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-5c4382bc6cbb67e0e5ec0f8ad69e2ab68a6d6e9fd06331a525134cfbb65533bb | What is the smallest natural number whose square ends with the digits 2017? | [
"No solution exists"
] | [
"What is the smallest natural number whose square ends with the digits 2017?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-254ded8af787c0de3ac9b41d314cb7bece45170ba3f6adf7fd3e3aa14c2db44a | What is the smallest number $c$ such that there is always at least one prime between $n$ and $cn$ for all sufficiently large $n$? | [
"\\( c > 1 \\)"
] | [
"What is the smallest number $c$ such that there is always at least one prime between $n$ and $cn$ for all sufficiently large $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e1f9d657dabeedc58f6a34b1600999d2f92bb0b49b5deefc2bc75fef6842a93 | What is the smallest number $d$ such that fewer than half of the positive integers with $d$-digits have all distinct digits? | [
"5"
] | [
"What is the smallest number $d$ such that fewer than half of the positive integers with $d$-digits have all distinct digits?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e4e94ccb8bc4fbecb5c9e5ca02a678a85b8bc29fb3292abb429594c39f5a3fca | What is the smallest number $k$ greater than $6$ for which the sum of the first $k$ positive integers equals the sum of the integers from $k$ to $n$, where $n$ is an integer greater than $k$? | [
"35"
] | [
"What is the smallest number $k$ greater than $6$ for which the sum of the first $k$ positive integers equals the sum of the integers from $k$ to $n$, where $n$ is an integer greater than $k$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-b430b84a961e68f80aae1993cd9327978842b2a9d8e95338b83a1c88c02813d5 | What is the smallest number (integer or integer + 0.5) over 200 that cannot be expressed as an USAMO index, where the index is calculated by \(10 \times \text{AIME Score} + \text{AMC 12 Score}\)? | [
"272.5"
] | [
"What is the smallest number (integer or integer + 0.5) over 200 that cannot be expressed as an USAMO index, where the index is calculated by \\(10 \\times \\text{AIME Score} + \\text{AMC 12 Score}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04e337c6f10a1354ebd5de72507952466fb0327b1808d471b2d649f0d3ea869e | What is the smallest number \( k \) such that a cube can be divided into \( k \) non-overlapping tetrahedrons? | [
"5"
] | [
"What is the smallest number \\( k \\) such that a cube can be divided into \\( k \\) non-overlapping tetrahedrons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-cb5ac77e53b2bb7f210b2482cbda41ce29b5b2ebce6bbecf1e135acd256293b2 | What is the smallest number \( n \) such that a square \( ABCD \) can be decomposed into \( n \) non-overlapping triangles, all of whose angles are acute? Construct one such decomposition for that \( n \). | [
"8"
] | [
"What is the smallest number \\( n \\) such that a square \\( ABCD \\) can be decomposed into \\( n \\) non-overlapping triangles, all of whose angles are acute? Construct one such decomposition for that \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d13f6406e862180e92aad42b057296357eef48f391335345c11e575b7779419c | What is the smallest number \( n \) such that no matter how you place \( n \) marks on a 3 by 3 tic-tac-toe board, you are guaranteed to get three in a row (either vertically, horizontally, or diagonally)? | [
"7"
] | [
"What is the smallest number \\( n \\) such that no matter how you place \\( n \\) marks on a 3 by 3 tic-tac-toe board, you are guaranteed to get three in a row (either vertically, horizontally, or diagonally)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a4506deeca9281f407f79560425c44015c24e1c3b96c4108e3d4fc45ec7c398f | What is the smallest number consisting only of the digits 1 and 0 that is divisible by 42? | [
"101010"
] | [
"What is the smallest number consisting only of the digits 1 and 0 that is divisible by 42?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-1fdce5b1d41116589a1c12a7c33a0c1c767a6fc82ea694c796ee83da9431f1e6 | What is the smallest number divisible by integers 1 through 9? | [
"2520"
] | [
"What is the smallest number divisible by integers 1 through 9?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e15cbf5f3641e1584debda45ae5e4f140617c53c7349e798ea0978a100b11e4 | What is the smallest number expressible as the sum of \(2\), \(3\), \(4\), or \(5\) distinct primes? | [
"28"
] | [
"What is the smallest number expressible as the sum of \\(2\\), \\(3\\), \\(4\\), or \\(5\\) distinct primes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0618a2a7d057e9bee910a44f08135d63106792e6704d48da48d647497fcebf7e | What is the smallest number of $1 \times 1$ squares needed to cover a $23 \times 23$ square when using squares of sizes $1 \times 1$, $2 \times 2$, and $3 \times 3$? | [
"1"
] | [
"What is the smallest number of $1 \\times 1$ squares needed to cover a $23 \\times 23$ square when using squares of sizes $1 \\times 1$, $2 \\times 2$, and $3 \\times 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e3457a14a42faab01cde5d8ee612713385bd1b116670cabdd14b51952ad153b1 | What is the smallest number of $3$-cell corners needed to be painted in a $6 \times 6$ square so that it is impossible to paint more than one corner of it? (The painted corners should not overlap.) | [
"9"
] | [
"What is the smallest number of $3$-cell corners needed to be painted in a $6 \\times 6$ square so that it is impossible to paint more than one corner of it? (The painted corners should not overlap.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-f7fbb961007d28733295cf77f78e6b9ba9acb03209658950ae15015107b2eea5 | What is the smallest number of $3$-cell corners that you need to paint in a $5 \times 5$ square so that no additional $3$-cell corner can be painted without overlapping with an already painted corner? | [
"4"
] | [
"What is the smallest number of $3$-cell corners that you need to paint in a $5 \\times 5$ square so that no additional $3$-cell corner can be painted without overlapping with an already painted corner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4589e4430750eee4aab95f94b07c29e47040cdf7466fefe4df2be5eedd866593 | What is the smallest number of 12-number selections needed to ensure that at least one of the selections matches all 12 numbers in any randomly drawn 20-number subset from a set of 21 numbers? This is equivalent to finding \( W(21, 20, 12, 12) \). | [
"3"
] | [
"What is the smallest number of 12-number selections needed to ensure that at least one of the selections matches all 12 numbers in any randomly drawn 20-number subset from a set of 21 numbers? This is equivalent to finding \\( W(21, 20, 12, 12) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-14545bff4a4b99c365342f08f2aa2794fcc670e2a009aeb2b1b4067a52592677 | What is the smallest number of 1x1 tiles needed to completely tile a 23x23 square using 1x1, 2x2, and 3x3 tiles? | [
"1"
] | [
"What is the smallest number of 1x1 tiles needed to completely tile a 23x23 square using 1x1, 2x2, and 3x3 tiles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d531bd58cc254b71b9b79c3fe7925970864296bb50243757ac14ae8e50f1cc1e | What is the smallest number of 2x2 corner blocks that can be removed from an 8x8 chessboard so that no more 2x2 corner blocks can be removed from the remaining part of the board? | [
"9"
] | [
"What is the smallest number of 2x2 corner blocks that can be removed from an 8x8 chessboard so that no more 2x2 corner blocks can be removed from the remaining part of the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-110d07b7f3592254d9f9de7e893373a9c3f7f6286a26c499e8c2d105c544312f | What is the smallest number of 2x2 corner blocks that can be removed from an 8x8 chessboard so that no more 2x2 corner blocks can be removed from the remaining part of the chessboard? | [
"9"
] | [
"What is the smallest number of 2x2 corner blocks that can be removed from an 8x8 chessboard so that no more 2x2 corner blocks can be removed from the remaining part of the chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-86a588e59c768ff3e2f59d837949ebc1bc8e0d87a218ef0c55935a4daa564551 | What is the smallest number of Bishops one can place on an $8 \times 8$ Chessboard so that all the cells are attacked by at least one Bishop? Provide an example. | [
"8"
] | [
"What is the smallest number of Bishops one can place on an $8 \\times 8$ Chessboard so that all the cells are attacked by at least one Bishop? Provide an example."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-ef7359c4b181d76efeccbefeb5baee3dcd37cc1b8e84924af2281cb909247130 | What is the smallest number of acute triangles that a regular five-pointed star can be cut into? | [
"10"
] | [
"What is the smallest number of acute triangles that a regular five-pointed star can be cut into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4fc50e750c3be908d4357a05ee09be017783e03576ba41ab499a14f9cdf38c7 | What is the smallest number of arbitrary points for which no 3 are collinear need to be chosen in the plane to ensure that you have an $n$-sided convex polygon? | [
"2^{n-2} + 1"
] | [
"What is the smallest number of arbitrary points for which no 3 are collinear need to be chosen in the plane to ensure that you have an $n$-sided convex polygon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c17994c5a595ac37fd7739500dc25ccf93071eea9d294dddebf184c807563b7 | What is the smallest number of balls of radius $1$ that can cover the entire cylinder, given that the height and radius of the base of the cylinder are both equal to $1$? | [
"3"
] | [
"What is the smallest number of balls of radius $1$ that can cover the entire cylinder, given that the height and radius of the base of the cylinder are both equal to $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-524efc509ece400be3ea9f04baf2fddd464cdb5a5b96d132f4520d8dc80931ca | What is the smallest number of blue points, \( n \), that can be marked on the plane so that each blue point has exactly 3 other blue points at a distance of 1 cm? | [
"6"
] | [
"What is the smallest number of blue points, \\( n \\), that can be marked on the plane so that each blue point has exactly 3 other blue points at a distance of 1 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-2fd451cbbcdc49cb3d667c6a8264513fd3a07a9a5b293ab96c4c968e41bce0e5 | What is the smallest number of cards a participant needs to use in order to ensure at least one entry matches the corresponding entry on the etalon-card in a lottery with cards containing 50 empty cells, where each participant writes the numbers 1 through 50 without repetition? | [
"26"
] | [
"What is the smallest number of cards a participant needs to use in order to ensure at least one entry matches the corresponding entry on the etalon-card in a lottery with cards containing 50 empty cells, where each participant writes the numbers 1 through 50 without repetition?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c8d415fd8b9c386479be10d0542e564f906588636e8a1d58aceceb7fd392b5c | What is the smallest number of cards a teacher can use to indicate the numbers of any four problems from a math problem book containing 300 problems? | [
"29"
] | [
"What is the smallest number of cards a teacher can use to indicate the numbers of any four problems from a math problem book containing 300 problems?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c91b9622a768860f1b838e54813f3143286af85192d1217f25513ba902be4a00 | What is the smallest number of cells of a $5 \times 5$ square that can be painted so that any four-cell rectangle contains at least one filled cell? Provide an example. | [
"7"
] | [
"What is the smallest number of cells of a $5 \\times 5$ square that can be painted so that any four-cell rectangle contains at least one filled cell? Provide an example."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-2f0b4e9d00792ab87ae37a8cc60c6035609c80beb4747203de8b542332e9b1ea | What is the smallest number of children that could have been at a picnic if each child had at least one brother and one sister there? | [
"4"
] | [
"What is the smallest number of children that could have been at a picnic if each child had at least one brother and one sister there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9672128764a3eb12b6f243d01bd087c1643bcc90c5d2229abd3d046f0b4850f9 | What is the smallest number of chips, $n$, that can be placed on a chessboard such that there are exactly 3 chips in any $3 \times 3$ square? | [
"12"
] | [
"What is the smallest number of chips, $n$, that can be placed on a chessboard such that there are exactly 3 chips in any $3 \\times 3$ square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-e9f430252f9b71912e95aeed713292a61aba615ad1e4fd2fcc1fd8b3fd2c0963 | What is the smallest number of circles of radius $\sqrt{2}$ needed to cover a rectangle of size $5 \times 3$? | [
"5"
] | [
"What is the smallest number of circles of radius $\\sqrt{2}$ needed to cover a rectangle of size $5 \\times 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45d135f2393e714d49a2409eedc23d0e53e6089be9f9811a498dd5ff9d006213 | What is the smallest number of circles of radius $\sqrt{2}$ needed to cover a rectangle of size $6 \times 3$? | [
"6"
] | [
"What is the smallest number of circles of radius $\\sqrt{2}$ needed to cover a rectangle of size $6 \\times 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-39c0820ed451c8c01d1dc33211648c648fc51c4c6e977d2c55f1874e4250c9b5 | What is the smallest number of circles that requires four colors to ensure that two circles tangent to each other are of different colors? | [
"19"
] | [
"What is the smallest number of circles that requires four colors to ensure that two circles tangent to each other are of different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4fac8f92e86aa28b87263809142d884c1f9df158ef82d35f4e6f5bd97376f00a | What is the smallest number of colors Vasya should use to paint a $5 \times 5$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors? | [
"4"
] | [
"What is the smallest number of colors Vasya should use to paint a $5 \\times 5$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6ec6a32b79fa1bcbbf956dbb400c934bc77ea96f72928a2afdf32367f4715665 | What is the smallest number of colors Vasya should use to paint a $6 \times 6$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors? | [
"4"
] | [
"What is the smallest number of colors Vasya should use to paint a $6 \\times 6$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1e47687ccb1ec44210c25a9ef4c6b433318855453e4986cb69b53a3ec2d4ae9 | What is the smallest number of colors Vasya should use to paint a $7 \times 7$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors? | [
"4"
] | [
"What is the smallest number of colors Vasya should use to paint a $7 \\times 7$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-240666ccbf04e0cff06b4c8a19a828eaf8c8e5f5a6df77e46587ec1533f16d68 | What is the smallest number of colors Vasya should use to paint an $8 \times 8$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors? | [
"4"
] | [
"What is the smallest number of colors Vasya should use to paint an $8 \\times 8$ square such that each cell has all its neighboring cells (sharing a common side) painted in different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f09d9d9741f73337147134fbe86ee73dfe1b65eef3c2bd6c04b06c526f8ecf86 | What is the smallest number of colors necessary to color the vertices and edges of a complete graph with \( n \) vertices such that no two edges with a common vertex have the same color and no edge has the same color as either of its vertices? | [
"n"
] | [
"What is the smallest number of colors necessary to color the vertices and edges of a complete graph with \\( n \\) vertices such that no two edges with a common vertex have the same color and no edge has the same color as either of its vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-134e6aa1d4ad76e18830b83373442adcb7dad01c47f888cc6039425b1f0ba96c | What is the smallest number of colors needed to color 8 boxes of 6 balls (one color for each ball), so that the balls in each box are all different colors and any pair of colors occurs in at most one box? | [
"16"
] | [
"What is the smallest number of colors needed to color 8 boxes of 6 balls (one color for each ball), so that the balls in each box are all different colors and any pair of colors occurs in at most one box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-04ee3130502e9452d3bbb3d113e9ed989e0d97c7c3404842aea593bae29573bd | What is the smallest number of colors needed to color the integers from 1 to 100 such that no integer is divisible by another integer of the same color? | [
"7"
] | [
"What is the smallest number of colors needed to color the integers from 1 to 100 such that no integer is divisible by another integer of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b7903550f5f5a9a2aa204f09372fceeadd676e5dccc22ad46d6628d483982357 | What is the smallest number of colors needed to distinguish \( n \) keys if all the keys are to be covered, considering that the keys are arranged in a circle and the key ring can be turned over? | [
"\\[ \\text{If } n = 1, \\text{ then } 1; \\text{ if } n = 3, 4, 5, \\text{ then } 3; \\text{ if } n > 5, \\text{ then } 2. \\]"
] | [
"What is the smallest number of colors needed to distinguish \\( n \\) keys if all the keys are to be covered, considering that the keys are arranged in a circle and the key ring can be turned over?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 11
} | 18 | 0 | 0 |
omr-ws-b53749fde9990b36fd9165ffb7695e595757b1366d4936f36d110b95df695c61 | What is the smallest number of colors needed to paint the cells of an $8 \times 8$ square so that no two cells of the same color are adjacent (neither on the side nor on the top)? | [
"4"
] | [
"What is the smallest number of colors needed to paint the cells of an $8 \\times 8$ square so that no two cells of the same color are adjacent (neither on the side nor on the top)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-d0f88d74fcc869847473cbe6d499cad2f5d3c9b4c84fba5d5ee87c33f75beec6 | What is the smallest number of colors that can be used to color a $3 \times 3$ square such that no two cells of the same color are in the same row, column, or on any of the two main diagonals? | [
"5"
] | [
"What is the smallest number of colors that can be used to color a $3 \\times 3$ square such that no two cells of the same color are in the same row, column, or on any of the two main diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-c1fe29f2947ffd42ebe72607b3a9feca2c6ebbba3bc5e04a7111c48cadec8e31 | What is the smallest number of colors that can be used to color a $4 \times 4$ square such that no two cells of the same color are in the same row, column, or on any of the two main diagonals? | [
"4"
] | [
"What is the smallest number of colors that can be used to color a $4 \\times 4$ square such that no two cells of the same color are in the same row, column, or on any of the two main diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96fa2f7efce1c3013fa6852a71566dedbc29f81ca6d0753e484c0b06b941e2c3 | What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered, considering the keys are arranged in a circle and the key ring can be turned over? | [
"\\( \\text{min colours} = \\begin{cases} \nn & \\text{if } n \\leq 2 \\\\\n3 & \\text{if } n = 3, 4, 5 \\\\\n2 & \\text{if } n \\geq 6 \n\\end{cases} \\)"
] | [
"What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered, considering the keys are arranged in a circle and the key ring can be turned over?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-17ef66ceda23f1dc550e2a77ea3816a6d42e636983bf199e56159d406bf56cd7 | What is the smallest number of colours needed to distinguish \( n \) keys if all the keys are to be covered, given that the keys are arranged in a circle and the key ring can be turned over? | [
"3"
] | [
"What is the smallest number of colours needed to distinguish \\( n \\) keys if all the keys are to be covered, given that the keys are arranged in a circle and the key ring can be turned over?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05864982235501cc747d610f4461bcbd5762352097a456de20e5ce469929f40c | What is the smallest number of convex sets whose union is the set $M$, where $M$ is formed from the plane by removing three points $A, B, C$, which are vertices of a triangle? | [
"3"
] | [
"What is the smallest number of convex sets whose union is the set $M$, where $M$ is formed from the plane by removing three points $A, B, C$, which are vertices of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-9abcd0eaf42615b92aadbce8f7a440dfd02a3e324e8510119859532be4aa5bb8 | What is the smallest number of copies of the same scalene triangle needed to form a rhombus? | [
"4"
] | [
"What is the smallest number of copies of the same scalene triangle needed to form a rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03b0ef308f597708363905889717d0c9cb085145958b2549a383b0a5e4291664 | What is the smallest number of copies of the same scalene triangle needed to form a square? | [
"8"
] | [
"What is the smallest number of copies of the same scalene triangle needed to form a square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-5b3bcfc58d4e8a03c6fe7e888876eda26e0c62a12e365bfb620c5ce1e9e6829d | What is the smallest number of cubes of various sizes (1cm, 2cm, 3cm, 4cm, 5cm, and 6cm on a side) needed to fill a hollow cube with interior measures of 7cm on each side? | [
"9"
] | [
"What is the smallest number of cubes of various sizes (1cm, 2cm, 3cm, 4cm, 5cm, and 6cm on a side) needed to fill a hollow cube with interior measures of 7cm on each side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-13b3368b361ea31ca8e799704bbb00c2a8d22d46ef236ee04f55a097b93b2efb | What is the smallest number of different numbers that can be selected such that each selected number is equal to the sum of any other three selected different numbers? | [
"7"
] | [
"What is the smallest number of different numbers that can be selected such that each selected number is equal to the sum of any other three selected different numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 8,
"tir": 14
} | 55 | 0 | 0 |
omr-ws-ccbe7659c75dfdeeae93963897ca0a5ea865f4f529e9bdc87ea2dd0b489296f4 | What is the smallest number of digits \( k \) such that the number consisting of \( k \) digits all being 1 (i.e., \( 111\ldots111 \)) is divisible by \( 3^{10} \)? | [
"59049"
] | [
"What is the smallest number of digits \\( k \\) such that the number consisting of \\( k \\) digits all being 1 (i.e., \\( 111\\ldots111 \\)) is divisible by \\( 3^{10} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-9588561789dbbe44d80f2119e09c95d378e2d12d2277c963adeb8238a80cdd53 | What is the smallest number of digits one can write in a row so that any 3-digit number can be obtained by striking out some of the digits and concatenating the rest? | [
"29"
] | [
"What is the smallest number of digits one can write in a row so that any 3-digit number can be obtained by striking out some of the digits and concatenating the rest?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-b5282085d64283411c62d934c3db4870764629a32393323119c20648f2bb8cbe | What is the smallest number of digits that a password can have if it is made up of the digits 1, 2, 3, and 4, and must contain all possible 4-digit numbers formed by these digits? | [
"259"
] | [
"What is the smallest number of digits that a password can have if it is made up of the digits 1, 2, 3, and 4, and must contain all possible 4-digit numbers formed by these digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-eeeeecdbfa88c0b6f0ae2134aa33c8990fd086414d372755040fb27daae9074f | What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from 100 to 999? | [
"29"
] | [
"What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from 100 to 999?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-59eba8dedeee1ecf3130af99be911ab9152e8fdc7e8671ce556724628602535d | What is the smallest number of discs that need to be selected from a box containing 20 yellow discs, 9 red discs, and 6 blue discs to ensure at least two discs of each color are selected? | [
"31"
] | [
"What is the smallest number of discs that need to be selected from a box containing 20 yellow discs, 9 red discs, and 6 blue discs to ensure at least two discs of each color are selected?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8faccafa059f507e437bb3ab9693b93c86ca59e5a1b565a1d791c56e6b3703e | What is the smallest number of elements of the set \( M \), where \( M \) consists of positive real numbers whose decimal representation digits are only 0 or 7, such that their sum is 1? | [
"8"
] | [
"What is the smallest number of elements of the set \\( M \\), where \\( M \\) consists of positive real numbers whose decimal representation digits are only 0 or 7, such that their sum is 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-782793b5c28c004311d43a21c2bf2aaa53b88895d72ac4b87bff64575a901b8d | What is the smallest number of elements that must be deleted from the set \(\{10, 20, 30, 40, 50, 60, 70, 80, 90\}\) so that the product of the remaining elements is a perfect square? | [
"2"
] | [
"What is the smallest number of elements that must be deleted from the set \\(\\{10, 20, 30, 40, 50, 60, 70, 80, 90\\}\\) so that the product of the remaining elements is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d33be325b47ecbee85b291919f495e4181eff238bbb6b63ee9c55d00d03a127a | What is the smallest number of equal parts that a 6x7 rectangle made of checkered paper can be cut into so that no T-tetromino can be cut out of each part? | [
"6"
] | [
"What is the smallest number of equal parts that a 6x7 rectangle made of checkered paper can be cut into so that no T-tetromino can be cut out of each part?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0eff245b672106e36188f5c48b386d7eb513a11b9972a0176a1e24da7b06898 | What is the smallest number of factors that should be removed from the product \(1 \times 2 \times 3 \times \cdots \times 26 \times 27\) so that the product of the remaining factors is a perfect square? | [
"4"
] | [
"What is the smallest number of factors that should be removed from the product \\(1 \\times 2 \\times 3 \\times \\cdots \\times 26 \\times 27\\) so that the product of the remaining factors is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-939c8d535f61b328e7d3fa8252020b511636c82982d694543b2e53740cf61cc5 | What is the smallest number of forecast variants needed to ensure that there exists at least one variant with all n results incorrect, given that each game can have one of three possible outcomes (winning of the hosts, winning of the guests, or a draw)? | [
"\\lceil (3/2)^n \\rceil"
] | [
"What is the smallest number of forecast variants needed to ensure that there exists at least one variant with all n results incorrect, given that each game can have one of three possible outcomes (winning of the hosts, winning of the guests, or a draw)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d4364cdc75e4f101f73bdda9ed2ecf57e0c5f9b5eba0ee4dd820bca434697623 | What is the smallest number of fuel dumping points required for Alexander to travel from space station $A$ to the abandoned space station $B$, which is 23 parsecs away, using a maximum of 45 fuel capsules, given that his spaceship can only carry a maximum of 15 fuel capsules at a time and can dump fuel capsules at any ... | [
"2"
] | [
"What is the smallest number of fuel dumping points required for Alexander to travel from space station $A$ to the abandoned space station $B$, which is 23 parsecs away, using a maximum of 45 fuel capsules, given that his spaceship can only carry a maximum of 15 fuel capsules at a time and can dump fuel capsules at... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d0fda3bff21b66126aa348186e9ce099186829487216c85735c52a3fc99b680f | What is the smallest number of gold coins \( m \) that can be arranged in both a triangular and a square form without residue? | [
"1225"
] | [
"What is the smallest number of gold coins \\( m \\) that can be arranged in both a triangular and a square form without residue?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abba044b24adb83a97eb46be9467b477aff34aa88d4736b5edd39e4de8b49f47 | What is the smallest number of integer squares whose sum is $2012$? | [
"4"
] | [
"What is the smallest number of integer squares whose sum is $2012$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-5fe881a8291426e95eb85dabcf8ae037eeda814600b2b0d2eb56cf7abbdf9281 | What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\%$ of the value of $x$? | [
"4"
] | [
"What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\\%$ of the value of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-867173305d2c150b3fe2633cd6121696ad802e694d9f5159bd87baba8e5a759c | What is the smallest number of integers that can be erased from the set $\{1, 2, 3, \ldots, 2011\}$ so that the product of the remaining integers ends in $2$? | [
"403"
] | [
"What is the smallest number of integers that can be erased from the set $\\{1, 2, 3, \\ldots, 2011\\}$ so that the product of the remaining integers ends in $2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1d438912b84376d5ae0843d38d561f37c49b74607b2cdc0b27af418bc74d6dd2 | What is the smallest number of integers that can be wiped off from the set $\{1, 2, 3, \ldots, 100\}$ so that the product of the remaining integers ends in $2$? | [
"21"
] | [
"What is the smallest number of integers that can be wiped off from the set $\\{1, 2, 3, \\ldots, 100\\}$ so that the product of the remaining integers ends in $2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-38d7d261f70f441ea31e63df1d8d04b6fa941d3f32fe10c55838a490deb78ae9 | What is the smallest number of integers that can be wiped off from the set \(\{1, 2, 3, \ldots, 100\}\) so that the product of the remaining integers ends in 2? | [
"21"
] | [
"What is the smallest number of integers that can be wiped off from the set \\(\\{1, 2, 3, \\ldots, 100\\}\\) so that the product of the remaining integers ends in 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-08629443da1e7ee959d5bde5d49f5bbf691f51e84624753a39fefea3b2560679 | What is the smallest number of integers that need to be selected from $\{1,2,\ldots,50\}$ to guarantee that two of the selected numbers are relatively prime? | [
"26"
] | [
"What is the smallest number of integers that need to be selected from $\\{1,2,\\ldots,50\\}$ to guarantee that two of the selected numbers are relatively prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-825b1ae0842be282298eeb11eace9d5e52b6eeeed05120b803e5b297f3a833ad | What is the smallest number of integers that need to be selected from the set \(\{1, 2, 3, \ldots, 50\}\) to guarantee that two of the selected numbers are relatively prime? | [
"26"
] | [
"What is the smallest number of integers that need to be selected from the set \\(\\{1, 2, 3, \\ldots, 50\\}\\) to guarantee that two of the selected numbers are relatively prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-867b809259520db4fcdd04da215d2fc0ca484912c6a2f102f057492382ee1989 | What is the smallest number of lines needed such that, by symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), the circle can cover any given point on the plane? | [
"3"
] | [
"What is the smallest number of lines needed such that, by symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), the circle can cover any given point on the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-9667fc9e5c67ca346ce060fe94ed0dd3f6d5619a155198eaac4915db8d5f3047 | What is the smallest number of lines required to divide the plane into at least 2019 regions using three families of parallel lines? | [
"674"
] | [
"What is the smallest number of lines required to divide the plane into at least 2019 regions using three families of parallel lines?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17bdde73aaefbfb29094b7ac50677afeeabaa85ed73dac0b55b370caf7941a62 | What is the smallest number of lines that must be drawn so that $2003$ different triangles can be found in the resulting drawing? | [
"24"
] | [
"What is the smallest number of lines that must be drawn so that $2003$ different triangles can be found in the resulting drawing?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-dc08cdddda335cf1191cd954177ccfa4ef5def8a59f479539a321a131d2d2307 | What is the smallest number of lines which can pass through an interior point of each square in a $3 \times 3$ square? | [
"4"
] | [
"What is the smallest number of lines which can pass through an interior point of each square in a $3 \\times 3$ square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-3d0febe900f5f106f35e1dceda5bcc16690ff0af7b35c6b92be377a85b27dc5d | What is the smallest number of lines which can pass through an interior point of each square in a $4 \times 4$ square? | [
"4"
] | [
"What is the smallest number of lines which can pass through an interior point of each square in a $4 \\times 4$ square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-a2f32737ec3480000d25d8e72946c2a788aa95886992120f1dd6203a6a7080b3 | What is the smallest number of markers that, when placed in rows of 4, leaves a remainder of 2; when placed in rows of 5, leaves a remainder of 3; and when placed in rows of 7, leaves a remainder of 5? | [
"138"
] | [
"What is the smallest number of markers that, when placed in rows of 4, leaves a remainder of 2; when placed in rows of 5, leaves a remainder of 3; and when placed in rows of 7, leaves a remainder of 5?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 49,
"tir": 29
} | 78 | 0 | 0 |
omr-ws-93c7f6d6964d72b57583eb0983424a5f7ceb5f246232ee6c26a69a8313a5a579 | What is the smallest number of nodes that can be marked in a rectangular $n \times k$ grid so that each cell contains at least two marked nodes? | [
"n + k"
] | [
"What is the smallest number of nodes that can be marked in a rectangular $n \\times k$ grid so that each cell contains at least two marked nodes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-c5215950548468a72069aa818f4b188c9213940ce3b6d34627e07e9742941b7e | What is the smallest number of non-intersecting segments that do not pass through a point $A$, such that each ray leaving $A$ intersects at least three of these segments? | [
"4"
] | [
"What is the smallest number of non-intersecting segments that do not pass through a point $A$, such that each ray leaving $A$ intersects at least three of these segments?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1397e559de6bcae794a010a76e2cfa9ea02e1b8d59fcccf14632674c1556c67f | What is the smallest number of parallelograms that a regular $2n$-gon can be cut into? | [
"n"
] | [
"What is the smallest number of parallelograms that a regular $2n$-gon can be cut into?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-ce994cf7dc5ff43f626a3f7bcc1f4da4220c0718cf0e5c9a4c62e9a19a96d23a | What is the smallest number of pennies that could be in a jar if when arranged in rows of 15, there are 7 left over, and when arranged in rows of 19, there are 8 left over? | [
"217"
] | [
"What is the smallest number of pennies that could be in a jar if when arranged in rows of 15, there are 7 left over, and when arranged in rows of 19, there are 8 left over?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-6944d2ee418168c658574ab067abc8565be95bdfcbf9347a7c5fcd0e5df3a45a | What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership? | [
"240"
] | [
"What is the smallest number of people that can be broken up both into 15 groups of equal membership and into 48 groups of equal membership?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f944ca62590623c561ac6795f35e462ed5d0ba09a17b0c6ae50d4d59d355e492 | What is the smallest number of pieces to cut a cube with edge length 6 so that cubes with edge lengths 3, 4, and 5 can be made from them? | [
"6"
] | [
"What is the smallest number of pieces to cut a cube with edge length 6 so that cubes with edge lengths 3, 4, and 5 can be made from them?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-651e7d42eac312c22e6157d0a4f3ab12b82af8106808c004954844d24b13c5ff | What is the smallest number of planes needed to divide a cube into congruent tetrahedra? | [
"3"
] | [
"What is the smallest number of planes needed to divide a cube into congruent tetrahedra?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 34,
"genselect": 8
} | 42 | 0 | 0 |
omr-ws-34aaa276bac51b118c84b157ce199a857bda02c79e5857005be4025817ab9973 | What is the smallest number of points that need to be marked so that 4 straight lines can be drawn through them? | [
"4"
] | [
"What is the smallest number of points that need to be marked so that 4 straight lines can be drawn through them?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52413165eba09f6311aae97c89502e85b6e71bb8eedc2647f41e6439827ad8cd | What is the smallest number of points that need to be marked so that 5 straight lines can be drawn through them? | [
"4"
] | [
"What is the smallest number of points that need to be marked so that 5 straight lines can be drawn through them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-7f23dafc18bd033bf933bce4477bd6079baa9a074fe7e2c45deff201a19daa75 | What is the smallest number of queens required to attack all of the squares of a $6$ by $6$ chessboard? | [
"4"
] | [
"What is the smallest number of queens required to attack all of the squares of a $6$ by $6$ chessboard?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-04f55648ca8b9de5cd3a6ed92591038f94598680989399ac7242b369822bd27c | What is the smallest number of questions needed to surely guess a number from 1 to 100, if the responder is allowed to lie once? The questions can only be answered with "Yes" or "No" and must be of the form "Does the intended number belong to such and such a numerical set?" | [
"11"
] | [
"What is the smallest number of questions needed to surely guess a number from 1 to 100, if the responder is allowed to lie once? The questions can only be answered with \"Yes\" or \"No\" and must be of the form \"Does the intended number belong to such and such a numerical set?\""
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-34a13ecbc2f98233ef1bf946e87d4e9987fd800d5f5320f11006da0ae5375dd5 | What is the smallest number of rays Semyon can draw inside a right angle with apex $O$ such that the sum of all the angles formed is $400$ degrees? How should these rays be positioned? | [
"4"
] | [
"What is the smallest number of rays Semyon can draw inside a right angle with apex $O$ such that the sum of all the angles formed is $400$ degrees? How should these rays be positioned?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07953cb519f900fcd7e3e115cfddfa3e22af811969e96a2ab920ecd1b8cab2fe | What is the smallest number of regular pentagon napkins with a side of $1$ required to cover a regular pentagon with a side of $2$ entirely? | [
"6"
] | [
"What is the smallest number of regular pentagon napkins with a side of $1$ required to cover a regular pentagon with a side of $2$ entirely?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-f04b4a7cf488d5fcd957143af7fe5fb77d181c558222c97b2e5ab7258aaabde8 | What is the smallest number of rooks that can control an $8\times8\times8$ chessboard? | [
"32"
] | [
"What is the smallest number of rooks that can control an $8\\times8\\times8$ chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99f645112f64fe8e352b6295e26db5989fd01a3f09bef55729a5f1838e2833d6 | What is the smallest number of shots needed to guarantee sinking a three-deck ship hidden on a $7 \times 7$ checkered field in a game of "battleship"? | [
"19"
] | [
"What is the smallest number of shots needed to guarantee sinking a three-deck ship hidden on a $7 \\times 7$ checkered field in a game of \"battleship\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-618bda8f4e25fa0315b51e41fa7633877d7c7bbb16b0d9a24c0b4da7f3cf9244 | What is the smallest number of sides of a polygon that can be placed inside a square, given that the perimeter of the polygon is $7$ times smaller than the perimeter of the square? | [
"3"
] | [
"What is the smallest number of sides of a polygon that can be placed inside a square, given that the perimeter of the polygon is $7$ times smaller than the perimeter of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-0ea91cc2157b1228d7888372b50f0860ea68fae85b8e6027b819a4d2c153acbb | What is the smallest number of small cubes used to build a large solid cube such that more than half of the small cubes are not visible from outside? | [
"1000"
] | [
"What is the smallest number of small cubes used to build a large solid cube such that more than half of the small cubes are not visible from outside?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2bd34d14e9cffe665369aafa25a78167bae099ca7100c033eb71815cab18954 | What is the smallest number of squares on an 8 × 8 chessboard that must be painted so that no 3 × 1 rectangle can be placed on the board without covering at least one painted square? | [
"21"
] | [
"What is the smallest number of squares on an 8 × 8 chessboard that must be painted so that no 3 × 1 rectangle can be placed on the board without covering at least one painted square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 6
} | 13 | 0 | 0 |
omr-ws-dc4be2a53577d2a4c9adf70d7ccd8a5fb6bbac761e02f1a8660f9f956b5b0461 | What is the smallest number of squares that Nils must color on an $M \times N$ board so that it is impossible to place a right zig-zag tile (covering four squares) on the board without covering a colored square? The tile can be freely rotated and mirrored. | [
"\\left\\lceil \\frac{MN}{3} \\right\\rceil"
] | [
"What is the smallest number of squares that Nils must color on an $M \\times N$ board so that it is impossible to place a right zig-zag tile (covering four squares) on the board without covering a colored square? The tile can be freely rotated and mirrored."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-4553d2865a01150da404203ffc25c89c0ec6458167dc57ca56cf9b3f37353a35 | What is the smallest number of squares that can be colored black on a $7 \times 7$ chessboard with its four corners deleted so that no uncolored $5$-square (Greek) cross can be found? | [
"5"
] | [
"What is the smallest number of squares that can be colored black on a $7 \\times 7$ chessboard with its four corners deleted so that no uncolored $5$-square (Greek) cross can be found?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-744daee2571b96582f7f926ab4781ed85483d18b681fe62e021472667ec02746 | What is the smallest number of squares that can be marked on a $5n \times 5n$ chessboard such that no row or column contains a block of $3n$ consecutive unmarked squares? | [
"\\(10n^2\\)"
] | [
"What is the smallest number of squares that can be marked on a $5n \\times 5n$ chessboard such that no row or column contains a block of $3n$ consecutive unmarked squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-935703f02e3df644eb3eb0c2021c6511554e8d0224c657c7228af9c702f32234 | What is the smallest number of squares that need to be colored on an \( n \times n \) square table so that every non-colored square shares at least one side with at least one colored square? | [
"\\left\\lceil \\dfrac{n^2}{3} \\right\\rceil"
] | [
"What is the smallest number of squares that need to be colored on an \\( n \\times n \\) square table so that every non-colored square shares at least one side with at least one colored square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-75a65067ab9c62f559269ade598f3b59efb4c34deafd8bdaabede9898026d41a | What is the smallest number of squares which need to be colored black on a $7 \times 7$ table with the four corner squares deleted so that a 5-square entirely uncolored Greek cross cannot be found on the table? | [
"7"
] | [
"What is the smallest number of squares which need to be colored black on a $7 \\times 7$ table with the four corner squares deleted so that a 5-square entirely uncolored Greek cross cannot be found on the table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6a979e3edf38351356fb4d699957463eca9765f531006a17a81ac5558c5acbb | What is the smallest number of students in a class such that no matter what 2-digit numbers they write, there will be at least three numbers whose digits have the same sum? | [
"35"
] | [
"What is the smallest number of students in a class such that no matter what 2-digit numbers they write, there will be at least three numbers whose digits have the same sum?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 4,
"tir": 16
} | 49 | 0 | 0 |
omr-ws-516feea82d29fb59fe7ac34a56314842fddcdde5852af6b653afd284cdc16bfb | What is the smallest number of tickets you need to purchase to guarantee that you win in 4-bit Binary Lottery? Justify your answer. | [
"4"
] | [
"What is the smallest number of tickets you need to purchase to guarantee that you win in 4-bit Binary Lottery? Justify your answer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-aeacb5435445e064ff85cf56bcd4acc3a65671150561c4f718f16ec2e7cb4af1 | What is the smallest number of unit segments that can be erased from the interior of a 2000*3000 rectangle grid so that no smaller rectangle remains intact? | [
"5995001"
] | [
"What is the smallest number of unit segments that can be erased from the interior of a 2000*3000 rectangle grid so that no smaller rectangle remains intact?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
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