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-55ce3871b0f123e3e3560f0b7da5e65ebac0b872e7bcafb993ea9d2d98ba807b | What is the greatest integer value of $x$ such that $\frac{x^2 + 2x + 5}{x-3}$ is an integer? | [
"23"
] | [
"What is the greatest integer value of $x$ such that $\\frac{x^2 + 2x + 5}{x-3}$ is an integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-486d5440151b9e1ee659c43e217183ef4abf720f80e6b1cd9a9d8057f6dd2455 | What is the greatest length \( n \) of a sequence \( S = (a_1, a_2, \ldots, a_n) \) of natural numbers such that each term is one more than the previous term and none of the numbers \( a_1, a_2, \ldots, a_n \) has a sum of digits not divisible by 5? | [
"8"
] | [
"What is the greatest length \\( n \\) of a sequence \\( S = (a_1, a_2, \\ldots, a_n) \\) of natural numbers such that each term is one more than the previous term and none of the numbers \\( a_1, a_2, \\ldots, a_n \\) has a sum of digits not divisible by 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-251d832c623a434b7da5d41e0dc2df38a82339e55c66f0f036b4fee4727e9797 | What is the greatest length of a block of consecutive positive integers each of which can be expressed as the sum of the squares of two positive integers? | [
"3"
] | [
"What is the greatest length of a block of consecutive positive integers each of which can be expressed as the sum of the squares of two positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a93765af84f3db9d55fc1188c1e18d7eacd561a04bb03b17ddaef0f817c4df3a | What is the greatest multiple of 3 that is less than 500 and is also a multiple of 4? | [
"492"
] | [
"What is the greatest multiple of 3 that is less than 500 and is also a multiple of 4?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eca370ecdca56900763e0e439320714332e8494528126abc9c1d771d3b6924d8 | What is the greatest multiple of 72 where the digits are distinct and arranged in ascending order? Provide a solution that clearly shows your approach and can be applied to similar problems, such as finding the greatest multiple of 70 with distinct and ascending digits. Avoid using trial and error or listing methods. | [
"134568"
] | [
"What is the greatest multiple of 72 where the digits are distinct and arranged in ascending order? Provide a solution that clearly shows your approach and can be applied to similar problems, such as finding the greatest multiple of 70 with distinct and ascending digits. Avoid using trial and error or listing metho... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-03f9b2241b376275085e5dd8b6716ab70f3f5dd6a12f511609e61ca34d56ac68 | What is the greatest number of Sundays that can occur in the first $49$ days of a year? | [
"7"
] | [
"What is the greatest number of Sundays that can occur in the first $49$ days of a year?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4de7d75d13f8f6428918d068c41a6277b921e43fe3fe91c90c818735ba6d6c9 | What is the greatest number of \(1 \times 1 \times 4\) boxes that can be placed inside a \(6 \times 6 \times 6\) box? | [
"52"
] | [
"What is the greatest number of \\(1 \\times 1 \\times 4\\) boxes that can be placed inside a \\(6 \\times 6 \\times 6\\) box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-99aa4c1666bc8e810ade4f587f3469451499706f39f1d33ae76466d03c72b6d8 | What is the greatest number of \(1 \times 1 \times 4\) boxes that can fit inside a \(6 \times 6 \times 6\) box? | [
"54"
] | [
"What is the greatest number of \\(1 \\times 1 \\times 4\\) boxes that can fit inside a \\(6 \\times 6 \\times 6\\) box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-07d669bf6e64aee39187421790890bfd5462c57c2ba23046c3fc7589c1e100d9 | What is the greatest number of acute angles that can occur in a convex n-gon? | [
"\\( n-3 \\)"
] | [
"What is the greatest number of acute angles that can occur in a convex n-gon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a1c05b8ff425fcbac3b72ff08da31ca8c904b7ca79e0e6a433cb6ad911d78964 | What is the greatest number of acute angles that can occur in a convex n-polygon? | [
"3"
] | [
"What is the greatest number of acute angles that can occur in a convex n-polygon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45d9ca38178080656fccfd69ff08dd5c745d98844146fbe0e1c531336eafd90a | What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1$? | [
"106"
] | [
"What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \\times 10 \\times 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d55282087aedf221ae1cfc400d8837607d016db926c994fdd329a941973dab5 | What is the greatest number of checkers that can be placed on the squares of a 26×26 checkerboard so that each unit square contains at most one checker, each checker lies entirely inside one unit square, and no four of them lie on the same 2×2 square? | [
"507"
] | [
"What is the greatest number of checkers that can be placed on the squares of a 26×26 checkerboard so that each unit square contains at most one checker, each checker lies entirely inside one unit square, and no four of them lie on the same 2×2 square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-ca0fe43e69bd0d14c53c939f4632263cd8f660868672544ae3a51db9fb96efb8 | What is the greatest number of circles of radius $\frac{1}{\sqrt{10}-1}$ that can be placed externally tangent to a circle of radius 1, if the small circles do not overlap? | [
"9"
] | [
"What is the greatest number of circles of radius $\\frac{1}{\\sqrt{10}-1}$ that can be placed externally tangent to a circle of radius 1, if the small circles do not overlap?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-22a7cf0bf6593ee53eac7cfc198bc7987578273b10e37da8a2d5e042666d8ddb | What is the greatest number of consecutive integers whose sum is 45? | [
"10"
] | [
"What is the greatest number of consecutive integers whose sum is 45?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9da0d693cd8f649ad8674b3ebff49a14328195bb6571748990c7ca885d254f1b | What is the greatest number of consecutive positive integers all of which are interesting, where an interesting positive integer is defined as a product of two (distinct or equal) prime numbers? | [
"3"
] | [
"What is the greatest number of consecutive positive integers all of which are interesting, where an interesting positive integer is defined as a product of two (distinct or equal) prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f8c1ceaae17c95992fc94ab1a22b395be579116e8e4b2340956d43bd98bf153 | What is the greatest number of consecutive terms of the sequence $x_n = n^4 + 2026$ whose greatest common divisor (gcd) is greater than 1? | [
"2"
] | [
"What is the greatest number of consecutive terms of the sequence $x_n = n^4 + 2026$ whose greatest common divisor (gcd) is greater than 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b0b7871d4d71271844e6f1a166de6b615e848f063a37266ee7ceed57814a3c2 | What is the greatest number of diagonals that can be drawn in a convex 2011-gon such that each newly drawn diagonal intersects no more than one of the already drawn diagonals? | [
"4016"
] | [
"What is the greatest number of diagonals that can be drawn in a convex 2011-gon such that each newly drawn diagonal intersects no more than one of the already drawn diagonals?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 4
} | 31 | 0 | 0 |
omr-ws-b5577e892b293a1eb4b91c342dd499f3a4a2f25c3d76abcf017c6ef307c871ec | What is the greatest number of diagonals that can be drawn in a convex 2015-gon such that each newly drawn diagonal intersects no more than one of the already drawn diagonals? | [
"4024"
] | [
"What is the greatest number of diagonals that can be drawn in a convex 2015-gon such that each newly drawn diagonal intersects no more than one of the already drawn diagonals?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-ceb0fdbd8fe158a9aa778e80c3f0c880a40b08eaec2e611ebdee1641e82ef562 | What is the greatest number of different noncongruent rectangles that can be bordered by 18 square tiles? | [
"3"
] | [
"What is the greatest number of different noncongruent rectangles that can be bordered by 18 square tiles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84708b17decf92951d7846facc0d4b2641cdf8dfcb88b98a94a0b856d2bb4f21 | What is the greatest number of different palindromic subsequences that a palindromic sequence of length $n$ can contain? | [
"\\( 2^n \\)"
] | [
"What is the greatest number of different palindromic subsequences that a palindromic sequence of length $n$ can contain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89d83989877036f9c46ca214acd3b6bb1d023a766f4ce567b5c20325cb90acd9 | What is the greatest number of dominoes that can be placed in a row such that the sum of all the numbers on the dominoes placed up to any moment is odd, and each domino can be flipped and added to either end of the row? | [
"16"
] | [
"What is the greatest number of dominoes that can be placed in a row such that the sum of all the numbers on the dominoes placed up to any moment is odd, and each domino can be flipped and added to either end of the row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3631184d321c982075a78866da2cbe005ec043d2ed1ca878f43cab17c73b4b6 | What is the greatest number of dots that can be placed on a $6$ by $6$ board such that all rows, columns, and both long diagonals have an odd number of dots? | [
"30"
] | [
"What is the greatest number of dots that can be placed on a $6$ by $6$ board such that all rows, columns, and both long diagonals have an odd number of dots?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-24aaca47bed1bb7b2f9baaab57878e86d5b74686876371aaaff7e038338bec11 | What is the greatest number of elements in a commutative ring that has exactly 2002 zero divisors? | [
"\\( 2003^2 \\)"
] | [
"What is the greatest number of elements in a commutative ring that has exactly 2002 zero divisors?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-5ee03e5e26c7049ac86d86d6f2a5b7709ee20931310ec855dd56ce1f9436e8f3 | What is the greatest number of elements in the set \(\{1, 2, 3, \ldots, 25\}\) such that no two numbers differ by a perfect square? | [
"10"
] | [
"What is the greatest number of elements in the set \\(\\{1, 2, 3, \\ldots, 25\\}\\) such that no two numbers differ by a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-391526640ed46c2de0e9f0af19123c567bdc82654a97679c4e351fbf3ca40ed4 | What is the greatest number of integers that can be selected from a set of 2015 consecutive numbers so that no sum of any two selected numbers is divisible by their difference? | [
"672"
] | [
"What is the greatest number of integers that can be selected from a set of 2015 consecutive numbers so that no sum of any two selected numbers is divisible by their difference?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-154e859c856b5101af00f53d4b7471e5b9d4fafd3d441f869de044e6dd0bf1cb | What is the greatest number of intersections formed by 10 of the same equilateral triangles? | [
"270"
] | [
"What is the greatest number of intersections formed by 10 of the same equilateral triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9afdc24d9624ae19f1cf065a7fcb540de0870ed95713e405e337b58d18f50754 | What is the greatest number of intersections formed by 10 of the same isosceles triangles? | [
"270"
] | [
"What is the greatest number of intersections formed by 10 of the same isosceles triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-5c8aba385047211a76cac83c73a4a3105e2a92f140689eafb0248993ec747e2a | What is the greatest number of intersections formed by 10 of the same rectangles? | [
"360"
] | [
"What is the greatest number of intersections formed by 10 of the same rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01fa221d3ec75909cc1ef21c68f56af86155c341205ab22fc4677fee2db32ab7 | What is the greatest number of intersections formed by 10 of the same right triangles? | [
"270"
] | [
"What is the greatest number of intersections formed by 10 of the same right triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-f0273ad0849ad6a1ddc1646e5b71f311916f1c436975af8977fee2b0044569fa | What is the greatest number of intersections formed by 10 of the same squares? | [
"360"
] | [
"What is the greatest number of intersections formed by 10 of the same squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-692d756ab5eea0faa3f3ad258a48f8876e5fe1ab27cbb9faa4ad4c59d3021ee5 | What is the greatest number of knights that can be arranged on an $8 \times 8$ chessboard such that no knight controls the square on which another lies? | [
"32"
] | [
"What is the greatest number of knights that can be arranged on an $8 \\times 8$ chessboard such that no knight controls the square on which another lies?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef0098a81814734b2e5b37240bd97f649255649b1c9337749eb6990bf77276d8 | What is the greatest number of odd integers in a sum of 335 pairwise distinct positive integers that equals 100000? | [
"314"
] | [
"What is the greatest number of odd integers in a sum of 335 pairwise distinct positive integers that equals 100000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 20
} | 73 | 0 | 0 |
omr-ws-8ac53cb5aae50632f4834cc8c7a6800090ff3ae76b31586d725cf6fcb96d9363 | What is the greatest number of parts in which three-dimensional space can be divided by $n$ planes and $n$ spheres? | [
"\\dfrac{5n^3 - 6n^2 + 7n + 2}{2}"
] | [
"What is the greatest number of parts in which three-dimensional space can be divided by $n$ planes and $n$ spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-e6c76687d626678216a251016083c6a13fdf445265105148d76d45a1c804136d | What is the greatest number of parts into which the plane can be cut by the edges of $n$ squares? | [
"4n^2 - 4n + 2"
] | [
"What is the greatest number of parts into which the plane can be cut by the edges of $n$ squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-58aca31912d118e2e649e79538056d38fd78cc6f7c8366fc52c68d4c98326940 | What is the greatest number of pieces of candy that cannot be represented using packs of 11, 14, and 15 pieces? | [
"49"
] | [
"What is the greatest number of pieces of candy that cannot be represented using packs of 11, 14, and 15 pieces?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01d7d3c24fcb22849c93409f9759c08a7fcb98b6c413803d310a9f0b106b5bea | What is the greatest number of points of intersection between 8 straight lines and 4 circles? | [
"104"
] | [
"What is the greatest number of points of intersection between 8 straight lines and 4 circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-d5cfc51107d38b98b3df5d96aff04afd49cedeb12fcb03b4dec1ba4eebb60243 | What is the greatest number of points of intersection that can be made with two circles and a square? | [
"18"
] | [
"What is the greatest number of points of intersection that can be made with two circles and a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-efb6cc39b2fba407c0e0e65fd572a58e3b393e0df656a436be83c2cd86297480 | What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper? | [
"11"
] | [
"What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9b98c5c573e95d01aabd2a17905aa21f84ecaf9f93724a0716963dd0ed32576 | What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6? | [
"576"
] | [
"What is the greatest number of positive integers less than or equal to 2016 that can be chosen such that no two of them differ by 1, 2, or 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-aa1f47ea508f7f2d12648e505792e71cfd2ad8673f094b7b74a546952ac224ff | What is the greatest number of regions into which 4 planes can divide a three-dimensional space? | [
"15"
] | [
"What is the greatest number of regions into which 4 planes can divide a three-dimensional space?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c3246189a00eed296d03b4f25bb0f938d84e3fc2b07e25196fca6705e5c1e89 | What is the greatest number of regions into which an equilateral triangle can be divided using exactly 3 straight lines? | [
"7"
] | [
"What is the greatest number of regions into which an equilateral triangle can be divided using exactly 3 straight lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64028b5f445da431474fec318cf53c8218fa7a471c4a68b1ebc2c64d47a29009 | What is the greatest number of regions into which four planes can divide three-dimensional space? | [
"15"
] | [
"What is the greatest number of regions into which four planes can divide three-dimensional space?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-786cf844a9f3524da16ed85b9e64bff67c83620c945716db3ca4e0f35b51493b | What is the greatest number of regions that $100$ ellipses can divide the plane into, including the unbounded region? | [
"19802"
] | [
"What is the greatest number of regions that $100$ ellipses can divide the plane into, including the unbounded region?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-767617c395b5f7a461b88762d4b95b56e32911eb96f6ec6b658486801fa6759c | What is the greatest number of regions that can be created by dividing the plane using a circle, a rectangle, and a triangle? | [
"22"
] | [
"What is the greatest number of regions that can be created by dividing the plane using a circle, a rectangle, and a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-a6edcc1d2c162793480b7a8aaff1c8ee239da0b558812aa00e135fb26bd54c01 | What is the greatest number of rooks that can be placed on an $n \times n$ chessboard such that each rook attacks an even number of other rooks? A rook is said to attack another rook if they are on the same row or column with no other rooks between them. | [
"n(n - 1)"
] | [
"What is the greatest number of rooks that can be placed on an $n \\times n$ chessboard such that each rook attacks an even number of other rooks? A rook is said to attack another rook if they are on the same row or column with no other rooks between them."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-efe0b043a225e055537edf5b829060c62924a86faaefbe00b5e5d1e3b49e8615 | What is the greatest number of sections into which a plane can be divided by \( n \) circles? | [
"n^2 - n + 2"
] | [
"What is the greatest number of sections into which a plane can be divided by \\( n \\) circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2180514ff8772a364c4f58c6724492cf4d6053ef2d4c5bb5894c25ec79220365 | What is the greatest number of three-digit numbers that can be arranged in one row such that any two adjacent numbers are not coprime and any two non-adjacent numbers are coprime? | [
"13"
] | [
"What is the greatest number of three-digit numbers that can be arranged in one row such that any two adjacent numbers are not coprime and any two non-adjacent numbers are coprime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-e9edb431db2171bf3d4731ac0fe709bd43488dbbbe3521e9794c8d26940b516c | What is the greatest number of unit cubes that can be seen from a single point on an $11 \times 11 \times 11$ wooden cube formed by gluing together $11^3$ unit cubes? | [
"331"
] | [
"What is the greatest number of unit cubes that can be seen from a single point on an $11 \\times 11 \\times 11$ wooden cube formed by gluing together $11^3$ unit cubes?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-eecfc0e2e230811e1c0c8894f614bf0d1ec2fef126c43899ad20517cebc48bbc | What is the greatest number of unit cubes that can be seen from a single point on an 11x11x11 wooden cube formed by gluing together $11^3$ unit cubes? | [
"331"
] | [
"What is the greatest number of unit cubes that can be seen from a single point on an 11x11x11 wooden cube formed by gluing together $11^3$ unit cubes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-faededa1f9c41cc3556bc7dec7d172f526534493cf5a6cfcec2638b59ba96675 | What is the greatest positive integer \( n \) such that \( 3^n \) is a factor of \( 200! \)? | [
"97"
] | [
"What is the greatest positive integer \\( n \\) such that \\( 3^n \\) is a factor of \\( 200! \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-1ad9e9230713172a37999ceefb2c4c067d07af80243e3150f7df7c7a758e1b75 | What is the greatest positive integer \( n \) such that \( n+5 \) divides \( n(n+1)(n+2)(n+3)(n+4) \)? | [
"115"
] | [
"What is the greatest positive integer \\( n \\) such that \\( n+5 \\) divides \\( n(n+1)(n+2)(n+3)(n+4) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-2cf3da0a3380408230d9b725a0289043294fd498ecc581898e06166c89536166 | What is the greatest positive integer that cannot be expressed as the sum of two composite odd numbers? | [
"38"
] | [
"What is the greatest positive integer that cannot be expressed as the sum of two composite odd numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-58c7be39ebb99357ab816c26786698af36ea3dbfdd6c7e15f92f001e6d6b4bbf | What is the greatest positive integer that is divisible by all primes less than its square root? | [
"30"
] | [
"What is the greatest positive integer that is divisible by all primes less than its square root?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-14f0fc220e09888207cec7d314adee6ee0068102203411125ae056b688c47a84 | What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers? | [
"5"
] | [
"What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e3022ea8995b04760415a2d6cabb191c2cd94930f00e095a1fcfb234a4100541 | What is the greatest positive value of \( x \) such that \(\frac{(x+1)^2}{x+23}\) is a positive integer? | [
"461"
] | [
"What is the greatest positive value of \\( x \\) such that \\(\\frac{(x+1)^2}{x+23}\\) is a positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-bf1568b7b4d77c22e867044cf4a1327bceb8a7d081cc4535c56ec2beb64b815d | What is the greatest positive value of the function $f(x) = \sqrt{10x - x^2} - \sqrt{18x - x^2 - 77}$ for all real $x$? | [
"\\(\\sqrt{21}\\)"
] | [
"What is the greatest positive value of the function $f(x) = \\sqrt{10x - x^2} - \\sqrt{18x - x^2 - 77}$ for all real $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-c7e29eb7e4e103253fa76f2d58654f1887cf6604925718b788ec762b22f23515 | What is the greatest possible ABBA-factor for a 100-letter word in the land of Microbablia, where the ABBA-factor is defined as the number of occurrences of the sequence ABBA (not necessarily consecutively) within the word? | [
"765625"
] | [
"What is the greatest possible ABBA-factor for a 100-letter word in the land of Microbablia, where the ABBA-factor is defined as the number of occurrences of the sequence ABBA (not necessarily consecutively) within the word?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-378d0bdb8b70dd6994b3a3e41eb415ef1176911a8bc7ddc5579709543ddf4e9c | What is the greatest possible GCD of \( n^3 + 2007 \) and \( (n+1)^3 + 2007 \)? | [
"\\( 3011 \\cdot 9030 + 1 \\)"
] | [
"What is the greatest possible GCD of \\( n^3 + 2007 \\) and \\( (n+1)^3 + 2007 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e3c8519b6ea403462ba06d9a2db9e7d2ac8e5382269c40f8da867057facb194 | What is the greatest possible \( n \) such that 73 coins can be placed on an \( n \times n \) square, with each house containing a non-negative integer number of coins, and any two adjacent houses differing by exactly one coin? All 73 coins must be placed on the board. | [
"11"
] | [
"What is the greatest possible \\( n \\) such that 73 coins can be placed on an \\( n \\times n \\) square, with each house containing a non-negative integer number of coins, and any two adjacent houses differing by exactly one coin? All 73 coins must be placed on the board."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-9ac6285d86b4456f8d7c8be6d61d63df5e12be9af959f1b18f56a9ad890bfdae | What is the greatest possible area between the graph of a non-increasing function passing through the points $(0,5)$, $(1,3)$, and $(2,2)$ and the x-axis from $x=0$ to $x=3$? | [
"10"
] | [
"What is the greatest possible area between the graph of a non-increasing function passing through the points $(0,5)$, $(1,3)$, and $(2,2)$ and the x-axis from $x=0$ to $x=3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-ec87d7b4d92312701cdab006a9e413ab515168a7895e8e1ae0465e34f780cfc8 | What is the greatest possible area of a kite with two 6 cm sides and two 10 cm sides? Provide a solution. | [
"60"
] | [
"What is the greatest possible area of a kite with two 6 cm sides and two 10 cm sides? Provide a solution."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 4
} | 24 | 0 | 0 |
omr-ws-6e3022d53b09c5e77e899279e1bd291a478002f53a7dbe4ac404e2c1c7951546 | What is the greatest possible area of a pentagon that can be inscribed in a unit circle? | [
"$\\frac{5\\sqrt{10+2\\sqrt{5}}}{8}$"
] | [
"What is the greatest possible area of a pentagon that can be inscribed in a unit circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-ed1cea9352867b7ed9acef35508695fe2f0533c340f476d95fad84e9aec82da8 | What is the greatest possible area of a triangle with vertices on or above the x-axis and on or below the parabola \( y = -(x - \frac{1}{2})^2 + 3 \)? Express your answer in simplest radical form. | [
"3\\sqrt{3}"
] | [
"What is the greatest possible area of a triangle with vertices on or above the x-axis and on or below the parabola \\( y = -(x - \\frac{1}{2})^2 + 3 \\)? Express your answer in simplest radical form."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-146a2a154a9ee5985b2573b4ca207a3fd0fc394d9ab9dcdbdefa2691679fe867 | What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the greatest possible area of a triangular region with one vertex at the center of a circle of radius 1 and the other two vertices on the circle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-ff8647c0065d730f75151ea699819c7cf289c34dd161c899d0ddbec11a9c54c4 | What is the greatest possible median of 10 nonnegative integers that have an average of 6? | [
"10"
] | [
"What is the greatest possible median of 10 nonnegative integers that have an average of 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97be7305e20cfcd1b61b9491a213551b58c45ec6bf491286132a69cbcb9460ec | What is the greatest possible number of Fridays that can fall on the 13th in a year? | [
"3"
] | [
"What is the greatest possible number of Fridays that can fall on the 13th in a year?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dfa0abbff035a9f9b0f4bc5a66ad1f4748b4936c4edc4a2c5603fe7f9d5d3641 | What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number? | [
"7"
] | [
"What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-760374659d0f4c1a613865281a0e7045d973179f0f0b45929a96dc146252d834 | What is the greatest possible number of parts that the plane \( xOy \) can be divided into by the graphs of \( n \) quadratic functions \( f_n(x) = a_nx^2 + b_nx + c \) where \( a_n \neq 0 \)? | [
"n^2 - n + 2"
] | [
"What is the greatest possible number of parts that the plane \\( xOy \\) can be divided into by the graphs of \\( n \\) quadratic functions \\( f_n(x) = a_nx^2 + b_nx + c \\) where \\( a_n \\neq 0 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-8d5ab098052a322d325688ca5fa0171987da8c443c8f25f475bd7862d9a3deba | What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12$? | [
"84"
] | [
"What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length $12$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-cd8d31520d601db7bff7f46d1875b0b49ab7cd528f7205858030a50df632d115 | What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12? | [
"84"
] | [
"What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 11
} | 42 | 0 | 0 |
omr-ws-724bf6caf3b1bcf6fbdd1deb2489d8d1b8bd8ea70c560c64c931b223a5fceaac | What is the greatest possible positive integer value of $x$ if $\displaystyle\frac{x^4}{x^2} < 10$? | [
"3"
] | [
"What is the greatest possible positive integer value of $x$ if $\\displaystyle\\frac{x^4}{x^2} < 10$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-369edb8069c2d62fa4c8a39ab7a6eaf2046d2001a1e4fcb2f4e758fd5ec746e2 | What is the greatest possible radius of a circle that passes through the points $(1, 2)$ and $(4, 5)$ and whose interior is contained in the first quadrant of the coordinate plane? | [
"\\(7 - 2\\sqrt{5}\\)"
] | [
"What is the greatest possible radius of a circle that passes through the points $(1, 2)$ and $(4, 5)$ and whose interior is contained in the first quadrant of the coordinate plane?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-60425bf529799bafac48d84cdae0ff814df388ea205361a0079f3dcb8813b990 | What is the greatest possible radius of a circle that passes through the points (1, 2) and (4, 5) and whose interior is contained in the first quadrant of the coordinate plane? Express your answer in simplest radical form. | [
"\\( 7 - 2\\sqrt{5} \\)"
] | [
"What is the greatest possible radius of a circle that passes through the points (1, 2) and (4, 5) and whose interior is contained in the first quadrant of the coordinate plane? Express your answer in simplest radical form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-f1902645ae50ede02d485be321e08d1901875d5990ff2f7b4417ce63b3749b2e | What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | [
"22"
] | [
"What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5471d285a8f92215004cb1f22f175edcb7652d77322a8a23a3b5a94e2e1a4718 | What is the greatest possible sum of two consecutive integers whose product is less than 400? | [
"39"
] | [
"What is the greatest possible sum of two consecutive integers whose product is less than 400?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa01ca3d06116a73c87a569d52eafe34cc39e47c6dc575c03ac0b05a7379a0a2 | What is the greatest possible three-digit number that is divisible by 3 and divisible by 6? | [
"996"
] | [
"What is the greatest possible three-digit number that is divisible by 3 and divisible by 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0a49f5ae809ba6b9d1c3cb92aa61f3bab6253592443a9d9c0456f795482c766c | What is the greatest possible total number of intersection points of \( n \) circles? | [
"\\( n(n-1) \\)"
] | [
"What is the greatest possible total number of intersection points of \\( n \\) circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-90786e847ea49e4eb96f317d8d03e394cef66b0111021d54260839502ef4e828 | What is the greatest possible value of $x$ for the equation $$\left(\frac{4x-16}{3x-4}\right)^2+\left(\frac{4x-16}{3x-4}\right)=12?$$ | [
"2"
] | [
"What is the greatest possible value of $x$ for the equation $$\\left(\\frac{4x-16}{3x-4}\\right)^2+\\left(\\frac{4x-16}{3x-4}\\right)=12?$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48b111ad7af11523db5bb97ab8cfe8eb17fdcb7eb542d5f4e19cb3e009d68933 | What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$? | [
"12"
] | [
"What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6df1c97bf5a552858499034d0753d9ace583ebd1e6608f4709b7bfdf91f9a637 | What is the greatest possible value of \( n \) such that a \( 4 \times 4 \) chessboard can be tiled by \( n \) pairwise incongruent rectangles with integer side lengths? | [
"5"
] | [
"What is the greatest possible value of \\( n \\) such that a \\( 4 \\times 4 \\) chessboard can be tiled by \\( n \\) pairwise incongruent rectangles with integer side lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2bb3ea8d001174fc7ec3afbe1652d1b54db98847b5c9af04e7281172623726bc | What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$? | [
"2^{1005}",
"\\(2^{1005}\\)"
] | [
"What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 50,
"genselect": 16,
"tir": 2
} | 68 | 0 | 0 |
omr-ws-1af719796637f18fcfd4135a8eaa6fbdfe1febad9ba7d19d1a60e4bd2d4f8a76 | What is the greatest power of $5$ that divides the coefficient of $x^{125}$ in the expansion of $$\left(1+x+x^2+x^3+\cdots+x^{125}\right)^{125}?$$ | [
"\\( 5^0 \\)"
] | [
"What is the greatest power of $5$ that divides the coefficient of $x^{125}$ in the expansion of \n$$\\left(1+x+x^2+x^3+\\cdots+x^{125}\\right)^{125}?$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef5c428462f57f9fa9ce0ddee3bfd1ee6bf5b6e06a2d4e9436ad90706296899c | What is the greatest power of 2 that is a factor of $10^{1002} - 4^{501}$? | [
"\\(2^{1005}\\)"
] | [
"What is the greatest power of 2 that is a factor of $10^{1002} - 4^{501}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-efeae573d195e4abf63ec7910224e388ea0c435d42a61d6389ffa6e5ef207f8b | What is the greatest power of 2 that is a factor of \(10^{1002} - 4^{501}\)? | [
"\\(2^{1005}\\)"
] | [
"What is the greatest power of 2 that is a factor of \\(10^{1002} - 4^{501}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-52adc3a7b0a22845ddbcf507a606a7e528c9cd6a01c74f1f17a6ee5a8d2a3ba5 | What is the greatest prime divisor of $3^{22} + 3$? | [
"2269"
] | [
"What is the greatest prime divisor of $3^{22} + 3$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 24
} | 47 | 0 | 0 |
omr-ws-8afecce9f82f2cf498978ab27d07f8a60f7b61372431f68d96df388e69b2ac38 | What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.) | [
"61"
] | [
"What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed562231ed7867ca5fa3e8a1463c81d9d466e1503a4c231debb96f517d2ed0c4 | What is the greatest prime factor of $2^{14} + 1$? | [
"113"
] | [
"What is the greatest prime factor of $2^{14} + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-39ca0c409db26b784d046457272618656bcab32df3c548dedf913aee6e2a7f45 | What is the greatest prime factor of $3^7+6^6$? | [
"67"
] | [
"What is the greatest prime factor of $3^7+6^6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-078164bd15061c4e989aa2a6ae361cc903d84f468056f7823e5930bf3e11bab0 | What is the greatest prime factor of 221? | [
"17"
] | [
"What is the greatest prime factor of 221?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45b558850d06e01a5522811a6246569d0f6b3235117549b6cc7eac89202baa5e | What is the greatest prime factor of 99? | [
"11"
] | [
"What is the greatest prime factor of 99?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7b4393110b6bef77ead5d21ab1f5a12bcf830a1124407d5becae285bf938b0f | What is the greatest prime factor of \(1 + 2 + 3 + \cdots + 2003 + 2004\)? | [
"401"
] | [
"What is the greatest prime factor of \\(1 + 2 + 3 + \\cdots + 2003 + 2004\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ce49e59e91e99b1df1ea821621aa3cf69e134810f428f4bc5db110ecc7c7fd0 | What is the greatest prime factor of \(29! - 1\)? | [
"594,278,556,271,609,021"
] | [
"What is the greatest prime factor of \\(29! - 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-9cc3530f9f0e052b8aaaa5e57edb4a094bfbc4cb48ff26ce216789380b6dcd0d | What is the greatest prime factor of \(541! - 23\)? | [
"23"
] | [
"What is the greatest prime factor of \\(541! - 23\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-290c845d9c4a2582adcb525b415782eaaa0ac84e3ea4ee4deed2f922f23a090d | What is the greatest prime factor of the number \(4^9 + 9^4\)? | [
"353"
] | [
"What is the greatest prime factor of the number \\(4^9 + 9^4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b52bc28a5a4864c3e38a8f2778b8cc40cda0d190295fef4246117dc6d21bcf4 | What is the greatest product obtainable from two integers whose sum is 1998? | [
"998001"
] | [
"What is the greatest product obtainable from two integers whose sum is 1998?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ccd3fa665575d53f0a2d760fb3cec3bf38f3e29d75608a2452e37d55cff365d | What is the greatest product obtainable from two integers whose sum is 246? | [
"15129"
] | [
"What is the greatest product obtainable from two integers whose sum is 246?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f52ef01de55b5291dd855f280e4845b9a58cb7edc971f501b3d2e6b6f0e01cec | What is the greatest real number $\alpha$ such that for any real numbers $0 = a_1 \leq a_2 \leq \dots \leq a_{11} = 1$, these numbers can be divided into two groups with arithmetic means $S_1$ and $S_2$ satisfying $|S_1 - S_2| \leq \alpha$? | [
"\\dfrac{1}{10}"
] | [
"What is the greatest real number $\\alpha$ such that for any real numbers $0 = a_1 \\leq a_2 \\leq \\dots \\leq a_{11} = 1$, these numbers can be divided into two groups with arithmetic means $S_1$ and $S_2$ satisfying $|S_1 - S_2| \\leq \\alpha$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-a0ecb296cebd3f49b8ca76ca3cdcfa364f3bcaee361c40dc03d9bdba8098176a | What is the greatest real number $a$ such that in a square with side length 1, which is colored using 3 colors, there can always be found two points of the same color at a distance $a$? | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"What is the greatest real number $a$ such that in a square with side length 1, which is colored using 3 colors, there can always be found two points of the same color at a distance $a$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 7
} | 10 | 0 | 0 |
omr-ws-77b541c045282a276641a8668f384099754d63d670cc1c66f318f826b2ee1d9e | What is the greatest three-digit multiple of 19? | [
"988"
] | [
"What is the greatest three-digit multiple of 19?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-104c7cd432f0ff9adef5ea6bd6ca5fc121f49e5eae60cfdb1333d207752e4942 | What is the greatest three-digit number that is one more than a multiple of 9 and three more than a multiple of 5? | [
"973"
] | [
"What is the greatest three-digit number that is one more than a multiple of 9 and three more than a multiple of 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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