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-53aaa900390ac42c565946f56538c5dff22b5439f3b83654aeab7a261f77c304 | What is the maximum local point of the function \( g(x) = x(10-2x)^2 \) that does not stretch to infinity? | [
"\\( \\left( \\frac{5}{3}, \\frac{2000}{27} \\right) \\)"
] | [
"What is the maximum local point of the function \\( g(x) = x(10-2x)^2 \\) that does not stretch to infinity?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3414e7f6a3de2288434ff14d97169b212a8f34e1c387ed3fd1eac7209bd0d753 | What is the maximum multiplicity of 1 as a root for a polynomial of the form $a_0 + a_1 x + \ldots + a_n x^n$ where $|a_n| = 1$ and $|a_i| \leq 1$ for all $i$, and how does it relate to $5\sqrt{n}$? | [
"\\(5\\sqrt{n}\\)"
] | [
"What is the maximum multiplicity of 1 as a root for a polynomial of the form $a_0 + a_1 x + \\ldots + a_n x^n$ where $|a_n| = 1$ and $|a_i| \\leq 1$ for all $i$, and how does it relate to $5\\sqrt{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2e97c146c7fb4646192331fe46315e6e5ab7fed2713f5039b2bd5e0081483b74 | What is the maximum natural number $n$ such that for an $n$-digit number $\overline{a_na_{n-1}...a_2a_1}$, the numbers $\overline{a_1}$, $\overline{a_2a_1}$, ..., $\overline{a_na_{n-1}...a_2a_1}$ are all perfect squares? | [
"2"
] | [
"What is the maximum natural number $n$ such that for an $n$-digit number $\\overline{a_na_{n-1}...a_2a_1}$, the numbers $\\overline{a_1}$, $\\overline{a_2a_1}$, ..., $\\overline{a_na_{n-1}...a_2a_1}$ are all perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 6
} | 27 | 0 | 0 |
omr-ws-26a7d14749c6d6772ac91630516cd2be33155e52bdfb519f7975e3ef52e519fd | What is the maximum natural number \( n \) such that \( \sin(2^n) \) is maximized, where the sine function is evaluated in degrees? | [
"6"
] | [
"What is the maximum natural number \\( n \\) such that \\( \\sin(2^n) \\) is maximized, where the sine function is evaluated in degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c439cab5c83e0d8070c36736021e8cc1ce4b556c24dadea38d45729577d16415 | What is the maximum number $N$ of rooks that can be placed on a chessboard such that no two rooks attack each other and no two of the pairwise distances between the rooks are equal? (The distance between two rooks is the distance between the centers of their cells.) | [
"6"
] | [
"What is the maximum number $N$ of rooks that can be placed on a chessboard such that no two rooks attack each other and no two of the pairwise distances between the rooks are equal? (The distance between two rooks is the distance between the centers of their cells.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0b359627f9701b1014cbb770225d5ff98a949697033045949722010270cc9d02 | What is the maximum number of $1 \times 3$ rectangular tiles that can be cut from a $10 \times 10$ squared rectangle? (Cuts are made along cell lines.) | [
"33"
] | [
"What is the maximum number of $1 \\times 3$ rectangular tiles that can be cut from a $10 \\times 10$ squared rectangle? (Cuts are made along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e7fce1ef441bbf88583ee8dbdfca4631cef90981d12f46667248af635ef58c4b | What is the maximum number of $1 \times 3$ rectangular tiles that can be cut from an $m \times n$ rectangle? (The answer will depend on $m$ and $n$.) | [
"\\left\\lfloor \\dfrac{mn - (m \\bmod 3)(n \\bmod 3)}{3} \\right\\rfloor"
] | [
"What is the maximum number of $1 \\times 3$ rectangular tiles that can be cut from an $m \\times n$ rectangle? (The answer will depend on $m$ and $n$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-2610107ffdf5c45f9e979be6fd281079d3608f4c179ad8deda6a8e66f38851eb | What is the maximum number of $1 \times 4$ pieces that can be placed on a $10 \times 10$ tray such that no two pieces touch? (Consider pieces to be touching if they share at least one vertex.) | [
"12"
] | [
"What is the maximum number of $1 \\times 4$ pieces that can be placed on a $10 \\times 10$ tray such that no two pieces touch? (Consider pieces to be touching if they share at least one vertex.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-023fd414a0a950e3ddb4256e1f115c388b9ccab5ec41c6c63ce4f188be16e3d7 | What is the maximum number of $1 \times 4$ rectangular tiles that can be cut from a $10 \times 10$ squared rectangle? (Cuts are made along cell lines.) | [
"20"
] | [
"What is the maximum number of $1 \\times 4$ rectangular tiles that can be cut from a $10 \\times 10$ squared rectangle? (Cuts are made along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-865fd644090cae92aafbeaa32f5d479edb49347820b1cd14a6f89a08502ea6d6 | What is the maximum number of $1 \times 4$ rectangular tiles that can be cut from an $m \times n$ rectangle? (The answer will depend on $m$ and $n$.) | [
"\\left\\lfloor \\dfrac{mn}{4} \\right\\rfloor"
] | [
"What is the maximum number of $1 \\times 4$ rectangular tiles that can be cut from an $m \\times n$ rectangle? (The answer will depend on $m$ and $n$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-219f09a6737c1cf578c2de558248a237aed2a77fa089a621330fed89f363a1be | What is the maximum number of $1 \times 5$ rectangular tiles that can be cut from a $10 \times 10$ squared rectangle? (Cuts are made along cell lines.) | [
"20"
] | [
"What is the maximum number of $1 \\times 5$ rectangular tiles that can be cut from a $10 \\times 10$ squared rectangle? (Cuts are made along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9ab1aea22e4f859d7ffcab4327f73ac2f8a8c9719129ab94a8bb9d1094c13d0 | What is the maximum number of $1 \times 5$ rectangular tiles that can be cut from an $m \times n$ rectangle? (The answer will depend on $m$ and $n$.) | [
"\\left\\lfloor \\frac{mn}{5} \\right\\rfloor - \\left\\lfloor \\frac{(m \\bmod 5)(n \\bmod 5)}{5} \\right\\rfloor"
] | [
"What is the maximum number of $1 \\times 5$ rectangular tiles that can be cut from an $m \\times n$ rectangle? (The answer will depend on $m$ and $n$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-ac82ad6c6297c2b38b7d504f5ed42ff2867a093667f3ed49c453d0457f34def0 | What is the maximum number of $1 \times 6$ rectangular tiles that can be cut from a $10 \times 10$ squared rectangle? (Cuts are made along cell lines.) | [
"16"
] | [
"What is the maximum number of $1 \\times 6$ rectangular tiles that can be cut from a $10 \\times 10$ squared rectangle? (Cuts are made along cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5da7bf74eedfb4710036e3c84037cd9cda2acfea322efb5457f49b9d3b00d34b | What is the maximum number of $1 \times 6$ rectangular tiles that can be cut from an $m \times n$ rectangle? (The answer will depend on $m$ and $n$.) | [
"\n\\begin{cases}\n\\left\\lfloor \\dfrac{mn}{6} \\right\\rfloor & \\text{if } m \\geq 6 \\text{ or } n \\geq 6, \\\\\n0 & \\text{otherwise}.\n\\end{cases}\n"
] | [
"What is the maximum number of $1 \\times 6$ rectangular tiles that can be cut from an $m \\times n$ rectangle? (The answer will depend on $m$ and $n$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c06f7b4b4292397b700b3b48506f5cb5393ec28386b561904f0db1c1dd9e5334 | What is the maximum number of $2 \times 3 \times 4$ blocks you can fill in a $5 \times 6 \times 7$ box, assuming that fractions of blocks are not allowed? | [
"6"
] | [
"What is the maximum number of $2 \\times 3 \\times 4$ blocks you can fill in a $5 \\times 6 \\times 7$ box, assuming that fractions of blocks are not allowed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-403ac718c355d52d7ba8b206c904f51c1405d7dac8d7ea315cf084de210d7729 | What is the maximum number of $3 \times 5$ rectangles that can be cut from a $16 \times 16$ sheet of paper? | [
"16"
] | [
"What is the maximum number of $3 \\times 5$ rectangles that can be cut from a $16 \\times 16$ sheet of paper?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-655335957a33b4855c92b65eba42f13cc8fb5dfc046c482ced1042ac2c157fdf | What is the maximum number of $L$-trominoes that can be placed on a $5 \times 5$ board such that no two $L$-trominoes touch by their sides? | [
"5"
] | [
"What is the maximum number of $L$-trominoes that can be placed on a $5 \\times 5$ board such that no two $L$-trominoes touch by their sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 24,
"genselect": 4
} | 28 | 0 | 0 |
omr-ws-664c021c525515fc7ff286b2a56aab4bda03765ebe1300d737b9691683b313b5 | What is the maximum number of $T$-shaped polyominos that can be placed into a $6 \times 6$ grid without any overlaps? The blocks can be rotated. | [
"8"
] | [
"What is the maximum number of $T$-shaped polyominos that can be placed into a $6 \\times 6$ grid without any overlaps? The blocks can be rotated."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-4fbbc966b43463d13d6e2a17569c190fbfc0b6c9468af98dc1124d72136590a4 | What is the maximum number of $a$-element subsets one can take from a set of $b$ elements such that the intersection of any two subsets has exactly $c$ elements? (Given $c < a < b$) | [
"\\dfrac{b(a - c)}{a^2 - b c}"
] | [
"What is the maximum number of $a$-element subsets one can take from a set of $b$ elements such that the intersection of any two subsets has exactly $c$ elements? (Given $c < a < b$)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-ffcaf503105aba84459ceb3df4412bad5ef28ab2bc8ef1ab2e552bb980d99e54 | What is the maximum number of $n$-element sets such that any two sets share exactly one element, and there is no element that appears in every set? | [
"n^2 - n + 1"
] | [
"What is the maximum number of $n$-element sets such that any two sets share exactly one element, and there is no element that appears in every set?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-97497a31b94e889234d848834132b94c8b5f5a3d26bc92765945fd03c012fd19 | What is the maximum number of $r$-subsets of an $n$-set such that the intersection of any two of these subsets has at most $p$ elements, where $0 \leq p < r < n$? | [
"\\dfrac{\\dbinom{n}{p+1}}{\\dbinom{r}{p+1}}"
] | [
"What is the maximum number of $r$-subsets of an $n$-set such that the intersection of any two of these subsets has at most $p$ elements, where $0 \\leq p < r < n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-9b5869e66bae9170d546820befc9680a6fe99f80e41feb74f39eba8bd673083d | What is the maximum number of 1 cm radius balls that can fit inside a sphere with a 20 cm radius? | [
"5075"
] | [
"What is the maximum number of 1 cm radius balls that can fit inside a sphere with a 20 cm radius?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-e801b13f63ade3fb6e479788905c93859b7858ed4fccf820ecdfb731cf666526 | What is the maximum number of 1 × 1 boxes that can be colored black in an n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes? | [
"\\( \\left\\lfloor \\frac{n^2}{2} \\right\\rfloor \\)"
] | [
"What is the maximum number of 1 × 1 boxes that can be colored black in an n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-689b7d35e767b0412b839f8391eb59e2d3aa889cc14f71644ae7ea5c59988f0b | What is the maximum number of 1-loss teams at the end of the season in a football league with 10 teams, where each team plays every other team exactly once? | [
"3"
] | [
"What is the maximum number of 1-loss 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",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-77e3a4edfbd5c44e9775157470770d9dda351660f4c6153dbedc8cf57532b9d5 | What is the maximum number of 1x1 squares that can be colored in a 7x7 grid such that no colored square has at least two adjacent (side-adjacent) colored squares? | [
"26"
] | [
"What is the maximum number of 1x1 squares that can be colored in a 7x7 grid such that no colored square has at least two adjacent (side-adjacent) colored squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-596e426a3bb32f6b864c08160b6e267d67daf49a515bbfa8376e5b4cb76e2726 | What is the maximum number of 2-digit numbers that can be in a non-constant geometric progression (GP)? | [
"5"
] | [
"What is the maximum number of 2-digit numbers that can be in a non-constant geometric progression (GP)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-88bb426461d2c1a467146e9219042899d1c094f4aa482221d328a49c110934a1 | What is the maximum number of 4 x 6 sheets that can be cut from a 20 x 26 paper without rearranging the paper? | [
"21"
] | [
"What is the maximum number of 4 x 6 sheets that can be cut from a 20 x 26 paper without rearranging the paper?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-5bbe7d88d32386856d29828755b6cb145447e25d4efbf9012677ce09b6639d6f | What is the maximum number of 6-digit phone numbers that can be in place at one time such that any two phone numbers differ in at least two places? | [
"100000"
] | [
"What is the maximum number of 6-digit phone numbers that can be in place at one time such that any two phone numbers differ in at least two places?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bddbca9a8edad51234a41f8723619a4849f3b8254711fe0ff3b8752fb35538f | What is the maximum number of ATM cards a bank can issue if each card is associated with a unique 6-digit password, and no three consecutive digits in the password are the same? | [
"963090"
] | [
"What is the maximum number of ATM cards a bank can issue if each card is associated with a unique 6-digit password, and no three consecutive digits in the password are the same?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3ea1873f6a8af09782ff2dc1cff30d24e4f689ad95a4561fd5f9c3d06026f72e | What is the maximum number of S-tetrominoes that can be tiled in an \( n \times n \) grid? | [
"\\left\\lfloor \\dfrac{n^2}{4} \\right\\rfloor"
] | [
"What is the maximum number of S-tetrominoes that can be tiled in an \\( n \\times n \\) grid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-20661f43d2626ed004939b46b095c310170e9f868bd4a49b9fbc524491413afd | What is the maximum number of Y Hexomino that can be placed on a 13×13 chessboard, where each Hexomino does not overlap? | [
"28"
] | [
"What is the maximum number of Y Hexomino that can be placed on a 13×13 chessboard, where each Hexomino does not overlap?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ee197a0bdbc1e9c3e23271c1dff46ce80200634cd03fd912e12398787a5a66a | What is the maximum number of acute angles a concave hexagon can have? | [
"5"
] | [
"What is the maximum number of acute angles a concave hexagon can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-c3c9b0daa7b6a7360776b3d4ef611d57e0badbfe7a67ef9f0b4c489935b9371c | What is the maximum number of acute angles a two-dimensional non-overlapping figure with 2017 corners can have? | [
"1345",
"3"
] | [
"What is the maximum number of acute angles a two-dimensional non-overlapping figure with 2017 corners can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-cc13f7448e58c228dd03a3045172f6190ccce4432cf522d403cee36fed39f983 | What is the maximum number of acute angles an $n$-sided shape can have? | [
"\\(\\left\\lfloor \\frac{2n+3}{3}\\right\\rfloor\\)"
] | [
"What is the maximum number of acute angles an $n$-sided shape can have?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-98b0d101f670e800cbc20e0dc7e05df1d477bb3106fd963c3f02d0adf3a35a81 | What is the maximum number of acute angles that a convex n-gon can have? | [
"3"
] | [
"What is the maximum number of acute angles that a convex n-gon can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c445460e8134dcb333bc6614c08d9ba1f988158f50993ab74735966750b5010 | What is the maximum number of acute angles that any convex polygon can have? | [
"3"
] | [
"What is the maximum number of acute angles that any convex polygon can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee963c50227142a92d99313b4bf3915bf7dbf7e102b310365e89427bca096313 | What is the maximum number of acute angles that can occur in a convex $n$-gon? | [
"3"
] | [
"What is the maximum number of acute angles that can occur in a convex $n$-gon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-9b1ae0c6d3f838f4bc935429820e7217d8849c9716698a4584fcb33d244b69fe | What is the maximum number of acute interior angles a non-overlapping planar 2017-gon can have? | [
"1345"
] | [
"What is the maximum number of acute interior angles a non-overlapping planar 2017-gon can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 7,
"tir": 7
} | 32 | 0 | 0 |
omr-ws-e7d5388adade1c74f114d3133b503c0e80e576ceae0cdf7903cc48c70969094e | What is the maximum number of acute interior angles that can be on a 2001-gon? | [
"1334"
] | [
"What is the maximum number of acute interior angles that can be on a 2001-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-0b28e8c5c88da2c9a0ac5a8776506f50dd2f5b391fe0ba18edbb61677847f9b9 | What is the maximum number of acute triangles that can be formed using 2018 points inside a circle as vertices? | [
"\\(\\frac{\\binom{2018}{3}}{2}\\)"
] | [
"What is the maximum number of acute triangles that can be formed using 2018 points inside a circle as vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-3c39e066e4ea7f3510810b3b225d28f6b1d1bff3bd1c79fc1bb5a87e2cfd3382 | What is the maximum number of admissible lines determined by $n$ points in the plane, where a line is admissible if its slope is equal to $0, 1, -1$, or $\infty$? | [
"$\\frac{3n}{2}$"
] | [
"What is the maximum number of admissible lines determined by $n$ points in the plane, where a line is admissible if its slope is equal to $0, 1, -1$, or $\\infty$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c7e54120d8a8d1fdc14935acc8879a2b2fe3ecd4ab3087f6fa2e274a86e5cae | What is the maximum number of angles greater than $\pi$ that a $45$-gon can have? | [
"42"
] | [
"What is the maximum number of angles greater than $\\pi$ that a $45$-gon can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cdac666783adf7d99019c792cb5d0ee3355d045e4facfea59915c3e8b8926a9d | What is the maximum number of areas that $8$ different parabolas in the form of $y = ax^2 + bx + c$ and $8$ different parabolas in the form of $x = a_1y^2 + b_1y + c_1$ can divide the Cartesian plane into? | [
"385"
] | [
"What is the maximum number of areas that $8$ different parabolas in the form of $y = ax^2 + bx + c$ and $8$ different parabolas in the form of $x = a_1y^2 + b_1y + c_1$ can divide the Cartesian plane into?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-e75cbfaaa51aa5b9ee38862c80a1de10245e3ccc62fc0e4e306cba8b2b24ef42 | What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? | [
"6"
] | [
"What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-7de8c8c6c139cb3760703bba8693ba4976529584bdba212ae55a7b4920c63ed7 | What is the maximum number of balls of diameter $10$ that can be placed in a box of size $50 \times 50 \times 60$? | [
"286"
] | [
"What is the maximum number of balls of diameter $10$ that can be placed in a box of size $50 \\times 50 \\times 60$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b223a9841fcd529b5bc4f2417befca0975c219d73228b79e6317fddf44e01e52 | What is the maximum number of binary \(n\)-words that differ in at least 3 places? | [
"\\(\\frac{2^n}{n+1}\\)"
] | [
"What is the maximum number of binary \\(n\\)-words that differ in at least 3 places?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5c03a7104c3be39476932258d096ae0f39a8ef131a8cb5c61519b2b55222cf1 | What is the maximum number of bishops that can be placed on a $101 \times 101$ chessboard such that none of them attack each other? A bishop “attacks” diagonally in all four directions. | [
"51"
] | [
"What is the maximum number of bishops that can be placed on a $101 \\times 101$ chessboard such that none of them attack each other? A bishop “attacks” diagonally in all four directions."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79d9e9140557b38535c339bf4034f05005144c74fb7e524239e172daa91b3a1e | What is the maximum number of bishops that can be placed on a 9x9 chessboard so that no two bishops attack each other? (Bishops always move diagonally.) | [
"16"
] | [
"What is the maximum number of bishops that can be placed on a 9x9 chessboard so that no two bishops attack each other? (Bishops always move diagonally.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1f398edf8970dd8c97c561bcf6989d4ab461d2d382991d02beaaa3e647e783f | What is the maximum number of bishops that can be placed on an $8 \times 8$ chessboard such that at most three bishops lie on any diagonal? | [
"38"
] | [
"What is the maximum number of bishops that can be placed on an $8 \\times 8$ chessboard such that at most three bishops lie on any diagonal?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-4b90b29c6d1ad47d6723c7f4eea6be84c8c2e41b361244cfe7f33f9a896df121 | What is the maximum number of bishops that can be placed on an $m \times n$ chessboard such that each bishop attacks no more than 2 other bishops? (A bishop attacks along diagonals.) | [
"\\( 2(m + n) - 4 \\)"
] | [
"What is the maximum number of bishops that can be placed on an $m \\times n$ chessboard such that each bishop attacks no more than 2 other bishops? (A bishop attacks along diagonals.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 2,
"tir": 9
} | 50 | 0 | 0 |
omr-ws-198cf4112cd31df9f72cc26fd685ad8ae041452d9668e50dee7d2f80e731c94b | What is the maximum number of bishops that can be placed on an $n \times n$ chessboard such that no two bishops attack each other? | [
"\\( 2n - 2 \\)"
] | [
"What is the maximum number of bishops that can be placed on an $n \\times n$ chessboard such that no two bishops attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 3
} | 16 | 0 | 0 |
omr-ws-90e39080b3ea66449e05c268080b32015911564f17bd5f3e588cc6b02112df3b | What is the maximum number of bounded regions that 6 overlapping ellipses can cut the plane into? | [
"61"
] | [
"What is the maximum number of bounded regions that 6 overlapping ellipses can cut the plane into?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ddc6fbb8fc1b1e24b360ac05f9f1cb88ab6abf853a7d4294003b7df5e9bb216 | What is the maximum number of bounded volumes that may be formed by five interpenetrating spheres, considering only the surfaces of the spheres as bounds and counting only volumes that are not further subdivided? | [
"29"
] | [
"What is the maximum number of bounded volumes that may be formed by five interpenetrating spheres, considering only the surfaces of the spheres as bounds and counting only volumes that are not further subdivided?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9fa15dd9480d5fd1b06b2578e8cdad8625433477c54638e87672f77efb78e2f6 | What is the maximum number of boys playing all five games, given that each boy plays at least two games out of five, with the distribution as follows: Cricket 18, Football 40, Volleyball 42, Basketball 45, Hockey 55? | [
"18"
] | [
"What is the maximum number of boys playing all five games, given that each boy plays at least two games out of five, with the distribution as follows: Cricket 18, Football 40, Volleyball 42, Basketball 45, Hockey 55?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-cacccd406ece66fad1f8a9d656d545919aeecd854f93b93f1e19fb3d6f32bb83 | What is the maximum number of cells that a line can cut on an $m \times n$ chessboard, where $m, n \in \mathbb{N}$? | [
"\\( m + n - \\gcd(m, n) \\)"
] | [
"What is the maximum number of cells that a line can cut on an $m \\times n$ chessboard, where $m, n \\in \\mathbb{N}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 4,
"tir": 18
} | 65 | 0 | 0 |
omr-ws-9251358fe9d058ed97fd16cfc33df8939e529efa5189a4637da069bc25b2cb74 | What is the maximum number of cells that can be marked in a 20 * 20 board such that every cell is adjacent to at most one marked cell? | [
"80"
] | [
"What is the maximum number of cells that can be marked in a 20 * 20 board such that every cell is adjacent to at most one marked cell?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-3bfa737e2570de133452077dff55dc98aa2f79c677166bd25058a2f1fbacece4 | What is the maximum number of checkers it is possible to put on a chessboard such that in every row and in every column there is an even number of checkers? | [
"32"
] | [
"What is the maximum number of checkers it is possible to put on a chessboard such that in every row and in every column there is an even number of checkers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bdf49c2bf2e81cf9d168f3d9371240b76b3542f904d3e435e7b698f0c24f5b9 | What is the maximum number of checkers that can be placed on an $8 \times 8$ chessboard such that each row and each column contains twice as many white checkers as black ones, with at most one checker on each cell? | [
"48"
] | [
"What is the maximum number of checkers that can be placed on an $8 \\times 8$ chessboard such that each row and each column contains twice as many white checkers as black ones, with at most one checker on each cell?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fbcd4fb0f74113648e39b659d1c45345c27a63ea9a2235e3f09c101c0cf2172a | What is the maximum number of checkers that can be placed on an $n \times n$ chessboard such that each row and each column contains an even number of checkers? | [
"\\( 2n \\cdot \\left\\lfloor \\dfrac{n}{2} \\right\\rfloor \\) or \\( n^2 \\) if \\( n \\) is even, and \\( n^2 - n \\) if \\( n \\) is odd."
] | [
"What is the maximum number of checkers that can be placed on an $n \\times n$ chessboard such that each row and each column contains an even number of checkers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 34,
"tir": 21
} | 55 | 0 | 0 |
omr-ws-76c926be8510013a153e46ea41df0467ef1baa4c6e6d1d0cb34ea7a950f8a402 | What is the maximum number of chess knights that can be placed on a $2013 \times 2013$ board so that no two knights attack each other? | [
"2026085"
] | [
"What is the maximum number of chess knights that can be placed on a $2013 \\times 2013$ board so that no two knights attack each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f0bd0802505174bb85e46486d6e9d35720cddc7e0d6a33b8292df5878b12fe49 | What is the maximum number of circles of radius $1$ that can be placed around a circle of radius $\sqrt{5}$ such that each smaller circle is tangent to the larger circle and no overlap occurs between any of the smaller circles? | [
"10"
] | [
"What is the maximum number of circles of radius $1$ that can be placed around a circle of radius $\\sqrt{5}$ such that each smaller circle is tangent to the larger circle and no overlap occurs between any of the smaller circles?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-61b75bb5cafb3f50fa6756003a827a282c7adf82b5e7d98e1058ccf81fa144e6 | What is the maximum number of circles of radius $\phi(n)$, at most tangential by twos, that can be included in a circle of radius $n$? | [
"\\left\\lfloor \\frac{\\pi}{\\arcsin\\left( \\frac{\\phi(n)}{n - \\phi(n)} \\right)} \\right\\rfloor"
] | [
"What is the maximum number of circles of radius $\\phi(n)$, at most tangential by twos, that can be included in a circle of radius $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b9e1a0a86609d25f554d51aa40f42bdf499d8c060c621adc77ff740df1e6281d | What is the maximum number of circles of radius \( r \) that can be tangent to one circle of radius \( nr \), where \( n \) is a natural number? | [
"\\(\\left\\lfloor \\dfrac{\\pi}{\\sin^{-1} \\dfrac{1}{n+1}} \\right\\rfloor\\)"
] | [
"What is the maximum number of circles of radius \\( r \\) that can be tangent to one circle of radius \\( nr \\), where \\( n \\) is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 18
} | 56 | 0 | 0 |
omr-ws-dd65af58efaaf1b9707b06745316207bbce54b240e4eff27ff9a5fea1781f781 | What is the maximum number of circles that can be drawn on a plane such that any two circles intersect at most at one point? | [
"Uncountably infinite (cardinality of the continuum)"
] | [
"What is the maximum number of circles that can be drawn on a plane such that any two circles intersect at most at one point?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 11
} | 32 | 0 | 0 |
omr-ws-f93e318d179786fffd17af0e9ebdd3a45f092882dfc0ec8537799ed9f6de5d59 | What is the maximum number of circles with radius 1 that can fit inside a circle with radius 3 without overlapping? | [
"7"
] | [
"What is the maximum number of circles with radius 1 that can fit inside a circle with radius 3 without overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-123643cd3a3d7e5a5854aa7844b3fcba3c3becbdd3080178b2a02db1611ee909 | What is the maximum number of circles with radius \( r \) that can be placed inside a larger circle with radius \( nr \), where \( n \) is a natural number? | [
"\\( p(n) = \\begin{cases} \n3k^2 + 3k + 1 & \\text{if } n = 2k + 1 \\\\\n3k^2 - k & \\text{if } n = 2k \n\\end{cases} \\)"
] | [
"What is the maximum number of circles with radius \\( r \\) that can be placed inside a larger circle with radius \\( nr \\), where \\( n \\) is a natural number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-05e58495f91dce545583c09e4e54713430ab9149ff04c7da2df6610c2be4b103 | What is the maximum number of circles, each with radius $1$, that can fit in a $50$ by $50$ square without overlapping? | [
"2450"
] | [
"What is the maximum number of circles, each with radius $1$, that can fit in a $50$ by $50$ square without overlapping?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbc7ab25b92b378b6ad880bd2b8f85c0197225fb12433cc4f7e8b829cf3b0c07 | What is the maximum number of coins that can be arranged in the cells of an $n \times n$ table (with at most one coin per cell) such that no coin is both below and to the right of any other coin? | [
"\\(2n-1\\)"
] | [
"What is the maximum number of coins that can be arranged in the cells of an $n \\times n$ table (with at most one coin per cell) such that no coin is both below and to the right of any other coin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 3,
"tir": 4
} | 23 | 0 | 0 |
omr-ws-1741088af64449713548d08a421a23dccea8e04e1318b2bb8595199167c970d7 | What is the maximum number of coins that can be placed in the cells of an \( n \times n \) table such that no coin is placed both directly below and directly to the right of any other coin? | [
"\\frac{n(n+1)}{2}"
] | [
"What is the maximum number of coins that can be placed in the cells of an \\( n \\times n \\) table such that no coin is placed both directly below and directly to the right of any other coin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-cc2f46291502b12e0739d30e429d418f06476af69b3c06054bf05d386d23521e | What is the maximum number of coins that can be placed on the cells of a $10 \times 10$ board such that no four coins form the vertices of a rectangle? | [
"34"
] | [
"What is the maximum number of coins that can be placed on the cells of a $10 \\times 10$ board such that no four coins form the vertices of a rectangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 2
} | 35 | 0 | 0 |
omr-ws-7006236e722e9f591bdf228b921b6491a44049a4a0fe310792e99751daa9d23b | What is the maximum number of colors that can be used to paint all the squares of a $10 \times 10$ squareboard such that each column and each row has at most five different colors? | [
"41"
] | [
"What is the maximum number of colors that can be used to paint all the squares of a $10 \\times 10$ squareboard such that each column and each row has at most five different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-230aafaff72e329d48ec498f70d61cbaf65fd18ceca060c44148a622ceba9686 | What is the maximum number of colors that can be used to paint an $8 \times 8$ chessboard such that each $1 \times 1$ square shares an edge with at least two squares of the same color? | [
"16"
] | [
"What is the maximum number of colors that can be used to paint an $8 \\times 8$ chessboard such that each $1 \\times 1$ square shares an edge with at least two squares of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aba073448146f98b2ec9af6ac71c3a7ebbf12e148ff7c75fb41293be2dff5b9b | What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard, such that every square is painted in a single colour and is adjacent (horizontally, vertically, or diagonally) to at least two other squares of its own colour? | [
"21"
] | [
"What is the maximum number of colours that can be used to paint an $8 \\times 8$ chessboard, such that every square is painted in a single colour and is adjacent (horizontally, vertically, or diagonally) to at least two other squares of its own colour?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-468882cad283e172a6382e8da1dce85ebc89bfb60bd7f06e316aa37e48ca70d9 | What is the maximum number of combinations of 5 balls chosen from 10 such that any two combinations share at most 2 numbers? | [
"6"
] | [
"What is the maximum number of combinations of 5 balls chosen from 10 such that any two combinations share at most 2 numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-46c6cd1ac02177389d28ffabc8e27b175659c253bc39528cac4cf65e57dfdeba | What is the maximum number of combinations of 5 balls chosen from 18 such that any two combinations share at most 2 numbers? | [
"68"
] | [
"What is the maximum number of combinations of 5 balls chosen from 18 such that any two combinations share at most 2 numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be4e5cc704d842a2e90c9588a3cb5778f26a50adb48b782da46f958cb0e8970d | What is the maximum number of committees that can be formed from $n$ people such that each committee has an odd number of people and any two committees have an even number of people in common (including zero)? | [
"\\( n \\)"
] | [
"What is the maximum number of committees that can be formed from $n$ people such that each committee has an odd number of people and any two committees have an even number of people in common (including zero)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d39181e91cda766e445c256d7049489c892498a999064110c3fcbfd951cc86ca | What is the maximum number of common tangents between at least two circles among 87 circles, where each circle is tangent to one or more other circles? | [
"14964"
] | [
"What is the maximum number of common tangents between at least two circles among 87 circles, where each circle is tangent to one or more other circles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89f7533ebc11f3c6df9b7d2a46b176181115d3eecaf2a1d84887f0a877cd20b2 | What is the maximum number of completely bounded areas, not further subdivided, that can be formed by three intersecting rectangles on a flat surface, considering only the sides of the rectangles as boundaries? | [
"25"
] | [
"What is the maximum number of completely bounded areas, not further subdivided, that can be formed by three intersecting rectangles on a flat surface, considering only the sides of the rectangles as boundaries?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-6cf06f23200e44b8cf82cf54aa729f746a064ace2cb1f02815cd605a4972ebe0 | What is the maximum number of congruent convex polytopes in a neighborly family in \(\mathbb{E}^3\)? | [
"8"
] | [
"What is the maximum number of congruent convex polytopes in a neighborly family in \\(\\mathbb{E}^3\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-ade0182a5fb395066c4fe23053e86b3bf9a0a6299596b3c850e239e894dc4328 | What is the maximum number of congruent spheres that can be placed on the surface of another sphere of the same radius? | [
"12"
] | [
"What is the maximum number of congruent spheres that can be placed on the surface of another sphere of the same radius?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8b948d0b137248aaa568cd9ace48555427af7688af6d6bd1ad0bf43c3d103ff | What is the maximum number of consecutive 'interesting' positive integers, where an 'interesting' integer is defined as one whose sum of digits is a perfect square? | [
"2"
] | [
"What is the maximum number of consecutive 'interesting' positive integers, where an 'interesting' integer is defined as one whose sum of digits is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-cce10972448a76fdcf242d51204a5742837129ec3cc167a84b782eac69be8215 | What is the maximum number of consecutive integers that can sum up to 2014? | [
"4028"
] | [
"What is the maximum number of consecutive integers that can sum up to 2014?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 8
} | 51 | 0 | 0 |
omr-ws-3c1e7bd4d9e493994bd788dab85a6df9a1de9e16ce63e8d91f3a62dde62bcf8a | What is the maximum number of consecutive positive integers that can be added together before the sum exceeds 400? | [
"27"
] | [
"What is the maximum number of consecutive positive integers that can be added together before the sum exceeds 400?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2eefd23dc3efc29db574c902809cefa0acf5972d7f4d6376f8640a1e328d441b | What is the maximum number of cycles in a graph of order \( n \) with girth \( d \)? | [
"\\left\\lfloor \\frac{n^2}{4} \\right\\rfloor"
] | [
"What is the maximum number of cycles in a graph of order \\( n \\) with girth \\( d \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-eaab2bbf4301d637420268f26fec7587e56c87b9f98f0f0c9d586b613f3d5fdb | What is the maximum number of cylindrical logs with a length of 2m and a diameter of 0.8m that can be placed in a covered rectangular container with dimensions 3.25m by 4.05m by 3m? | [
"30"
] | [
"What is the maximum number of cylindrical logs with a length of 2m and a diameter of 0.8m that can be placed in a covered rectangular container with dimensions 3.25m by 4.05m by 3m?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e94e05c616c0110a59e4e98b10a682c96fa888cdf291e5011337b3c07ed4aa7c | What is the maximum number of diagonals of a regular $12$-gon that can be selected such that no two of the chosen diagonals are perpendicular? Note that sides are not considered diagonals, and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular. | [
"3"
] | [
"What is the maximum number of diagonals of a regular $12$-gon that can be selected such that no two of the chosen diagonals are perpendicular? Note that sides are not considered diagonals, and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f261c21fb07062d056d9b333dc010ac23ebf20d14307ed1441b0cea850965456 | What is the maximum number of diagonals that can be drawn on a chessboard such that no two diagonals share any common points? | [
"16"
] | [
"What is the maximum number of diagonals that can be drawn on a chessboard such that no two diagonals share any common points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-ed7ee0db4427023c80ae51a983a303c744a61ebd136451529b8a6c9b7f030b45 | What is the maximum number of direction changes in a stroll through the lines of a 2004 x 2004 grid divided into 2004 x 2004 unit squares with lines parallel to the four sides of the square, if the stroll does not pass through the same point twice? | [
"4008007"
] | [
"What is the maximum number of direction changes in a stroll through the lines of a 2004 x 2004 grid divided into 2004 x 2004 unit squares with lines parallel to the four sides of the square, if the stroll does not pass through the same point twice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cb850e962d5d5eb5433f041cd9c9dd0e19ce7328c1d3dc4d2aa5e4e83ffd6df0 | What is the maximum number of distinct colors we can use to color each square of a $100$ by $100$ square grid such that no row or column contains more than 7 distinct colors? | [
"49"
] | [
"What is the maximum number of distinct colors we can use to color each square of a $100$ by $100$ square grid such that no row or column contains more than 7 distinct colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-3d80f4f5ecbc80efb8f0dc7233892cc41cea1513a73e9c9a4a4dc2391d3af87c | What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101? | [
"22"
] | [
"What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a93c67c5f9b1e6d4af7b3a7b1c9c717f6901a59b363cfb5af6a2900530356ae0 | What is the maximum number of distinct integers that can be written in a row such that the sum of any 11 consecutive numbers is either 100 or 101? | [
"22"
] | [
"What is the maximum number of distinct integers that can be written in a row such that the sum of any 11 consecutive numbers is either 100 or 101?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-0203d358d79d33636204bc80ea80c61e439540d0b3d46366c6a58b53fedc4a89 | What is the maximum number of distinct prime factors under 100 that the function \( f(x) = x^4 + x^3 + x^2 + x + 1 \) can have? | [
"6"
] | [
"What is the maximum number of distinct prime factors under 100 that the function \\( f(x) = x^4 + x^3 + x^2 + x + 1 \\) can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b98ff74516532065c2863f630b116dfa714451151699384033354b41ad68515f | What is the maximum number of distinct regions that 50 lines and 60 circles can divide a rectangle into? | [
"10816"
] | [
"What is the maximum number of distinct regions that 50 lines and 60 circles can divide a rectangle into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-fb1ad52422ae87f4cadcad26fd1ed947fe5239780f7558eb5f7d66d6517be3c9 | What is the maximum number of distinct roots a monic polynomial of degree 2013 in $\mathbb{Z}[X]$ can have in the interval $(0,3)$? | [
"2013"
] | [
"What is the maximum number of distinct roots a monic polynomial of degree 2013 in $\\mathbb{Z}[X]$ can have in the interval $(0,3)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2493d3ecf99d161493037059609f773ef1182d1544b9086f38882d681f1f0c07 | What is the maximum number of divisors of $20!^{20!}$ that can be drawn to a board such that no subset of these divisors (containing at least one number) has a product that is a perfect square? | [
"256"
] | [
"What is the maximum number of divisors of $20!^{20!}$ that can be drawn to a board such that no subset of these divisors (containing at least one number) has a product that is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bee8ab38e4a7bec9270992c3ff6e68c579dafab6256bdfeafa1ded54c4ae478b | What is the maximum number of edges a graph can have so that it doesn't contain $K_{2,3}$ as a subgraph? | [
"2n - 3"
] | [
"What is the maximum number of edges a graph can have so that it doesn't contain $K_{2,3}$ as a subgraph?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-ef0a284291c3df818f78755c6197968c38dc186efefc28595beef0c230f5a91f | What is the maximum number of edges a graph can have so that it doesn't contain $K_{3,3}$ as a subgraph? | [
"12"
] | [
"What is the maximum number of edges a graph can have so that it doesn't contain $K_{3,3}$ as a subgraph?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-e5ab462d99d8a1be6446a112d69a6c4e89aea55067d4388abb8185f3cb29bf0e | What is the maximum number of edges in a bipartite graph that contains no matching with $k$ edges and no vertex of degree $d$? | [
"(k-1)(d-1)"
] | [
"What is the maximum number of edges in a bipartite graph that contains no matching with $k$ edges and no vertex of degree $d$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-f560c5ff51ef5c35ddb7ea7cfdc107087210652b63e2a5b0921857702913f2d5 | What is the maximum number of edges in a triangle-free graph on $6$ vertices? | [
"9"
] | [
"What is the maximum number of edges in a triangle-free graph on $6$ vertices?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c948a164eb0ec2ae734e1a9587fdb7f66203b7bd0e6733fbda392b2a7b5bcba5 | What is the maximum number of edges in a unipathic subgraph of the tournament $T_n$ on $n$ vertices? | [
"\\( n-1 \\)"
] | [
"What is the maximum number of edges in a unipathic subgraph of the tournament $T_n$ on $n$ vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
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