integer \\( k \\) such that \\( k^2 \\) is a factor of \\( 10! \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-8ecf50552215a878944c5a8dd080a2656791e9ac04a582d16521d01fd0727a0e | What is the largest integer \( k \) such that a configuration made of congruent regular hexagons, where each hexagon shares a side with another hexagon, cannot have \( k \) vertices? | [
"11",
"8"
] | [
"What is the largest integer \\( k \\) such that a configuration made of congruent regular hexagons, where each hexagon shares a side with another hexagon, cannot have \\( k \\) vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-8ad8eebdd012178d4e8cb4a88f3a23dc8f7bca6d53014fbb7d23ca710ca11440 | What is the largest integer \( k \) such that a configuration made up of congruent regular hexagons, where each hexagon shares a side with another hexagon, cannot have \( k \) vertices? | [
"15"
] | [
"What is the largest integer \\( k \\) such that a configuration made up of congruent regular hexagons, where each hexagon shares a side with another hexagon, cannot have \\( k \\) vertices?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f8aa9ea0eb796f0e82a25f48c8ed5faed377644d0f0e24de95963708fd36b9e | What is the largest integer \( n < 9613 \) such that \( 9613^2 - n^2 \) is a perfect square? | [
"9595"
] | [
"What is the largest integer \\( n < 9613 \\) such that \\( 9613^2 - n^2 \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-eaf1096ee9d8e13b925c24069115d12cec475db328a48b1681c9c8dd6be55669 | What is the largest integer \( n \) for which \( n^2 + 24n + 16 \) is a perfect square? | [
"21"
] | [
"What is the largest integer \\( n \\) for which \\( n^2 + 24n + 16 \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-de50053c012a8b1782555ed7c0555e598c3d6b15aeb9df64d384d3c776899901 | What is the largest integer \( n \) such that \( (100^2 - 99^2)(99^2 - 98^2) \cdots (3^2 - 2^2)(2^2 - 1^2) \) is divisible by \( 3^n \)? | [
"49"
] | [
"What is the largest integer \\( n \\) such that \\( (100^2 - 99^2)(99^2 - 98^2) \\cdots (3^2 - 2^2)(2^2 - 1^2) \\) is divisible by \\( 3^n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-12cc9c1324ff4f5c2dde8f34df459adc776aba284b9702447e9b082a3bf3992a | What is the largest integer \( n \) such that \( 2^n + 65 \) is equal to the square of an integer? | [
"10"
] | [
"What is the largest integer \\( n \\) such that \\( 2^n + 65 \\) is equal to the square of an integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ec3a8da29d05c58b67272f0b00b439751dc6d9fa26b2d158c849db51bdf64bba | What is the largest integer \( n \) such that \( 4^{37} + 4^{1000} + 4^n \) is a perfect square? | [
"1962"
] | [
"What is the largest integer \\( n \\) such that \\( 4^{37} + 4^{1000} + 4^n \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50a73ff097d4a9e3371b8d26f721facdeab0b605172d404c09f6c2a83d12bfcf | What is the largest integer \( n \) such that \( 5^n \) divides \( \frac{2006!}{(1003!)^2} \)? | [
"2"
] | [
"What is the largest integer \\( n \\) such that \\( 5^n \\) divides \\( \\frac{2006!}{(1003!)^2} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de46b196d5a29f19635f8013121293ec1ba51f935e172c9c3fec187349bc3fe4 | What is the largest integer \( n \) such that \( S_n \) is a solvable group? | [
"4"
] | [
"What is the largest integer \\( n \\) such that \\( S_n \\) is a solvable group?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3d4601257d57f6eaac47e713b16fb236726c4e6fedc701e8baf4888318a4e7a3 | What is the largest integer \( n \) such that it is impossible to buy exactly \( n \) chicken nuggets using packages of 11 or 7? | [
"59"
] | [
"What is the largest integer \\( n \\) such that it is impossible to buy exactly \\( n \\) chicken nuggets using packages of 11 or 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-794a813f447b9553049213258cfba909baf187ebbbc759aebb49f78de49f0602 | What is the largest integer \( n \) such that the equation \( 13a + 18b = n \) has no solutions for positive integers \( a \) and \( b \)? | [
"234"
] | [
"What is the largest integer \\( n \\) such that the equation \\( 13a + 18b = n \\) has no solutions for positive integers \\( a \\) and \\( b \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 20
} | 42 | 0 | 0 |
omr-ws-e20b9abde6167045e3e38743552ff1b2268d9df2aafc491e58e5cef4e490db45 | What is the largest integer \( n \) such that the number \( 4^{37} + 4^{1000} + 4^n \) is a perfect square? | [
"1962"
] | [
"What is the largest integer \\( n \\) such that the number \\( 4^{37} + 4^{1000} + 4^n \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c63f8e8395c8fac06fb52ec1300e728f2861f6f8b276b545610fbd85f9bb2dcc | What is the largest integer \( n \) such that the quantity \( \frac{50!}{(5!)^n} \) is an integer? | [
"12"
] | [
"What is the largest integer \\( n \\) such that the quantity \\( \\frac{50!}{(5!)^n} \\) is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-7ede22b84c4cb813d0b9ac9a6be80e053ddbacd3ca4e15bb711aecaae23ff50d | What is the largest integer \( n \) that makes \( 2^n + 65 \) equal to an integer square? | [
"10"
] | [
"What is the largest integer \\( n \\) that makes \\( 2^n + 65 \\) equal to an integer square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-bffe00fe7813baf79a56c4b1106bd747d365fa71f588d52b6287ac6779029a9b | What is the largest integer \( x \) such that \( \frac{40n-40}{\frac{n}{2}-18} \) is an integer? | [
"2836"
] | [
"What is the largest integer \\( x \\) such that \\( \\frac{40n-40}{\\frac{n}{2}-18} \\) is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-ba87559361ce572d36873720e33d70afc697e4ea8db13f20b785065e73efac11 | What is the largest integer \(n\) such that \(\frac{2009!}{7^n}\) is an integer? | [
"333"
] | [
"What is the largest integer \\(n\\) such that \\(\\frac{2009!}{7^n}\\) is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
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