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-06f35275d5f14a297aa43841b037090db497b9b7a6b243027885af6d6b79f119 | What is the largest number of good angles in a convex $n$-gon, where an angle is called good if its sine is equal to the product of the sines of all other angles of the $n$-gon? | [
"2"
] | [
"What is the largest number of good angles in a convex $n$-gon, where an angle is called good if its sine is equal to the product of the sines of all other angles of the $n$-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-bdf0ff8c37ecaccaccf68fe7eea93e4090f6ff9b8fa3ef8024609f42b9cf0f05 | What is the largest number of good diagonals a convex pentagon can have, where a good diagonal is defined as one that divides the pentagon's area in half? | [
"2"
] | [
"What is the largest number of good diagonals a convex pentagon can have, where a good diagonal is defined as one that divides the pentagon's area in half?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-0eb10dbc06545389373cbe0c51be17846da59c1f0cf408ebf4f38aa547f95746 | What is the largest number of halfmarks that cannot be expressed using coins of denominations 8, 9, and 10 halfmarks? | [
"31"
] | [
"What is the largest number of halfmarks that cannot be expressed using coins of denominations 8, 9, and 10 halfmarks?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d631ae2835290f42841625d96f775b2244a213a1ecfb05deb63f1414ff1f32e0 | What is the largest number of integer points that can be chosen in space so that none of the segments with ends in them contains two more integer points? | [
"27"
] | [
"What is the largest number of integer points that can be chosen in space so that none of the segments with ends in them contains two more integer points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-e4b34d27f70ccff398ea3c5ccbadf69de82d2f36e5f4c821667aaa3f964ad70e | What is the largest number of integers from $1$ to $2022$ that can be chosen so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers? | [
"674"
] | [
"What is the largest number of integers from $1$ to $2022$ that can be chosen so that no sum of any two different chosen integers is divisible by any difference of two different chosen integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30c922a2eb075ad2a0b736c5f423602228f90be281f8aa3cc277f59a7524e87c | What is the largest number of integers that can be chosen from 1 to 100 such that no chosen number is thrice any of the other chosen integers? | [
"76"
] | [
"What is the largest number of integers that can be chosen from 1 to 100 such that no chosen number is thrice any of the other chosen integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-d0629ef9d3538487fe235ffb33c0fa8d01a4216ad4e96900e61c278ee251538b | What is the largest number of integers that may be selected from $1$ to $2016$ inclusive such that the least common multiple of any number of integers selected is also selected? | [
"40"
] | [
"What is the largest number of integers that may be selected from $1$ to $2016$ inclusive such that the least common multiple of any number of integers selected is also selected?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 9
} | 26 | 0 | 0 |
omr-ws-03de43dffc4c13837526d0ad5b7d3136a0e0245d657d8f8479736d5f2a8c2193 | What is the largest number of integers that may be selected from 1 to 2016 inclusive such that the least common multiple of any number of integers selected is also selected? | [
"40"
] | [
"What is the largest number of integers that may be selected from 1 to 2016 inclusive such that the least common multiple of any number of integers selected is also selected?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-d59a91a19f31c329f9d9cc8a414c2109c3dd8056411e54c6c379294d1231feeb | What is the largest number of isosceles triangles that can be formed with sides lying on 12 straight lines drawn on a plane, given that no two lines are parallel? | [
"60"
] | [
"What is the largest number of isosceles triangles that can be formed with sides lying on 12 straight lines drawn on a plane, given that no two lines are parallel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-01735e3e83b3f4e2b2306a9df4a040ca661377f382ae2c836a0043a51e170b44 | What is the largest number of kings that can be placed on a chessboard so that no two of them put each other in check? | [
"16"
] | [
"What is the largest number of kings that can be placed on a chessboard so that no two of them put each other in check?"
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 49
} | 49 | 0 | 0 |
omr-ws-83468e26b168a4d180b1ba821635b9a93f7c388882a77ce87eaae487800f95e0 | What is the largest number of kings that can be placed on a chessboard such that no two of them attack each other? | [
"16"
] | [
"What is the largest number of kings that can be placed on a chessboard such that no two of them attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc667157b5e34dd71bb90f04a55d346fca2f58dec68c98d3b266ed19fb5916e9 | What is the largest number of knights that can be placed on a $5 \times 5$ chessboard so that each knight attacks exactly two other knights? | [
"16"
] | [
"What is the largest number of knights that can be placed on a $5 \\times 5$ chessboard so that each knight attacks exactly two other knights?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 1,
"tir": 8
} | 46 | 0 | 0 |
omr-ws-fcfeaf9c584b269a88ed0c27ebb7c1ca246e7f9f2333325ce3a77c9de1ffbd84 | What is the largest number of lines that can be drawn in a plane so that each line intersects exactly four others? | [
"8"
] | [
"What is the largest number of lines that can be drawn in a plane so that each line intersects exactly four others?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 28,
"genselect": 6
} | 34 | 0 | 0 |
omr-ws-8903204bca393c47537ae6f208034a379f5adf0042596c3033c6c4ca180cd6b2 | What is the largest number of links that a closed polyline can have, all the vertices of which lie on the boundary of a triangle, and each link is parallel to one of the sides of this triangle? | [
"6"
] | [
"What is the largest number of links that a closed polyline can have, all the vertices of which lie on the boundary of a triangle, and each link is parallel to one of the sides of this triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28f37b64ce117a7bfd49ea397345266c2517c88493c192b6d990a2b9d2e31d6a | What is the largest number of members of one family, whose elements have three digits, if two or more numbers are considered to be in the same family if they share at least one digit in common? | [
"270"
] | [
"What is the largest number of members of one family, whose elements have three digits, if two or more numbers are considered to be in the same family if they share at least one digit in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a58b0535451780682f6ef1ecb61fe5678462ed44bd08cf6d2d3a667c673f7d5 | What is the largest number of non-attacking rooks that can be placed on the surface of a $50 \times 50 \times 50$ cube, where a rook can attack cells in the same row and through continuations of this row across edges of the cube? | [
"75"
] | [
"What is the largest number of non-attacking rooks that can be placed on the surface of a $50 \\times 50 \\times 50$ cube, where a rook can attack cells in the same row and through continuations of this row across edges of the cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7565d7563edef766044554be0eb3f444020bc12d3648fe88e13ef0673dcdefd2 | What is the largest number of non-congruent trapezoids that can be formed using four segments of different lengths? | [
"3"
] | [
"What is the largest number of non-congruent trapezoids that can be formed using four segments of different lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-98de94d974573ade5fe06be52034d98495d63352ee5e9a7ca7ec1aecaaef1bc2 | What is the largest number of non-intersecting diagonals that can be drawn in a convex $n$-gon (diagonals with a common vertex are allowed)? | [
"$n-3$"
] | [
"What is the largest number of non-intersecting diagonals that can be drawn in a convex $n$-gon (diagonals with a common vertex are allowed)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3507b23175df50f4ab7aede5c19d9b6bae48b678dbca4eca503ad1e1758af3b3 | What is the largest number of obtuse angles possible in triangles formed by the vertices of a 15-gon, where all vertices of the 15-gon lie on the circumference of a circle with the center inside the 15-gon? | [
"442"
] | [
"What is the largest number of obtuse angles possible in triangles formed by the vertices of a 15-gon, where all vertices of the 15-gon lie on the circumference of a circle with the center inside the 15-gon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bb1b95f950ed16e0635b5ec0c2ba1042556a7d62b2fb134fbd44730a8ca4d1c | What is the largest number of obtuse triangles that can be composed of 16 different segments, where each triangle is composed of three segments, and the largest segment does not exceed twice the smallest? | [
"5"
] | [
"What is the largest number of obtuse triangles that can be composed of 16 different segments, where each triangle is composed of three segments, and the largest segment does not exceed twice the smallest?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-e5e65e3211506cae16a98e55291aa222f3363d417d62618a398bf751088e9a95 | What is the largest number of obtuse triangles that can be formed using the vertices of a 15-sided polygon, where all vertices lie on the circumference of a circle? | [
"455"
] | [
"What is the largest number of obtuse triangles that can be formed using the vertices of a 15-sided polygon, where all vertices lie on the circumference of a circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5d75dd0f01459d77308691fcdc59427f94a0d81d46f228e4b9a50c8a3184e4a0 | What is the largest number of parts a plane can be divided into by $2007$ third-degree polynomial graphs? | [
"6041071"
] | [
"What is the largest number of parts a plane can be divided into by $2007$ third-degree polynomial graphs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-6560e4f5d8116c6339b0f8691b3004ebbad2fc3f3d96a47827ad6bcdfa501560 | What is the largest number of parts that \( n \) rays can cut a plane into? | [
"\\dfrac{n(n + 1)}{2} + 1"
] | [
"What is the largest number of parts that \\( n \\) rays can cut a plane into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-903bea3b5a23e8110a10a6d031cbc91355ed5bf15599e42f7543e1b5f622d5d7 | What is the largest number of pawns that can be placed on a $19 \times 19$ chessboard such that each pawn can move to a neighboring square (horizontally or vertically) in each move, no two pawns move to the same square, and no pawn moves along the same line in two successive moves, allowing the game to continue indefin... | [
"324"
] | [
"What is the largest number of pawns that can be placed on a $19 \\times 19$ chessboard such that each pawn can move to a neighboring square (horizontally or vertically) in each move, no two pawns move to the same square, and no pawn moves along the same line in two successive moves, allowing the game to continue i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-9b3c99bec9cb718f54b9291e476b62874ca6e6ebf076540011948492a7c7b561 | What is the largest number of points in which the graphs of a third-degree polynomial and a fifth-degree polynomial can meet? | [
"5"
] | [
"What is the largest number of points in which the graphs of a third-degree polynomial and a fifth-degree polynomial can meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06903fadf7b93bda42b8d53d06da38ff21acb63e14d9565a7ad588ddf4765c1a | What is the largest number of points that can be arranged on a given circle so that in all the triangles they form, none of the angles is less than $25^\circ$? | [
"7"
] | [
"What is the largest number of points that can be arranged on a given circle so that in all the triangles they form, none of the angles is less than $25^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-03d790b3ece18854be32c400b9b435cf7adbf9adcb15c88d9f90d8c851327e05 | What is the largest number of points that can be arranged on a plane so that among their pairwise distances there would be only two different ones? | [
"7"
] | [
"What is the largest number of points that can be arranged on a plane so that among their pairwise distances there would be only two different ones?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca206fdde34fc39c2fec97155ab5988de14fe03f1f28019a824aa7f83cb4c66f | What is the largest number of points that can be placed in the ring $\{(x,y) \mid 1 \le x^2 + y^2 \le 2\}$ such that the distance between any two points is at least 1? | [
"8"
] | [
"What is the largest number of points that can be placed in the ring $\\{(x,y) \\mid 1 \\le x^2 + y^2 \\le 2\\}$ such that the distance between any two points is at least 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-073439746870499d302a2800a8499c684530e42aba2520d2ec1a3da8f208c877 | What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer? | [
"3"
] | [
"What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ce85377a1fbfeedffa5e1207b1656158d3eb87175bdb5c3d8cb3a4bf57b3777 | What is the largest number of positive integers less than or equal to $2016$ such that no two chosen numbers have a difference of $1$, $2$, or $6$? | [
"576"
] | [
"What is the largest number of positive integers less than or equal to $2016$ such that no two chosen numbers have a difference of $1$, $2$, or $6$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-4e1cf4831f9c894215db9e7e1cd35bbb86501332258e1367250e6a96d4172566 | What is the largest number of primes that could divide an integer \( n \) which has 40 positive divisors, including 1 and \( n \)? | [
"4"
] | [
"What is the largest number of primes that could divide an integer \\( n \\) which has 40 positive divisors, including 1 and \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2336caccd5108cbc7eb113ffed2f7d16ed7f0b276c1007bd311931e10cea3918 | What is the largest number of primes that sum to 125? | [
"42"
] | [
"What is the largest number of primes that sum to 125?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b05e784eea6bf0165fa2aa5e796dcc2ff6e5afe26595af22e7030bc728cc8a6d | What is the largest number of queens that can be placed on a chessboard such that no two of them attack each other? | [
"8"
] | [
"What is the largest number of queens that can be placed on a chessboard such that no two of them attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f97e0e725b842981a898f7dac17b2bb809faaf6e269c2bc02049daeec71cc6d8 | What is the largest number of queens that can be placed on an $n \times n$ chessboard such that no two queens can attack each other? | [
"\\( n \\)"
] | [
"What is the largest number of queens that can be placed on an $n \\times n$ chessboard such that no two queens can attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-21f61288a4987748c0f90fd274d2d7227d43732cb306ddfcf8f22f88a8ddf35c | What is the largest number of rays emerging from one point that can be drawn in space so that the angles between any two rays are equal to either $90^\circ$ or $120^\circ$? For this highest number, specify all possible ray locations. | [
"4"
] | [
"What is the largest number of rays emerging from one point that can be drawn in space so that the angles between any two rays are equal to either $90^\\circ$ or $120^\\circ$? For this highest number, specify all possible ray locations."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-6d1e8701697bb64f71f66d67fe7948bec36f2bc1cfc320f1874ab82e6e77d0ed | What is the largest number of real roots that a polynomial $p(x)$ with real coefficients can have, given that $p(x^2 - 1) = p(x) p(-x)$? | [
"4"
] | [
"What is the largest number of real roots that a polynomial $p(x)$ with real coefficients can have, given that $p(x^2 - 1) = p(x) p(-x)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-c4818b4dedb4def16938f7e7aac67bfe3cde910f16df7739e7ac8b1ebaa591ef | What is the largest number of real roots that a polynomial \( p(x) \) with real coefficients can have, given that \( p(x^2 - 1) = p(x) p(-x) \)? | [
"4"
] | [
"What is the largest number of real roots that a polynomial \\( p(x) \\) with real coefficients can have, given that \\( p(x^2 - 1) = p(x) p(-x) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-fdc1b53d310bb710d0c28cdc3186a5602899407482c34ddd977d41ce0f0ec015 | What is the largest number of regions into which the plane could be divided by a planar diagram composed of three circles and two lines? | [
"23"
] | [
"What is the largest number of regions into which the plane could be divided by a planar diagram composed of three circles and two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-bba1f6b73df2fd7e882b745e13d0451cf6939ce9daadf5cf5fd3690e3dec6600 | What is the largest number of rooks that can be placed on an $n \times n$ chessboard such that each rook is being threatened by at most 1 other rook? | [
"\\( n \\)"
] | [
"What is the largest number of rooks that can be placed on an $n \\times n$ chessboard such that each rook is being threatened by at most 1 other rook?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31d2ec3024b576954b6e64835ff32758351e73ff13d37d77cc88fe8f1ba03b12 | What is the largest number of rooks that can be placed on an $n \times n$ chessboard such that each rook is being threatened by at most 2 other rooks? | [
"\\( 2n - 2 \\)"
] | [
"What is the largest number of rooks that can be placed on an $n \\times n$ chessboard such that each rook is being threatened by at most 2 other rooks?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d724c5976a1dd26b31329e085dd21a65ac78edb13b1a63832d162813c9b739a | What is the largest number of self-intersection points a closed broken line consisting of 7 links (none of which lie on the same line) can have? The vertex of the polyline does not lie on a link unless it is one of its ends. | [
"14"
] | [
"What is the largest number of self-intersection points a closed broken line consisting of 7 links (none of which lie on the same line) can have? The vertex of the polyline does not lie on a link unless it is one of its ends."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-f528d59502201ba3d6880a472065c7ba1f2198b89eb823f0d88d2616e969ee81 | What is the largest number of sides a convex polygon can have if it contains exactly three internal obtuse angles? | [
"6"
] | [
"What is the largest number of sides a convex polygon can have if it contains exactly three internal obtuse angles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a6c1080c39902acddcc875eff169633e43bcf6f05ec354f86ef56aaa2b016bc | What is the largest number of sides a figure can have that is a common part of a triangle and a convex quadrilateral? Draw an example. | [
"7"
] | [
"What is the largest number of sides a figure can have that is a common part of a triangle and a convex quadrilateral? Draw an example."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0daffdf4735c1a6bd30ad26bc7dc104edb3e02ebc3978c32404b79bb5614485 | What is the largest number of solid $2\text{-in} \times 2\text{-in} \times 1\text{-in}$ blocks that can fit in a $3\text{-in} \times 2\text{-in} \times 3\text{-in}$ box? | [
"4"
] | [
"What is the largest number of solid $2\\text{-in} \\times 2\\text{-in} \\times 1\\text{-in}$ blocks that can fit in a $3\\text{-in} \\times 2\\text{-in} \\times 3\\text{-in}$ box?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9b4ec29817043d8665acd6f8101e4a4149651520c81bcdee4ef0ea10d6020710 | What is the largest number of squares on a $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces? | [
"64"
] | [
"What is the largest number of squares on a $9 \\times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 3
} | 20 | 0 | 0 |
omr-ws-dc335cd3f4ee596c2f20597667673c6a5b0258e852fb0ff7c772e80b4764913e | What is the largest number of squares that can be colored in an $n \times n$ grid such that no row or column contains more than 2 colored squares? | [
"\\( 2n \\)"
] | [
"What is the largest number of squares that can be colored in an $n \\times n$ grid such that no row or column contains more than 2 colored squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-20a5422a0a91598b0a3105515646f7840e9582986469acbeee97c904813cfc05 | What is the largest number of squares that can be colored in an $n \times n$ grid such that no row or column contains more than 3 colored squares? | [
"\\( 3n \\)"
] | [
"What is the largest number of squares that can be colored in an $n \\times n$ grid such that no row or column contains more than 3 colored squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-171b061a2fdd96cafa30bf000f63c87e88d0dcd007aa5672561f2b7c7812980b | What is the largest number of strips $1 \times 5$ that can be cut from a sheet of checkered paper measuring $12 \times 13$? (You can only cut along cell lines.) | [
"31"
] | [
"What is the largest number of strips $1 \\times 5$ that can be cut from a sheet of checkered paper measuring $12 \\times 13$? (You can only cut along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3d4cc05c8549d106a4ff997e2339d252e9d1b2cadb54d4640b834a1d80a42b1a | What is the largest number of strips $1 \times 6$ that can be cut from a sheet of checkered paper measuring $12 \times 13$? (You can only cut along cell lines.) | [
"26"
] | [
"What is the largest number of strips $1 \\times 6$ that can be cut from a sheet of checkered paper measuring $12 \\times 13$? (You can only cut along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a384f0bc69d52e31664f25efe07b402bf5d37eae1d206761de903c28c176640 | What is the largest number of subsequences of the form $n, n+1, n+2$ that a sequence of 2001 positive integers can have? | [
"296740963"
] | [
"What is the largest number of subsequences of the form $n, n+1, n+2$ that a sequence of 2001 positive integers can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-4b07acd95fe7f5379cfd140d29d8db4173b4034c2e4cc640f3222d3784cbd15d | What is the largest number of subsets that can be selected from the set $\{1,2,3,...,n\}$ such that any two selected subsets have at least one element in common? | [
"$2^{n-1}$"
] | [
"What is the largest number of subsets that can be selected from the set $\\{1,2,3,...,n\\}$ such that any two selected subsets have at least one element in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24aa86822da75c7a27b0af7ac22bc077a5f71fc50d12388957cdf71d4cad8951 | What is the largest number of subsets that can be selected from the set [n] such that any two selected subsets have at least one element in common? | [
"\\(2^{n-1}\\)"
] | [
"What is the largest number of subsets that can be selected from the set [n] such that any two selected subsets have at least one element in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-e67b87a0617ecd41cffc5345ca2c2b09aef8e9edd6ec888d7bfd9629f0a7bdbb | What is the largest number of subsets you can choose from $\{1, \dots , n\}$ such that each subset has an odd number of elements and the intersection of any two distinct subsets has an even number of elements? | [
"\\(2^{\\left\\lfloor \\frac{n}{2} \\right\\rfloor}\\)"
] | [
"What is the largest number of subsets you can choose from $\\{1, \\dots , n\\}$ such that each subset has an odd number of elements and the intersection of any two distinct subsets has an even number of elements?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad21a73678343ab84f3c4b552229c8f168768ac3d933d17e22125fadbb5a20ba | What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such t... | [
"4"
] | [
"What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are s... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-ea3692bcc066662d7825a87179c68ae5a83e706273642e1d6d588f8b4544b757 | What is the largest number of triangles that can be created using the sides of a $10$-gon (using triplets of its sides)? | [
"0"
] | [
"What is the largest number of triangles that can be created using the sides of a $10$-gon (using triplets of its sides)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-59f31b8075178f2ea73889082029d6200887c6eb0474837816f3937ae25d199c | What is the largest number of vertices a polygon can have, all sides which lie on six lines? | [
"12"
] | [
"What is the largest number of vertices a polygon can have, all sides which lie on six lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-23642dad42b3fa5b9101957f6aa568fd3a0e237a0979a76bc59cfd6ae702c0b5 | What is the largest number of vertices of a regular $12$-gon that can be chosen so that no three form an isosceles triangle? | [
"4"
] | [
"What is the largest number of vertices of a regular $12$-gon that can be chosen so that no three form an isosceles triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5597260f5535a3e62df5633ff778a392781fad17741798171990655f23f43241 | What is the largest number such that if you remove the \(n\)th digit from the number, the resulting number is divisible by \(n\) for each \(n\) from 1 to the length of the number? | [
"999960"
] | [
"What is the largest number such that if you remove the \\(n\\)th digit from the number, the resulting number is divisible by \\(n\\) for each \\(n\\) from 1 to the length of the number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-ca7fae36685252f9d23511db9f2bd4f7fa23be9acbb15920b2336c78e28281f3 | What is the largest number such that in any given triangle, the ratio between the lengths of an arbitrary altitude and the inradius of the triangle is still greater than this number? | [
"2"
] | [
"What is the largest number such that in any given triangle, the ratio between the lengths of an arbitrary altitude and the inradius of the triangle is still greater than this number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8e6ef74b3f7d45bcbda0a92367fd03374f41a03836a669de37aecab69a941ea | What is the largest number that can be made by inserting two multiplication signs between the digits $2, 2, 3, 3,$ and $3$ in that order? | [
"2178"
] | [
"What is the largest number that can be made by inserting two multiplication signs between the digits $2, 2, 3, 3,$ and $3$ in that order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-66f49192ee5db2b3d316053cfb00fdb3499b949e209401b9ee1cf0bdc0628590 | What is the largest number that can be obtained using 4 twos? | [
"\\( 2^{2^{22}} \\)"
] | [
"What is the largest number that can be obtained using 4 twos?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-b079fca62396b88d4bfdb0c99753ac7853ae9df2dfca58f53aa5bff7e4aff65d | What is the largest number that cannot be expressed in the form \(5x + 8y\), where \(x\) and \(y\) are natural numbers? | [
"27"
] | [
"What is the largest number that cannot be expressed in the form \\(5x + 8y\\), where \\(x\\) and \\(y\\) are natural numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d0ffc39dcb06c8dde96c0b509a1e4b1bb16a61e04ee023bcd72a672464853ab | What is the largest number that cannot be made using the numbers 5, 8, and 9? | [
"12"
] | [
"What is the largest number that cannot be made using the numbers 5, 8, and 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-818b5a0eccb27918803ba6342d8f2d85bd4709db84d1834b86fc7d1afd9cdc20 | What is the largest number that cannot be written as $9x + 9y + 16z$ where $x, y,$ and $z$ are natural numbers and $x, y$ are multiples of 9 and $z$ is a multiple of 16? | [
"20561"
] | [
"What is the largest number that cannot be written as $9x + 9y + 16z$ where $x, y,$ and $z$ are natural numbers and $x, y$ are multiples of 9 and $z$ is a multiple of 16?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3f45cf1e6cab0da20e2e5e9b4d36479fbf8be4fbd9231fc5662588289be1261 | What is the largest number that cannot be written as the sum of distinct squares? | [
"128"
] | [
"What is the largest number that cannot be written as the sum of distinct squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c9fa262c04e6d26dcb059ac4e8092c7da709483e69fac12d87ad46e9a4b414a | What is the largest number that is divisible by all numbers less than its square root? | [
"24"
] | [
"What is the largest number that is divisible by all numbers less than its square root?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-9d22b22167e0042a7195b86b405c0874ac69f8c4c1eaf63cb8eedc2771331644 | What is the largest number that leaves a remainder of 1 when divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10? | [
"2521"
] | [
"What is the largest number that leaves a remainder of 1 when divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-76e289cc2e3c1bc9dc9b451c57f1b84ca3a2d2f986714c806536e30203e74a44 | What is the largest number which, when divided into each of 1905, 2587, 3951, 7020, and 8725, leaves the same remainder each time? | [
"341"
] | [
"What is the largest number which, when divided into each of 1905, 2587, 3951, 7020, and 8725, leaves the same remainder each time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-640523f19b27a470463fc46f2804360b3f5d650704f4a8595fd887199de5ed8e | What is the largest number, all of whose digits are 3 or 2, and whose digits add up to $11$? | [
"32222"
] | [
"What is the largest number, all of whose digits are 3 or 2, and whose digits add up to $11$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-717f7fd01b3376d854158454e73cf6c295c4fea927f6150ebb45b10c01c1ae72 | What is the largest number, with its digits all different, whose digits add up to 16? | [
"643210"
] | [
"What is the largest number, with its digits all different, whose digits add up to 16?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-36d276b5079e5c9bd652487e5a345b592d50ec1633abe1e9e433f8b021c06b74 | What is the largest odd square-free integer $n$ such that $\phi(n) = 1200$? | [
"2013"
] | [
"What is the largest odd square-free integer $n$ such that $\\phi(n) = 1200$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef6d4d83dac521f8ee7da574ecb86acd6d2fa2a4bff6abd10caea66d7c7ee27e | What is the largest odd square-free integer \( n \) such that \( \varphi(n) = 1200 \)? Here, \( \varphi(n) \) represents the number of positive integers less than or equal to \( n \) that are relatively prime to \( n \). | [
"2265"
] | [
"What is the largest odd square-free integer \\( n \\) such that \\( \\varphi(n) = 1200 \\)?\n\nHere, \\( \\varphi(n) \\) represents the number of positive integers less than or equal to \\( n \\) that are relatively prime to \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-7a2974e90c78aa6124dc38cd7f2f58e1f33d9c1a4b051837f49b1adee3bbe389 | What is the largest perfect square factor of 1512? | [
"36"
] | [
"What is the largest perfect square factor of 1512?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c850239221fa1c1fc29273bd76ecc3a12bdfe8403db47b928464696cdb15ed5a | What is the largest perfect square that contains each of the digits 0 through 9 at most once? | [
"9814072356"
] | [
"What is the largest perfect square that contains each of the digits 0 through 9 at most once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0eb56c284844bbe7b4dff332b084125e5569b99b1a5588df818244263a15231f | What is the largest perfect square that divides the expression $2014^3 - 2013^3 + 2012^3 - 2011^3 + 2010^3 - \cdots + 2^3 - 1^3$? | [
"\\(1007^2\\)"
] | [
"What is the largest perfect square that divides the expression $2014^3 - 2013^3 + 2012^3 - 2011^3 + 2010^3 - \\cdots + 2^3 - 1^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 28
} | 51 | 0 | 0 |
omr-ws-cb09c51f59e1d0a79b9be7016e415f3594e75836137ced2944dca0e24e6f6de0 | What is the largest perimeter of a rectangle whose vertices are placed inside or on the surface of a unit cube? | [
"\\( 4\\sqrt{2} \\)"
] | [
"What is the largest perimeter of a rectangle whose vertices are placed inside or on the surface of a unit cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-550c8abe5ffaed861a8a3e117f674ebab921188d32d30a20dd649f9a2af9fdaa | What is the largest perimeter that a face of a tetrahedron with edge lengths $5, 5, 8, 9, 13, 13$ can have? | [
"34"
] | [
"What is the largest perimeter that a face of a tetrahedron with edge lengths $5, 5, 8, 9, 13, 13$ can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-c331dedea61af7545c6b913e2acde750907436f90d0beabf42de6655a2f7f40e | What is the largest positive integer $abcdef$ that can be formed from the digits $1, 2, 3, 4, 5, 6$, each used exactly once, if $abcdef$ is divisible by 6, $abcde$ is divisible by 5, $abcd$ is divisible by 4, $abc$ is divisible by 3, and $ab$ is divisible by 2? | [
"321654"
] | [
"What is the largest positive integer $abcdef$ that can be formed from the digits $1, 2, 3, 4, 5, 6$, each used exactly once, if $abcdef$ is divisible by 6, $abcde$ is divisible by 5, $abcd$ is divisible by 4, $abc$ is divisible by 3, and $ab$ is divisible by 2?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-e37deb7c8f26909a3347c3ae129c7a246b8e6b70bc62ecfa9095dde36b061903 | What is the largest positive integer $abcdef$ that can be formed from the digits 1, 2, 3, 4, 5, 6, each used exactly once, if $abcdef$ is divisible by 6, $abcde$ is divisible by 5, $abcd$ is divisible by 4, $abc$ is divisible by 3, and $ab$ is divisible by 2? | [
"321654"
] | [
"What is the largest positive integer $abcdef$ that can be formed from the digits 1, 2, 3, 4, 5, 6, each used exactly once, if $abcdef$ is divisible by 6, $abcde$ is divisible by 5, $abcd$ is divisible by 4, $abc$ is divisible by 3, and $ab$ is divisible by 2?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-f29a32f4a2ebe7609ebe29cc02a73211306be8e91d60c1b30b01ad40adb26b87 | What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$? | [
"48"
] | [
"What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e62d06a6cfa3dcdc74904d4d05268406a33e5d3459decfa665dc9c2b8e927d8e | What is the largest positive integer $n$ for which there are no positive integers $a$ and $b$ such that $8a + 11b = n$? | [
"88"
] | [
"What is the largest positive integer $n$ for which there are no positive integers $a$ and $b$ such that $8a + 11b = n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-b4e729fb6fffdd4a3ecf12775bc917faa95913aac022e46e4d4de656103a3f14 | What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$? | [
"112"
] | [
"What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\\frac{8}{15} < \\frac{n}{n + k} < \\frac{7}{13}$?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-d9345bf4c51227fddcf333ecf68c716ae6cb9fce31f67b90eeb6cde72c13a50f | What is the largest positive integer $n$ for which there is a unique integer $k$ such that \[ \frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}? \] | [
"97"
] | [
"What is the largest positive integer $n$ for which there is a unique integer $k$ such that\n\\[\n\\frac{8}{15} < \\frac{n}{n+k} < \\frac{7}{13}?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ae4a6a771ad9d0e553bb2c80e0a25d0c755331ff1a60833c1f042bc168cd8034 | What is the largest positive integer $n$ less than or equal to $2000$ for which $3^7$ divides $\binom{2n}{n}$? | [
"1943"
] | [
"What is the largest positive integer $n$ less than or equal to $2000$ for which $3^7$ divides $\\binom{2n}{n}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 15
} | 25 | 0 | 0 |
omr-ws-3e9047a2f1dabe384e9902f31b52859fbb31e8f7a0e6d176ee7053611427674c | What is the largest positive integer $n$ such that $$\frac{a^2}{\frac{b}{29} + \frac{c}{31}} + \frac{b^2}{\frac{c}{29} + \frac{a}{31}} + \frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a + b + c)$$ holds for all positive real numbers $a, b, c$? | [
"\\(\\frac{899}{60}\\)"
] | [
"What is the largest positive integer $n$ such that $$\\frac{a^2}{\\frac{b}{29} + \\frac{c}{31}} + \\frac{b^2}{\\frac{c}{29} + \\frac{a}{31}} + \\frac{c^2}{\\frac{a}{29} + \\frac{b}{31}} \\ge n(a + b + c)$$ holds for all positive real numbers $a, b, c$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dab2d4fbd4d1b5cb0a596ff1750e8716121db813c2b9ba5bd1ecdd61689d428a | What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times \cdots \times 50$ is divisible by $10^n$? | [
"11"
] | [
"What is the largest positive integer $n$ such that $10 \\times 11 \\times 12 \\times \\cdots \\times 50$ is divisible by $10^n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06bb5425e0eb6f2652a312c2ea4a729a3a2b5dcd5cda339d5a05d6351dc55913 | What is the largest positive integer $n$ such that $n!$ divides $\frac{(5!)!}{(4!)!}$? | [
"97"
] | [
"What is the largest positive integer $n$ such that $n!$ divides $\\frac{(5!)!}{(4!)!}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 15
} | 32 | 0 | 0 |
omr-ws-ed8fa757d7e31b8d254fe3f0f7813eb6d0f3a9f199093c54d0e0361e35f0c175 | What is the largest positive integer \( n \) for which a square cannot be divided into \( n \) square sections (not necessarily equal in area)? | [
"5"
] | [
"What is the largest positive integer \\( n \\) for which a square cannot be divided into \\( n \\) square sections (not necessarily equal in area)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a45c08bbcc2729574956b6e427972948e839e3aa490e5205968c6df21a978b66 | What is the largest positive integer \( n \) so that \( n \) is not the area of a union of two squares with corners on lattice points and sides parallel to the \( x \)- and \( y \)-axes? | [
"3"
] | [
"What is the largest positive integer \\( n \\) so that \\( n \\) is not the area of a union of two squares with corners on lattice points and sides parallel to the \\( x \\)- and \\( y \\)-axes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36b8bcf89a5217fbbcb3bd87034d22e9c644ff5a8875c269d9ad0adda9467172 | What is the largest positive integer \( n \) such that \( 4^n + 4^{27} + 4^{1000} \) is a perfect square? | [
"514"
] | [
"What is the largest positive integer \\( n \\) such that \\( 4^n + 4^{27} + 4^{1000} \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-ce10b6e657ffdbf7635d0c6b14599e353f8bc833b880d067c8a6e8d5c0ae8508 | What is the largest positive integer \( n \) such that \( n^2 + 100 \) is divisible by \( n + 10 \)? | [
"190"
] | [
"What is the largest positive integer \\( n \\) such that \\( n^2 + 100 \\) is divisible by \\( n + 10 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-d64321b7cec3d009df9518da7f38684c02f8434cebf08740264e6f81c66caddd | What is the largest positive integer \( n \) such that \[ \frac{a^2}{\frac{b}{29} + \frac{c}{31}} + \frac{b^2}{\frac{c}{29} + \frac{a}{31}} + \frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c) \] holds for all positive real numbers \( a, b, c \)? | [
"14"
] | [
"What is the largest positive integer \\( n \\) such that \n\\[\n\\frac{a^2}{\\frac{b}{29} + \\frac{c}{31}} + \\frac{b^2}{\\frac{c}{29} + \\frac{a}{31}} + \\frac{c^2}{\\frac{a}{29} + \\frac{b}{31}} \\ge n(a+b+c)\n\\]\nholds for all positive real numbers \\( a, b, c \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-993a7d87a7628b062b8e4fb7b0ee1b31ee68cac6bfcfa3630d6d94f2fe622bbf | What is the largest positive integer \( n \) such that there exist \( n \) distinct positive integers where the sum of any two of these integers is always a power of 2? | [
"2"
] | [
"What is the largest positive integer \\( n \\) such that there exist \\( n \\) distinct positive integers where the sum of any two of these integers is always a power of 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a535b5c61d2111c65ffa52a5e3a6999f942c93883910dd21d1d708e7c55d359d | What is the largest positive integer \( n \) which does not contain the digit 0, such that the sum of its digits is 15 and the sum of the digits of \( 2n \) is less than 20? | [
"5511111"
] | [
"What is the largest positive integer \\( n \\) which does not contain the digit 0, such that the sum of its digits is 15 and the sum of the digits of \\( 2n \\) is less than 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-600e2d8ca43145dd8b8269516297caacf91538916781a75fab38a7734f379140 | What is the largest positive integer \(n\) for which \((n+10)\) divides \((n^3 + 100)\)? | [
"890"
] | [
"What is the largest positive integer \\(n\\) for which \\((n+10)\\) divides \\((n^3 + 100)\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-a8996e494c3742bcfa15934af125cea2047256b0d14f8196c8d21c172c17cd03 | What is the largest positive integer \(n\) for which \(n^3 + 100\) is divisible by \(n + 10\)? | [
"890"
] | [
"What is the largest positive integer \\(n\\) for which \\(n^3 + 100\\) is divisible by \\(n + 10\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1462bc38dd8bbbb33451e224da4fcdd04af0340411593f85b952aef2bf943a30 | What is the largest positive integer less than \((\sqrt{6} + \sqrt{5})^{6}\)? | [
"10581"
] | [
"What is the largest positive integer less than \\((\\sqrt{6} + \\sqrt{5})^{6}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-865ef8915f03f9cbc129a420e1284e4612d42161c97cf6e161fa5482c86c9afa | What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiples of $13$, $17$, and $23$? | [
"84"
] | [
"What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiples of $13$, $17$, and $23$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 35,
"genselect": 7
} | 42 | 0 | 0 |
omr-ws-9e75bd4935b36a6103e0ad9fe8ef563da2753efa8f3ba0c227be3c2dbb8d1162 | What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiples of 13, 17, and 23? | [
"84"
] | [
"What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiples of 13, 17, and 23?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2a76b6484c767e6e058ac1cee5c51655a2640c8d3b5e214473746c65ca84beaa | What is the largest positive integer that cannot be expressed as the sum of a positive multiple of 42 and a positive composite integer? | [
"215"
] | [
"What is the largest positive integer that cannot be expressed as the sum of a positive multiple of 42 and a positive composite integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.