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-957271abc02c0cc14768e2333876b55183423ac6dde21daacd14232c77bf99c0 | What is the maximum number of edges of a regular $n$-gonal prism that can intersect a plane, that does not pass through the vertices of the prism? | [
"\\( n+2 \\)"
] | [
"What is the maximum number of edges of a regular $n$-gonal prism that can intersect a plane, that does not pass through the vertices of the prism?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 7,
"tir": 3
} | 28 | 0 | 0 |
omr-ws-eb1116736963f21b8722d6427851371b0d65c7e1fbf5caeda32b279d335df387 | What is the maximum number of elements in a set \( S \) containing integers from 1 to 2009 such that for any two elements \( a \) and \( b \) in \( S \), the sum \( a + b \) is not equal to the absolute difference \( |a - b| \)? | [
"2009"
] | [
"What is the maximum number of elements in a set \\( S \\) containing integers from 1 to 2009 such that for any two elements \\( a \\) and \\( b \\) in \\( S \\), the sum \\( a + b \\) is not equal to the absolute difference \\( |a - b| \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 44,
"genselect": 8
} | 52 | 0 | 0 |
omr-ws-486d636a57723620a2cee4e095d49e223633c9b9a08b15d20420e4ee982969c5 | What is the maximum number of elements in a subset of $\{1, 2, 3, \ldots, 2017, 2018\}$ such that no element is 5 times another element? | [
"1682"
] | [
"What is the maximum number of elements in a subset of $\\{1, 2, 3, \\ldots, 2017, 2018\\}$ such that no element is 5 times another element?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-f70d8c5b71a57427d3f63d8b340969ba89f5fe35504cf6dc9bcf5c0808b2fa00 | What is the maximum number of elements that can be chosen from the set $\{1, 2, \ldots, 2001\}$ such that the difference between any two chosen numbers is neither 5 nor 8? | [
"924"
] | [
"What is the maximum number of elements that can be chosen from the set $\\{1, 2, \\ldots, 2001\\}$ such that the difference between any two chosen numbers is neither 5 nor 8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 19
} | 64 | 0 | 0 |
omr-ws-ba310ca3606118a4f0cc15f7fc0af251966ab6ad411ca0eaabed75afd64b6c8c | What is the maximum number of elements that can be selected from the set $\{1, 2, 3, \dots, 100\}$ such that no sum of any three selected numbers is equal to a selected number? | [
"68"
] | [
"What is the maximum number of elements that can be selected from the set $\\{1, 2, 3, \\dots, 100\\}$ such that no sum of any three selected numbers is equal to a selected number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-8c9552a1bdd0307c70f33418fe7e8b5bf767f676e5621efc626873754023d42f | What is the maximum number of elements that can be selected from the set $\{1, 2, 3, \ldots, 100\}$ such that no sum of any three selected numbers is equal to a selected number? | [
"67"
] | [
"What is the maximum number of elements that can be selected from the set $\\{1, 2, 3, \\ldots, 100\\}$ such that no sum of any three selected numbers is equal to a selected number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-b14e9f16edefff8b47df67bf3eb3167d9fbb2a48998f5abd084191a626582259 | What is the maximum number of elements that can be selected from the set \( S = \{1, 2, \ldots, n\} \) such that no sum of any three selected numbers is equal to a selected number? | [
"n - \\left\\lfloor \\dfrac{n}{3} \\right\\rfloor + 1"
] | [
"What is the maximum number of elements that can be selected from the set \\( S = \\{1, 2, \\ldots, n\\} \\) such that no sum of any three selected numbers is equal to a selected number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-ec2fd82cdcb2ddccd4a21cfe93cb04eb91fe18203521118af6a4e342f23d602c | What is the maximum number of elements that can be selected from the set \(\{1, 2, 3, \ldots, 31\}\) such that the sum of any two selected elements is not a perfect square? | [
"13"
] | [
"What is the maximum number of elements that can be selected from the set \\(\\{1, 2, 3, \\ldots, 31\\}\\) such that the sum of any two selected elements is not a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-215210381bb501f1811ac7364ad9c0f9c2b624040017287bbb72a8e130086252 | What is the maximum number of equilateral triangles with side length 1 cm that can be traced using one hundred long segments on a sheet of paper? | [
"66"
] | [
"What is the maximum number of equilateral triangles with side length 1 cm that can be traced using one hundred long segments on a sheet of paper?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-d7a8a7e47cab1f9d510b811ef39a88a589f432479c84e941dfb353d30868785e | What is the maximum number of games needed to determine the winner in a double elimination tournament with 10 teams? | [
"19"
] | [
"What is the maximum number of games needed to determine the winner in a double elimination tournament with 10 teams?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5db0716ca05a42754f4d8030afa958192f36b396100c04a1c4857d8a112ec534 | What is the maximum number of groups of cards that can be obtained if each group must have a sum of 9, given that there are 600 cards in total, with 200 cards each having the numbers 1, 2, and 5 written on them? | [
"150"
] | [
"What is the maximum number of groups of cards that can be obtained if each group must have a sum of 9, given that there are 600 cards in total, with 200 cards each having the numbers 1, 2, and 5 written on them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 13
} | 50 | 0 | 0 |
omr-ws-0c96bbe62f119ad58876dc9c1f32f2d914c87e5b12b8c83b86e10140f405913f | What is the maximum number of groups that n different people can be separated into? For example, with 3 people (ABC), the possible groupings are: A B C; AB C; AC B; BC A; ABC. | [
"B_n"
] | [
"What is the maximum number of groups that n different people can be separated into? For example, with 3 people (ABC), the possible groupings are: A B C; AB C; AC B; BC A; ABC."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09229bbbe391c10dbcc8312608e1c7012e8731cec12c0028677489600557c31e | What is the maximum number of integer lattice points that can lie on an arc of length $R^{\frac{1}{3}}$ of a circle centered at the origin with radius $R$? | [
"2"
] | [
"What is the maximum number of integer lattice points that can lie on an arc of length $R^{\\frac{1}{3}}$ of a circle centered at the origin with radius $R$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-ab6f9319930d7560b8684458b20147984206a6bac9ca76be81657eaaa1a66c85 | What is the maximum number of integer solutions for the equation $p(x) = x^3$ given that the polynomial $p(x)$ satisfies $p(100) = 100$? | [
"3"
] | [
"What is the maximum number of integer solutions for the equation $p(x) = x^3$ given that the polynomial $p(x)$ satisfies $p(100) = 100$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ef4d7d39ed1367c8d62b0ddc0512aada69990559ab1a77ddb435fffe5ab63d9 | What is the maximum number of integers that can be chosen from $1, 2, \dots, 99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighboring integers is a 3-digit number? | [
"59"
] | [
"What is the maximum number of integers that can be chosen from $1, 2, \\dots, 99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighboring integers is a 3-digit number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9114f2c5f2bc92156ac8b87496c84ec97b88f75f512346629c32c1265759ebc1 | What is the maximum number of integers that can be selected from the set \(\{1, 2, 3, \ldots, 2021\}\) such that the sum of any two selected numbers is not a multiple of their difference? | [
"674"
] | [
"What is the maximum number of integers that can be selected from the set \\(\\{1, 2, 3, \\ldots, 2021\\}\\) such that the sum of any two selected numbers is not a multiple of their difference?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-01e30afc72085f02e57471fbd555989f3e0cc983829812717fd8df17ce76e276 | What is the maximum number of intersection points between 3 circles and 3 triangles on a plane? | [
"54"
] | [
"What is the maximum number of intersection points between 3 circles and 3 triangles on a plane?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66394cd262e44b9c1ab501aca8b55806df507e575148e3f2241dbef395111809 | What is the maximum number of intersection points of 3 squares and 5 lines in the plane, assuming the number is finite? | [
"64"
] | [
"What is the maximum number of intersection points of 3 squares and 5 lines in the plane, assuming the number is finite?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 5
} | 60 | 0 | 0 |
omr-ws-dc94f51de802f62c5ddee8749e0c23afd2cb2d5d79f23ce7d1362df10409832e | What is the maximum number of intersection points of 6 circles and 5 straight lines? | [
"95"
] | [
"What is the maximum number of intersection points of 6 circles and 5 straight lines?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-defea834bdb822f1ca94975ffc68b814480427be85c5668dcab07d7fca2643e3 | What is the maximum number of intersections between a 3rd degree polynomial and a 4th degree polynomial? | [
"4"
] | [
"What is the maximum number of intersections between a 3rd degree polynomial and a 4th degree polynomial?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9c00c93861665c39f85e7116d77fce68ec78dc89de5a8ccc55cc05c4c124df79 | What is the maximum number of intersections possible when you have 2 circles, 3 ellipses, and 3 lines in a plane? | [
"71"
] | [
"What is the maximum number of intersections possible when you have 2 circles, 3 ellipses, and 3 lines in a plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-8a28033f77e984fb254aff1919c8b77092444c9b80d9b4aea6553f4c3486e528 | What is the maximum number of isosceles triangles that can be formed using 100 points on a straight line and one additional point outside the line as vertices? | [
"50"
] | [
"What is the maximum number of isosceles triangles that can be formed using 100 points on a straight line and one additional point outside the line as vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-cf254bc5a193484c45c54d525e5cfebbc4b4441dd771ce2d915be37665bd6b13 | What is the maximum number of joints that can be created by \( n \) lines in 3-space, where a joint is defined as a point where at least three lines, not all on the same plane, meet? | [
"\\Theta(n^{3/2})"
] | [
"What is the maximum number of joints that can be created by \\( n \\) lines in 3-space, where a joint is defined as a point where at least three lines, not all on the same plane, meet?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-4680c071eb3d4d73d3a1e59f035675cb6191015fde17cb28f95bf147a305ebfe | What is the maximum number of knights that can be placed on a $2 \times 2015$ chessboard such that no two knights attack each other? | [
"2015"
] | [
"What is the maximum number of knights that can be placed on a $2 \\times 2015$ chessboard such that no two knights attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-287a3b5a3e06228ab9fafe0bea3aebd13ee7a5eabd097c2ad9cac0517028b16a | What is the maximum number of knights that can be placed on a $5 \times 5$ chessboard such that no two knights can attack each other? | [
"13"
] | [
"What is the maximum number of knights that can be placed on a $5 \\times 5$ chessboard such that no two knights can attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-fe2b866404a0729e5462f88810e501dd60d7462eedd9d66665bb9dfc74f8c6e4 | What is the maximum number of knights that can be placed on a $9 \times 9$ chessboard such that no two knights attack each other? | [
"41"
] | [
"What is the maximum number of knights that can be placed on a $9 \\times 9$ chessboard such that no two knights attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-68a51445f0e8333f8e970146016e5950c6b452044d7764a91ff7a19a749e8510 | What is the maximum number of knights that can be placed on a 2015×2 chessboard such that no two knights attack each other? | [
"2016"
] | [
"What is the maximum number of knights that can be placed on a 2015×2 chessboard such that no two knights attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-9c90dd0b63eac9dd07dfb2f65d3bed848085ed097a0be7dba62e4d329006a4b5 | What is the maximum number of knights that can be placed on a 2x2015 chessboard such that no two knights attack each other? | [
"2016"
] | [
"What is the maximum number of knights that can be placed on a 2x2015 chessboard such that no two knights attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-bc69136fb41eb9e7e1b4fe7ff820117423f89fca8c9b875e0ed39e71e113fdd6 | What is the maximum number of knights that can be placed on an $8 \times 8$ chessboard such that each knight attacks at most seven other knights? | [
"62"
] | [
"What is the maximum number of knights that can be placed on an $8 \\times 8$ chessboard such that each knight attacks at most seven other knights?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-bb4320917f9e2d62403d4cacb3b872618b8b18d82992bee2bfc458c10071eda6 | What is the maximum number of knights that can be placed on an $n \times n$ chessboard such that no knight attacks another? | [
"\\left\\lceil \\dfrac{n^2}{2} \\right\\rceil"
] | [
"What is the maximum number of knights that can be placed on an $n \\times n$ chessboard such that no knight attacks another?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c19809548731fbe7b4d98e8933d804ca668484fbef618586a84bf7eeed89833d | What is the maximum number of length \( k \) cycles in a graph of order \( n \) with girth \( d \)? (Of particular interest is the case \( k = d \).) | [
"\\dbinom{\\left\\lfloor \\frac{n}{2} \\right\\rfloor}{\\frac{d}{2}} \\dbinom{\\left\\lceil \\frac{n}{2} \\right\\rceil}{\\frac{d}{2}}"
] | [
"What is the maximum number of length \\( k \\) cycles in a graph of order \\( n \\) with girth \\( d \\)? (Of particular interest is the case \\( k = d \\).)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-b27c421e94696f39706b68f079d2bee689a38367a4f20fecdba01016ba2ced83 | What is the maximum number of lines of symmetry a tetrahedron can have such that each line has a different length? The tetrahedron does not have to be regular. | [
"3"
] | [
"What is the maximum number of lines of symmetry a tetrahedron can have such that each line has a different length? The tetrahedron does not have to be regular."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-5f4fe0e112334b647771f9541999d55b5a29fb14e2ceaa3da646effa78cf3468 | What is the maximum number of lines on the plane such that there exist 8 points with at least 3 points on every line? | [
"8"
] | [
"What is the maximum number of lines on the plane such that there exist 8 points with at least 3 points on every line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-267826b2d87a2c47dd4bbd3430bb59b1e18223daebd602b559cdd8a17203df68 | What is the maximum number of members in a subset of \(\{1, 2, 3, \ldots, 2014\}\) such that no two members differ by 4 or 5? | [
"896"
] | [
"What is the maximum number of members in a subset of \\(\\{1, 2, 3, \\ldots, 2014\\}\\) such that no two members differ by 4 or 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-6bd637bc94a0f05e9aa3ae5959ca8e026fc70fada79ff0be8af475b97d4e5328 | What is the maximum number of moves needed to reach a point $(p,q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down? | [
"\\max(|p|, |q|)"
] | [
"What is the maximum number of moves needed to reach a point $(p,q)$ from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-17eac92705ea56d29f3e4d34e6a89c5d9815d8449b85cc2e6254a3c41fe93237 | What is the maximum number of natural numbers greater than 1, all of which are pairwise coprime, that can sum to 96? | [
"7",
"8"
] | [
"What is the maximum number of natural numbers greater than 1, all of which are pairwise coprime, that can sum to 96?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 6
} | 23 | 0 | 0 |
omr-ws-54c132c9a8680d603466bb4d32dbf757b00b443558094a0b64be1073e1b533f7 | What is the maximum number of natural numbers that one can select from the first 99 natural numbers and be sure that no three of the selected numbers add up to 99? | [
"67"
] | [
"What is the maximum number of natural numbers that one can select from the first 99 natural numbers and be sure that no three of the selected numbers add up to 99?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-55aa03a50588b71bf86d7513d225ebcea51092c35b299727048af10befe34b6f | What is the maximum number of neighborly convex polytopes in \(\mathbb{E}^3\), all combinatorially equivalent to a cube? This maximum is between 24 and 64. | [
"24"
] | [
"What is the maximum number of neighborly convex polytopes in \\(\\mathbb{E}^3\\), all combinatorially equivalent to a cube? This maximum is between 24 and 64."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f75b649872b3055e708ff1fe7543f4aa6a5f90b8a50dc284db301d45c69d49de | What is the maximum number of non-congruent cubes that can be obtained by cutting up a cube of side length 2016? | [
"424"
] | [
"What is the maximum number of non-congruent cubes that can be obtained by cutting up a cube of side length 2016?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 15
} | 28 | 0 | 0 |
omr-ws-5b2e9afabd73fdeba93481ef58c07991780571bca9352504511ee56b58fe0f11 | What is the maximum number of non-overlapping crosses that can be placed within the boundary of a $10 \times 11$ chessboard, where each cross covers exactly five unit squares and is formed by removing the unit squares at the four corners from a $3 \times 3$ square? | [
"15"
] | [
"What is the maximum number of non-overlapping crosses that can be placed within the boundary of a $10 \\times 11$ chessboard, where each cross covers exactly five unit squares and is formed by removing the unit squares at the four corners from a $3 \\times 3$ square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8922bc9aaff765d1e92236a2d544e82284560a6a7802fb287bfbd65e2c917599 | What is the maximum number of noncongruent triangles that can be formed in a rectangular prism? | [
"8"
] | [
"What is the maximum number of noncongruent triangles that can be formed in a rectangular prism?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-f536552c8afd1a65254421662237b9594bbd2afb24c5d29cf23f65f817b0f765 | What is the maximum number of numbers from the set \(\{1!, 2!, 3!, 4!, 5!, 6!, 7!, 8!, 9!\}\) that can be chosen such that their product is the square of an integer? | [
"7"
] | [
"What is the maximum number of numbers from the set \\(\\{1!, 2!, 3!, 4!, 5!, 6!, 7!, 8!, 9!\\}\\) that can be chosen such that their product is the square of an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-b3479dfbe5a0caea0642de9d112bc8ce8372e6955eb27d301dd91b4c8fadfed0 | What is the maximum number of numbers that can be removed from an 11-inch ruler if one still wants to be able to measure each distance from 1 inch up to 11 inches? | [
"6"
] | [
"What is the maximum number of numbers that can be removed from an 11-inch ruler if one still wants to be able to measure each distance from 1 inch up to 11 inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c925f2eb8392f56e95953dee162cb12cde1498ca7bc80cd083c1c9fe7556967d | What is the maximum number of numbers you can choose from the list $1, 2, 3, \ldots, 160$ so that no two numbers differ by $4$? | [
"80"
] | [
"What is the maximum number of numbers you can choose from the list $1, 2, 3, \\ldots, 160$ so that no two numbers differ by $4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-c13fdc32828f077e41b69c60b780642f11013d356fb0a390d353eae8c7331e7e | What is the maximum number of objects that can be placed on a circle with 'n' places such that no two objects are adjacent to each other? | [
"For even \\( n \\): \\( \\frac{n}{2} \\), For odd \\( n \\): \\( \\frac{n-1}{2} \\)"
] | [
"What is the maximum number of objects that can be placed on a circle with 'n' places such that no two objects are adjacent to each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-edfeec4d80b9954ca485d2590a400eacb848a9ba02456fe2fda2cef3a66f4668 | What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point, where the angle between any two rays is taken to be less than or equal to $180^\circ$? | [
"60"
] | [
"What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point, where the angle between any two rays is taken to be less than or equal to $180^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-e76da970f99c80a668df638c053f440dd1346854ea4cec38932e658080ac7947 | What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point, where the angle between any two rays is taken to be less than or equal to 180°? | [
"60"
] | [
"What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point, where the angle between any two rays is taken to be less than or equal to 180°?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-554755587ce0e6962b935978ec6973a52123d0343b20dc29cc57b8aa4124ef86 | What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point? | [
"105"
] | [
"What is the maximum number of obtuse angles that can be formed by 15 rays originating from a single point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71c07304fd0fd283c3fa923357e50d773209f0810e3d7002fbd5e321dec72876 | What is the maximum number of packets with dimensions \( x = 30 \), \( y = 20 \), and \( z = 10 \) that can be placed in a box with dimensions length = 100, width = 80, and height = 120? | [
"160"
] | [
"What is the maximum number of packets with dimensions \\( x = 30 \\), \\( y = 20 \\), and \\( z = 10 \\) that can be placed in a box with dimensions length = 100, width = 80, and height = 120?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-5ecdd67623e163b9080e69dc712cf3b79dac83aa0a1c71d0c341e51698f3e2c1 | What is the maximum number of pairs of acquaintances in a group of 2n (n>1) people among whom there are no cliques? | [
"\\( n^2 \\)"
] | [
"What is the maximum number of pairs of acquaintances in a group of 2n (n>1) people among whom there are no cliques?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 8
} | 29 | 0 | 0 |
omr-ws-5d0df0629824ec5f3c9f0e81616c397f586bc6cddda37fe5acef1ac2071f54d3 | What is the maximum number of pairwise incomparable subsets of a given set $X$ with $n$ elements? The answer is $\displaystyle \binom{n}{\left\lfloor \frac{n}{2} \right\rfloor}$. | [
"\\(\\binom{n}{\\left\\lfloor \\frac{n}{2} \\right\\rfloor}\\)"
] | [
"What is the maximum number of pairwise incomparable subsets of a given set $X$ with $n$ elements? The answer is $\\displaystyle \\binom{n}{\\left\\lfloor \\frac{n}{2} \\right\\rfloor}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-964540a51de1d5649814672a413241dd0c37d33eefcc92294b8616eb46afa67e | What is the maximum number of parts a plane can be divided into by N lines and M circles? | [
"\\( 1 + \\frac{N^2 + N}{2} + 2NM + M(M-1) \\)"
] | [
"What is the maximum number of parts a plane can be divided into by N lines and M circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 9
} | 20 | 0 | 0 |
omr-ws-8d204743617ef7b0befcfd54746f580d39236f8c7d62cdeafaf7d9424311e798 | What is the maximum number of parts a plane can be split into using $n$ zigzags, where a zigzag is defined as a polyline formed from two parallel rays and a segment connecting the origins of these rays? | [
"2n^2 - n + 1"
] | [
"What is the maximum number of parts a plane can be split into using $n$ zigzags, where a zigzag is defined as a polyline formed from two parallel rays and a segment connecting the origins of these rays?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-d75388a6bfd790eab1b6fe3581b133f495d316e16b4451dec427d1ab97846e7f | What is the maximum number of parts in which an n-dimensional ball can be divided by k hyperplanes? | [
"\\sum_{i=0}^n \\binom{k}{i}"
] | [
"What is the maximum number of parts in which an n-dimensional ball can be divided by k hyperplanes?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9f6c1564fa4ed0851fed0db5f97ba071d171a450b1c3479b6b6c4ce4244d32be | What is the maximum number of parts in which an n-sphere (the boundary of an n + 1-dimensional ball) can be divided by k hyperplanes? | [
"2 \\sum_{i=0}^n \\binom{k-1}{i}"
] | [
"What is the maximum number of parts in which an n-sphere (the boundary of an n + 1-dimensional ball) can be divided by k hyperplanes?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-8f5ab98d42cff4e393fe7487ca65883c65523e1af709ec34f0fb843f323f6cd2 | What is the maximum number of parts into which a 2D plane can be divided using 20 "$M$" shapes, where each "$M$" is composed of 2 semilines and 2 segments? | [
"3121"
] | [
"What is the maximum number of parts into which a 2D plane can be divided using 20 \"$M$\" shapes, where each \"$M$\" is composed of 2 semilines and 2 segments?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-62b670c62d7ccbad457ef5fcf72eb748a77c298a2885e80d6036d9cdb811fa36 | What is the maximum number of parts into which a plane can be divided by n circles? | [
"n^2 - n + 2"
] | [
"What is the maximum number of parts into which a plane can be divided by n circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca80050eda37f290faed7d70727b40f49c504308946006700aff5a2e4e32e3b4 | What is the maximum number of pawns that can be placed on an $n \times n$ chessboard such that no two pawns attack each other? (A pawn attacks the two squares diagonally in front of it.) | [
"\\(\\frac{n(n+1)}{2} \\text{ if } n \\text{ is odd, and } \\frac{n^2}{2} \\text{ if } n \\text{ is even}\\)"
] | [
"What is the maximum number of pawns that can be placed on an $n \\times n$ chessboard such that no two pawns attack each other? (A pawn attacks the two squares diagonally in front of it.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 2,
"tir": 15
} | 34 | 0 | 0 |
omr-ws-5aa173eaa96a980f642c1baa503e3616d93556822ec8855425032e55dfe8a94e | What is the maximum number of pegs that can be positioned on a triangular board with 5 rows (1 peg in the first row, 2 in the second, and so on) so that no legal moves are possible? | [
"10"
] | [
"What is the maximum number of pegs that can be positioned on a triangular board with 5 rows (1 peg in the first row, 2 in the second, and so on) so that no legal moves are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-1bc01edad2e096182b1975e2e0f368c77aa6aa35251f9ad5551d4a2a1d85be57 | What is the maximum number of permutations $f(n)$ for the existence of an n-tuple of all different integers such that the scalar product $a_1 \cdot b_1 + a_2 \cdot b_2 + \cdots + a_n \cdot b_n$ is always the same number? (Clearly, $1 < f(n) < n!$) | [
"2"
] | [
"What is the maximum number of permutations $f(n)$ for the existence of an n-tuple of all different integers such that the scalar product $a_1 \\cdot b_1 + a_2 \\cdot b_2 + \\cdots + a_n \\cdot b_n$ is always the same number? (Clearly, $1 < f(n) < n!$)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-52d6a93249718c1f1ef1c7a82b7658b845d5c269d8f8367bf1124f81bfc2aefe | What is the maximum number of pieces (i.e., completely bounded volumes) that can be formed when two right circular cones and one right circular cylinder mutually interpenetrate, with the base of each cone and both bases of the cylinder sealed by precisely fitting flat circular surfaces? Count only pieces that are not f... | [
"12"
] | [
"What is the maximum number of pieces (i.e., completely bounded volumes) that can be formed when two right circular cones and one right circular cylinder mutually interpenetrate, with the base of each cone and both bases of the cylinder sealed by precisely fitting flat circular surfaces? Count only pieces that are ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-92937e30c2751be704142628ddfb50799fc63d027fe511b900e4547467ed0f8f | What is the maximum number of pieces into which a cylinder can be divided by 5 oblique cuts? | [
"26"
] | [
"What is the maximum number of pieces into which a cylinder can be divided by 5 oblique cuts?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e3e5558367cabcccc5cb65501c772c47347506ddc0f6b67557c8c878fcb7f24 | What is the maximum number of pieces that a spherical watermelon can be divided into with four straight planar cuts? | [
"15"
] | [
"What is the maximum number of pieces that a spherical watermelon can be divided into with four straight planar cuts?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7b9c5577ecc7399cce3c401998bb2f6a52a3d821fe61f73e4fd440e069ff97a | What is the maximum number of pieces that can be created when a sphere is cut with 4 straight slices? | [
"15"
] | [
"What is the maximum number of pieces that can be created when a sphere is cut with 4 straight slices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-cf33a433189763fdca968d9df7487ee8588c1e42301f4c1ebddfb72ca2b5468c | What is the maximum number of pieces that can be made by slicing a pie with 81 cuts, assuming stacking is not allowed and centerpieces without the crust are permissible? | [
"3322"
] | [
"What is the maximum number of pieces that can be made by slicing a pie with 81 cuts, assuming stacking is not allowed and centerpieces without the crust are permissible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-735a925442a9e7ee0d4fc78e1decd14093d19ccbbaba9ec6f4f05250fe333ceb | What is the maximum number of pieces that can be obtained by cutting a sphere with \( n \) planar cuts, where no pieces are moved between cuts? | [
"\\(\\frac{n^3 + 5n + 6}{6}\\)"
] | [
"What is the maximum number of pieces that can be obtained by cutting a sphere with \\( n \\) planar cuts, where no pieces are moved between cuts?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-2c5e603034f3fcdb331d18dcc08ea69577ac50d85b1a98807979e3b8017c389d | What is the maximum number of pieces that can be obtained by cutting a spherical cake ten times with a knife in the shape of an infinite plane? | [
"176"
] | [
"What is the maximum number of pieces that can be obtained by cutting a spherical cake ten times with a knife in the shape of an infinite plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-0160141727a6bfa35b04b037ab4a131e7314e64b104540dd9f658a608d9aa7bd | What is the maximum number of pieces that can be used to tile an $8 \times 8$ chessboard using $1 \times 3$ rectangles and T-shaped tetrominoes? | [
"20"
] | [
"What is the maximum number of pieces that can be used to tile an $8 \\times 8$ chessboard using $1 \\times 3$ rectangles and T-shaped tetrominoes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-b2abc62d052e41232f196226e4160fad66c7539f668c667d4ea16d817015178e | What is the maximum number of pieces you can cut from a cube using 4 straight cuts, where each piece has flat faces? | [
"15"
] | [
"What is the maximum number of pieces you can cut from a cube using 4 straight cuts, where each piece has flat faces?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-95b93d65195c247d3d924878ca5c6cee07e6077ccd9339c849707c313167992f | What is the maximum number of points in $\mathbb{R}^3$ such that every three points form an acute-angled triangle? | [
"5"
] | [
"What is the maximum number of points in $\\mathbb{R}^3$ such that every three points form an acute-angled triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-cc820de466f46165579f8adac1fe424cdfa58c484aed88a016fd36f17774e34e | What is the maximum number of points in common between a line segment and a plane? | [
"\\infty"
] | [
"What is the maximum number of points in common between a line segment and a plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-041148dff10c7e3ce04cef431c059fc6d25c8056595d64c10e41720ce2da299b | What is the maximum number of points in the disk \( D = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\} \) such that the distance between any two points is at least 1? | [
"7"
] | [
"What is the maximum number of points in the disk \\( D = \\{(x,y) \\in \\mathbb{R}^2 : x^2 + y^2 \\leq 1\\} \\) such that the distance between any two points is at least 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51dac73953083bd2e57252a36cadca6bf4a0ab3bdff6cf395c491cf144820e87 | What is the maximum number of points in the intersection of the graphs of two different third-degree polynomial functions \( y = a(x) \) and \( y = b(x) \)? | [
"2"
] | [
"What is the maximum number of points in the intersection of the graphs of two different third-degree polynomial functions \\( y = a(x) \\) and \\( y = b(x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a052b265a3c0713f30c0b938a2ffda0b8ff1862c61ed073467c4c40494a2572 | What is the maximum number of points of intersection between 4 straight lines and 4 circles? | [
"39"
] | [
"What is the maximum number of points of intersection between 4 straight lines and 4 circles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e4d65f118e7f10ff25bfd997d552abec4a01ce7056ec948cddf86643e6a8ec9 | What is the maximum number of points of intersection between a straight line, a circle, a parabola, an ellipse, and a hyperbola? | [
"32"
] | [
"What is the maximum number of points of intersection between a straight line, a circle, a parabola, an ellipse, and a hyperbola?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-5579cc9008a9626ff892109ca3902e9e958863a99308d72b5cdc72b45c922f25 | What is the maximum number of points of intersection of $n$ circles? | [
"\\( \\frac{n(n-1)}{2} \\)"
] | [
"What is the maximum number of points of intersection of $n$ circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4bd839a0e8306c45f0fb9e5edd98157eff68c84cf45f669971361d8e779274e | What is the maximum number of points of intersection of 8 straight lines and 4 circles? | [
"104"
] | [
"What is the maximum number of points of intersection of 8 straight lines and 4 circles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-6c1c1ada8ab7858956cb82619753c2b13873e85315eaf3512a59070a52667d04 | What is the maximum number of points of intersection of the contours of two rectangles? | [
"16"
] | [
"What is the maximum number of points of intersection of the contours of two rectangles?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3cb01e60b5f836e361c37b4b052035ac0716d5d779fb3ca0dc15c99236ceaa5 | What is the maximum number of points of intersection of the graphs of two different fourth-degree polynomial functions \( y = P(x) \) and \( y = Q(x) \), each with leading coefficient 1? | [
"3"
] | [
"What is the maximum number of points of intersection of the graphs of two different fourth-degree polynomial functions \\( y = P(x) \\) and \\( y = Q(x) \\), each with leading coefficient 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b54773c566a7e80a04526c4bf44280394a7e5872b8efbaa0cfe4bb6ce678cd1f | What is the maximum number of points of intersection that two quadrilaterals can have on a 5 by 5 pin geoboard? (The quadrilaterals are not necessarily convex.) | [
"16"
] | [
"What is the maximum number of points of intersection that two quadrilaterals can have on a 5 by 5 pin geoboard? (The quadrilaterals are not necessarily convex.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-a3a3dcb7d0228e38f755ac5a95b34c73abd289c7b451da75f2ba380202c0cff8 | What is the maximum number of points of intersection when two distinct circles and three distinct lines intersect each other? | [
"17"
] | [
"What is the maximum number of points of intersection when two distinct circles and three distinct lines intersect each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-90f1c0141bc7101f4c16c5df82c2bf06adb03b37bca6d0edf564692c07a03677 | What is the maximum number of points that can be chosen from an $N$ by $N$ grid such that no 4 of the chosen points form a rectangle with sides parallel to the axes of the grid? | [
"N^{3/2}"
] | [
"What is the maximum number of points that can be chosen from an $N$ by $N$ grid such that no 4 of the chosen points form a rectangle with sides parallel to the axes of the grid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-5d177f6f67d504fda7d824f7b7ac630d0677f402b74a80cc2ab93ee76838f24e | What is the maximum number of points that can be chosen from an $N$ by $N$ grid such that no 4 of the chosen points form any rectangle (with sides parallel or not parallel to the axes of the grid)? | [
"2N - 1"
] | [
"What is the maximum number of points that can be chosen from an $N$ by $N$ grid such that no 4 of the chosen points form any rectangle (with sides parallel or not parallel to the axes of the grid)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-62606a81fba10b53a3f6ae941abaccb57982b263f2282492c80f74010efbd9a7 | What is the maximum number of points that can be placed in a regular hexagon with side $1$ so that the distance between any two is not less than $\sqrt{2}$? | [
"3"
] | [
"What is the maximum number of points that can be placed in a regular hexagon with side $1$ so that the distance between any two is not less than $\\sqrt{2}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-df3c6aad772796f44eb4eb0c2202928b697b7f5bf046d13ba6b8bb1bd0132bb2 | What is the maximum number of points that can be placed in an \(X \times Y\) rectangle such that the distance between any two points is at least \(d\)? | [
"\\(\\left \\lfloor \\frac {2(X + d)(Y + d)}{d^2 \\sqrt{3}} \\right \\rfloor\\) (Upper bound)"
] | [
"What is the maximum number of points that can be placed in an \\(X \\times Y\\) rectangle such that the distance between any two points is at least \\(d\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78f6ceecc83e142d04b07b80e8c74f34e6e43b633c2b52caf71ce3ac7382999a | What is the maximum number of points that can be placed on the Euclidean unit sphere in $\mathbb{R}^n$ such that the distance between any two points is at least $\sqrt{2}$? | [
"\\(2n\\)"
] | [
"What is the maximum number of points that can be placed on the Euclidean unit sphere in $\\mathbb{R}^n$ such that the distance between any two points is at least $\\sqrt{2}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-551cf986f1a7965d6eb438d033eec4b4469f6bc4bf5f05ffb2e89b4a966eb405 | What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\angle POQ$ is a multiple of $20$ degrees? | [
"18"
] | [
"What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\\angle POQ$ is a multiple of $20$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b8ed42e8c6762254be0d6e09db9745098947281f5fbde8538120467dac330eb8 | What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\angle POQ$ is a multiple of $60$ degrees? | [
"12"
] | [
"What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\\angle POQ$ is a multiple of $60$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e58766591c28b8658642a2a979a6d0f7458b07fbc72b6e1c9c9ae6178a9c5d3 | What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\angle POQ$ is a multiple of $90$ degrees? | [
"6"
] | [
"What is the maximum number of points that can be placed on the surface of a sphere with center $O$ such that for any two of these points $P$ and $Q$, $\\angle POQ$ is a multiple of $90$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8035060ff10b317c4f5b965b0d6cd01c8956c13c36369ba53fd67a168b7249cf | What is the maximum number of points where at least two of four distinct circles intersect in a plane? | [
"12"
] | [
"What is the maximum number of points where at least two of four distinct circles intersect in a plane?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-228d6ef64eb4c65cfb30543c1d3d09d949a9a99efc072cc589c83a15f82d0eb2 | What is the maximum number of points with integer coordinates that can lie on a circle with center at $(\sqrt{2}, \sqrt{3})$? | [
"1"
] | [
"What is the maximum number of points with integer coordinates that can lie on a circle with center at $(\\sqrt{2}, \\sqrt{3})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-d2bf42d2407b7e18de22012e2ef5b351aaa2ec31a22241e1b8e9c73e3abb086f | What is the maximum number of positive integers one can choose between $1$ and $100$ such that no two chosen integers differ by a factor of less than $1.5$? | [
"11"
] | [
"What is the maximum number of positive integers one can choose between $1$ and $100$ such that no two chosen integers differ by a factor of less than $1.5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66e8dcbfe26eac29111532ec30abcc55c2ed4a8ab3427ec95c7f54738ed722cf | What is the maximum number of positive integers that can be found such that for any two of them \( A \) and \( B \), the following inequality holds: \( |A - B| \geq \frac{AB}{100} \)? | [
"18"
] | [
"What is the maximum number of positive integers that can be found such that for any two of them \\( A \\) and \\( B \\), the following inequality holds: \\( |A - B| \\geq \\frac{AB}{100} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-3909c00608512a93f52c642833e1a4ba2ef9052fc8fa49254bcbec06f9a3bae1 | What is the maximum number of positive integers that can be listed such that none of the numbers and none of the sums of any number of distinct numbers from the list are divisible by any perfect square other than $1$? | [
"3"
] | [
"What is the maximum number of positive integers that can be listed such that none of the numbers and none of the sums of any number of distinct numbers from the list are divisible by any perfect square other than $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-1bc2e41a8047bfe50b9eb6dbea5097ff3d3540bd9b0cba85757f9e848b96d4aa | What is the maximum number of positive integers you can choose such that none of them are greater than $2019$ and no two of them differ by $2$, $1$, or $9$? | [
"606"
] | [
"What is the maximum number of positive integers you can choose such that none of them are greater than $2019$ and no two of them differ by $2$, $1$, or $9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-d7ce02cf5fb98b6e765a9d394f6eda8b07a670cfb3918b912be393ffe8e75ba3 | What is the maximum number of possible changes of direction in a path traveling on the edges of a $2004 \times 2004$ rectangular array, if the path does not cross the same place twice? | [
"4018019"
] | [
"What is the maximum number of possible changes of direction in a path traveling on the edges of a $2004 \\times 2004$ rectangular array, if the path does not cross the same place twice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5c9ac218fcf1e4c3c233d817803893e784b2330b4a0b827a3d429e60aba16f61 | What is the maximum number of possible intersection points among 5 lines and 2 circles in a plane? | [
"30"
] | [
"What is the maximum number of possible intersection points among 5 lines and 2 circles in a plane?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99da7d578b42c32059cae09547dfc2244bd7781870942b22ca6903891561697b | What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer? | [
"1"
] | [
"What is the maximum number of primes that divide both the numbers $n^3+2$ and $(n+1)^3+2$ where $n$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16d2f1c3596da52074913750c90a6b56f0bd7f892bf2d1ef1b06ecf2376c954b | What is the maximum number of quadrilaterals that can be cut off from a convex $n$-gon by a single diagonal, such that each of these quadrilaterals can be circumscribed? | [
"2"
] | [
"What is the maximum number of quadrilaterals that can be cut off from a convex $n$-gon by a single diagonal, such that each of these quadrilaterals can be circumscribed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-bc6881decd5dfff6d640bbb415cbeb14eae2ac51c436926779b81e2bd352c61a | What is the maximum number of quartets that can be placed in a $5 \times 5$ square? | [
"225"
] | [
"What is the maximum number of quartets that can be placed in a $5 \\times 5$ square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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