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-180bcc41a86ef52e6347b6eb4ddfe2ee4a92a381d574153a7a168d17794220c3 | What is the maximal number of colors that can be used to color the squares of a 10x10 board such that each row and each column contains at most 5 different colors? | [
"25"
] | [
"What is the maximal number of colors that can be used to color the squares of a 10x10 board such that each row and each column contains at most 5 different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 2
} | 45 | 0 | 0 |
omr-ws-95d3637730fdfe28e586a66a0f125313d0759119d4d3fdd2a625e6d59332fa2c | What is the maximal number of crosses that can fit in a $10 \times 11$ board without overlapping? | [
"14"
] | [
"What is the maximal number of crosses that can fit in a $10 \\times 11$ board without overlapping?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94e4798d9141f09858d1378b8a85d26c319eb2a76647964148077bed49a59e00 | What is the maximal number of distinct $n \times n$ real matrices such that their pairwise products are zero, i.e., $A_{i} A_{j} = 0$ for all $i \neq j$? | [
"n + 1"
] | [
"What is the maximal number of distinct $n \\times n$ real matrices such that their pairwise products are zero, i.e., $A_{i} A_{j} = 0$ for all $i \\neq j$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-c10dbe4fe8c49019fa8e18bcaa381d8e21b8c7006190c87c4bac0371e68440da | What is the maximal number of elements a finite subgroup of $GL_2(\mathbb{Z})$ could have? | [
"12"
] | [
"What is the maximal number of elements a finite subgroup of $GL_2(\\mathbb{Z})$ could have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-3792227b9b11c602ecc6f02fdba76c553a46b8a8bd6a4f3b47a0d14aa9fdb0ee | What is the maximal number of elements a finite subgroup of $GL_3(\mathbb{Z})$ could have? | [
"48"
] | [
"What is the maximal number of elements a finite subgroup of $GL_3(\\mathbb{Z})$ could have?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-138b4adcd1d514e7e1e74a87ec333775fe0ab3336aa543bee17065394624d075 | What is the maximal number of elements that can be chosen from the set $\{1, 2, \ldots, 31\}$ such that the sum of any two of them is not a perfect square? | [
"3"
] | [
"What is the maximal number of elements that can be chosen from the set $\\{1, 2, \\ldots, 31\\}$ such that the sum of any two of them is not a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c1d5a9387b1350c605aab739fd7f47c4c36315e16ae78e7b73bc892051a4a38 | What is the maximal number of knights that can be placed on an 8x8 chessboard so that each knight threatens at most 7 others? | [
"60"
] | [
"What is the maximal number of knights that can be placed on an 8x8 chessboard so that each knight threatens at most 7 others?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-0e798f7f3d576e94fab13e84775a402b0a5bbf88a3f92dd21e39405d655d79e2 | What is the maximal number of matrices \( A_1, \ldots, A_k \in M_n(\mathbb{C}) \) such that they pairwise anticommute, i.e., \( A_i A_j = -A_j A_i \) for all \( i \neq j \)? | [
"\\( 1 + 2 \\cdot v_2(n) \\)"
] | [
"What is the maximal number of matrices \\( A_1, \\ldots, A_k \\in M_n(\\mathbb{C}) \\) such that they pairwise anticommute, i.e., \\( A_i A_j = -A_j A_i \\) for all \\( i \\neq j \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-092f355133194d45e86b1ca3f19cc804c4f78d352c1e72e26d728bd3c6093f9a | What is the maximal number of pieces of two shapes, a right-angled triangle and a trapezoid, that can be used to tile a $7 \times 7$ square? | [
"98"
] | [
"What is the maximal number of pieces of two shapes, a right-angled triangle and a trapezoid, that can be used to tile a $7 \\times 7$ square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-a8965bd3f6d02059d6986ac84dbdfd3af63f90e9878a9bc3c1e6a331fcdf93fb | What is the maximal number of points in $R^n$ such that the distance between any two points is an odd integer? | [
"n + 1"
] | [
"What is the maximal number of points in $R^n$ such that the distance between any two points is an odd integer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-2c419f823312757e57c90f75f028c82640eabeae53c5a0e493613c96bab95fb0 | What is the maximal number of points in a set such that only two distinct Manhattan distances occur between all pairs of distinct points in the set? | [
"9"
] | [
"What is the maximal number of points in a set such that only two distinct Manhattan distances occur between all pairs of distinct points in the set?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-b7b7c9bdbb39b4f133a807d7a236cb712cc8768bc84f70b279d9e4b21198cd5a | What is the maximal number of points of intersection of a fourth-degree polynomial and a fifth-degree polynomial? | [
"4"
] | [
"What is the maximal number of points of intersection of a fourth-degree polynomial and a fifth-degree polynomial?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3142fa1e215ee965065cbb4b1985e9b8ad171e94e66ae446d484239677c5cda | What is the maximal number of regions a circle can be divided into by segments joining \( n \) points on the boundary of the circle? | [
"\\(\\binom{n}{4} + \\binom{n}{2} + 1\\)"
] | [
"What is the maximal number of regions a circle can be divided into by segments joining \\( n \\) points on the boundary of the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-999d0eb2c90a97de72d906ca6010c084244e904b3354782f4bc996eebf175f3e | What is the maximal number of regions into which 3 circles and 2 triangles can divide the plane? | [
"26"
] | [
"What is the maximal number of regions into which 3 circles and 2 triangles can divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd6d4898a4ec6cdc4ab39eb96f3813e654e78b94e0400db781b4d5d6ea8a8bf1 | What is the maximal number of regions into which \( n \) circles and \( m \) triangles can divide the plane? | [
"\\dfrac{9m^2 + 12mn + 2n^2 + 3m - 2n + 2}{2}"
] | [
"What is the maximal number of regions into which \\( n \\) circles and \\( m \\) triangles can divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-87bdc2a6fe541b4f327e2b287bacff2225d727b77dea18c943826a2925652f72 | What is the maximal number of short-sighted rooks that can be placed on a $100 \times 100$ chessboard such that no rook can beat another? A short-sighted rook beats all squares in the same row and column within a distance of 60 steps. | [
"178"
] | [
"What is the maximal number of short-sighted rooks that can be placed on a $100 \\times 100$ chessboard such that no rook can beat another? A short-sighted rook beats all squares in the same row and column within a distance of 60 steps."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-0bd17f234bd25bc4c8af5e9e4ca6f7d6a981d68ebd0c0b665d4ed7a900dcd05a | What is the maximal number of solutions the equation $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$ can have, where $a_1, b_1, a_2, b_2, \ldots, a_{10}, b_{10}$ are real numbers and all $a_i \neq 0$? | [
"2"
] | [
"What is the maximal number of solutions the equation $$\\max \\{a_1x+b_1, a_2x+b_2, \\ldots, a_{10}x+b_{10}\\}=0$$ can have, where $a_1, b_1, a_2, b_2, \\ldots, a_{10}, b_{10}$ are real numbers and all $a_i \\neq 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-178f5fb02d4fdc288b04639aebe9a4433797cc5c68af64ea9e21f5be7366a498 | What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common edge or a common vertex. | [
"\\(\\begin{cases}\\frac{mn}2&(\\text{when }mn\\text{ is even})\\\\\\frac{mn+1}2&(\\text{when }mn\\text{ is odd})\\end{cases}\\)"
] | [
"What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common edge or a common vertex."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66240e3f14484b9e195699c9e882ac638c519f9fd8cfc437335dd18c7cd06e6f | What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common edge. | [
"\\(\\begin{cases}\\frac{mn}2&(\\text{when }mn\\text{ is even})\\\\\\frac{mn+1}2&(\\text{when }mn\\text{ is odd})\\end{cases}\\)"
] | [
"What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common edge."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-03f2968bc4032419548472ba991fd96d9ca69999afce6d49e9a3cf1259eb3359 | What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common vertex. | [
"\\(\\begin{cases}\\frac{mn}2&(\\text{when }mn\\text{ is even})\\\\\\frac{mn+1}2&(\\text{when }mn\\text{ is odd})\\end{cases}\\)"
] | [
"What is the maximal number of squares that can be selected in an $m$ by $n$ grid so that no two squares are adjacent? Here, two squares are said to be adjacent if they share a common vertex."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b1886a4c78438f589d0659ca5bfc29f16d4cf92c4cc6c01a205daaea38b2595 | What is the maximal number of trees with an age of 2010 years in the original group, given that the average age of the group was 41 years and after the destruction of one such tree, the average age of the remaining trees is 40 years? | [
"39"
] | [
"What is the maximal number of trees with an age of 2010 years in the original group, given that the average age of the group was 41 years and after the destruction of one such tree, the average age of the remaining trees is 40 years?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-25b2ccb6c413fd953aedee32db03573fe1bd219536e5318ee6e039240090ecd6 | What is the maximal number of vectors in $\mathbb{R}^n$ such that the scalar product of any two of them is at most 0? | [
"\\(2n\\)"
] | [
"What is the maximal number of vectors in $\\mathbb{R}^n$ such that the scalar product of any two of them is at most 0?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-fb558c06f1b8ce6e0815469b36abae3d0e12a23ae116576245e9a0891ba72060 | What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square? | [
"24"
] | [
"What is the maximal number of vertices of a convex polyhedron whose each face is either a regular triangle or a square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39a549fb45397592c663ff446e3dbb442ab153dfc128d5a4fc7ebaef992f11db | What is the maximal order among the elements in PGL(2, F_31) with an order divisible by 31? | [
"31"
] | [
"What is the maximal order among the elements in PGL(2, F_31) with an order divisible by 31?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-de8521ac5f27664c6d39d1976d6e0cbd8dc429df9c99d5ad987a494c2c9dcb89 | What is the maximal order of an element in PGL(2, F_31)? | [
"32"
] | [
"What is the maximal order of an element in PGL(2, F_31)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ded0cd2573a75d07f528e838d5fcbaa134dd357d6fda73a3fb5155b919570b61 | What is the maximal order of an element of the group $\mbox{GL}_n\left(\mathbb{F}_q\right)$? | [
"\\( q^n - 1 \\)"
] | [
"What is the maximal order of an element of the group $\\mbox{GL}_n\\left(\\mathbb{F}_q\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e260ba402d5dcd94460fdd3eea3781722dd1b312948b6606651e66359810e8e8 | What is the maximal perimeter of a triangle inscribed in a circle of radius $R$? | [
"\\( 3R \\sqrt{3} \\)"
] | [
"What is the maximal perimeter of a triangle inscribed in a circle of radius $R$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c2f3ded877aef2bf32553af0169176a45a7a88fbe7478ec1a7073e8e66af1bf4 | What is the maximal possible number of acute angles in a convex polygon? | [
"3"
] | [
"What is the maximal possible number of acute angles in a convex polygon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83325954e4ee9cf192f2042d5a7ed52d417641efd50a74de93fa86bd620167eb | What is the maximal possible number of roots in the interval (0,1) for a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1? | [
"2021"
] | [
"What is the maximal possible number of roots in the interval (0,1) for a polynomial of degree 2022 with integer coefficients and a leading coefficient of 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f15058a4672ad2503b801e0b01f8200f6fb9a747e64a9212f6ee6ab215bacf05 | What is the maximal possible number of terms when the number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, each greater than $1$? | [
"8"
] | [
"What is the maximal possible number of terms when the number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, each greater than $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-baaa0c2cd8932538e187d9ec91cf114045fee0d48220b2eec0b200105bb3be1a | What is the maximal possible number of terms when the number 125 is written as the sum of several pairwise distinct and relatively prime numbers, all greater than 1? | [
"9"
] | [
"What is the maximal possible number of terms when the number 125 is written as the sum of several pairwise distinct and relatively prime numbers, all greater than 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-3e2568de5122d4c2f2e012fb43db37ac2e955151ca4f3afb6e5722e3f53703e5 | What is the maximal possible number that can be written on a blackboard, given that the sum of any two distinct positive integers on the blackboard is a power of $2$? | [
"1"
] | [
"What is the maximal possible number that can be written on a blackboard, given that the sum of any two distinct positive integers on the blackboard is a power of $2$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-137e5e50e73bc06ec175cf17cfc7a7d6457727857d856220cf9655f1b1707a50 | What is the maximal possible perimeter of a triangle contained within an 11-dimensional unit cube in $\mathbb{R}^{11}$? | [
"\\sqrt{5} + \\sqrt{6} + \\sqrt{11}"
] | [
"What is the maximal possible perimeter of a triangle contained within an 11-dimensional unit cube in $\\mathbb{R}^{11}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a1fbc106635fc572768d52320e32e157187db782eefee1e424ac1454bb379932 | What is the maximal possible value of $n$ such that no matter how 25 squares are selected in an infinite chessboard, one can find $n$ squares in which none of them share a common corner? | [
"7"
] | [
"What is the maximal possible value of $n$ such that no matter how 25 squares are selected in an infinite chessboard, one can find $n$ squares in which none of them share a common corner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-059589ac2c42fd0cac608235e6a5427ba19c9795880e14aed9335b2a329d0e36 | What is the maximal power of $2$ that divides $(2^n)!$? | [
"\\( 2^n - 1 \\)"
] | [
"What is the maximal power of $2$ that divides $(2^n)!$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 4,
"tir": 10
} | 37 | 0 | 0 |
omr-ws-07cc5f458fc2d8bac63f29d55f13c444cc081c797dc79a33973cd55a2ecb60d1 | What is the maximal proportion of a square that can be covered by two disks placed inside it, if overlap is allowed? | [
"\\frac{\\pi}{2} (2 - \\sqrt{3})"
] | [
"What is the maximal proportion of a square that can be covered by two disks placed inside it, if overlap is allowed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c54e83e538abf500dad0d318701df9e0e8fef40f20c373509c279863e34b0714 | What is the maximal proportion of a square that can be covered by two non-overlapping disks placed inside it? | [
"\\dfrac{\\pi (9 - 6\\sqrt{2})}{2}"
] | [
"What is the maximal proportion of a square that can be covered by two non-overlapping disks placed inside it?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-804a1160b86b26283a8885095e9e426abd7c2c3addb48299e17f446f56841426 | What is the maximal size of an abelian subgroup of $S_n$? | [
"\\( 3^{n/3} \\)"
] | [
"What is the maximal size of an abelian subgroup of $S_n$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa318e09ebcd0557f1be8997d471196c7dbd606c7d30fa1da5a91b90d7d70b7e | What is the maximal size of the symmetry group of a collection of $ n$ points in the plane? (Can one do better than the dihedral group?) | [
"\\( 2n \\)"
] | [
"What is the maximal size of the symmetry group of a collection of $ n$ points in the plane? (Can one do better than the dihedral group?)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-7b75fa0d960f189dd30b37a3d20272b25b78b8a1dd4d681ad800f4ba387e26b1 | What is the maximal tamely abelian extension of $\mathbb{Q}_p$? | [
"$\\mathbb{Q}_p^{nr}(\\zeta_p)$"
] | [
"What is the maximal tamely abelian extension of $\\mathbb{Q}_p$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-62916e0410ec4b9aa859c7d01075c4468f78dde60dbcefc0acd87a96677cabc5 | What is the maximal value of \( x \) such that the expression \( 4^{27} + 4^{1000} + 4^x \) is a perfect square? | [
"1972"
] | [
"What is the maximal value of \\( x \\) such that the expression \\( 4^{27} + 4^{1000} + 4^x \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d92322d93fa091573a4b1fce1bac912b2c266dfcf0c3dbe94efcea1b6c431c2a | What is the maximal value of the expression \( x_1 + \cdots + x_n - a x_1 x_2 \cdots x_n \) where \( x_1, \ldots, x_n, a \) are real numbers and \( x_1^2 + \cdots + x_n^2 = 1 \)? | [
"\\(\\max(n - a, \\sqrt{n(n-1)})\\)"
] | [
"What is the maximal value of the expression \\( x_1 + \\cdots + x_n - a x_1 x_2 \\cdots x_n \\) where \\( x_1, \\ldots, x_n, a \\) are real numbers and \\( x_1^2 + \\cdots + x_n^2 = 1 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b881b4bfc613d10d294d9a9ab3b4e05c68d36845681bda751ace62c11fbf679b | What is the maximal volume of a pyramid that can be inscribed inside a cube? | [
"\\(\\frac{1}{3} s^3\\)"
] | [
"What is the maximal volume of a pyramid that can be inscribed inside a cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 4
} | 15 | 0 | 0 |
omr-ws-b1feb3742d7219272c7dd236ea5cf76e3b5049f3995466e5698c9694c4e50871 | What is the maximum $k$ for which it is always possible to choose a $100 \times 100$ checkered square containing at least $k$ whole dominoes on a domino-covered checkered plane? | [
"4950"
] | [
"What is the maximum $k$ for which it is always possible to choose a $100 \\times 100$ checkered square containing at least $k$ whole dominoes on a domino-covered checkered plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-711688aea4537f05f512f97b290d72969843a86ea9fa03c67644bedfa9efc522 | What is the maximum $k$ such that the set $\{1, 2, \ldots, 2021\}$ can be partitioned into $k$ sets such that the sum of the elements of one set is not divisible by the sum of the elements of any other set? | [
"1010"
] | [
"What is the maximum $k$ such that the set $\\{1, 2, \\ldots, 2021\\}$ can be partitioned into $k$ sets such that the sum of the elements of one set is not divisible by the sum of the elements of any other set?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-45f6a6902f037ec69fa9bc4becb3ee09c605f0667ac3d41c5496a348ef9fe38d | What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots, a_n$ for a polynomial $P$ with integer coefficients and degree $2022$, where $P(a_i) = i$ for all $1 \le i \le n$? | [
"2022"
] | [
"What is the maximum $n$ such that there exist integers $a_1, a_2, \\cdots, a_n$ for a polynomial $P$ with integer coefficients and degree $2022$, where $P(a_i) = i$ for all $1 \\le i \\le n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 7
} | 34 | 0 | 0 |
omr-ws-6def003906c5a025648126d55ba15f3b3ae4805c9739c78fae4f136c5ccea17f | What is the maximum $n$ such that there exist rational numbers $a_1, a_2, \cdots, a_n$ for a polynomial $P$ with rational coefficients and degree $2022$, where $P(a_i) = i$ for all $1 \le i \le n$? | [
"2023"
] | [
"What is the maximum $n$ such that there exist rational numbers $a_1, a_2, \\cdots, a_n$ for a polynomial $P$ with rational coefficients and degree $2022$, where $P(a_i) = i$ for all $1 \\le i \\le n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-08cd7012de855ec3897d54373839ea22791fd74bf6b8411253141b719d178084 | What is the maximum \( n \) such that you can color all integers in the set \([1, 2, 3, \ldots, n]\) with only 2 colors such that no three integers forming a geometric progression are all the same color? | [
"8"
] | [
"What is the maximum \\( n \\) such that you can color all integers in the set \\([1, 2, 3, \\ldots, n]\\) with only 2 colors such that no three integers forming a geometric progression are all the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-167992e2293be1f7d17d099cf8f58adf3c2180d15067f2fd634b402f1bd8d19f | What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$, and $15\%$ solutions? | [
"\\(2\\frac{1}{7} \\text{L}\\)"
] | [
"What is the maximum amount of a $12\\%$ acid solution that can be obtained from $1$ liter of $5\\%$, $10\\%$, and $15\\%$ solutions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c81b69c2230ae7a02d3332dae2a39116dad2191c749c80bb0eea96aafe7d4545 | What is the maximum amount of grass that you can get to Damascus if the distance between Cairo and Damascus is 1000 miles, the camel consumes 1 kilogram of grass per mile, and the camel's maximum load is 1000 kilograms? | [
"1399.766"
] | [
"What is the maximum amount of grass that you can get to Damascus if the distance between Cairo and Damascus is 1000 miles, the camel consumes 1 kilogram of grass per mile, and the camel's maximum load is 1000 kilograms?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af1d6abd9f2c8ef71124e3884841bd97057bfa4eff3a181f203e81737b204739 | What is the maximum and minimum area of a triangle given that the radius of its circumcircle is $9$ cm and the radius of its incircle is $4$ cm? | [
"Maximum area \\( 64\\sqrt{2} \\) or \\( \\sqrt{8096} \\), Minimum area \\( 40\\sqrt{5} \\) or \\( \\sqrt{8000} \\)"
] | [
"What is the maximum and minimum area of a triangle given that the radius of its circumcircle is $9$ cm and the radius of its incircle is $4$ cm?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 20
} | 22 | 0 | 0 |
omr-ws-b7be34a112de557bc8fe7b45df030f34dd56794f2f954fa814ca88a04fcaa2e8 | What is the maximum and minimum radius of a circle that can be inscribed in a quadrilateral with sides of lengths \(a\), \(b\), \(c\), and \(d\)? | [
"0"
] | [
"What is the maximum and minimum radius of a circle that can be inscribed in a quadrilateral with sides of lengths \\(a\\), \\(b\\), \\(c\\), and \\(d\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-fc9b7619a6a2426e4c78dd8053a5755ba0e68ea72488f6a21dd9ea27697efda5 | What is the maximum and minimum value of $\cos A + \cos B + \cos C$ for a triangle $ABC$ with $A + B + C = 180^\circ$? | [
"Maximum = \\(\\frac{3}{2}\\), Minimum > 1"
] | [
"What is the maximum and minimum value of $\\cos A + \\cos B + \\cos C$ for a triangle $ABC$ with $A + B + C = 180^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-c51df1eaf13877fc2d22b89516d2595ce5d37b77f07020779afc6f5ae570918e | What is the maximum and minimum value of $\frac{x^4+y^4+z^4}{(x+y+z)^4}$ if $(x+y+z)^3=32xyz$, and $x, y, z$ are positive real numbers? | [
"\\(\\min = \\frac{383-165\\sqrt 5}{256}, \\max = \\frac{9}{128}\\)"
] | [
"What is the maximum and minimum value of $\\frac{x^4+y^4+z^4}{(x+y+z)^4}$ if $(x+y+z)^3=32xyz$, and $x, y, z$ are positive real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-c4b6da33f351ff1829cdf4bd543740d1af26484bc7292eb147ba1bf213562803 | What is the maximum and minimum value of $\sin A + \sin B + \sin C$ for a triangle $ABC$ with $A + B + C = 180^\circ$? | [
"\\(\\frac{3\\sqrt{3}}{2}\\) (for the maximum value) and \\(0\\) (for the minimum value)."
] | [
"What is the maximum and minimum value of $\\sin A + \\sin B + \\sin C$ for a triangle $ABC$ with $A + B + C = 180^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 25
} | 26 | 0 | 0 |
omr-ws-eb966ee274eb8f94eded0d9480c48b8cc66751711bdfec4375f902b9d1d5ec4c | What is the maximum angle \( \angle BAC \) when one of its trisectors is the diameter of the circumcircle? | [
"\\( 135^\\circ \\)"
] | [
"What is the maximum angle \\( \\angle BAC \\) when one of its trisectors is the diameter of the circumcircle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-e4bb5102e44164fe61cfcefe708b376a381a1333decca80c42dc6a5107c63bb2 | What is the maximum area of a good polygon (a polygon with a center of symmetry) whose vertices are inside or on the sides of a triangle with area \( S \)? | [
"\\dfrac{S}{2}"
] | [
"What is the maximum area of a good polygon (a polygon with a center of symmetry) whose vertices are inside or on the sides of a triangle with area \\( S \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1367090878f20572b4d9f9c00a9c92291777950c4d316876fe79fa594bb1267a | What is the maximum area of a pentagon whose sides are $\sqrt{3}, 2, 2, 3, 3$ in that order? (Without using Wolfram Alpha) | [
"9"
] | [
"What is the maximum area of a pentagon whose sides are $\\sqrt{3}, 2, 2, 3, 3$ in that order? (Without using Wolfram Alpha)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-62179c81da7c4f3fcc2ffd441d97f358a294c13090239bcb162f0ae2153592d9 | What is the maximum area of a quadrilateral that can be formed using four straight fences with lengths 1, 4, 7, and 8 units? | [
"18"
] | [
"What is the maximum area of a quadrilateral that can be formed using four straight fences with lengths 1, 4, 7, and 8 units?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55e5e1c8f2cfb99721f60c5336f4d360355dbb0e1f38e32a3276e06e11900047 | What is the maximum area of a quadrilateral with side lengths $18, 22, 39,$ and $41$? | [
"798"
] | [
"What is the maximum area of a quadrilateral with side lengths $18, 22, 39,$ and $41$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-78d9f5f79ed28bd0f73c156214ae7499ace9f58c4940d573a4709af5bab7cee7 | What is the maximum area of a quadrilateral with sides of lengths 1, 4, 7, and 8? | [
"18"
] | [
"What is the maximum area of a quadrilateral with sides of lengths 1, 4, 7, and 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71d8a4f3ae2986beea37cefd6583f1ff67d9152b41e74d34672f905f97aff012 | What is the maximum area of a rectangle circumscribed about a fixed rectangle of length 8 and width 4? | [
"72"
] | [
"What is the maximum area of a rectangle circumscribed about a fixed rectangle of length 8 and width 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 1
} | 46 | 0 | 0 |
omr-ws-60c0ad038db11cc049a455e188ba07627c4f7ab8fe9d7762a71d736c6ba88ad5 | What is the maximum area of a rectangle in the xy-plane with its bottom edge on the x-axis between \( x = 0 \) and \( x = \pi \), and its top corners on the graph of \( y = \sin x \)? Provide the answer accurate to two decimal places. | [
"1.11"
] | [
"What is the maximum area of a rectangle in the xy-plane with its bottom edge on the x-axis between \\( x = 0 \\) and \\( x = \\pi \\), and its top corners on the graph of \\( y = \\sin x \\)? Provide the answer accurate to two decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-6dcdebcc2da4f72128f33a3ed04188ab7152608768e254f3a0f18cd9e6c4d3d3 | What is the maximum area of a rectangle that can be inscribed in a triangle with area \( M \)? | [
"\\dfrac{M}{2}"
] | [
"What is the maximum area of a rectangle that can be inscribed in a triangle with area \\( M \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-160bfa69a0828473293967f6b2cb5ba7df908ee3620ea9f7802482781f878533 | What is the maximum area of a regular $n$-gon ${\cal P}$ with side lengths $A_1A_2 = A_2A_3 = \cdots = A_nA_1$? | [
"\\dfrac{n s^2}{4 \\tan\\left(\\dfrac{\\pi}{n}\\right)}"
] | [
"What is the maximum area of a regular $n$-gon ${\\cal P}$ with side lengths $A_1A_2 = A_2A_3 = \\cdots = A_nA_1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-fd6fd11095e8af180a773cd315082f99ccd47d770d3064a0c2bf16050464ea72 | What is the maximum area of a right-angled triangle with legs \( x \) and \( y \) and hypotenuse \( z \)? | [
"\\dfrac{z^2}{4}"
] | [
"What is the maximum area of a right-angled triangle with legs \\( x \\) and \\( y \\) and hypotenuse \\( z \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4367fdb39915d5c1c59a0af61254b8c49fe4426ed7ca410232d7b38ee6f25657 | What is the maximum area of a semicircle that can be cut out from a triangular cardboard with sides 13 inches, 14 inches, and 15 inches? | [
"\\(\\frac{1568\\pi}{81}\\)"
] | [
"What is the maximum area of a semicircle that can be cut out from a triangular cardboard with sides 13 inches, 14 inches, and 15 inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 7,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-0904961896aa8fd6600c7c6472589642ff1e6794d1bb5c3a7ae4ca6604ab44ad | What is the maximum area of a triangle formed by three points on an arc of length $R^{\frac{1}{3}}$ of a circle centered at the origin with radius $R$? | [
"\\dfrac{1}{16}"
] | [
"What is the maximum area of a triangle formed by three points on an arc of length $R^{\\frac{1}{3}}$ of a circle centered at the origin with radius $R$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-ceb3dc4baf371eaff32bf72f150e6f558fb6f8be92617fdea624c0db29e7209f | What is the maximum area of a triangle inscribed in an ellipse with a semimajor axis of length 4 and a semiminor axis of length 2? | [
"6\\sqrt{3}"
] | [
"What is the maximum area of a triangle inscribed in an ellipse with a semimajor axis of length 4 and a semiminor axis of length 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01bbe138e187f45d9810bf9dfa13b0dec2a0b7612f6067d037b095d305e1f2e8 | What is the maximum area of a triangle that can be enclosed within a $10$ by $10$ unit square, given that the centroid of the triangle coincides with the center of the square? | [
"\\dfrac{75}{2}"
] | [
"What is the maximum area of a triangle that can be enclosed within a $10$ by $10$ unit square, given that the centroid of the triangle coincides with the center of the square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8c62aeffb72d8fb921d72382e86892bb5c7f4c2d2045196099692fa1a0da094 | What is the maximum area of a triangle that can be inscribed in the ellipse given by the equation $\frac{x^2}{9} + \frac{y^2}{1} = 1$? | [
"$\\frac{9\\sqrt{3}}{4}$"
] | [
"What is the maximum area of a triangle that can be inscribed in the ellipse given by the equation $\\frac{x^2}{9} + \\frac{y^2}{1} = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f0d134980d351e9837062d393ea8bf6f358685f13657a5ecdd39e6a458f3e039 | What is the maximum area of a triangle whose sides \(a\), \(b\), and \(c\) satisfy the conditions \(0 \le a \le 1\), \(1 \le b \le 2\), and \(2 \le c \le 3\)? | [
"1"
] | [
"What is the maximum area of a triangle whose sides \\(a\\), \\(b\\), and \\(c\\) satisfy the conditions \\(0 \\le a \\le 1\\), \\(1 \\le b \\le 2\\), and \\(2 \\le c \\le 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7329e722d25cfc388ec622f93ab8f944c5cb1761bb52f525d363028b8c5467d1 | What is the maximum area of a triangle whose sides \(a\), \(b\), and \(c\) satisfy the inequalities \(0 \leq a \leq 1\), \(1 \leq b \leq 2\), and \(2 \leq c \leq 3\)? | [
"1"
] | [
"What is the maximum area of a triangle whose sides \\(a\\), \\(b\\), and \\(c\\) satisfy the inequalities \\(0 \\leq a \\leq 1\\), \\(1 \\leq b \\leq 2\\), and \\(2 \\leq c \\leq 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5fe65b22ac5299ef76b37d30128797fe4b86fd9c155026f044020cd357ad6f1 | What is the maximum area of a triangle with a given circumradius $R$ and a given median length $m$? | [
"m \\sqrt{m(2R - m)}"
] | [
"What is the maximum area of a triangle with a given circumradius $R$ and a given median length $m$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd6cfec92a47ec5424c127d473f15bbd8cd04f10422437a4b235e1837409684e | What is the maximum area of a triangle with a perimeter of 16? | [
"\\(\\frac{64\\sqrt{3}}{9}\\)"
] | [
"What is the maximum area of a triangle with a perimeter of 16?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5545e58671418ae29155a7c766521fed8cf952d7d5920aa3d88a5d0c12ecfcc2 | What is the maximum area of a triangle with vertices in a given square (or on its boundary)? | [
"\\( \\frac{s^2}{2} \\)"
] | [
"What is the maximum area of a triangle with vertices in a given square (or on its boundary)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-3c9f198e99e77ec1462ad7bcea3d6aa1e5a18eea43926a4bf620c7038f3a091b | What is the maximum area of an equilateral triangle that can be enclosed within a $10$ by $10$ unit square? | [
"\\(200\\sqrt{3} - 300\\)"
] | [
"What is the maximum area of an equilateral triangle that can be enclosed within a $10$ by $10$ unit square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 18
} | 64 | 0 | 0 |
omr-ws-a538a6677f83051db386a5cd0f3c8401a461c781292ee200e6d67203c48d95c8 | What is the maximum area of an equilateral triangle whose vertices all lie on the edges of a square with side length 1 unit? The area is expressed in the form $a\sqrt{b} - c$ units squared. Find the value of $a + b + c$. | [
"8"
] | [
"What is the maximum area of an equilateral triangle whose vertices all lie on the edges of a square with side length 1 unit? The area is expressed in the form $a\\sqrt{b} - c$ units squared. Find the value of $a + b + c$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-df93cbdc2f22f7bcf45b4108dfe9025a00f26a81fa02c9223fa9d1847725d64a | What is the maximum area of intersection between a square and a triangle, both inscribed in a circle of radius 10? | [
"100"
] | [
"What is the maximum area of intersection between a square and a triangle, both inscribed in a circle of radius 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-e761a8241b16a596d5c24eaf57720bc9e96204b50c8979dd827f6afe09c1005c | What is the maximum area of quadrilateral $ABCD$ with $AB=BC=3$ and $CD=7$? | [
"16\\sqrt{2}"
] | [
"What is the maximum area of quadrilateral $ABCD$ with $AB=BC=3$ and $CD=7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f23e455a7fc529437f40343178a5492bf779c579a2f92c55d42fade42dc40314 | What is the maximum area of the bounding rectangle of a triangle with side lengths 7, 8, and 9? | [
"\\( 24\\sqrt{5} \\)"
] | [
"What is the maximum area of the bounding rectangle of a triangle with side lengths 7, 8, and 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-5138f4ef5b095155bb0973651add97d226c19817438628bd72db47a72b3fc6c8 | What is the maximum area of the quadrilateral that can be enclosed using four straight fences with lengths of 1, 4, 7, and 8 meters? | [
"18"
] | [
"What is the maximum area of the quadrilateral that can be enclosed using four straight fences with lengths of 1, 4, 7, and 8 meters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f1656109c50ea0d83bf1a87e6d2b7d8fd97cf983558a41471907b30c419ad65b | What is the maximum area of triangle \(ABC\) given the side lengths satisfy \(0 \le AB \le 1 \le BC \le 2 \le CA \le 3\)? | [
"1"
] | [
"What is the maximum area of triangle \\(ABC\\) given the side lengths satisfy \\(0 \\le AB \\le 1 \\le BC \\le 2 \\le CA \\le 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0d93675c6a4519b73e9a3fdaef73824e06cf186b80c8f8c93234ab395aa1970c | What is the maximum area that a parallelogram inscribed in a triangle can have? (The vertices of the parallelogram belong to the sides of the triangle.) | [
"\\dfrac{1}{2}"
] | [
"What is the maximum area that a parallelogram inscribed in a triangle can have? (The vertices of the parallelogram belong to the sides of the triangle.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-441829893e818be5fff7633be3c9b081e13b21a6291d2010492610a62d1f7962 | What is the maximum area that can be enclosed by a fence of length $2\ell$ and an infinitely long wall? What is the shape of this optimal fence? The fence can be any continuous curve. | [
"\\(\\frac{2\\ell^2}{\\pi}\\)"
] | [
"What is the maximum area that can be enclosed by a fence of length $2\\ell$ and an infinitely long wall? What is the shape of this optimal fence? The fence can be any continuous curve."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-4840ac0ad4340305e0f815720c469322f4770f40e1657fb1243fd225ad769a63 | What is the maximum area that the image of a cube with edge length $1$ can occupy when projected parallel onto a plane? How must the projection plane then lie relative to the cube? | [
"\\(\\frac{3\\sqrt{3}}{4}\\)"
] | [
"What is the maximum area that the image of a cube with edge length $1$ can occupy when projected parallel onto a plane? How must the projection plane then lie relative to the cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24e7e6268c6f9082257274871636f766086f43d451284c78673a898fa864f4a2 | What is the maximum average BAC (in g/dL) during the first 5 hours for the function $C(t) = 0.135t \cdot e^{-2.802t}$? (Round your answer to four decimal places.) | [
"0.0177"
] | [
"What is the maximum average BAC (in g/dL) during the first 5 hours for the function $C(t) = 0.135t \\cdot e^{-2.802t}$? (Round your answer to four decimal places.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-9e34cdad9b37d6f9ce4eedbf1cdc692c435657bb0be1f45e6267f242edd15bb5 | What is the maximum cardinality of a subset \( S \) contained in \(\{1, 2, \ldots, 2021\}\) such that there do not exist \( a, b, c, d \) in \( S \) for which \( ab - cd = 1 \)? | [
"1011"
] | [
"What is the maximum cardinality of a subset \\( S \\) contained in \\(\\{1, 2, \\ldots, 2021\\}\\) such that there do not exist \\( a, b, c, d \\) in \\( S \\) for which \\( ab - cd = 1 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-61cfe74fb43ad4b621cf77f852ddff07ee8cd8df222623bf225d8d5f73d9f5e7 | What is the maximum concentration (in µg/mL) of the antibiotic during the first 12 hours for the function $C(t) = 2(e^{-0.4t} - e^{-0.6t})$? (Round your answer to four decimal places.) | [
"0.2963"
] | [
"What is the maximum concentration (in µg/mL) of the antibiotic during the first 12 hours for the function $C(t) = 2(e^{-0.4t} - e^{-0.6t})$? (Round your answer to four decimal places.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-72fc37bba312eb2f683ef013de4eea7c91fdca3a9d8cac134b44cef9bc8ad9d9 | What is the maximum determinant of a $3 \times 3$ matrix where the entries are $1$, $0$, and $-1$? | [
"4"
] | [
"What is the maximum determinant of a $3 \\times 3$ matrix where the entries are $1$, $0$, and $-1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6386f51ff30294c3b85b35a933096419e3973ba8f312df4c5cdc8df93ab308d0 | What is the maximum determinant of an $n \times n$ matrix where the entries are $1$, $0$, and $-1$? | [
"n^{n/2}"
] | [
"What is the maximum determinant of an $n \\times n$ matrix where the entries are $1$, $0$, and $-1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-47107bd4da59f8b79f089fc24ff7fda2ba016cda7b39b85168f6c519ed14f9f0 | What is the maximum dimension of a linear subspace \( V \) of \( n \times n \) matrices such that for all \( A, B \in V \), \( \text{trace}(AB) = 0 \)? | [
"\\frac{n(n-1)}{2}"
] | [
"What is the maximum dimension of a linear subspace \\( V \\) of \\( n \\times n \\) matrices such that for all \\( A, B \\in V \\), \\( \\text{trace}(AB) = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-6e56105ef7b6973e571537c4765207262908d6bdc71c364ae8cf9bef67ebf7ca | What is the maximum dimension of a subspace of \( M_n(\mathbb{C}) \) generated by a finite subgroup of \( GL_n(\mathbb{C}) \) (not necessarily Abelian)? | [
"\\( n^2 \\)"
] | [
"What is the maximum dimension of a subspace of \\( M_n(\\mathbb{C}) \\) generated by a finite subgroup of \\( GL_n(\\mathbb{C}) \\) (not necessarily Abelian)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-816c9987b2478b9101a76f58518ff6ae8df25521ad5beae982e0f054576f000a | What is the maximum element, if any, of the set \( A = \mathbb{N} - \bigcup_{m \in \mathbb{N}} S_m \), where \( S_m = \{ n \in \mathbb{N} \mid 138m \leq n \leq 139m \} \)? | [
"18905"
] | [
"What is the maximum element, if any, of the set \\( A = \\mathbb{N} - \\bigcup_{m \\in \\mathbb{N}} S_m \\), where \\( S_m = \\{ n \\in \\mathbb{N} \\mid 138m \\leq n \\leq 139m \\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-2967f72efed8db95899776f655e1fb24eac1f47032832320ab40a50020175765 | What is the maximum error on a 1000 term approximation of $\sum_{n=1}^{\infty}\frac{1}{n^2}$? To find the maximum error, consider the Lagrange Remainder $R_{1000}(1)$ for the series $\sum_{n=1}^{\infty}\frac{1}{n^2}$, where $R_{1000}(1)\le \left| \frac{\max(|f^{(1001)}(z)|)}{1001!}\right|$ for $0 \le z \le 1$. Determin... | [
"\\(\\frac{1}{1000}\\)"
] | [
"What is the maximum error on a 1000 term approximation of $\\sum_{n=1}^{\\infty}\\frac{1}{n^2}$?\n\nTo find the maximum error, consider the Lagrange Remainder $R_{1000}(1)$ for the series $\\sum_{n=1}^{\\infty}\\frac{1}{n^2}$, where $R_{1000}(1)\\le \\left| \\frac{\\max(|f^{(1001)}(z)|)}{1001!}\\right|$ for $0 \\l... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-cf35f0ec7e697cc69015536729777aa285f8386d569173fec2ba9f76ee664c1d | What is the maximum length of a boat that can turn from a river of width \( a \) into a channel of width \( b \) that is perpendicular to the river? | [
"\\((a^{\\frac{2}{3}}+b^{\\frac{2}{3}})^{\\frac{3}{2}}\\)"
] | [
"What is the maximum length of a boat that can turn from a river of width \\( a \\) into a channel of width \\( b \\) that is perpendicular to the river?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b355a828505d42f5e462dbfaa6fb23def8665187579bcef2120d424f9dcf2ad6 | What is the maximum length of a geometric sequence with all terms in a nonconstant arithmetic sequence with length $k$? | [
"\\lfloor \\log_2 k \\rfloor + 1"
] | [
"What is the maximum length of a geometric sequence with all terms in a nonconstant arithmetic sequence with length $k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92756ccd8cda4ea50ca941da82d7bb1efee0ddcf7c18ca7fe4665f9b052e90ff | What is the maximum length of a pole that can be carried around the corner where a 10-foot wide hallway and a 5-foot wide hallway intersect without tilting it? | [
"\\(\\left(10^{\\frac{2}{3}} + 5^{\\frac{2}{3}}\\right)^{\\frac{3}{2}}\\)"
] | [
"What is the maximum length of a pole that can be carried around the corner where a 10-foot wide hallway and a 5-foot wide hallway intersect without tilting it?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-bb2efeb7ec35f465c7657d6df9997dcd5a2da45b1d165e07c2c1921cbd237b2f | What is the maximum length of an arithmetic progression (A.P.) of natural numbers such that each term is of the form $2^m + 3^n$, where $m$ and $n$ are nonnegative integers? | [
"6"
] | [
"What is the maximum length of an arithmetic progression (A.P.) of natural numbers such that each term is of the form $2^m + 3^n$, where $m$ and $n$ are nonnegative integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4622f070e051675a8f4ef51a6557602c235965521b61f254a3437584ae25716 | What is the maximum length of the block of repeating digits in the decimal expansion of \( \frac{1}{n} \)? | [
"\\( n-1 \\)"
] | [
"What is the maximum length of the block of repeating digits in the decimal expansion of \\( \\frac{1}{n} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
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