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-ba4ffb3948a004f7d4ffe7c4e80dca08bd286a7c4185316cadb635ffb31fe803 | What is the largest positive integer that cannot be expressed as the sum of distinct prime numbers (possibly one)? | [
"6"
] | [
"What is the largest positive integer that cannot be expressed as the sum of distinct prime numbers (possibly one)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-206268aa57dfe700c468588b7b427e1b2fc4a3eeee505adf2c9651d5f47d31fd | What is the largest positive integer that cannot be expressed in the form \(7x + 11y\), where \(x\) and \(y\) are positive integers? | [
"42"
] | [
"What is the largest positive integer that cannot be expressed in the form \\(7x + 11y\\), where \\(x\\) and \\(y\\) are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7dc7f4551aa13d78da4c986e77b16b74c958e025e4550193856fadd23ef7d165 | What is the largest positive integer that cannot be written as $42a + b$, where $a$ and $b$ are positive integers and $b$ is composite? | [
"215"
] | [
"What is the largest positive integer that cannot be written as $42a + b$, where $a$ and $b$ are positive integers and $b$ is composite?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9157952156576ef0c7e2db7ba182c7d359066c20e871d4e9934e9cb113488ddd | What is the largest positive integer that cannot be written as \( 42a + b \) where \( a \) and \( b \) are positive integers and \( b \) is composite? | [
"215"
] | [
"What is the largest positive integer that cannot be written as \\( 42a + b \\) where \\( a \\) and \\( b \\) are positive integers and \\( b \\) is composite?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a3ce4e9815ccda0aabd38d4a2a82fad6fc80ffaee7acc28c2eecb79a7d2aa9c6 | What is the largest positive integer that divides all numbers of the form \( n^5 - n \) for \( n = 1, 2, 3, \ldots \)? | [
"30"
] | [
"What is the largest positive integer that divides all numbers of the form \\( n^5 - n \\) for \\( n = 1, 2, 3, \\ldots \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9c3a3148bc971a494aba9ed797267a5b79dcaaf8385410dbfecc781616d93e80 | What is the largest positive integer that divides all the numbers \( 1^5 - 1, 2^5 - 2, \ldots, n^5 - n, \ldots \)? | [
"30"
] | [
"What is the largest positive integer that divides all the numbers \\( 1^5 - 1, 2^5 - 2, \\ldots, n^5 - n, \\ldots \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-df5666f5fad7bfd2c911b479c63d33d88988d07b08dd792e8fbb6fba1b14e45a | What is the largest positive integer that divides exactly two elements of the set \( M = \{1^2, 2^2, 3^2, \ldots, 99^2, 100^2\} \)? | [
"2500"
] | [
"What is the largest positive integer that divides exactly two elements of the set \\( M = \\{1^2, 2^2, 3^2, \\ldots, 99^2, 100^2\\} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-1b10c26c55189142ba7c5922797a2ec3d7c90628f4e44dacb81d392591457d0a | What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? | [
"215"
] | [
"What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-7af53689349bda7c578d0f0b9006b367e981a17488d5cb37b3df4f89aa510b5f | What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer? | [
"215"
] | [
"What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 28,
"genselect": 8
} | 36 | 0 | 0 |
omr-ws-cd2e3a7f17a53afa3a7cef7411d986276779db1a8707cf6733dad4863527d42c | What is the largest positive integer value of $n$ such that $5^n$ divides the product $101 \times 103 \times 105 \times \ldots \times 2011 \times 2013$? | [
"241"
] | [
"What is the largest positive integer value of $n$ such that $5^n$ divides the product $101 \\times 103 \\times 105 \\times \\ldots \\times 2011 \\times 2013$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a8089198919775180730aaa953ece4bd3ac818038192dcf669cdf2c7bd34ea3 | What is the largest positive integer with only even digits that is less than $10,000$ and is a multiple of $9$? | [
"8820"
] | [
"What is the largest positive integer with only even digits that is less than $10,000$ and is a multiple of $9$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-284a0aa8efc1df67abcdf5f6e23446e472530f26bf03618064cb841723f39a9e | What is the largest positive multiple of $12$ that is less than $350?$ | [
"348"
] | [
"What is the largest positive multiple of $12$ that is less than $350?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e56efbfad716e1e77915b343baf9feef3d43600f1f27a3c16f5da1977a8c61e | What is the largest positive number that cannot be expressed in the form $10a + 11b + 12c$, where $a$, $b$, and $c$ are positive integers? | [
"49"
] | [
"What is the largest positive number that cannot be expressed in the form $10a + 11b + 12c$, where $a$, $b$, and $c$ are positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eab8119f77e36181849ddabdad3b3c8c80366c88ebbc6d1b100e7ebdc3af3eb2 | What is the largest positive number that cannot be expressed in the form $11x + 12y + 13z$ where $x, y, z$ are positive integers? | [
"90"
] | [
"What is the largest positive number that cannot be expressed in the form $11x + 12y + 13z$ where $x, y, z$ are positive integers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 25
} | 35 | 0 | 0 |
omr-ws-468b4e233bd666c94a9c8f9903af344f2d752503c9c0f4baf26ef2781a4430cf | What is the largest positive number that cannot be expressed in the form $11x + 13y + 15z$ where $x, y, z$ are positive integers? | [
"103"
] | [
"What is the largest positive number that cannot be expressed in the form $11x + 13y + 15z$ where $x, y, z$ are positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-48e459f6c187f596b81f1d85c618fe33060cee6ace99539460ff62e17a672219 | What is the largest positive number that cannot be expressed in the form $11x + 15y + 18z$ where $x, y, z$ are positive integers? | [
"123"
] | [
"What is the largest positive number that cannot be expressed in the form $11x + 15y + 18z$ where $x, y, z$ are positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-d10d99967795c8cf608474372ea18986b3b5796c609a7eb9cda973381b477727 | What is the largest positive number that cannot be expressed in the form $12x + 13y + 14z$ where $x, y, z$ are positive integers? | [
"62"
] | [
"What is the largest positive number that cannot be expressed in the form $12x + 13y + 14z$ where $x, y, z$ are positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-2c45ea5f447b1a8caea5abcd5186b4c720fa09f5c73f6c16484bb1c65f581add | What is the largest possible area for the shadow of a cube with an edge length of 1, assuming the sun is directly overhead and the cube can be in any position? | [
"1"
] | [
"What is the largest possible area for the shadow of a cube with an edge length of 1, assuming the sun is directly overhead and the cube can be in any position?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aee3d1077f2eab2748277e04eee8e649f1dd8282f7c31a7d18c67d5dee0e4dc0 | What is the largest possible area of a magic octagon, where a magic octagon is defined as an octagon whose sides follow the lines of a checkerboard and have side lengths of $1, 2, 3, 4, 5, 6, 7, 8$ (in any order)? | [
"49"
] | [
"What is the largest possible area of a magic octagon, where a magic octagon is defined as an octagon whose sides follow the lines of a checkerboard and have side lengths of $1, 2, 3, 4, 5, 6, 7, 8$ (in any order)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-71b82c897a9cf64199469b2c882ac284ab924353d8b16aeabbf71e95c6ab1504 | What is the largest possible area of a parallelogram if an interior point has distances of 1, 4, 7, and 8 from the corners? | [
"36"
] | [
"What is the largest possible area of a parallelogram if an interior point has distances of 1, 4, 7, and 8 from the corners?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 16
} | 20 | 0 | 0 |
omr-ws-20c7390b6e139a45847edffeefb57d780db9c56e57a781e1edd44a78b08bc727 | What is the largest possible area of a right triangle \(ABC\) with \(A \in \omega_a\), \(B \in \omega_b\), \(C \in \omega_c\) and a right angle at \(A\), given that the radii of the concentric circles \(\omega_a\), \(\omega_b\), and \(\omega_c\) are 4, 6, and 8 respectively? | [
"\\(12\\sqrt{5}\\)"
] | [
"What is the largest possible area of a right triangle \\(ABC\\) with \\(A \\in \\omega_a\\), \\(B \\in \\omega_b\\), \\(C \\in \\omega_c\\) and a right angle at \\(A\\), given that the radii of the concentric circles \\(\\omega_a\\), \\(\\omega_b\\), and \\(\\omega_c\\) are 4, 6, and 8 respectively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-f377296940bbb818d1ae4aed79225c787773fa2bec1d6d184b74a2d2652ab666 | What is the largest possible area of a triangle that lies completely within a $97$-dimensional hypercube of side length $1$, where its vertices are three of the vertices of the hypercube? | [
"28"
] | [
"What is the largest possible area of a triangle that lies completely within a $97$-dimensional hypercube of side length $1$, where its vertices are three of the vertices of the hypercube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-a68868ffd80dab88a828416d54e5892bee3f53cfcbbdd7c2eb540ed0e6354d9a | What is the largest possible area of a triangle with sides \(a\), \(b\), and \(c\) satisfying \(0 \leq a \leq 1 \leq b \leq 2 \leq c \leq 3\)? | [
"1"
] | [
"What is the largest possible area of a triangle with sides \\(a\\), \\(b\\), and \\(c\\) satisfying \\(0 \\leq a \\leq 1 \\leq b \\leq 2 \\leq c \\leq 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e972208f20e11932d472657f17f1627520ddbb590d1ce646c0f11c92a1de1ff9 | What is the largest possible area of a triangle with the largest side length $39$ and inradius $10$? | [
"540"
] | [
"What is the largest possible area of a triangle with the largest side length $39$ and inradius $10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-642faaf6be71ad9b265b156ef15c6eda840d2c33d625ce2fb1a305c4f97788e0 | What is the largest possible area of the orthogonal projection of a cube with side length $1$ onto a plane? | [
"$\\sqrt{3}$"
] | [
"What is the largest possible area of the orthogonal projection of a cube with side length $1$ onto a plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 42,
"tir": 19
} | 61 | 0 | 0 |
omr-ws-4ce5c7159d692f259be5a891b0f23b950ffc0b05d2e4a9f8a1c0e2e299333089 | What is the largest possible area of the orthogonal projection of a cube with side length 1 onto a plane? | [
"1"
] | [
"What is the largest possible area of the orthogonal projection of a cube with side length 1 onto a plane?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f3a32823e73b293cc6c98055c36e7f1e41833e987d218948f8fa3df568bad40 | What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5)$ and the other on the sphere of radius 87 with center $(12,8,-16)$? | [
"137"
] | [
"What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5)$ and the other on the sphere of radius 87 with center $(12,8,-16)$?"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-521552eeb3e068d64dbee780e9e4665f3f871846dcdfe3fae2dfd353d8403e43 | What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5),$ and the other on the sphere of radius 87 with center $(12,8,-16)$? | [
"137"
] | [
"What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2,-10,5),$ and the other on the sphere of radius 87 with center $(12,8,-16)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cd2e909cc30b0abab22a403237daf5f7037d92f039fc2b9b6d56208253504ed | What is the largest possible greatest common divisor (gcd) of \(7n + 4\) and \(19n - 1\) given that \(n\) can represent any positive integer? | [
"83"
] | [
"What is the largest possible greatest common divisor (gcd) of \\(7n + 4\\) and \\(19n - 1\\) given that \\(n\\) can represent any positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7087722045642a8bff5c31ec4c658a8c6ee956c16f83d3ae850709d9e956f922 | What is the largest possible integer \( n \) such that \( n! \) (n factorial) has 12 terminal zeroes when multiplied out? | [
"\\( n = 54 \\)"
] | [
"What is the largest possible integer \\( n \\) such that \\( n! \\) (n factorial) has 12 terminal zeroes when multiplied out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-a6704f3c451d013850aaf70af1d284e63d973c64a1d0781130f1e32d9a3dfb2e | What is the largest possible least common multiple of some positive integers whose sum is 5? | [
"6"
] | [
"What is the largest possible least common multiple of some positive integers whose sum is 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9331d954b58dd41de70a49d644892f5e6351c73bfcdc78db61e7d836d34e74d2 | What is the largest possible least common multiple of some positive integers whose sum is 9? | [
"20"
] | [
"What is the largest possible least common multiple of some positive integers whose sum is 9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-3a7b578a896c6a918597e5aac88b6214bb326675e2172dd99b7d3ed97b527fa5 | What is the largest possible length of a route you can follow along the edges of an $n \times n$ grid from the point $A$ in the lower left corner to the point $B$ in the upper right corner, without returning to any previously visited point? | [
"\\( 2(n-1) \\)"
] | [
"What is the largest possible length of a route you can follow along the edges of an $n \\times n$ grid from the point $A$ in the lower left corner to the point $B$ in the upper right corner, without returning to any previously visited point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c31357afc2fdd772cf787ed4bbc2556a8d6d7a36a09df7a41e594d8c26a4457 | What is the largest possible length of an arithmetic progression of positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) with a common difference of 2, such that \( a_{k}^2 + 1 \) is prime for \( k = 1, 2, \ldots, n \)? | [
"3"
] | [
"What is the largest possible length of an arithmetic progression of positive integers \\( a_{1}, a_{2}, \\cdots, a_{n} \\) with a common difference of 2, such that \\( a_{k}^2 + 1 \\) is prime for \\( k = 1, 2, \\ldots, n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2bee872627e2a422d3f96b192970f6c67957e6cbb84fd16b80021755f3d59485 | What is the largest possible median for the five number set $\{x, 2x, 3, 2, 5\}$ if $x$ can be any integer? | [
"5"
] | [
"What is the largest possible median for the five number set $\\{x, 2x, 3, 2, 5\\}$ if $x$ can be any integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab1b95cd726a180292c53ae35d56c8c0d60a9fc55326d5112af3c334cdab38db | What is the largest possible number of 2-dimensional faces of an $n$-dimensional cube whose orientation agrees with the orientations of its four sides? | [
"n(n-1)2^{n-3}"
] | [
"What is the largest possible number of 2-dimensional faces of an $n$-dimensional cube whose orientation agrees with the orientations of its four sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-016668c3370f908c07c79f1087313b4e54a956ca341ee0b4daa17a4be7b6972d | What is the largest possible number of acute angles a 2001-gon (polygon with 2001 sides) can have if no two sides cross each other? | [
"1600"
] | [
"What is the largest possible number of acute angles a 2001-gon (polygon with 2001 sides) can have if no two sides cross each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9fb84bff15d9ec612a50ae872a0989c8b46b7edae51246dfdc3aafb54caa1e28 | What is the largest possible number of acute angles a 2001-gon can have if no two sides cross each other? | [
"3"
] | [
"What is the largest possible number of acute angles a 2001-gon can have if no two sides cross each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 7
} | 11 | 0 | 0 |
omr-ws-84fc128dad0f6b524dd756b7807f795ca599b5770491583ee0c317ce1c85d0ac | What is the largest possible number of binary sequences of length 10 that can be used such that any two sequences differ in more than five places? | [
"6"
] | [
"What is the largest possible number of binary sequences of length 10 that can be used such that any two sequences differ in more than five places?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0651bf9f33417fccdc08c0f86f2c27510e634b71a2da0a9e3091aa80bada9c44 | What is the largest possible number of distinct positive integers such that no four of them have a sum divisible by 4? | [
"6"
] | [
"What is the largest possible number of distinct positive integers such that no four of them have a sum divisible by 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-53a856e3a86ee0986b44c0ab0b79866c972ddecd2596727cbefc8c87a6058e0f | What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge? | [
"\\( n^2 \\)"
] | [
"What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 3,
"tir": 1
} | 28 | 0 | 0 |
omr-ws-ef599e7117fc4bfad56c259bbe20a01a4a58f410b210bb81d520a660b03fe69b | What is the largest possible number of edges of an $n$-dimensional cube whose orientation agrees with the orientations of all 2-dimensional faces containing the edge? | [
"n \\cdot 2^{n-1}"
] | [
"What is the largest possible number of edges of an $n$-dimensional cube whose orientation agrees with the orientations of all 2-dimensional faces containing the edge?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-e8313d938c38c18202fef6a17c6b9c73a7e010bb45df351e457e303fc865fcab | What is the largest possible number of elements in a subset $A$ of positive integers, where the sum of any three distinct elements in $A$ results in a prime number? | [
"4"
] | [
"What is the largest possible number of elements in a subset $A$ of positive integers, where the sum of any three distinct elements in $A$ results in a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6aea06e73a87692e2ab06d5a5a1f6e7f101048b917b31de98dae04d5e4e39326 | What is the largest possible number of elements in a subset of \(\{1, 2, 3, \ldots, 9\}\) such that the sum of every pair (of distinct elements) in the subset is different? | [
"5"
] | [
"What is the largest possible number of elements in a subset of \\(\\{1, 2, 3, \\ldots, 9\\}\\) such that the sum of every pair (of distinct elements) in the subset is different?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e3c89deb7197ebad6dd857714b89bebe8eb4e526281a128aa47deec97da245d | What is the largest possible number of elements in the intersection of the set \( M \) (which is a geometric progression) and the set \( \{2^n + 1 \mid n \in \mathbb{Z}\} \)? | [
"2"
] | [
"What is the largest possible number of elements in the intersection of the set \\( M \\) (which is a geometric progression) and the set \\( \\{2^n + 1 \\mid n \\in \\mathbb{Z}\\} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a5d30ea2598a7f5f183d44640253a3a294eafba36fe03bdab15d4c21c8a4a106 | What is the largest possible number of elements in the intersection of the set \( M \) (which is a geometric progression) and the set \( \{2^n - 1 \mid n \in \mathbb{Z}\} \)? | [
"2"
] | [
"What is the largest possible number of elements in the intersection of the set \\( M \\) (which is a geometric progression) and the set \\( \\{2^n - 1 \\mid n \\in \\mathbb{Z}\\} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e09aa5a2b55c3698bd611afad5b962397a9c5e81092f73536a1693522d27f394 | What is the largest possible number of equidistant planes from five points, no four of which are coplanar? | [
"10"
] | [
"What is the largest possible number of equidistant planes from five points, no four of which are coplanar?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c4f52df71beb2cdb2f1b1ffbd3683c053cacb751f7e56965623365e6bf8cb19 | What is the largest possible number of integers in a set \( S \) of positive integers less than or equal to 2004, such that no number in \( S \) is double the value of another number in \( S \)? | [
"1333"
] | [
"What is the largest possible number of integers in a set \\( S \\) of positive integers less than or equal to 2004, such that no number in \\( S \\) is double the value of another number in \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e866fd8e5d60017d3f690e93d470a9ca14683c123d1376d64c6f03e2ce2fa61 | What is the largest possible number of intersections between a circle and two distinct lines? | [
"5"
] | [
"What is the largest possible number of intersections between a circle and two distinct lines?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-886384aeee317bdc90be6f8f336040fb69b7b5497a50643c5d6d8a2f1796025d | What is the largest possible number of obtuse-angled triangles that can be formed using the vertices of a 15-gon, where all vertices lie on the circumference of a circle and the center of the circle is inside the 15-gon? | [
"315"
] | [
"What is the largest possible number of obtuse-angled triangles that can be formed using the vertices of a 15-gon, where all vertices lie on the circumference of a circle and the center of the circle is inside the 15-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-87e942ef33d70ca1c019aacb51df1b20e405f03e31cb1fbbb4a5f570d7372f39 | What is the largest possible number of points $n$ that can be marked inside an equilateral triangle with side length $1$ such that no two marked points are within a distance of $\frac{1}{3}$ from each other? | [
"10"
] | [
"What is the largest possible number of points $n$ that can be marked inside an equilateral triangle with side length $1$ such that no two marked points are within a distance of $\\frac{1}{3}$ from each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-b83927711e99f09ad7f726b9f8c2d42afc1e71518c96d6a94d798710c4710a90 | What is the largest possible number of shuffles it could take to return 53 cards to their original order, given that the machine permutes the cards in the same way every time? | [
"235"
] | [
"What is the largest possible number of shuffles it could take to return 53 cards to their original order, given that the machine permutes the cards in the same way every time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d213d3a9e33e33b28bbd8fdf8ad5dd17ea538baad65212882c52b2112e9696aa | What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers? | [
"(n+1)^2",
"\\((n+1)^2\\)"
] | [
"What is the largest possible number of subsets of the set $\\{1, 2, \\dots , 2n+1\\}$ such that the intersection of any two subsets consists of one or several consecutive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 53,
"genselect": 13
} | 66 | 0 | 0 |
omr-ws-1e746da4f535ab9efda8e91fea5a606dc89eeeadcc6525d475cabeda6aa2a991 | What is the largest possible number of subsets of the set \(\{1, 2, \ldots, 2n+1\}\) such that the intersection of any two subsets consists of one or several consecutive integers? | [
"\\( (n+1)^2 \\)"
] | [
"What is the largest possible number of subsets of the set \\(\\{1, 2, \\ldots, 2n+1\\}\\) such that the intersection of any two subsets consists of one or several consecutive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-e4f26ca091c67f5ca70b9234c385092fa927d2249cd4c27573d9abf5c4af0e53 | What is the largest possible number of three-element sets such that each pair of sets has exactly one common element, and no element belongs to all of the sets? | [
"7"
] | [
"What is the largest possible number of three-element sets such that each pair of sets has exactly one common element, and no element belongs to all of the sets?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bec3daa80daca67555b80ab1a6e68b4a31be288acf2324b4e7fd0d62b886bacf | What is the largest possible product of positive integers whose sum is 20? | [
"1458"
] | [
"What is the largest possible product of positive integers whose sum is 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ef217a9b560b4d74ee651f3f486a186c05fa0a2d4894d3692b0b334e4c8249b | What is the largest possible product of two factors that together use each of the digits $1$ to $5$ exactly once? | [
"22412"
] | [
"What is the largest possible product of two factors that together use each of the digits $1$ to $5$ exactly once?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-66ff78015ac225b66dd00054d423757046fda10ff8dffef9d95278ab42dbc798 | What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? | [
"\\(\\frac{\\sqrt{2}}{2}\\)",
"\\(\\sqrt{2}/2\\)"
] | [
"What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 60,
"genselect": 3,
"tir": 56
} | 119 | 0 | 0 |
omr-ws-9291f14731fb68b0f6255d05e13c590e5095391d19428ad3692912349c4419e0 | What is the largest possible radius of a semicircle that can be drawn in an isosceles trapezoid with parallel bases of lengths 10 and 20, and legs of length 13? | [
"\\(\\frac{120}{13}\\)"
] | [
"What is the largest possible radius of a semicircle that can be drawn in an isosceles trapezoid with parallel bases of lengths 10 and 20, and legs of length 13?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-eb99c95bca6f18a4e7683bdeb2bd023d29b3edb5bb691d69360e1649ea497d0d | What is the largest possible ratio of the area of a kite inscribed in a regular octagon to the area of the octagon? | [
"\\(\\frac{1 + \\sqrt{2}}{4}\\)"
] | [
"What is the largest possible ratio of the area of a kite inscribed in a regular octagon to the area of the octagon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b9d77adecd02ce5f5a09e9b0644f171baf26644e1badbc671c1d48f2879aff3 | What is the largest possible rational root of the equation $ax^2 + bx + c = 0$ where $a, b,$ and $c$ are positive integers that do not exceed $100$? | [
"\\(-\\frac{1}{99}\\)"
] | [
"What is the largest possible rational root of the equation $ax^2 + bx + c = 0$ where $a, b,$ and $c$ are positive integers that do not exceed $100$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 26
} | 57 | 0 | 0 |
omr-ws-7e0e58e7236c31a1fb9248d8d0ff4405662dcf38511237d3b0997bf654891b5d | What is the largest possible root of the equation $x^{20} - 7x^{19} + \ldots + 17 = 0$? | [
"7"
] | [
"What is the largest possible root of the equation $x^{20} - 7x^{19} + \\ldots + 17 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-7e330cadcd4dbe3f1bd859ebddb8a3c70446fda3c2bc2574a67d90e07d3380f0 | What is the largest possible size of an angle of a triangle formed by joining the midpoints of three edges of a cube? | [
"120 degrees",
"120°"
] | [
"What is the largest possible size of an angle of a triangle formed by joining the midpoints of three edges of a cube?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-00d14c61df7cdd5b2c51b4f4fe80069dceb2aab4033c4fb16c7aabed303c0e40 | What is the largest possible sum of all elements in a nondegenerate 2021×2021 square matrix where each element is either 0 or 1? | [
"4082421"
] | [
"What is the largest possible sum of all elements in a nondegenerate 2021×2021 square matrix where each element is either 0 or 1?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-db0cb76791df076bed38d1898c0268351f26467c39737df0eb9bc880d2bfc2dc | What is the largest possible sum of the digits in the display of a digital watch that shows hours and minutes with AM and PM? Note: The possible answers are given as choices, but the problem statement itself does not require those choices to be included. | [
"23"
] | [
"What is the largest possible sum of the digits in the display of a digital watch that shows hours and minutes with AM and PM?\n\nNote: The possible answers are given as choices, but the problem statement itself does not require those choices to be included."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b071a9a9c3055736aa3cdcf251a6d407ec5033e4c7658cc1b2fca51ee64bf133 | What is the largest possible sum of the numbers of the cards that are left on the table when nine cards numbered from 1 to 9 are distributed such that Ana, Jhon, and Celio each take two cards, leaving three cards in the pile? | [
"24"
] | [
"What is the largest possible sum of the numbers of the cards that are left on the table when nine cards numbered from 1 to 9 are distributed such that Ana, Jhon, and Celio each take two cards, leaving three cards in the pile?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-43326f75970ec5a00250de8e374ca275b9e02465622c9adf099948d8417875d5 | What is the largest possible sum of two five-digit palindromes with no repeating digits? | [
"184047"
] | [
"What is the largest possible sum of two five-digit palindromes with no repeating digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 13
} | 19 | 0 | 0 |
omr-ws-fa5bdda83a727abe6b8b38da15c4dd1cc2b4d6d47a32e603ef3406bb917315d8 | What is the largest possible sum of two relatively prime factors of 4680 (excluding 1 and 4680)? | [
"941"
] | [
"What is the largest possible sum of two relatively prime factors of 4680 (excluding 1 and 4680)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-04f4441a3963b5e0923c3011268aeae22ecc23b7a2db1ad2bd58ba19f64cb61e | What is the largest possible value of $\dfrac{x^2 + 2x + 6}{x^2 + x + 5}$ where $x$ is a positive real number? | [
"\\dfrac{20 + 2\\sqrt{5}}{19}"
] | [
"What is the largest possible value of $\\dfrac{x^2 + 2x + 6}{x^2 + x + 5}$ where $x$ is a positive real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-311fc1a7b8db915b61cf08b3c0bcafd9c57f84a55697d38d212e200cccbd84c0 | What is the largest possible value of $\dfrac{x^2+2x+6}{x^2+x+5}$ where $x$ is a positive real number? | [
"None of the preceding"
] | [
"What is the largest possible value of $\\dfrac{x^2+2x+6}{x^2+x+5}$ where $x$ is a positive real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e2adf5c691c4b05dadafa3babe92c50d374e0c9bffe0fe434430844e4a06eaa | What is the largest possible value of \( A \) in the cryptarithm shown, where each instance of a particular letter represents the same digit (0-9) and different letters represent different digits? \[ \begin{array}{c@{}c@{}c} & A & B & C \\ + & C & A & B \\ \hline & B & B & D & A \\ \end{array} \] | [
"6"
] | [
"What is the largest possible value of \\( A \\) in the cryptarithm shown, where each instance of a particular letter represents the same digit (0-9) and different letters represent different digits?\n\n\\[\n\\begin{array}{c@{}c@{}c}\n & A & B & C \\\\\n+ & C & A & B \\\\\n\\hline\n & B & B & D & A \\\\\n\\end{arra... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-d50f7a39b85872788b05b031175b163e7c3304161f46dcda9189d1f66c08700d | What is the largest possible value of \( n \) such that any ordering of points \( A_1, A_2, \ldots, A_n \) in the plane results in a broken line \( B_1B_2 \ldots B_n \) that does not intersect itself? | [
"4"
] | [
"What is the largest possible value of \\( n \\) such that any ordering of points \\( A_1, A_2, \\ldots, A_n \\) in the plane results in a broken line \\( B_1B_2 \\ldots B_n \\) that does not intersect itself?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6afacfa79bf5fe0d40451ab08dff7f30b1a7a7b64e37ce847b75470fda9e0985 | What is the largest possible value of \(a_5\) such that \(a_1, a_2, a_3, a_4, a_5\) form an arithmetic sequence where all terms are prime numbers? | [
"29"
] | [
"What is the largest possible value of \\(a_5\\) such that \\(a_1, a_2, a_3, a_4, a_5\\) form an arithmetic sequence where all terms are prime numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7059bd0693b8b068c9748565b441ba84b78d476f9218489467f832068b5f9222 | What is the largest possible value of the expression $$\gcd(n^2 + 3, (n + 1)^2 + 3)$$ for natural numbers $n$? | [
"13"
] | [
"What is the largest possible value of the expression $$\\gcd(n^2 + 3, (n + 1)^2 + 3)$$ for natural numbers $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-530d710c52c16ffff9b0363f9687c26f63b79860e5577af42ec603d2008614b4 | What is the largest possible value of the four-digit number ABCD, given the multiplication: \[ \begin{array}{r} AB \\ \times \ C \\ \hline CD \\ \end{array} \] | [
"1919"
] | [
"What is the largest possible value of the four-digit number ABCD, given the multiplication:\n\n\\[ \\begin{array}{r}\nAB \\\\\n\\times \\ C \\\\\n\\hline\nCD \\\\\n\\end{array} \\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-749831002385219033c746c3aefbe77e8fb2b2ea99437b766513a661133db986 | What is the largest possible value of the greater of the two non-zero integers for which the difference of their cubes equals the square of their difference? | [
"2"
] | [
"What is the largest possible value of the greater of the two non-zero integers for which the difference of their cubes equals the square of their difference?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15f4bd2c0284921f37b427ac0c85e84d08f0e5bae43b8ca1b4ed9ea52175c23a | What is the largest possible value of the greatest common divisor of \(a + 201b\) and \(201a + b\) for relatively prime integers \(a\) and \(b\)? | [
"40400"
] | [
"What is the largest possible value of the greatest common divisor of \\(a + 201b\\) and \\(201a + b\\) for relatively prime integers \\(a\\) and \\(b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-9c4f8d30292534d4d4d3bed212fb583eae1daddb07003e3e0593b72319eeb12d | What is the largest possible value of the greatest common factor (GCF) of $n^2+n+1$ and $n^2-4n+16$ for a positive integer $n$? | [
"13"
] | [
"What is the largest possible value of the greatest common factor (GCF) of $n^2+n+1$ and $n^2-4n+16$ for a positive integer $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-5ac5db12b9350b9178ec652ef1208a8006be821dc5a01ab9d7a7a74185aa92b9 | What is the largest possible value of the product of a set of positive integers whose sum is $31$? | [
"78732"
] | [
"What is the largest possible value of the product of a set of positive integers whose sum is $31$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-44ff8464b643ff8e57236bd386a82cf9441d274ce8e4eebe77ae904d958b29f2 | What is the largest possible value of the ratio of the area of triangle A'B'C' to the area of triangle ABC, where A', B', and C' are defined as the points of intersection of the trisectors of angles A, B, and C, respectively, in triangle ABC? Is there a minimum value for this ratio? | [
"\\(\\frac{2}{15}\\)"
] | [
"What is the largest possible value of the ratio of the area of triangle A'B'C' to the area of triangle ABC, where A', B', and C' are defined as the points of intersection of the trisectors of angles A, B, and C, respectively, in triangle ABC? Is there a minimum value for this ratio?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d82f8084b2578fb637ed9011db34b4ab98d9207c890ce9052d45072d99ddaaa2 | What is the largest possible volume of a tetrahedron inscribed in a cube of side length 1, such that one face of the tetrahedron is coplanar with one face of the cube? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the largest possible volume of a tetrahedron inscribed in a cube of side length 1, such that one face of the tetrahedron is coplanar with one face of the cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-78946284ad0ac37f7d783aa4dbad63289ce7dd7a8cde7a7c9af17e7b3f912a90 | What is the largest power of $2$ that divides $2014!$? | [
"2005"
] | [
"What is the largest power of $2$ that divides $2014!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 1,
"tir": 28
} | 40 | 0 | 0 |
omr-ws-3fe9800b1f2f75c926c3d3a437deec36edcbfd622d1a2322027ecbaef6ef5549 | What is the largest power of $2$ that divides the number $[(3+\sqrt{7})^{2023}]$? | [
"\\(2^0\\) or simply \\(1\\)."
] | [
"What is the largest power of $2$ that divides the number $[(3+\\sqrt{7})^{2023}]$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ff631b7f3978fe52d4b3ba992714735a3aa35a3afa2e08320cac95bed019217 | What is the largest power of $2012$ that can divide $2010^{2011^{2012}} + 2012^{2011^{2010}}$? | [
"2011"
] | [
"What is the largest power of $2012$ that can divide $2010^{2011^{2012}} + 2012^{2011^{2010}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bf57ecaa1c48398c0fdd28c3e8b18162e4798283e219ad92eda57665f03a174 | What is the largest power of $225$ that divides $69!$? | [
"7"
] | [
"What is the largest power of $225$ that divides $69!$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-43301f7c7fc3cf0115ea279395a448208995b63b210c0fb095b4cb18037b999e | What is the largest power of 2 that can divide $(2^{2005})!$? | [
"\\(2^{2005} - 1\\)"
] | [
"What is the largest power of 2 that can divide $(2^{2005})!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-2d1f97e8f512db613aa6b866579c694ca0f847e544b77d11947d3d04c742e74b | What is the largest power of 2 that is a Fibonacci number? | [
"8"
] | [
"What is the largest power of 2 that is a Fibonacci number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02a32e32954de05e78dd34c5cc5814d7b331cf1910368e9780b620eaa4999fc8 | What is the largest power of 2011 that divides \(2010^{2011^{2012}} + 2012^{2011^{2010}}\)? | [
"2011^{2011}"
] | [
"What is the largest power of 2011 that divides \\(2010^{2011^{2012}} + 2012^{2011^{2010}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-9c2b6f3d6230d8e18ee706c8fe2fce7a33f47984698af4abdafeb6b5230f8dd3 | What is the largest power of 3 that divides $10^{3^{1999}}$? | [
"\\(3^{2001}\\)"
] | [
"What is the largest power of 3 that divides $10^{3^{1999}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-aa8ed068c90d77cadfb13908649937236febd9638caf238239df2d3ad3898ec5 | What is the largest power of 3 that divides $9001!$? | [
"4498"
] | [
"What is the largest power of 3 that divides $9001!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 29
} | 51 | 0 | 0 |
omr-ws-d2a4ca9e5509759700162349b4b8abde1c6cdca0101ab3a93faf6759ff686e59 | What is the largest power of 3 that divides \( 2^{3^k} + 1 \) for any nonnegative integer \( k \)? | [
"\\( k+1 \\)"
] | [
"What is the largest power of 3 that divides \\( 2^{3^k} + 1 \\) for any nonnegative integer \\( k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67c073ca8a151fb9021b5a63afa752b716f912ff8bf90b2d338c8cf083fa63b4 | What is the largest power of \( \sqrt{3} - \sqrt{2} \) that divides \( (2 - \sqrt{6} + \sqrt{3} + 3 \sqrt{2})^{2023} \)? | [
"2023"
] | [
"What is the largest power of \\( \\sqrt{3} - \\sqrt{2} \\) that divides \\( (2 - \\sqrt{6} + \\sqrt{3} + 3 \\sqrt{2})^{2023} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-6144d4c2f2da0840d01e967997e1bce53b80988ada68f7978632605f5964742e | What is the largest power of \(3\) that divides \(10^k - 1\), where \(k\) is a positive integer? | [
"3^{2 + v_3(k)}"
] | [
"What is the largest power of \\(3\\) that divides \\(10^k - 1\\), where \\(k\\) is a positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-080fc97923c6734c93601ea4b004499e2b2d18144e2b211c871ecbcde6888117 | What is the largest prime $p$ such that $\cos\frac{5p\pi}{6}=\sin\frac{5p\pi}{12}$? | [
"2"
] | [
"What is the largest prime $p$ such that $\\cos\\frac{5p\\pi}{6}=\\sin\\frac{5p\\pi}{12}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6e2f77dd2ad0ca9dc6266ca716b7c220b60af998e299b3c701a7bf4223bcfb7 | What is the largest prime \( p \) for which the numbers \( p^2 - 8 \), \( p^2 - 2 \), and \( p^2 + 10 \) are all prime as well? | [
"7"
] | [
"What is the largest prime \\( p \\) for which the numbers \\( p^2 - 8 \\), \\( p^2 - 2 \\), and \\( p^2 + 10 \\) are all prime as well?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-6969b1784bf30dcd474d2af37f3598a6dd263e58cdda690ebb5bf977de2b3694 | What is the largest prime \( p \) for which the numbers \( p^2 - 8 \), \( p^2 - 2 \), and \( p^2 + 10 \) are all primes as well? | [
"7"
] | [
"What is the largest prime \\( p \\) for which the numbers \\( p^2 - 8 \\), \\( p^2 - 2 \\), and \\( p^2 + 10 \\) are all primes as well?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-261a6211a8e2215c49e9508e2de75e1b247e5783094da90c06752d5361cf5790 | What is the largest prime \( p \) that makes \( \sqrt{17p + 625} \) an integer? | [
"67"
] | [
"What is the largest prime \\( p \\) that makes \\( \\sqrt{17p + 625} \\) an integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-15d1234eb6b43c403cea003b1e4200cb4ee78d4b5d6b648d767ec376e73a9be4 | What is the largest prime divisor of \(3^{15} + 1\)? | [
"271"
] | [
"What is the largest prime divisor of \\(3^{15} + 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f3ad3f454f4dee8817e7201338562971d9939051fd703d2fa216517741a160da | What is the largest prime factor of $$(25!)^3 - (24!)^3$$? | [
"31"
] | [
"What is the largest prime factor of $$(25!)^3 - (24!)^3$$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3e367f93dcdc2b1992c1efb3fcfe35ffeba520f50ea0db6955a4a10b88e2ae58 | What is the largest prime factor of $1337$? | [
"191"
] | [
"What is the largest prime factor of $1337$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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