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-767f824b7415efd5c0c2627b11c14ddbbbbe2d9d08ef6c9924d518fc44014272 | What is the maximum number of quartets that can be placed in an $m \times n$ rectangle? | [
"\\(\\frac{m(m-1)n(n-1)}{4}\\)"
] | [
"What is the maximum number of quartets that can be placed in an $m \\times n$ rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-1c89465daaafbc5dfc58289c30feae3b6d76fa57feebfd80aebcec7a507ded5b | What is the maximum number of queens that can be placed on a chessboard such that not every square is under attack by at least one queen? | [
"42"
] | [
"What is the maximum number of queens that can be placed on a chessboard such that not every square is under attack by at least one queen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-a182771c0eae93326fa17dabfd09e75fd0e6980eebcf0d4382e8c1990c041915 | What is the maximum number of real roots of the equation $x^6 + ax^4 + bx^2 + c = 0$, given that $a > 0$, $b \in \mathbb{R}$, and $c < 0$? | [
"2"
] | [
"What is the maximum number of real roots of the equation $x^6 + ax^4 + bx^2 + c = 0$, given that $a > 0$, $b \\in \\mathbb{R}$, and $c < 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 22
} | 22 | 0 | 0 |
omr-ws-a8ac572cd8749d2ae2213312d824aad39afdf99281e1db9ac4aecd67ff7f6e47 | What is the maximum number of regions a convex n-gon is divided into by its diagonals if no three diagonals intersect at a single point? | [
"\\( \\binom{n}{4} + \\frac{n(n-3)}{2} + 1 \\)"
] | [
"What is the maximum number of regions a convex n-gon is divided into by its diagonals if no three diagonals intersect at a single point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 11
} | 28 | 0 | 0 |
omr-ws-3cb2a8b34e9675577275e70334f1a4a1e9f033852103973b949d498c7fd98842 | What is the maximum number of regions defined by 5 rectangles in the plane? | [
"81"
] | [
"What is the maximum number of regions defined by 5 rectangles in the plane?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a275ddc5225ab46c49ea78f4b4993e725ae822310acc85ab530ea2e8c2beb4b4 | What is the maximum number of regions determined by $n$ zigzags, where a zigzag is defined as a group of two parallel rays with opposite direction and a segment connecting their endpoints? | [
"\\( n^2 - n + 2 \\)"
] | [
"What is the maximum number of regions determined by $n$ zigzags, where a zigzag is defined as a group of two parallel rays with opposite direction and a segment connecting their endpoints?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a51895e684a9c4f5557b45e555181eee278325015ddcccd62b4e2a5414c6205 | What is the maximum number of regions formed by \( n \) lines in the plane if \( c \) of the lines are concurrent? | [
"\\dfrac{n^2 + n - c^2 + 3c}{2}"
] | [
"What is the maximum number of regions formed by \\( n \\) lines in the plane if \\( c \\) of the lines are concurrent?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-432ee15cb6551154784c2b3a44a62009fa6dbcee9aff0d3c93710c8272607d31 | What is the maximum number of regions formed by \( n \) lines in the plane if \( k \) of the lines are parallel? | [
"\\dfrac{n^2 + n - k^2 + k + 2}{2}"
] | [
"What is the maximum number of regions formed by \\( n \\) lines in the plane if \\( k \\) of the lines are parallel?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6017a55598a10b206828f574efbc0907de46091b4f751afde20ddf0dd29a4b12 | What is the maximum number of regions formed by \( n \) planes in space? | [
"\\dfrac{n^3 + 5n + 6}{6}"
] | [
"What is the maximum number of regions formed by \\( n \\) planes in space?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa13c62da7d36494302d7ed5e833a7f01683b6151ec4ecc7d436745006ae6797 | What is the maximum number of regions formed inside a decagon by joining each pair of its ten vertices with chords? | [
"246"
] | [
"What is the maximum number of regions formed inside a decagon by joining each pair of its ten vertices with chords?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62393cd6ddaffb749a70ddf34beae48a422a8ca010b8775b90d1c3a21a4443ba | What is the maximum number of regions into which \( n \) ellipses can divide the plane? | [
"\\( 2^{n+1} - 2 \\)"
] | [
"What is the maximum number of regions into which \\( n \\) ellipses can divide the plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9fa73bb5a1b7d4d72e6f9cabd066f2278f0dff62ae146ffc34968119423e7d93 | What is the maximum number of regions into which a circle can be divided by drawing diagonals between a number of points inscribed on the circle? | [
"\\dbinom{n}{4} + \\dbinom{n}{2} + 1"
] | [
"What is the maximum number of regions into which a circle can be divided by drawing diagonals between a number of points inscribed on the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-8fc0244b9d2d500e9fefb9ac23960fd68cb103624b4a1d8e2b43d49cbd943929 | What is the maximum number of regions that can be created on a sphere $S$ by placing $n$ circles $w_1, w_2, \ldots, w_n$ on it? | [
"\\( 2 + n(n-1) \\)"
] | [
"What is the maximum number of regions that can be created on a sphere $S$ by placing $n$ circles $w_1, w_2, \\ldots, w_n$ on it?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-a38c34d0935e05c7f044fc1c5fba5d16f9e69c34048fe19d4fce6c74a97f2cc2 | What is the maximum number of regions that can be formed by cutting a sphere with 5 great circles? | [
"22"
] | [
"What is the maximum number of regions that can be formed by cutting a sphere with 5 great circles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9b06760307ae82b99d1ee604db3127a075c69ee555d9ead9fce88bdd7d22cad7 | What is the maximum number of regions that can be formed when $n=7$ dots are drawn on the circumference of a circle and all the dots are joined to one another by straight lines? | [
"57"
] | [
"What is the maximum number of regions that can be formed when $n=7$ dots are drawn on the circumference of a circle and all the dots are joined to one another by straight lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f0c59234f473acfdab63e54c0fa628e92d3300fbf1b075686e3db972b813f15 | What is the maximum number of regions that can be formed when a circle and a rhombus are drawn on a flat surface? | [
"9"
] | [
"What is the maximum number of regions that can be formed when a circle and a rhombus are drawn on a flat surface?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e908871acee96816a9e298cc57070f254bb7e131f3ff65b90841551c79523d64 | What is the maximum number of regions that n circles can divide a plane into? | [
"\\( n(n-1) + 2 \\)"
] | [
"What is the maximum number of regions that n circles can divide a plane into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-b3908530ef84d9cda2d9cca8a06f77116fca9b97d5ca2a796e16c9588cfe0a62 | What is the maximum number of regions that n lines and m circles can divide the plane into? | [
"\\(\\frac{n^2 + n + 2m^2 - 2m + 4 + 4nm}{2}\\)"
] | [
"What is the maximum number of regions that n lines and m circles can divide the plane into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 3
} | 12 | 0 | 0 |
omr-ws-c5254edbfa210a8f92a2aa33d05605d43c544bcd56574d34b7e2158b96928474 | What is the maximum number of regions that n planes can divide the 3-dimensional space into? | [
"\\(\\frac{1}{6}(n^3 + 5n + 6)\\)"
] | [
"What is the maximum number of regions that n planes can divide the 3-dimensional space into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-91bbd39e9d55d913f60def3409f1b75bee59e4d93c504fcff0b9b90368d98fa2 | What is the maximum number of regions that n spheres can divide the 3-dimensional space into? | [
"\\(\\frac{n}{3}(n^2 - 3n + 8)\\)"
] | [
"What is the maximum number of regions that n spheres can divide the 3-dimensional space into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 11
} | 59 | 0 | 0 |
omr-ws-8bda0805e62c957582fedf109e869931e4b8746ff09490dcb79f3d91dd3ea60b | What is the maximum number of regions that n straight lines can divide a plane into? | [
"\\(\\frac{n(n+1)}{2} + 1\\)"
] | [
"What is the maximum number of regions that n straight lines can divide a plane into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-9467db84e4343636cb04937666cbeb8996ffe054f8eed83ad09a38da5bce9109 | What is the maximum number of regions that the diagonals of a convex n-gon can create? | [
"\\(\\frac{(n-1)(n-2)(n^2-3n+12)}{24}\\)"
] | [
"What is the maximum number of regions that the diagonals of a convex n-gon can create?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-82fb7181f54dd2bcbfb6eeb825b7d1b0dd6c1ab7149c2f3012209fbdef573793 | What is the maximum number of regions you can create using 2 equal-sized circles and 3 straight lines? | [
"15"
] | [
"What is the maximum number of regions you can create using 2 equal-sized circles and 3 straight lines?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13632cb51595771dc97485693e4717c9e69b7e115b8d44985de92429476ad5e9 | What is the maximum number of regular hexagons of side 1 unit that can be cut from a circular disk of radius \( r \) units? | [
"1 + 3\\left\\lfloor \\frac{r - 1}{\\sqrt{3}} \\right\\rfloor \\left( \\left\\lfloor \\frac{r - 1}{\\sqrt{3}} \\right\\rfloor + 1 \\right)"
] | [
"What is the maximum number of regular hexagons of side 1 unit that can be cut from a circular disk of radius \\( r \\) units?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-46354030be42128c3ee651b26f118cddb4cd25b512a7cefd4689cf36d1795ba9 | What is the maximum number of regular m-sided polygons that can be present on an n-hedron? | [
"\\left\\lfloor \\dfrac{3n - 12}{m - 3} \\right\\rfloor"
] | [
"What is the maximum number of regular m-sided polygons that can be present on an n-hedron?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-5c6df7b0ff3018e0b011ec21a0a918258aaf7ec2e242091a691431144e7f56ed | What is the maximum number of regular pentagons that can be present on a 97-hedron? | [
"96"
] | [
"What is the maximum number of regular pentagons that can be present on a 97-hedron?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-e5cee1942c2e2a1bdc594cac5a363c871f55236339cd9e887a9a9c410d6c0729 | What is the maximum number of right-angle triangles with integer sides that can have the same area? | [
"\\( +\\infty \\)"
] | [
"What is the maximum number of right-angle triangles with integer sides that can have the same area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bc42ad50e819efdd52a662db181ff6470a533a575df8b5951371efce851626e | What is the maximum number of rooks that can be placed on a $10 \times 10$ chessboard such that among any 5 rooks, there are at least 2 rooks that attack each other? | [
"40"
] | [
"What is the maximum number of rooks that can be placed on a $10 \\times 10$ chessboard such that among any 5 rooks, there are at least 2 rooks that attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-028eb5abe48f28fe1edbbe737c0a67b537a3f12151139f66200eb29ac36d8eb2 | What is the maximum number of rooks that can be placed on a $5 \times 5 \times 5$ 3D chessboard such that none of them attack each other? | [
"25"
] | [
"What is the maximum number of rooks that can be placed on a $5 \\times 5 \\times 5$ 3D chessboard such that none of them attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-077c826ebdc23a4013a19ff4c4af0173beed26cab00acafbacf45128beeb8efe | What is the maximum number of rooks that can be placed on a $6 \times 6$ chessboard such that each rook is attacking at most one other rook? | [
"6"
] | [
"What is the maximum number of rooks that can be placed on a $6 \\times 6$ chessboard such that each rook is attacking at most one other rook?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcf9eb0d87c15697dfda4484685230499ade5120f51e6c9e1263558227cb62d4 | What is the maximum number of rooks that can be placed on a 5 × 5 × 5 3D chessboard such that none of them attack each other? | [
"25"
] | [
"What is the maximum number of rooks that can be placed on a 5 × 5 × 5 3D chessboard such that none of them attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-d8706d8c8193544e28b80c8e26e29cfffd220365a9c6f70e148ad23698bb475b | What is the maximum number of rooks that can be placed on a chessboard so that each rook attacks exactly two other rooks? | [
"16"
] | [
"What is the maximum number of rooks that can be placed on a chessboard so that each rook attacks exactly two other rooks?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-452dc7a67734964b948ccca13a5e3427c65a21069ece86b42e0f931391eea885 | What is the maximum number of rooks that can be placed on a chessboard such that each rook endangers at most 2 other rooks? | [
"16"
] | [
"What is the maximum number of rooks that can be placed on a chessboard such that each rook endangers at most 2 other rooks?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-80511f02a7585ec11ba1ed4d6a775d9754e48db9a2f514511195d50d26b39645 | What is the maximum number of rooks that can be placed on an $8 \times 8$ chessboard such that each rook, except the first one, attacks an odd number of rooks already on the board? A rook attacks to the left, to the right, above, and below, and only the first rook in each direction. | [
"32"
] | [
"What is the maximum number of rooks that can be placed on an $8 \\times 8$ chessboard such that each rook, except the first one, attacks an odd number of rooks already on the board? A rook attacks to the left, to the right, above, and below, and only the first rook in each direction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6c247824b41633928a43d7be2daa527ee363e7fe63afc31b6e5143de3e4f843e | What is the maximum number of rooks that can be placed on an $n \times n$ chessboard such that every two rooks threaten at least one empty square? | [
"2n - 2"
] | [
"What is the maximum number of rooks that can be placed on an $n \\times n$ chessboard such that every two rooks threaten at least one empty square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e164cfcdb30f06e28a0b74a327fdc53b2548192681ef36f3f1d0181f625cc4c2 | What is the maximum number of rooks that can be placed on an \( n \times n \) chessboard such that each rook is attacking exactly one other rook? | [
"\\( \\frac{4n}{3} \\) if \\( n \\equiv 0 \\pmod{3} \\), \\( 4\\lfloor \\frac{n}{3} \\rfloor + 1 \\) if \\( n \\equiv 1 \\pmod{3} \\), and \\( 4\\lfloor \\frac{n}{3} \\rfloor + 3 \\) if \\( n \\equiv 2 \\pmod{3} \\)."
] | [
"What is the maximum number of rooks that can be placed on an \\( n \\times n \\) chessboard such that each rook is attacking exactly one other rook?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-8ebdbae8efd44d85e8d49c4685dead734c6ef0e4e5c4fa404d895e989ec168fc | What is the maximum number of roots that the polynomial $(ax^3+bx+c)(bx^3+cx+a)(cx^3+ax+b)$ can have, given that $a$, $b$, and $c$ are distinct non-zero real numbers? | [
"7"
] | [
"What is the maximum number of roots that the polynomial $(ax^3+bx+c)(bx^3+cx+a)(cx^3+ax+b)$ can have, given that $a$, $b$, and $c$ are distinct non-zero real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-12cc7bc4d95008aa6928c3aff8fdf29aa8897cc2976ba149e64abb46535f6c3d | What is the maximum number of shuffles required for 13 distinct cards to return to their original positions if the magician shuffles them in the same manner each time? | [
"60"
] | [
"What is the maximum number of shuffles required for 13 distinct cards to return to their original positions if the magician shuffles them in the same manner each time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-d168e5d49f977a44ff9b9bb2347267a364c4ca9568eb4eacc17f3d302119f529 | What is the maximum number of sides of a convex polygon if exactly three of its interior angles are obtuse? | [
"6"
] | [
"What is the maximum number of sides of a convex polygon if exactly three of its interior angles are obtuse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-47420718da0f020cbd8cb053c2ab2f432a2c28cf0724f9c76a44e6af377ee13c | What is the maximum number of sides that Alice's convex polygons can have, given that they can be divided into right triangles with angles measuring $30$ degrees and $60$ degrees? | [
"6"
] | [
"What is the maximum number of sides that Alice's convex polygons can have, given that they can be divided into right triangles with angles measuring $30$ degrees and $60$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-3e4fb14df3fe12d3d90d46e5fd44a4f9a23bbd319b6030bd794741bd785fc4aa | What is the maximum number of sides that Bart's convex polygons can have, given that they can be divided into right triangles with angles measuring $30$ degrees and $60$ degrees? | [
"6"
] | [
"What is the maximum number of sides that Bart's convex polygons can have, given that they can be divided into right triangles with angles measuring $30$ degrees and $60$ degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-d8f9af5b5db3d59fe0808b25a450ceb23033d208d24ac5c264c841b88f27de95 | What is the maximum number of solid regular tetrahedrons that can be placed against each other so that one of their edges coincides with a given line segment in space? | [
"5"
] | [
"What is the maximum number of solid regular tetrahedrons that can be placed against each other so that one of their edges coincides with a given line segment in space?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-76213df09f6070eff885e19af6ae09691bb878af836604e6fef442569df9aff2 | What is the maximum number of solid spheres with a radius of 1 that can fit inside a sphere with a radius of $n$, where $n \in \mathbb{N}$? | [
"n^3"
] | [
"What is the maximum number of solid spheres with a radius of 1 that can fit inside a sphere with a radius of $n$, where $n \\in \\mathbb{N}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-0ead0ffd9c7b629634ee20ab1078eafc9147792f091c44880baa5a80ae8a9ed3 | What is the maximum number of space diagonals that a polyhedron with 10 vertices can have? A space diagonal is a diagonal with endpoints as vertices of the polyhedron but lies completely inside the polyhedron. | [
"35"
] | [
"What is the maximum number of space diagonals that a polyhedron with 10 vertices can have? A space diagonal is a diagonal with endpoints as vertices of the polyhedron but lies completely inside the polyhedron."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70272e1d42f7c69d6d3989960e8a4a867930e1a3c461043e5322c56e3c597750 | What is the maximum number of spheres of radius $r$ that can be placed inside a sphere of radius $nr$, where $n$ is a natural number? | [
"\\( s(n) = \\begin{cases} \n28k^3 - 6k^2 + 2k & \\text{if } n = 4k \\\\\n28k^3 + 18k^2 + 6k + 1 & \\text{if } n = 4k + 1 \\\\\n28k^3 + 36k^2 + 16k + 2 & \\text{if } n = 4k + 2 \\\\\n28k^3 + 60k^2 + 44k + 11 & \\text{if } n = 4k + 3 \n\\end{cases} \\)"
] | [
"What is the maximum number of spheres of radius $r$ that can be placed inside a sphere of radius $nr$, where $n$ is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-330c6067fcbdd3c8d715c0e0998433270bc47b6643f76508cc3378f9c0e3d50b | What is the maximum number of spheres of radius \( r \) that can be tangent to one sphere of radius \( nr \), where \( n \) is a natural number? | [
"12n^2"
] | [
"What is the maximum number of spheres of radius \\( r \\) that can be tangent to one sphere of radius \\( nr \\), where \\( n \\) is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-46160f7ba339861190b220e68b32f74e81c303beebf8ed27f816535df7b7b087 | What is the maximum number of spheres with radius 1 that can fit into a sphere with radius 2? | [
"2"
] | [
"What is the maximum number of spheres with radius 1 that can fit into a sphere with radius 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-651f969b2a762dd8eb356215de7d56506e8d507037d02d0d81d568cf07efcbc0 | What is the maximum number of square values that a quadratic polynomial (which is not a square of a linear polynomial) might assume at consecutive integer values of its variable? | [
"3"
] | [
"What is the maximum number of square values that a quadratic polynomial (which is not a square of a linear polynomial) might assume at consecutive integer values of its variable?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 28
} | 28 | 0 | 0 |
omr-ws-6135b019af8cba1a9dec42195387cd14aa3d6745b99cac30420b5533020414b8 | What is the maximum number of squares in an $8 \times 8$ board that can be colored so that for each square in the board, at most one adjacent square is colored? | [
"18"
] | [
"What is the maximum number of squares in an $8 \\times 8$ board that can be colored so that for each square in the board, at most one adjacent square is colored?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d92ab434aa09b5ce1e389c9a51b4433ea236200b3c4370bfa2f7b96dc3c03cf5 | What is the maximum number of squares that can be colored on a $2018 \times 2018$ board such that no four colored squares form the vertices of a rectangle? | [
"90810"
] | [
"What is the maximum number of squares that can be colored on a $2018 \\times 2018$ board such that no four colored squares form the vertices of a rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-87bb949be71900a339fb6f05b3ad9c3c2341ac602dc75a74fb3018ce536095a3 | What is the maximum number of squares that can be covered on a $15 \times 36$-checkerboard using square tiles of side lengths $7$ or $5$, where the tiles must cover whole squares of the board and be non-overlapping? | [
"666"
] | [
"What is the maximum number of squares that can be covered on a $15 \\times 36$-checkerboard using square tiles of side lengths $7$ or $5$, where the tiles must cover whole squares of the board and be non-overlapping?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02d2dac65ef5476cb4a8e08e7bf40d54870812af86f40d29c23b1da3bfbf46f5 | What is the maximum number of squares that can be marked on an $n \times n$ chessboard such that for any three columns, at most three marked squares belong to the same row? | [
"3n"
] | [
"What is the maximum number of squares that can be marked on an $n \\times n$ chessboard such that for any three columns, at most three marked squares belong to the same row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-dd05fdfb219318f2de5b51a9d62467e6716ac64036d3bee004c243933c9386ac | What is the maximum number of squares, each with a side length of $5.5$ cm, that can fit into a larger rectangle with dimensions $10$ cm by $15.5$ cm? | [
"2"
] | [
"What is the maximum number of squares, each with a side length of $5.5$ cm, that can fit into a larger rectangle with dimensions $10$ cm by $15.5$ cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c23e6e5df90f464d34e32d88cc115b07408e225d645b28ec4ea1bd0bcb8356dd | What is the maximum number of squares, each with a side length of $5.5$ cm, that can fit into a larger square with a side length of $10$ cm? | [
"1"
] | [
"What is the maximum number of squares, each with a side length of $5.5$ cm, that can fit into a larger square with a side length of $10$ cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f531544fc0e932a4ebb3e29d9e5c1406f4b06ac6376fa8c9fbbac3cf3273a31 | What is the maximum number of students who play at least one game, given that there are 200 students in total, with 100 playing cricket, 50 playing hockey, and 60 playing basketball, and the overlaps are 30 students playing both cricket and hockey, 35 students playing both hockey and basketball, and 45 students playing... | [
"130"
] | [
"What is the maximum number of students who play at least one game, given that there are 200 students in total, with 100 playing cricket, 50 playing hockey, and 60 playing basketball, and the overlaps are 30 students playing both cricket and hockey, 35 students playing both hockey and basketball, and 45 students pl... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-094cc023c8c57ef792e676805ad440fa1ebe4e637514ae256f30cfa0ffb26322 | What is the maximum number of subsets from the set $\{1, 2, \ldots, n\}$ such that if for two of them like $A$ and $B$ we have $A \subset B$, then $|B - A| \geq 3$? | [
"\\left\\lceil \\dfrac{2^n}{3} \\right\\rceil"
] | [
"What is the maximum number of subsets from the set $\\{1, 2, \\ldots, n\\}$ such that if for two of them like $A$ and $B$ we have $A \\subset B$, then $|B - A| \\geq 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-d0a871d631e5b309d4a3828291a6d27e34b661ed29dd2e49db62eaecbb04dba3 | What is the maximum number of subsets of \( S = \{1, 2, \ldots, 2n\} \) such that no subset is contained in another and no two subsets cover the whole set \( S \)? | [
"\\(\\binom{2n}{n-1}\\)"
] | [
"What is the maximum number of subsets of \\( S = \\{1, 2, \\ldots, 2n\\} \\) such that no subset is contained in another and no two subsets cover the whole set \\( S \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 2
} | 32 | 0 | 0 |
omr-ws-01b54c7e8c043560a93eb913e9b083124b31f583fd87b4ad65e580b96073a88c | What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$, such that any $2$ of them share exactly $1$ element? | [
"16"
] | [
"What is the maximum number of subsets of size $5$, taken from the set $A=\\{1,2,3,...,20\\}$, such that any $2$ of them share exactly $1$ element?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-27aa0f9f70a80b99ead52f03b2c5ab953a6155e1527ddc660e90d5693872054d | What is the maximum number of subsets that can be chosen from a 20-member set such that each subset has at least one member and any two subsets have at most two common members? | [
"1350"
] | [
"What is the maximum number of subsets that can be chosen from a 20-member set such that each subset has at least one member and any two subsets have at most two common members?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-fe8d225dc69018408f9f480c4df7796e707d56617335a2240801c268dead631d | What is the maximum number of subsets that can be chosen from a set of 20 members so that every subset has at least one member and every pair of subsets has at most two members in common? | [
"1350"
] | [
"What is the maximum number of subsets that can be chosen from a set of 20 members so that every subset has at least one member and every pair of subsets has at most two members in common?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48cec1870d3ab98af8fab92e6e7017eb58ab89d734d1549aace8c5b44755c0e1 | What is the maximum number of subsets that can be chosen from a set of 20 members such that every subset has at least one member and every pair of subsets has at most two members in common? | [
"1350"
] | [
"What is the maximum number of subsets that can be chosen from a set of 20 members such that every subset has at least one member and every pair of subsets has at most two members in common?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2621132a54e7d301019cb7c5373adc442711d42dbe38020d2ce1228b28339b27 | What is the maximum number of subsets, having the property that none of them is a subset of another, can a set with 10 elements have? | [
"252"
] | [
"What is the maximum number of subsets, having the property that none of them is a subset of another, can a set with 10 elements have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-d968a349e78c062573cb525486ba9739f9a205969e6ee685beafa14ce3507eef | What is the maximum number of symmetries that an $n$-sided concave polygon can have? | [
"\\(\\frac{n}{2}\\) if \\(n\\) is even, and \\(1\\) if \\(n\\) is odd."
] | [
"What is the maximum number of symmetries that an $n$-sided concave polygon can have?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-180971071e0ecaa0f5f523169d2df45b776350092885c4f1fdd2c90c3e29d134 | What is the maximum number of symmetries that an 11-sided concave polygon can have? (A symmetry of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.) | [
"22"
] | [
"What is the maximum number of symmetries that an 11-sided concave polygon can have? (A symmetry of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-40e10e64ff7effb29059f6e1e4eacd653e019e2647f696f4c19462b80497469b | What is the maximum number of terms in a geometric progression (GP) that can be obtained from the set $\{100, 101, 102, \ldots, 1000\}$? | [
"6"
] | [
"What is the maximum number of terms in a geometric progression (GP) that can be obtained from the set $\\{100, 101, 102, \\ldots, 1000\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0adfdad4423f84c81b0788aa0b1fc3138e221b0cefdd10b2bbfb8e0071aa700 | What is the maximum number of terms in a geometric progression with a common ratio greater than 1, where all terms are integers between 100 and 1000 inclusive? | [
"6"
] | [
"What is the maximum number of terms in a geometric progression with a common ratio greater than 1, where all terms are integers between 100 and 1000 inclusive?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-da860148935e2332af48a327cc7c8a897d4444f436d7098fd7e6fbbf24f88428 | What is the maximum number of terms in the sequence \( P(1), P(2), P(3), \dots \) that can be equal to the sum of the two preceding terms, where \( P(x) \) is a quadratic trinomial? | [
"2"
] | [
"What is the maximum number of terms in the sequence \\( P(1), P(2), P(3), \\dots \\) that can be equal to the sum of the two preceding terms, where \\( P(x) \\) is a quadratic trinomial?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e095d1fd115c1d8543fb562d576574912b353f7d9fdbc56b3247028cfe20045c | What is the maximum number of three-element subsets of \(\{1,2,3,4,5,6,7,8\}\) such that any two subsets have at most one element in common? | [
"8"
] | [
"What is the maximum number of three-element subsets of \\(\\{1,2,3,4,5,6,7,8\\}\\) such that any two subsets have at most one element in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de36c379dd2c2212939746121566b4b55ddd0b6578e82c098071b4cb7408505f | What is the maximum number of three-element subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that any two subsets have at most one element in common? | [
"8"
] | [
"What is the maximum number of three-element subsets of the set $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ such that any two subsets have at most one element in common?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f04387874bb8106ca7da9072d49010934fdf5c4f68564f9f60daf47eaedcf9bf | What is the maximum number of three-element subsets of the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) such that any two subsets have at most one element in common? | [
"8"
] | [
"What is the maximum number of three-element subsets of the set \\(\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\) such that any two subsets have at most one element in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 29
} | 29 | 0 | 0 |
omr-ws-12c6686edb5efd80073b34894e7b64ebc0dbe42db0ee934f25fa2bddf3533022 | What is the maximum number of triangles formed by the $\frac{n(n-3)}{2}$ diagonals in a convex $n$-gon, given that no three diagonals intersect at a common point? | [
"\\dfrac{n(n-1)(n-2)(n-3)}{24}"
] | [
"What is the maximum number of triangles formed by the $\\frac{n(n-3)}{2}$ diagonals in a convex $n$-gon, given that no three diagonals intersect at a common point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-4e41a63bdbc29d9fb6afdb332ea273a6cb40f2d2aae9719a5e65dcbd07ff9e0a | What is the maximum number of triangles that can be formed with vertices at twelve points placed on the sides of a triangle? | [
"208"
] | [
"What is the maximum number of triangles that can be formed with vertices at twelve points placed on the sides of a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-06801c3c3fd4634ae0d55d15ab173a0906c909235ead9b7bd9aed8b3a216f450 | What is the maximum number of turns you can make when you move through every single cube exactly once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at. | [
"2[(a-1)(b-1) + (b-1)(c-1) + (c-1)(a-1)]"
] | [
"What is the maximum number of turns you can make when you move through every single cube exactly once in an $a$ by $b$ by $c$ cuboid? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-5d7d5c77770afe4e060928a7872e2dee8aebb7c60134aa3a4b2e533a7c396685 | What is the maximum number of turns you can make when you move through every single cube exactly once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at. | [
"3n(n - 1)"
] | [
"What is the maximum number of turns you can make when you move through every single cube exactly once in an $n$ by $n$ by $n$ cube? You can choose which 1 by 1 by 1 cube to start at."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-7a74fac842181541b1415f84cb384e87c9b59f7600a4ae1598a5829dbfab98e9 | What is the maximum number of two-dimensional intersecting flat planes possible that can be formed by interconnecting 14 points in a three-dimensional space? | [
"5"
] | [
"What is the maximum number of two-dimensional intersecting flat planes possible that can be formed by interconnecting 14 points in a three-dimensional space?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e13345c17f9c23737a28a77a86d850b89d58570eccf461d3e8eaf31a239a24a | What is the maximum number of undefeated teams at the end of the season in a football league with 10 teams, where each team plays every other team exactly once? | [
"1"
] | [
"What is the maximum number of undefeated teams at the end of the season in a football league with 10 teams, where each team plays every other team exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-03298173953d1948107b14d6439f94fd4d779abe81c8b00dab0cafcd1236283e | What is the maximum number of unit cubes that can be seen from one viewpoint in a $13 \times 13 \times 13$ cube? | [
"469"
] | [
"What is the maximum number of unit cubes that can be seen from one viewpoint in a $13 \\times 13 \\times 13$ cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-35c368c7d232aa9d815d70aa7842cd93b0063b9319119757f06c1833a9652891 | What is the maximum number of unit distances in the plane for $n$ points? Denote this function as $m(n)$. Find constructive upper and lower bounds for $m(n)$. | [
"O\\left(n^{4/3}\\right)"
] | [
"What is the maximum number of unit distances in the plane for $n$ points? Denote this function as $m(n)$. Find constructive upper and lower bounds for $m(n)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3829204f741c26d478c8a8c4a8761014ea523b479e5e35896173b663e456add9 | What is the maximum number of values we can take from the list $1!, 2!, 3!, 4!, \ldots, 28!$ such that their product is a perfect square? | [
"27"
] | [
"What is the maximum number of values we can take from the list $1!, 2!, 3!, 4!, \\ldots, 28!$ such that their product is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-50b59122f767ca22b8c6776c5c423a0410319df697b49a5f5a5fe8dc17bf8ea9 | What is the maximum number of vectors that can be released from a given point in the Euclidean space $\mathbb{R}^n$ such that all the pairwise angles between them are obtuse? | [
"\\( n+1 \\)"
] | [
"What is the maximum number of vectors that can be released from a given point in the Euclidean space $\\mathbb{R}^n$ such that all the pairwise angles between them are obtuse?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8bb6ef38c8ac5bb8bbd71525c466f4e037d8923f3eda733dfa80f78bff531ecd | What is the maximum number of vertices Peter can paint on a $2 \times n$ board so that every square has exactly one painted vertex? | [
"\\(\\left\\lfloor \\frac{2n}{3} \\right\\rfloor\\)"
] | [
"What is the maximum number of vertices Peter can paint on a $2 \\times n$ board so that every square has exactly one painted vertex?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db004104a00377b7d038884375de5893eefa9247bb0acd17a3c3613a2148e173 | What is the maximum number of vertices of a regular $120$-gon that can be marked so that no three marked vertices form an isosceles triangle with angles $18^\circ, 81^\circ, 81^\circ$? | [
"60"
] | [
"What is the maximum number of vertices of a regular $120$-gon that can be marked so that no three marked vertices form an isosceles triangle with angles $18^\\circ, 81^\\circ, 81^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-402451059e0114b737b3bc86dc4b0cf4bf11a48166fba9b4a0d08e282fb9c72d | What is the maximum number of white squares that can be surrounded by $n$ black squares? | [
"\\left\\lfloor \\dfrac{n^2}{16} \\right\\rfloor"
] | [
"What is the maximum number of white squares that can be surrounded by $n$ black squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-174d15d4f84d66f6bdd62c6f9e9ebdcb2e3f5ade00eb142d1878194c00743f8f | What is the maximum number of white squares that can be surrounded by \( n \) black squares? | [
"\\left\\lfloor \\frac{\\left\\lfloor \\frac{n}{2} \\right\\rfloor^2}{4} \\right\\rfloor"
] | [
"What is the maximum number of white squares that can be surrounded by \\( n \\) black squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-c235a48bd23b9d4fd27fc44a5de7580741424b0e3ee2d4cd5ca24b34fb5a1bb2 | What is the maximum number of zeroes that the quantity $1^n + 2^n + 3^n + 4^n$ can end in, where $n$ is a natural number? | [
"2"
] | [
"What is the maximum number of zeroes that the quantity $1^n + 2^n + 3^n + 4^n$ can end in, where $n$ is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b18d8212f081e50f321c8ec063cccc4cc355e9424439b462e53b2f48015c1fc7 | What is the maximum number such that all of its digits are different and when you add any three adjacent digits of this number, the digits of the result are not found in the number? | [
"9873650"
] | [
"What is the maximum number such that all of its digits are different and when you add any three adjacent digits of this number, the digits of the result are not found in the number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d463ad7be9640e23252d47470de01673c27cce8f88053a5fdd7af019404bc522 | What is the maximum parabolic area in terms of \( R \), \( H \), and \( e \) (the distance of the cutting plane from the circular base)? | [
"\\dfrac{4H(R^2 - e^2)^{3/2}}{3R^2}"
] | [
"What is the maximum parabolic area in terms of \\( R \\), \\( H \\), and \\( e \\) (the distance of the cutting plane from the circular base)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-c8d02416e38f30dfc00069bc36fb202c5729b11e63c9d9829cf253a69fbb5967 | What is the maximum perimeter of a rectangle inscribed in a semicircle of radius 1? | [
"\\(2\\sqrt{5}\\)"
] | [
"What is the maximum perimeter of a rectangle inscribed in a semicircle of radius 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d761828b6282847081ac25ba488070796b74e6f47ce9ce5e4ab1c4f8d8a73dda | What is the maximum perimeter of a triangle with an area of $\frac{1}{2}$ square unit, where each of its three vertices coincide with a different dot in a square lattice where neighboring points horizontally and vertically are one unit apart? | [
"\\( \\sqrt{41} + \\sqrt{2} + 5 \\)"
] | [
"What is the maximum perimeter of a triangle with an area of $\\frac{1}{2}$ square unit, where each of its three vertices coincide with a different dot in a square lattice where neighboring points horizontally and vertically are one unit apart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-2378060ae2451582694b91089a9dce4a01ec344eac1556bd6bc01e95502b09b0 | What is the maximum perimeter of all rectangles that can be inscribed in the ellipse given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$? | [
"\\( 4 \\sqrt{a^2 + b^2} \\)"
] | [
"What is the maximum perimeter of all rectangles that can be inscribed in the ellipse given by $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-d8e72029de77ab386818801ec324f6b12c373be243afaae2ce0b884cf5863000 | What is the maximum portion of any circle that can lie on any parabola? Consider a parabola as a "cup" and fit a circle inside it such that the bottom of the circle matches the vertex of the parabola. Determine if there is a fixed value for this maximum portion. | [
"\\dfrac{1}{2}"
] | [
"What is the maximum portion of any circle that can lie on any parabola? Consider a parabola as a \"cup\" and fit a circle inside it such that the bottom of the circle matches the vertex of the parabola. Determine if there is a fixed value for this maximum portion."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-4c93dcd9b0719a10fac3a37fa300a50046e60589d83a00875a9e27b1b2dfbea3 | What is the maximum positive integer $n$ such that we can fill an $8 \times n$ grid with the letters $A, B, C, D$ so that for every pair of rows, there exists at most one column where the two intersection squares contain the same letter? | [
"7"
] | [
"What is the maximum positive integer $n$ such that we can fill an $8 \\times n$ grid with the letters $A, B, C, D$ so that for every pair of rows, there exists at most one column where the two intersection squares contain the same letter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c876ab05c6da1430af5c988fefe4c8a81789540ab76fce51f18e869eb15c44ff | What is the maximum positive integer \( n \) such that for positive integers \( A \), \( B \), and \( C \) with \( A + B = C \), the inequality \( C > 10^n \cdot \text{rad}(A \cdot B \cdot C) \) holds? | [
"2"
] | [
"What is the maximum positive integer \\( n \\) such that for positive integers \\( A \\), \\( B \\), and \\( C \\) with \\( A + B = C \\), the inequality \\( C > 10^n \\cdot \\text{rad}(A \\cdot B \\cdot C) \\) holds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-74c5e1cb6f43010634a9a0c23bce03470d03d03c34f4a016188c19884113a741 | What is the maximum possible area for an isosceles triangle with two sides equal to \(a\)? | [
"\\(\\frac{a^2}{2}\\)"
] | [
"What is the maximum possible area for an isosceles triangle with two sides equal to \\(a\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-652364bc5e5eee29667a5d868d1dfaa297905f8685b8c5ea3dd54109a4871ee1 | What is the maximum possible area of a semicircle inscribed in a square of side length 1? Express your answer in the form $(a - b\sqrt{c}) \pi$, where $a, b, c$ are positive integers and $c$ is square-free. | [
"(3 - 2\\sqrt{2}) \\pi"
] | [
"What is the maximum possible area of a semicircle inscribed in a square of side length 1? Express your answer in the form $(a - b\\sqrt{c}) \\pi$, where $a, b, c$ are positive integers and $c$ is square-free."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1a1f34c0d383845b5ab48f999c0f0ce07444e5f442cf823f6e2b7d6a9a730828 | What is the maximum possible area of a set of points in a plane where the maximum distance between any two points is less than or equal to 1? | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"What is the maximum possible area of a set of points in a plane where the maximum distance between any two points is less than or equal to 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 31
} | 48 | 0 | 0 |
omr-ws-9fefddc9a346e8b1417ae7d9028f582e52448172a136d538cb671798370c6c1d | What is the maximum possible area of a triangle with integer side lengths and a perimeter of 7? | [
"\\(\\frac{3\\sqrt{7}}{4}\\)"
] | [
"What is the maximum possible area of a triangle with integer side lengths and a perimeter of 7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1dd70eb1404a1aa657142a5dc5c9869c0f67fd1a6d86560b020a7f2ba36b6c7e | What is the maximum possible area of an equilateral triangle that can fit inside a 10x11 rectangle? | [
"\\(221\\sqrt{3}-330\\)"
] | [
"What is the maximum possible area of an equilateral triangle that can fit inside a 10x11 rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-bbd6f5ee10af1c811d785345b38049a1150e9a30985e111704887942448c0580 | What is the maximum possible area of an octagon inscribed in a circle of radius $20$? | [
"\\( 800\\sqrt{2} \\)"
] | [
"What is the maximum possible area of an octagon inscribed in a circle of radius $20$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e4512226161d3a65fda6252a87812111b88afbc852fe374e223d1672cc73f47e | What is the maximum possible area that can be enclosed using a completely flexible piece of wood of length $10$ as one side and a straight river as another side? | [
"\\(\\frac{50}{\\pi}\\)"
] | [
"What is the maximum possible area that can be enclosed using a completely flexible piece of wood of length $10$ as one side and a straight river as another side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
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