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-ed83c66be6482061c758ad9b488a57ba1aafc41912da7f106cd04ae0d6220bce | An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers? | [
"4"
] | [
"An $\\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a7834dbfeeaeab62b52c2cc5226b8ae85cbe7389664cc8923382a9fddfb0c32 | An $\textit{ortho-polygon}$ is a non-crossed polygon where every pair of consecutive sides is perpendicular. We know that there is an $\textit{ortho-polygon}$ of $n$ sides such that, numbering its sides clockwise from $1$ to $n$, the length of each side is equal to the number given to it. Also, $n$ is the nearest integ... | [
"2016"
] | [
"An $\\textit{ortho-polygon}$ is a non-crossed polygon where every pair of consecutive sides is perpendicular. We know that there is an $\\textit{ortho-polygon}$ of $n$ sides such that, numbering its sides clockwise from $1$ to $n$, the length of each side is equal to the number given to it. Also, $n$ is the neares... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-4313ece2699c45d7bf9f7af76058ba2fcfc3f981222eaa3b8c24918fc1c8db20 | An $a \times b \times c$ box has half the volume of an $(a+2) \times (b+2) \times (c+2)$ box, where $a \leq b \leq c$. What is the largest possible value of $c$? | [
"130"
] | [
"An $a \\times b \\times c$ box has half the volume of an $(a+2) \\times (b+2) \\times (c+2)$ box, where $a \\leq b \\leq c$. What is the largest possible value of $c$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-ec5346cb798ec0b98ced65071fe10c4e80292376b0e3cc5f502940eb1d4d79bc | An $a \times b \times c$ box is painted red, where $a \le b \le c$ are positive integers. It is then cut into $abc$ unit cubes. The number of cubes that do not have any red faces is equal to one-third the total number of cubes. Find the number of possible triples $(a, b, c)$. | [
"9"
] | [
"An $a \\times b \\times c$ box is painted red, where $a \\le b \\le c$ are positive integers. It is then cut into $abc$ unit cubes. The number of cubes that do not have any red faces is equal to one-third the total number of cubes. Find the number of possible triples $(a, b, c)$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a18713f43e7fe88c2da802903c5691e379a60859071f2da6f182d901efb79f12 | An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Ea... | [
"180"
] | [
"An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yell... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02206201d064f21b6bef995835e11add3394aa6bb608615d280472cd4f5d6a6c | An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$-faces of the box contains exactly 9 red cubes, exactly 12 green cubes, and some yellow cubes. Each o... | [
"180"
] | [
"An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$-faces of the box contains exactly 9 red cubes, exactly 12 green cubes, and some yellow c... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b19c9d28f8e884878c85c91c5c0569a896f0e55af0d705ea9e9b07968677f39 | An $a$-foot-tall statue is placed on top of a $b$-foot-tall building. A $c$-foot-tall person wants to stand at a distance from the building such that the statue occupies the largest possible angle in the observer's field of vision. How far away from the building should the person stand? | [
"\\( \\sqrt{(a+b-c)(b-c)} \\)"
] | [
"An $a$-foot-tall statue is placed on top of a $b$-foot-tall building. A $c$-foot-tall person wants to stand at a distance from the building such that the statue occupies the largest possible angle in the observer's field of vision. How far away from the building should the person stand?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a88569311665b0e9284b327d86ca805fe04fecc022b7fe12c13cc14d36aaf3e3 | An $i$-curve is the set of pairs of complex numbers $(x,y)$ such that $P(x,y) = 0$, where $P$ is a polynomial with complex coefficients of degree $i$. It is called irreducible if $P$ cannot be factored as $Q \cdot R$, where $Q, R$ are nonconstant polynomials with complex coefficients. An $i$-curve is called tangent to ... | [
"3264"
] | [
"An $i$-curve is the set of pairs of complex numbers $(x,y)$ such that $P(x,y) = 0$, where $P$ is a polynomial with complex coefficients of degree $i$. It is called irreducible if $P$ cannot be factored as $Q \\cdot R$, where $Q, R$ are nonconstant polynomials with complex coefficients. An $i$-curve is called tange... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-fd71c691700aa12fadcef2f19fffeaf3ba85eef9d329a83c3f13f14722007d01 | An $m \times n \times p$ cuboid is painted red on the outside and then cut into smaller $1 \times 1 \times 1$ cubes. Determine how many of these smaller cubes are painted on (i) no sides, (ii) one side, (iii) two sides, (iv) three sides. | [
"(i) $(m-2)(n-2)(p-2)$, (ii) $2\\sum_{cyc} (m-2)(n-2)$, (iii) $4\\sum_{cyc} (m-2)$, (iv) $8$"
] | [
"An $m \\times n \\times p$ cuboid is painted red on the outside and then cut into smaller $1 \\times 1 \\times 1$ cubes. Determine how many of these smaller cubes are painted on (i) no sides, (ii) one side, (iii) two sides, (iv) three sides."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-ddfed2e46d90b33b4347bf7cbcd7f5319fbd04fa44da3af123c6784d4118d472 | An $m \times n$ checkerboard is colored randomly such that each square is independently assigned red or black with probability $\frac{1}{2}$. Two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q$, where succes... | [
"400"
] | [
"An $m \\times n$ checkerboard is colored randomly such that each square is independently assigned red or black with probability $\\frac{1}{2}$. Two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q$, where... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95771fc0f3fbab7990e1f81b3ea10f3170f870dd1afe8ee050c85d8581a15184 | An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still ensuring no two rows have the same pattern. | [
"\\( n - m + 1 \\)"
] | [
"An $m \\times n$ chessboard where $m \\le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still ensuring no two rows have the same pattern."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-7a8c3e7e5f17ed32cd128e29583d3e9706e6280b5d6325c78ac9d9e9ee15b4c8 | An $m \times n$ chessboard with $m, n \geq 2$ is given. Some dominoes are placed on the chessboard such that: (i) Each domino occupies two adjacent squares of the chessboard; (ii) It is not possible to place another domino without overlapping; (iii) It is not possible to slide a domino horizontally or vertically withou... | [
"0"
] | [
"An $m \\times n$ chessboard with $m, n \\geq 2$ is given. Some dominoes are placed on the chessboard such that:\n(i) Each domino occupies two adjacent squares of the chessboard;\n(ii) It is not possible to place another domino without overlapping;\n(iii) It is not possible to slide a domino horizontally or vertica... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-ca96feea8b6485b910b0a008cb7bbd7481a6a69a5426b1f417fe2bd0ac729cde | An $m \times n$ rectangle $(m, n > 1)$ is divided into $mn$ unit squares. In how many ways can two of the squares be cut off so that the remaining part of the rectangle can be covered with dominoes $2 \times 1$? | [
"\\left( \\dfrac{mn}{2} \\right)^2"
] | [
"An $m \\times n$ rectangle $(m, n > 1)$ is divided into $mn$ unit squares. In how many ways can two of the squares be cut off so that the remaining part of the rectangle can be covered with dominoes $2 \\times 1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-9f5836db82cc5bba5047a5e367f898a8123699d16eb35e1a9829b2733479f8b6 | An $m \times n$ rectangle is tiled with $\frac{mn}{2}$ $1 \times 2$ dominoes. The tiling is such that whenever the rectangle is partitioned into two smaller rectangles, there exists a domino that is part of the interior of both rectangles. Given $mn \geq 3$, what is the minimum possible value of $mn$? | [
"48"
] | [
"An $m \\times n$ rectangle is tiled with $\\frac{mn}{2}$ $1 \\times 2$ dominoes. The tiling is such that whenever the rectangle is partitioned into two smaller rectangles, there exists a domino that is part of the interior of both rectangles. Given $mn \\geq 3$, what is the minimum possible value of $mn$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a1d905aad7f70d05e4ae0608e9e39db383a7b511fc398ff7b888b03bffa5bff | An $m \times n$ rectangle, where $m, n \in \mathbb{N}^*$, is divided into smaller squares such that the sides of each square are parallel to the corresponding sides of the rectangle and the length of each side is an integer. Determine the minimum of the sum of the sides of these squares. | [
"\\( n + m - \\gcd(n, m) \\)"
] | [
"An $m \\times n$ rectangle, where $m, n \\in \\mathbb{N}^*$, is divided into smaller squares such that the sides of each square are parallel to the corresponding sides of the rectangle and the length of each side is an integer. Determine the minimum of the sum of the sides of these squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-9f30d588b0cf3ff4c3adbf82ab8f11e49a6f3151495eb8fbeba7c05d944a22e8 | An $m \times n$ rectangular board is given where $m$ and $n$ are odd integers. The board is covered with 2 × 1 dominoes, so that no two dominoes overlap and only the bottom left square of the board is empty. At any point in time, it is allowed to slide a domino so that it covers an empty square and still stays on the b... | [
"4"
] | [
"An $m \\times n$ rectangular board is given where $m$ and $n$ are odd integers. The board is covered with 2 × 1 dominoes, so that no two dominoes overlap and only the bottom left square of the board is empty. At any point in time, it is allowed to slide a domino so that it covers an empty square and still stays on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-ce068f83b99fc4d1ec58ecf4b51556b6537eef4908e8d2476a7d2be3b847d63a | An $m$ by $n$ array of unit squares contains 360 rectangles of any size. What is a value of $m + n$? | [
"19"
] | [
"An $m$ by $n$ array of unit squares contains 360 rectangles of any size. What is a value of $m + n$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1112eeefe688ed1992c2b6bbc36446ca90efb9e04240d7a20a485e84fd936315 | An $n \times n \times n$ cube is divided into unit cubes. A closed non-self-intersecting polygon in space is formed such that each side of the polygon joins the centers of two unit cubes sharing a common face. The faces of unit cubes that intersect the polygon are called distinguished. Determine the minimum number of c... | [
"1"
] | [
"An $n \\times n \\times n$ cube is divided into unit cubes. A closed non-self-intersecting polygon in space is formed such that each side of the polygon joins the centers of two unit cubes sharing a common face. The faces of unit cubes that intersect the polygon are called distinguished. Determine the minimum numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-b3ac845a076049e24e17bf6f8b930afbbb6e9759b4cc61e141ad96ff0613e8c2 | An $n \times n \times n$ cube is painted red on the outside and then cut into $n^3$ smaller $1 \times 1 \times 1$ cubes. Determine how many of these smaller cubes are painted on (i) no sides, (ii) one side, (iii) two sides, (iv) three sides. | [
"(i) $(n-2)^3$, (ii) $(n-2)^2 \\cdot 6$, (iii) $(n-2) \\cdot 12$, (iv) $8$"
] | [
"An $n \\times n \\times n$ cube is painted red on the outside and then cut into $n^3$ smaller $1 \\times 1 \\times 1$ cubes. Determine how many of these smaller cubes are painted on (i) no sides, (ii) one side, (iii) two sides, (iv) three sides."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-bc96aa0349d078c721af8915b756231c7fee22d0920abb7005ed7f280fd23231 | An $n \times n$ chessboard, where $n \geq 2$, is numbered with the numbers $1, 2, \ldots, n^2$ (each number used exactly once). Let $D(n)$ be the minimum possible value of the largest difference between the numbers of any two neighboring (sharing a common edge) squares. Find $D(5)$. | [
"5"
] | [
"An $n \\times n$ chessboard, where $n \\geq 2$, is numbered with the numbers $1, 2, \\ldots, n^2$ (each number used exactly once). Let $D(n)$ be the minimum possible value of the largest difference between the numbers of any two neighboring (sharing a common edge) squares. Find $D(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fa9b346ec2267efe72ec6e75d3ae09d76889d715ef6182966a43110a211c306 | An $n \times n$ grid has the first column and first row completely filled with the integers from $1$ to $n$, with one integer in each box. In how many ways can the remainder of the grid be filled with the integers from $1$ to $n$ so that each row and each column contains each integer exactly once? | [
"(n-2)!"
] | [
"An $n \\times n$ grid has the first column and first row completely filled with the integers from $1$ to $n$, with one integer in each box. In how many ways can the remainder of the grid be filled with the integers from $1$ to $n$ so that each row and each column contains each integer exactly once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-1dd04f550fe37949791a33f938c1651f4db9bae5d052d8e8200b15e8a2794b1a | An $n \times n$ grid is filled with $0$s and $1$s such that for any $n$ squares, no two of which lie in the same row or column, at least one square is filled with $1$. Let $f(n)$ be the minimum value of $r + c$ over all such grids, where $r$ is the number of rows and $c$ is the number of columns such that all squares i... | [
"6"
] | [
"An $n \\times n$ grid is filled with $0$s and $1$s such that for any $n$ squares, no two of which lie in the same row or column, at least one square is filled with $1$. Let $f(n)$ be the minimum value of $r + c$ over all such grids, where $r$ is the number of rows and $c$ is the number of columns such that all squ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-ced48d38796e369149ecbf1ba81e34b1cd19b9f10aaf0b9a0a4c3a949bb1c1d3 | An $n \times n$ grid is filled with 0s and 1s such that any selection of $n$ cells, with no two cells in the same row or column, contains at least one 1. Find the minimum value of $k$ such that there exist $i$ rows and $j$ columns with $i + j = k$ and the intersection of these rows and columns contains only 1s. Calcula... | [
"11"
] | [
"An $n \\times n$ grid is filled with 0s and 1s such that any selection of $n$ cells, with no two cells in the same row or column, contains at least one 1. Find the minimum value of $k$ such that there exist $i$ rows and $j$ columns with $i + j = k$ and the intersection of these rows and columns contains only 1s. C... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e543adde1cddbfae2ef4759c6f6e8299f3c2fc5991abdc0ac7b01b330547b046 | An $n \times n$ matrix $A$ with complex entries has all eigenvalues either $0$ or $1$. Under what conditions is $A$ idempotent? | [
"\\( A \\) must be diagonalizable."
] | [
"An $n \\times n$ matrix $A$ with complex entries has all eigenvalues either $0$ or $1$. Under what conditions is $A$ idempotent?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-1857ce6a724b43cf6ef4b0b18b0569f3445f23cfff828d772422fddab4dac939 | An $n \times n$ square consists of $n^2$ unit squares. A monotonous path in this square is a path of length $2n$ starting at the left lower corner and ending at the right upper corner, moving along the sides of unit squares. For each $k$, where $0 \leq k \leq 2n-1$, let $S_k$ be the set of all monotonous paths such tha... | [
"0"
] | [
"An $n \\times n$ square consists of $n^2$ unit squares. A monotonous path in this square is a path of length $2n$ starting at the left lower corner and ending at the right upper corner, moving along the sides of unit squares. For each $k$, where $0 \\leq k \\leq 2n-1$, let $S_k$ be the set of all monotonous paths ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-df9be59d9b9f5a818d4b00014182c6f774dc8db4d739493939ebb4ad246474f5 | An $n \times n$ square grid ($n \ge 3$) is rolled into a cylinder. Some of the cells are then colored black. Let $k$ be the maximum number of black cells found in any line (horizontal, vertical, or diagonal) among all possible configurations. Find the minimum possible value of $k$ for $n = 5$. | [
"3"
] | [
"An $n \\times n$ square grid ($n \\ge 3$) is rolled into a cylinder. Some of the cells are then colored black. Let $k$ be the maximum number of black cells found in any line (horizontal, vertical, or diagonal) among all possible configurations. Find the minimum possible value of $k$ for $n = 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-dd1e7310be8798351a922a8f91135cc4ddcbaea717b12681f96893e403806d3d | An $n \times n$ square grid ($n \geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Let $f(n)$ be the minimum number of black cells needed to ensure that there exist two parallel lines (horizontal, vertical, or diagonal) containing the same number of black cells. Find $f(5)$. | [
"16"
] | [
"An $n \\times n$ square grid ($n \\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Let $f(n)$ be the minimum number of black cells needed to ensure that there exist two parallel lines (horizontal, vertical, or diagonal) containing the same number of black cells. Find $f(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-630514998e40d308dcdb732ef23aff3eed172b35d0adddd8bf542a2926dd2af0 | An $n \times n$ square table is divided into $n^2$ unit cells. Some unit segments of the grid (i.e., the sides of any unit cell) are colored black so that each unit cell has exactly one black side. Find the greatest possible number of black unit segments. | [
"\\(\\frac{n^2 + 4n - 5}{2}\\) for odd \\(n\\) and \\(\\frac{n^2 + 4n - 4}{2}\\) for even \\(n\\)."
] | [
"An $n \\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the grid (i.e., the sides of any unit cell) are colored black so that each unit cell has exactly one black side. Find the greatest possible number of black unit segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 11
} | 16 | 0 | 0 |
omr-ws-f03d79557da085f8f12e4369468794a8da9e160ebfdc4e3dd8a2fd5995d82563 | An $n \times n$ square table is divided into $n^2$ unit cells. Some unit segments of the grid (i.e., the sides of any unit cell) are colored black so that each unit cell has exactly one black side. Find the smallest possible number of black unit segments. | [
"$\\frac{n^2}{2}$ if $n$ is even, $\\frac{n^2+1}{2}$ if $n$ is odd."
] | [
"An $n \\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the grid (i.e., the sides of any unit cell) are colored black so that each unit cell has exactly one black side. Find the smallest possible number of black unit segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 4,
"tir": 18
} | 66 | 0 | 0 |
omr-ws-0213c3a30ca31357a4ffb6ba089482d3d8b4902a1e4462fa4236c6ee8864d960 | An $n \times n$ table is filled with the numbers $1, 2, \ldots, n^2$ such that for all $1 \le i < n^2$, the integers $i$ and $i+1$ are in squares that share a common edge. What is the minimum possible sum of the $\frac{1}{2}n(n-1)$ numbers filled in the shaded squares? (The figure shows the case when $n=5$.) | [
"80"
] | [
"An $n \\times n$ table is filled with the numbers $1, 2, \\ldots, n^2$ such that for all $1 \\le i < n^2$, the integers $i$ and $i+1$ are in squares that share a common edge. What is the minimum possible sum of the $\\frac{1}{2}n(n-1)$ numbers filled in the shaded squares? (The figure shows the case when $n=5$.)"
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-14a8db61a97de1bdc163e161bad2404786cd6ce57ff9d638c470c3d53dbbe711 | An $n \times n$ table is written on a square piece of cardboard. Knuffle draws some diagonals in some of the $n^2$ cells, then uses a knife to cut along the marked diagonals. To Knuffle's surprise, the resulting piece of cardboard is still connected. If $k$ represents the minimum number of cells that must be left uncut... | [
"5"
] | [
"An $n \\times n$ table is written on a square piece of cardboard. Knuffle draws some diagonals in some of the $n^2$ cells, then uses a knife to cut along the marked diagonals. To Knuffle's surprise, the resulting piece of cardboard is still connected. If $k$ represents the minimum number of cells that must be left... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-d45cf8ef084ec7a31260bd835ae449b25b991c702e16a36b55db33ca680c3785 | An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. a.) Find the smallest $s \i... | [
"n + 1"
] | [
"An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key.\na.) Find the smallest... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7492d478ccd75fd5b2e8e5929d1542e22ba667c90a1671db422845f9801288cd | An $n$ by $n$ table has an integer in each cell, such that no two cells within a row share the same number. After permuting the elements within each row to obtain a table that has $n$ distinct numbers in each column, let $S$ be the sum of the numbers in the main diagonal of this new table. If $n=5$ and the initial tabl... | [
"75"
] | [
"An $n$ by $n$ table has an integer in each cell, such that no two cells within a row share the same number. After permuting the elements within each row to obtain a table that has $n$ distinct numbers in each column, let $S$ be the sum of the numbers in the main diagonal of this new table. If $n=5$ and the initial... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94e2f9fcd47f659ee8b5fa3da8fc7b94e62231fe41c95ef07972b2d73abe67a4 | An $n$-by-$n$-by-$n$ cube is formed from $n^3$ unit cubes. The removal of some of the unit cubes leaves a solid $S$ such that: 1. The projection of $S$ onto each face of the original cube is an $n$-by-$n$ square. 2. From each unit cube in $S$, one can reach any other unit cube in $S$ along a chain of cubes each of whic... | [
"n^2"
] | [
"An $n$-by-$n$-by-$n$ cube is formed from $n^3$ unit cubes. The removal of some of the unit cubes leaves a solid $S$ such that:\n1. The projection of $S$ onto each face of the original cube is an $n$-by-$n$ square.\n2. From each unit cube in $S$, one can reach any other unit cube in $S$ along a chain of cubes each ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-668977b13143bf0be544fee46c92252c20380ea2be18e9a1dc7219f5dea1dcd0 | An $n$-die is a die with $n$ surfaces. Consider a $2012$-die whose opposite sides are paired as $\{1,2\}, \{3,4\}, \ldots, \{2011,2012\}$. Let these pairs be denoted as $A_1, A_2, \ldots, A_{1006}$. For any $1 \leq a \leq 2012$, define $f(a)$ as the element in the set $A_{\left\lfloor \frac{a+1}{2} \right\rfloor} - \{a... | [
"0"
] | [
"An $n$-die is a die with $n$ surfaces. Consider a $2012$-die whose opposite sides are paired as $\\{1,2\\}, \\{3,4\\}, \\ldots, \\{2011,2012\\}$. Let these pairs be denoted as $A_1, A_2, \\ldots, A_{1006}$. For any $1 \\leq a \\leq 2012$, define $f(a)$ as the element in the set $A_{\\left\\lfloor \\frac{a+1}{2} \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a32efa3509a36bc59bd941f64ae57197e5588f18caf6550c1fbd26fb05283df | An $n$-digit number $\alpha$ is said to be special if $\alpha$ is equal to the arithmetic mean of the $n!$ numbers one obtains by rearranging the digits of $\alpha$ in all possible ways, and the digits of $\alpha$ are not all equal. The first six special numbers are $370, 407, 481, 518, 592,$ and $629$. Find the next l... | [
"370370",
"703"
] | [
"An $n$-digit number $\\alpha$ is said to be special if $\\alpha$ is equal to the arithmetic mean of the $n!$ numbers one obtains by rearranging the digits of $\\alpha$ in all possible ways, and the digits of $\\alpha$ are not all equal. The first six special numbers are $370, 407, 481, 518, 592,$ and $629$. Find t... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 2
} | 24 | 0 | 0 |
omr-ws-9265b375ca60bd14787655a84f41970bc50015b0903f506ffb8e38797f1a02e1 | An $n$-digit positive integer $N$ is a Kaprekar number if the sum of the number formed by the last $n$ digits in $N^2$, and the number formed by the first $n$ or $n-1$ digits in $N^2$ equals $N$. Find all three-digit Kaprekar numbers. | [
"297, 703, 999"
] | [
"An $n$-digit positive integer $N$ is a Kaprekar number if the sum of the number formed by the last $n$ digits in $N^2$, and the number formed by the first $n$ or $n-1$ digits in $N^2$ equals $N$. Find all three-digit Kaprekar numbers."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-48fd5717a0219436e64e95bc34e347c721b04bde5d7b0df165381e5f62af3363 | An $n$-dimensional cube $C$ is decomposed into $2^n$ smaller rectangular boxes by $n$ planes $P_1, P_2, \ldots, P_n$ such that each edge of $C$ is perpendicular to exactly one of those planes. The $2^n$ boxes are colored black and white such that any two neighboring boxes have different colors. Suppose that the sum of ... | [
"1"
] | [
"An $n$-dimensional cube $C$ is decomposed into $2^n$ smaller rectangular boxes by $n$ planes $P_1, P_2, \\ldots, P_n$ such that each edge of $C$ is perpendicular to exactly one of those planes. The $2^n$ boxes are colored black and white such that any two neighboring boxes have different colors. Suppose that the s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a6c5ffb7d856fbd585f0f4644af67f1477f9630e2f81c22daee0285cf5014ff | An $n$-gon $A_1A_2\cdots A_n$ is triangulated into $n-2$ triangles. Among those triangles, let there be $a_i$ of them that have $A_i$ as one of its vertices. Calculate the value of $$\displaystyle a_1-\frac{1}{a_2-\frac{1}{\cdots -\frac{1}{a_{n-1}}}}$$ for a given triangulation where $n=8$, $a_1=5$, $a_2=3$, $a_3=2$, $... | [
"\\dfrac{9}{2}"
] | [
"An $n$-gon $A_1A_2\\cdots A_n$ is triangulated into $n-2$ triangles. Among those triangles, let there be $a_i$ of them that have $A_i$ as one of its vertices. Calculate the value of \n$$\\displaystyle a_1-\\frac{1}{a_2-\\frac{1}{\\cdots -\\frac{1}{a_{n-1}}}}$$\nfor a given triangulation where $n=8$, $a_1=5$, $a_2=... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e12be001101ca4174f733e1099b4f6ac5f3cb55b3ecad0f2e07692b45d95df37 | An $n$-sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals. | [
"\\dfrac{n(n-1)(n-2)(n-3)}{6}"
] | [
"An $n$-sided convex polygon is such that there is no common point for any three of its diagonals. Determine the number of triangles that are formed such that two of their vertices are vertices of the polygon and the third is an intersection of two diagonals."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-a087c3130a35be2d729f48a6d4d82b7ccd42a3eb8c0fb2f9ac4788991971d290 | An $n$-sided regular polygon is inscribed into a circle of radius $R$. The point $M$ moves along this circle. For each position of $M$, consider the sum of distances from $M$ to the lines containing the sides of the polygon. Find all the positions of point $M$ for which this sum is minimal. | [
"\\( M \\) is at the vertices of the polygon."
] | [
"An $n$-sided regular polygon is inscribed into a circle of radius $R$. The point $M$ moves along this circle. For each position of $M$, consider the sum of distances from $M$ to the lines containing the sides of the polygon. Find all the positions of point $M$ for which this sum is minimal."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-763761d5833fc6a433a97b157efde426846863559acc1cc56316a436e7ee3ece | An $n$-term sequence $a_0, a_1, \ldots, a_n$ will be called sweet if, for each $0 \le i \le n - 1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1, 0$ is a sweet sequence with $4$ terms. Given that $a_0, a_1, \ldots, a_{2013}$ is a sweet sequence, find the value of $a_0^... | [
"4040106"
] | [
"An $n$-term sequence $a_0, a_1, \\ldots, a_n$ will be called sweet if, for each $0 \\le i \\le n - 1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1, 0$ is a sweet sequence with $4$ terms. Given that $a_0, a_1, \\ldots, a_{2013}$ is a sweet sequence, find the value... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-ee055e85b5c90d0b1925f32171d91b81c5a78f6b61a74a6ec2caea76fb7cb2ec | An $n$-tuplet of positive integers $1<a_1<a_2<\ldots<a_n$ is called a good $n$-tuplet if and only if any permutation $b_1, b_2, \ldots, b_n$ of $a_1, a_2, \ldots, a_n$ satisfies $b_1 \mid b_2 \cdot b_3 \cdot \ldots \cdot b_n + 1$. Find all $n$ such that there exists a unique good $n$-tuplet. | [
"4"
] | [
"An $n$-tuplet of positive integers $1<a_1<a_2<\\ldots<a_n$ is called a good $n$-tuplet if and only if any permutation $b_1, b_2, \\ldots, b_n$ of $a_1, a_2, \\ldots, a_n$ satisfies $b_1 \\mid b_2 \\cdot b_3 \\cdot \\ldots \\cdot b_n + 1$. Find all $n$ such that there exists a unique good $n$-tuplet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-8b0eedc73317bf3528efdbaf07cd05be1aea8bbc7097edb1dd7a64fb9e8f7909 | An 'unfair' coin has a 2/3 probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even? | [
"\\(\\frac{3^{50} + 1}{2 \\cdot 3^{50}}\\)"
] | [
"An 'unfair' coin has a 2/3 probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-4c6f9cbf2a99b8c724d7dd9331ed2ce87be32c0c2eff3bd3240e634edb110434 | An 11 x 11 x 11 wooden cube is formed by gluing together \(11^3\) unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | [
"331"
] | [
"An 11 x 11 x 11 wooden cube is formed by gluing together \\(11^3\\) unit cubes. What is the greatest number of unit cubes that can be seen from a single point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-48abb46c826dcf48d940dec903bc878cc11026dbf62ed112cfd4e1f9045d6525 | An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms? | [
"2"
] | [
"An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1200a4c421ca017b253f4ffefb26d9f5bb6aa36606290e32bdb42b9b202b6e3c | An 8 by 8 board (with 64 1 by 1 squares) is painted white. We are allowed to choose any rectangle consisting of 3 of the 64 squares and paint each of the 3 squares in the opposite colour (the white ones black, the black ones white). After performing a series of such operations, what is the minimum number of white squar... | [
"1"
] | [
"An 8 by 8 board (with 64 1 by 1 squares) is painted white. We are allowed to choose any rectangle consisting of 3 of the 64 squares and paint each of the 3 squares in the opposite colour (the white ones black, the black ones white). After performing a series of such operations, what is the minimum number of white ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-f1a9a18c77a2a4fac39ecffa02ff27ce2400acb468b154887209a9d88a05de82 | An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard? [asy] draw((0,0)--(8,0)--(8,8)--(0,8)--cycle); draw((1,8)--(1,0)); draw((7,8)--... | [
"73"
] | [
"An 8 by 8 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 5 black squares, can be drawn on the checkerboard?\n\n[asy]\ndraw((0,0)--(8,0)--(8,8)--(0,8)--cycle);\ndraw((1,8)--(1,0));\nd... | [
"MATH_training_set"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-88ee3987ac6f8daced517c7ff3caf4e06d36a82b396408023ca4bfeb0c1307a9 | An 8 by 8 checkerboard is placed with corners at (0, 0), (8, 0), (0, 8), and (8, 8). Find the number of different rectangles on the checkerboard with corners at points with integer coordinates. | [
"1296"
] | [
"An 8 by 8 checkerboard is placed with corners at (0, 0), (8, 0), (0, 8), and (8, 8). Find the number of different rectangles on the checkerboard with corners at points with integer coordinates."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01484c00da317dcba74da045a6325d48a0939792e5a30252a53e4f4dd85f4013 | An 8 by 8 grid is colored with black and white squares. A coloring is called balanced if each 2 by 2 subgrid contains exactly two black and two white squares. Counting rotations and reflections as different, how many balanced colorings of the 8 by 8 grid are there? | [
"510"
] | [
"An 8 by 8 grid is colored with black and white squares. A coloring is called balanced if each 2 by 2 subgrid contains exactly two black and two white squares. Counting rotations and reflections as different, how many balanced colorings of the 8 by 8 grid are there?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-5716c592986919754b44d0c814869bfc78577398e8eea7235326cfc396bb43fd | An 8 by 8 grid is divided such that the top left 4 by 4 grid is black and the rest is white. The grid is cut along the grid lines into parts where each part has three times as many white squares as black squares. Determine the maximum number of pieces required and provide the configuration of these pieces. Calculate th... | [
"16"
] | [
"An 8 by 8 grid is divided such that the top left 4 by 4 grid is black and the rest is white. The grid is cut along the grid lines into parts where each part has three times as many white squares as black squares. Determine the maximum number of pieces required and provide the configuration of these pieces. Calcula... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-de26cb6fe90a2f847d7f5a85b32cabe961ff38f96614dc32b7f2777b1440340e | An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the ith row and jth column equals the sum of the numbers in the (i − 1) by (j − 1) sub-grid with row less than i and column less than j. What is the number in the 8th row and 8th column? | [
"2508"
] | [
"An 8 by 8 grid of numbers obeys the following pattern: \n1) The first row and first column consist of all 1s.\n2) The entry in the ith row and jth column equals the sum of the numbers in the (i − 1) by (j − 1) sub-grid with row less than i and column less than j.\nWhat is the number in the 8th row and 8th column?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-fe99f0c256936e58c40d45d760011cc6dedce47eae15c3b04faee09863fe0d27 | An 8 cm square paper has a 1 cm square cut out from one corner. Mariana must divide the remaining paper into triangles of equal area. What is the minimum number of such triangles? | [
"18"
] | [
"An 8 cm square paper has a 1 cm square cut out from one corner. Mariana must divide the remaining paper into triangles of equal area. What is the minimum number of such triangles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-221cdb56df4c0421982cc499bad9dd2e44b5e4d3777bb5d3f74c4dc6989c1260 | An 8 foot by 8 foot area has been tiled by one foot square tiles. Two of the tiles were defective. The probability that the two defective tiles share an edge can be expressed as \( \frac{m}{n} \), where \( (m, n) = 1 \). Find \( m+n \). | [
"19"
] | [
"An 8 foot by 8 foot area has been tiled by one foot square tiles. Two of the tiles were defective. The probability that the two defective tiles share an edge can be expressed as \\( \\frac{m}{n} \\), where \\( (m, n) = 1 \\). Find \\( m+n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-9ec1cc1156a4bef682874fad500d26d5dea2d6bb38f536a32232ec862e6ff672 | An 8 x 8 checkerboard has 20 unit squares colored red and the rest colored white. When the board is folded in half along a line parallel to one edge, exactly seven pairs of red unit squares coincide. Compute the number of pairs of white unit squares that coincide. | [
"19"
] | [
"An 8 x 8 checkerboard has 20 unit squares colored red and the rest colored white. When the board is folded in half along a line parallel to one edge, exactly seven pairs of red unit squares coincide. Compute the number of pairs of white unit squares that coincide."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a24d37f2a367d1412d42e131133dc1c8f5a8b3e8e439f19db5c2ba16ac294d13 | An 8 × 8 chessboard has been painted such that if two squares are adjacent diagonally or are a knight's move apart, they must be of different colors. What is the minimal number of colors required to satisfy this condition? | [
"2"
] | [
"An 8 × 8 chessboard has been painted such that if two squares are adjacent diagonally or are a knight's move apart, they must be of different colors. What is the minimal number of colors required to satisfy this condition?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d10b644ba237a4f103cc1269274b9c42c6578a58cc0a3e8e1fa53191c0d5aeb7 | An 8-digit telephone number contains only two different digits and starts with $77$. Find all possible telephone numbers. | [
"576"
] | [
"An 8-digit telephone number contains only two different digits and starts with $77$. Find all possible telephone numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d1c6ae493f2ff8e402157ee2db6729bf7f44f0af001adc10cefb9efae7db891 | An 8-inch by 11-inch sheet of paper is laid flat so that the top and bottom edges are 8 inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold? | [
"8"
] | [
"An 8-inch by 11-inch sheet of paper is laid flat so that the top and bottom edges are 8 inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-b6dd5a840b8b8c760a609f2392fc89e0a56399a3e86c870d7b6c1485b29a0681 | An 8-sided die, a 12-sided die, and a 20-sided die are rolled. Compute the probability that one of the values rolled is equal to the sum of the other two values rolled. | [
"\\(\\frac{23}{240}\\)"
] | [
"An 8-sided die, a 12-sided die, and a 20-sided die are rolled. Compute the probability that one of the values rolled is equal to the sum of the other two values rolled."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 8,
"tir": 58
} | 95 | 0 | 0 |
omr-ws-51a4ec45edd13cfe6bfbe8fd254f6c64426099af792e3f4dc6af47863e4f0936 | An 8.5'' by 11'' paper $ABCD$ with $AB > BC$ is folded in half horizontally to form rectangle $AB_{1}C_{1}D$, vertically to form $AB_{1}C_{2}D_{1}$, and then horizontally again to form $AB_{2}C_{3}D_{1}$. This folded piece of paper is then cut across line $MN$, where $M$ and $N$ are on sides $AB_{2}$ and $C_{3}D_{1}$, ... | [
"195"
] | [
"An 8.5'' by 11'' paper $ABCD$ with $AB > BC$ is folded in half horizontally to form rectangle $AB_{1}C_{1}D$, vertically to form $AB_{1}C_{2}D_{1}$, and then horizontally again to form $AB_{2}C_{3}D_{1}$. This folded piece of paper is then cut across line $MN$, where $M$ and $N$ are on sides $AB_{2}$ and $C_{3}D_{... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd5e41c46758a91588644bd2e7600200f2ff95d28151f414489b11dde1f6efc8 | An 8.5-meter ladder is leaning against a vertical wall. How many meters is its base from the wall if the ladder reaches 7.5 meters up the wall? | [
"4"
] | [
"An 8.5-meter ladder is leaning against a vertical wall. How many meters is its base from the wall if the ladder reaches 7.5 meters up the wall?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54e4d7fcf061b2e10ee93950ed5012f7d2a508eef25a456405684ed795b0f1ea | An 8x8 chess board is filled in with a number in each square such that each number is equal to the average of all the numbers adjacent to it. If the lower left-hand corner square has the number 10, what are all the possible values of the upper right-hand square? | [
"10"
] | [
"An 8x8 chess board is filled in with a number in each square such that each number is equal to the average of all the numbers adjacent to it. If the lower left-hand corner square has the number 10, what are all the possible values of the upper right-hand square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0fed1336434f3b2c81f5483db940731117e06948110c3b0527ae66d42ac8f5b | An 8x8 square is divided into 64 unit squares, each covered by two isosceles triangles (one black and one white). A "fine" coloring requires that any two neighboring triangles are of different colors. How many different "fine" colorings are there? | [
"2"
] | [
"An 8x8 square is divided into 64 unit squares, each covered by two isosceles triangles (one black and one white). A \"fine\" coloring requires that any two neighboring triangles are of different colors. How many different \"fine\" colorings are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-f98c084ff3427db2e85053287faa6e6da31e4b43dadb28aa9b6d38a4fb50f1c1 | An 8×8 table has a figure in each square. In each move, a figure can be moved to a diagonally adjacent square. i) At least how many squares can be occupied at any moment? ii) At least how many squares can be free at any moment? | [
"32"
] | [
"An 8×8 table has a figure in each square. In each move, a figure can be moved to a diagonally adjacent square. \ni) At least how many squares can be occupied at any moment?\nii) At least how many squares can be free at any moment?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-0fc577b3b7c8c5a8e37684354e01ec413013f316282eade5bfb72636e9f15eb7 | An ABC triangle is divided into 4 parts by 2 lines. The area of each of the smaller triangles is 1, 3, and 3. What is the area of the initial triangle? | [
"12"
] | [
"An ABC triangle is divided into 4 parts by 2 lines. The area of each of the smaller triangles is 1, 3, and 3. What is the area of the initial triangle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f58145eb2e42d4ae82abee0204f9ea56e809b04e7fd1d10f85f2336b3d12ab2d | An AIME has 15 questions, with 5 easy, 5 medium, and 5 hard questions. Let \( e(X) \) denote the number of easy questions up to question \( X \) (including question \( X \)), \( m(X) \) the number of medium questions, and \( h(X) \) the number of hard questions. Find the number of ways to arrange the questions such tha... | [
"429"
] | [
"An AIME has 15 questions, with 5 easy, 5 medium, and 5 hard questions. Let \\( e(X) \\) denote the number of easy questions up to question \\( X \\) (including question \\( X \\)), \\( m(X) \\) the number of medium questions, and \\( h(X) \\) the number of hard questions. Find the number of ways to arrange the que... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aab6739f32be0e349e542ae41c9ca49a8e571005b08a050d6a3c6cea78ea112a | An Akash set is a set of integers that does not contain two integers such that one divides the other. Compute the minimum positive integer $n$ such that the set $\{1, 2, 3, ..., 2017\}$ can be partitioned into $n$ Akash subsets. | [
"11"
] | [
"An Akash set is a set of integers that does not contain two integers such that one divides the other. Compute the minimum positive integer $n$ such that the set $\\{1, 2, 3, ..., 2017\\}$ can be partitioned into $n$ Akash subsets."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-8dd38f993791aea5c0ae0bbcec498e5352ce9a2320201da700f2c7585f494386 | An American football field is $100$ yards long, and its width is half the average of its length and its diagonal. Find its area. | [
"\\(\\frac{16000}{3}\\)"
] | [
"An American football field is $100$ yards long, and its width is half the average of its length and its diagonal. Find its area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5d2fcee35157475e7f377c49847d2719e60436f03e168b22d1fa8706e17c48b8 | An Amoeba starts at position 0 on the number line. After each second, each Amoeba splits into two, one moving to the position +1 and the other to -1 from its current position. If two Amoebas occupy the same position, they immediately annihilate each other. Let \( f(n) \) be the number of Amoebas after \( n \) seconds. ... | [
"\\( f(n) = 2^{w(n)} \\) where \\( w(n) \\) denotes the number of ones in the binary expansion of \\( n \\)."
] | [
"An Amoeba starts at position 0 on the number line. After each second, each Amoeba splits into two, one moving to the position +1 and the other to -1 from its current position. If two Amoebas occupy the same position, they immediately annihilate each other. Let \\( f(n) \\) be the number of Amoebas after \\( n \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-85cd61367c34c78617b7787645d1dcc4b881cd889b0ed7169e2d29b0e167a082 | An AoPS classroom starts with 25 students. Every 0.5 seconds, a student either joins or leaves with probabilities $\frac{1}{5}$ and $\frac{4}{5}$, respectively. Ddot1 posts boxed text every 0.5 seconds, with the number of boxes given by $70 - \text{(number of students in the classroom)}$. The classroom crashes if there... | [
"\\(\\frac{25}{6}\\) seconds"
] | [
"An AoPS classroom starts with 25 students. Every 0.5 seconds, a student either joins or leaves with probabilities $\\frac{1}{5}$ and $\\frac{4}{5}$, respectively. Ddot1 posts boxed text every 0.5 seconds, with the number of boxes given by $70 - \\text{(number of students in the classroom)}$. The classroom crashes ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b416a3fb64d3e01cc68b9b256c706f9fd526f6702ffe677e37d03024814677f5 | An Egyptian king has taken 100 prisoners and arranges them in a straight column, placing a red or black hat on each prisoner's head. Each prisoner can see the hats of all the people in front of them but not their own or those behind them. Starting from the 100th prisoner (the one at the back), each prisoner must correc... | [
"99"
] | [
"An Egyptian king has taken 100 prisoners and arranges them in a straight column, placing a red or black hat on each prisoner's head. Each prisoner can see the hats of all the people in front of them but not their own or those behind them. Starting from the 100th prisoner (the one at the back), each prisoner must c... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f3613ee9151fef0bad6ef0bfc251f0065f422afbd34ac94f09c5fee8d42b324 | An Emperor invited 2015 wizards to a festival. Each wizard knows who is good and who is evil, but the Emperor does not. A good wizard always tells the truth, while an evil wizard can lie or tell the truth. The Emperor can ask each wizard a yes/no question and then expel one wizard, who is then revealed as good or evil.... | [
"1007"
] | [
"An Emperor invited 2015 wizards to a festival. Each wizard knows who is good and who is evil, but the Emperor does not. A good wizard always tells the truth, while an evil wizard can lie or tell the truth. The Emperor can ask each wizard a yes/no question and then expel one wizard, who is then revealed as good or ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-431de8f986d14a58ff13c56ba03c3bfa676ffd0e38e0f6ba8f344a22570d04bf | An English tourist has four ways to travel to Tombstone: 1. Ride the stagecoach all the way with a thirty-minute stopover at a way-house. 2. Walk all the way, with the coach beating him by one mile. 3. Walk to the way-house and then take the coach, arriving at the way-house just in time to catch the coach. 4. Take the ... | [
"\\dfrac{4}{3}"
] | [
"An English tourist has four ways to travel to Tombstone:\n1. Ride the stagecoach all the way with a thirty-minute stopover at a way-house.\n2. Walk all the way, with the coach beating him by one mile.\n3. Walk to the way-house and then take the coach, arriving at the way-house just in time to catch the coach.\n4. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-7f03d85f5f63ec5cbfeb4b56c494bffa737c7bbddef61c970603a3464d4eef6d | An Exooter student starts from the top right cell of a $3n \times 3n$ grid tessellated into $L$ trominos. Each $L$ tromino has two one-way doors between adjacent cells, and doors are removed from the center of an $L$ piece that aren't to the two other squares of the $L$ piece. Calculate the number of distinct cells the... | [
"25"
] | [
"An Exooter student starts from the top right cell of a $3n \\times 3n$ grid tessellated into $L$ trominos. Each $L$ tromino has two one-way doors between adjacent cells, and doors are removed from the center of an $L$ piece that aren't to the two other squares of the $L$ piece. Calculate the number of distinct cel... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7feba0e0772746d810dfabb6b14aab52d3241ecde280ce85c505eec12c95bcf0 | An Olympiad in mathematics is held at the Intellect Center. The building has 9 classrooms measuring $9$ m $\times 9$ m. Due to anti-COVID restrictions, participants in the same classroom must be at least $4.3$ m apart while solving problems. What is the maximum number of people who can write this Olympiad? | [
"81"
] | [
"An Olympiad in mathematics is held at the Intellect Center. The building has 9 classrooms measuring $9$ m $\\times 9$ m. Due to anti-COVID restrictions, participants in the same classroom must be at least $4.3$ m apart while solving problems. What is the maximum number of people who can write this Olympiad?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-9390104ee9475f612e01354417933073c088ce8cab9d1b32869330d5802926fd | An Orexes tribe sets up headquarters 30 miles away from Santa's North Pole workshop. They plan to intercept Santa's path after traveling 70 miles from their starting point at a 60-degree angle relative to the imaginary line between their headquarters and the workshop. When they intercept, Santa is already a mile beyond... | [
"\\( 1 + 10\\sqrt{37} \\)"
] | [
"An Orexes tribe sets up headquarters 30 miles away from Santa's North Pole workshop. They plan to intercept Santa's path after traveling 70 miles from their starting point at a 60-degree angle relative to the imaginary line between their headquarters and the workshop. When they intercept, Santa is already a mile b... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-856662221a61298e4ac40bf260db4c462de8e4cf244931d463bf2b4c7e245f3a | An \( 8 \times 8 \) chessboard is to be tiled with pieces of the following shapes: - \( 1 \times 3 \) rectangle (covers 3 squares) - T-shaped tetromino (covers 4 squares) (a) What is the maximum number of pieces that can be used? (b) How many ways are there to tile the chessboard using this maximum number of pieces? | [
"0"
] | [
"An \\( 8 \\times 8 \\) chessboard is to be tiled with pieces of the following shapes:\n- \\( 1 \\times 3 \\) rectangle (covers 3 squares)\n- T-shaped tetromino (covers 4 squares)\n\n(a) What is the maximum number of pieces that can be used?\n(b) How many ways are there to tile the chessboard using this maximum num... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-509bf4083bc64555ac28105c68b2bfa9e516a68dab9f5c529695ebda42efdf6f | An \( a \times b \times c \) box has half the volume of an \( (a+2) \times (b+2) \times (c+2) \) box, where \( a \leq b \leq c \). What is the largest possible value of \( c \)? | [
"130"
] | [
"An \\( a \\times b \\times c \\) box has half the volume of an \\( (a+2) \\times (b+2) \\times (c+2) \\) box, where \\( a \\leq b \\leq c \\). What is the largest possible value of \\( c \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-196602cf9a578ccbb4c8802aadca3c1fc6e59c28d8172dff56db5a844e335bbe | An \( m \) by \( n \) by \( p \) rectangular box has half the volume of an \( m + 2 \) by \( p + 2 \) by \( n + 2 \) rectangular box, where \( m \), \( n \), and \( p \) are integers, and \( m \le n \le p \). What is the largest possible value of \( p \)? | [
"130"
] | [
"An \\( m \\) by \\( n \\) by \\( p \\) rectangular box has half the volume of an \\( m + 2 \\) by \\( p + 2 \\) by \\( n + 2 \\) rectangular box, where \\( m \\), \\( n \\), and \\( p \\) are integers, and \\( m \\le n \\le p \\). What is the largest possible value of \\( p \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 6
} | 27 | 0 | 0 |
omr-ws-ae0b4ce9048c049ffe0b46bf56eb28bef5e7754a57e0c2483854dab182cb6a6c | An \( n \)-gon (which is not self-intersecting) is triangulated using \( m \) interior vertices. The triangulation consists of \( N \) triangles such that: (1) their union is the original \( n \)-gon; (2) the union of their vertices is the set consisting of the \( n \) vertices of the \( n \)-gon and the \( m \) interi... | [
"n + 2m - 2"
] | [
"An \\( n \\)-gon (which is not self-intersecting) is triangulated using \\( m \\) interior vertices. The triangulation consists of \\( N \\) triangles such that: (1) their union is the original \\( n \\)-gon; (2) the union of their vertices is the set consisting of the \\( n \\) vertices of the \\( n \\)-gon and t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b625a604e9acab867d0110c764fa1d7510b708e53d8d69dfc817a06f2337de7a | An \( n \times n \) grid has a unit square removed from one of its corners, where \( n \geq 2 \). Determine the minimum number of figures, each consisting of either 3 or 5 unit squares as depicted below, required to cover the remaining part of the grid without overlap and without extending beyond the grid boundaries. [... | [
"21"
] | [
"An \\( n \\times n \\) grid has a unit square removed from one of its corners, where \\( n \\geq 2 \\). Determine the minimum number of figures, each consisting of either 3 or 5 unit squares as depicted below, required to cover the remaining part of the grid without overlap and without extending beyond the grid bo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-9d8144c71e15ec41543b8b88d2c0ef091cd054d75242a48f79bb201f9a6a4f81 | An \( n \times n \) square grid (\( n \geqslant 3 \)) is rolled into a cylinder. Some of the cells are then colored black. Let \( f(n) \) be the minimum number of black cells needed to ensure that there exist two parallel lines (horizontal, vertical, or diagonal) of cells containing the same number of black cells. Find... | [
"11"
] | [
"An \\( n \\times n \\) square grid (\\( n \\geqslant 3 \\)) is rolled into a cylinder. Some of the cells are then colored black. Let \\( f(n) \\) be the minimum number of black cells needed to ensure that there exist two parallel lines (horizontal, vertical, or diagonal) of cells containing the same number of blac... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-649420bdf647171ebe8479cbe40b65eab7149306f1572916b9f43feed1be5497 | An \( n \times n \times n \) cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Find the number of ways to color ... | [
"2^{n^3 + 3n^2 + 3n} \\mod (10^9 + 7)"
] | [
"An \\( n \\times n \\times n \\) cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Find the number of ways ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-832e25c314ba7f57b2848d26b04ad2e77c42b5ba15da3e1d70440c2582a4dca1 | An \(a \times b \times c\) box has half the volume of an \((a+2) \times (b+2) \times (c+2)\) box, where \(a \leq b \leq c\). What is the largest possible value of \(c\)? | [
"130"
] | [
"An \\(a \\times b \\times c\\) box has half the volume of an \\((a+2) \\times (b+2) \\times (c+2)\\) box, where \\(a \\leq b \\leq c\\). What is the largest possible value of \\(c\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90e80ab857274fc8bdbda2f1027fdf99cd2561609dd714f0c14204d2b7da581d | An \(m \times n\) chessboard has the property that a knight, starting in a corner of the chessboard, can visit all \(mn\) squares without ever landing on a square twice. What is the smallest possible area of such a board, given the area is greater than 1? | [
"12"
] | [
"An \\(m \\times n\\) chessboard has the property that a knight, starting in a corner of the chessboard, can visit all \\(mn\\) squares without ever landing on a square twice. What is the smallest possible area of such a board, given the area is greater than 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a8403422e7ac461143744b606d5368b3e54dda49bff2da5849150cc73fedff4f | An \(n \times n\) table is filled with numbers \(-1, 0, 1\) such that every row and column contains exactly one \(1\) and one \(-1\). After reordering the rows and columns, the table is transformed such that each number is replaced with its negative. Determine the minimum number of reordering steps required to achieve ... | [
"2"
] | [
"An \\(n \\times n\\) table is filled with numbers \\(-1, 0, 1\\) such that every row and column contains exactly one \\(1\\) and one \\(-1\\). After reordering the rows and columns, the table is transformed such that each number is replaced with its negative. Determine the minimum number of reordering steps requir... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-379bcd38854f078b644273212ff54bdbddb97c62a3db4c163bdd68ec28eb9c43 | An absent-minded professor wrote $n$ letters and sealed them in envelopes without writing the addresses on the envelopes. He then wrote the $n$ addresses on the envelopes at random. What is the probability that at least one of the letters was addressed correctly? | [
"\\( 1 - \\frac{1}{e} \\)"
] | [
"An absent-minded professor wrote $n$ letters and sealed them in envelopes without writing the addresses on the envelopes. He then wrote the $n$ addresses on the envelopes at random. What is the probability that at least one of the letters was addressed correctly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3b080d3e0b1a2ea21249850df11b0d055e92d3e08ef718ec38bd55d61176db94 | An absent-minded professor wrote 5 letters and sealed them in envelopes without writing the addresses on the envelopes. Having forgotten which letter she had put into which envelope, she wrote the 5 addresses at random. (a) What is the total number of ways in which the addresses can be assigned? (b) How many of these w... | [
"76"
] | [
"An absent-minded professor wrote 5 letters and sealed them in envelopes without writing the addresses on the envelopes. Having forgotten which letter she had put into which envelope, she wrote the 5 addresses at random.\n(a) What is the total number of ways in which the addresses can be assigned?\n(b) How many of ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6cafb5b5b096070b8aab7af5594fee347432032cc14d386cd04c980e9d3671b | An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6? | [
"20"
] | [
"An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-840abd6ab6b38798d9c700202cdab49a47bbf1137c03b83c4ce8d0558ae1a7bf | An accountant and her boss have to travel to a nearby town. The accountant catches a train traveling at 50 mi/hr while the boss leaves 1 hour later in a car traveling at 60 mi/hr. They meet at the train station at the exact same time. If the train and the car traveled in straight lines on parallel paths, how far is it ... | [
"300 miles"
] | [
"An accountant and her boss have to travel to a nearby town. The accountant catches a train traveling at 50 mi/hr while the boss leaves 1 hour later in a car traveling at 60 mi/hr. They meet at the train station at the exact same time. If the train and the car traveled in straight lines on parallel paths, how far i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-e80678ef955c941eae9969290376ca319d170f450f96922d8322f2f10b05b3e4 | An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o'clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. Let \( f(t), g(t), h(t) \) be the respective absolute deviations of the separating angles from \( \frac{2 \cdot \pi}{3} \) at \( ... | [
"\\dfrac{\\pi}{6}"
] | [
"An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o'clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. Let \\( f(t), g(t), h(t) \\) be the respective absolute deviations of the separating angles from \\( \\frac{2 \\cdot \\pi}{3... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5f1f3ac6ac7bc1decc434a78d1b432d898ab0b5b19c98217f926fd740153bd5a | An accurate clock has an hour hand of length $a$ and a minute hand of length $b$ (where $b > a$), both measured from the pivot at the centre of the clock face. Let $x$ be the distance between the ends of the hands when the angle between the hands is $\theta$, where $0 \leq \theta < \pi$. Find the value of $x$ when the ... | [
"\\sqrt{b^2 - a^2}"
] | [
"An accurate clock has an hour hand of length $a$ and a minute hand of length $b$ (where $b > a$), both measured from the pivot at the centre of the clock face. Let $x$ be the distance between the ends of the hands when the angle between the hands is $\\theta$, where $0 \\leq \\theta < \\pi$. Find the value of $x$ ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42de5f03ef5f3d3fa3449ec4c17ca23532dd2b067fbd7d5df0556e1327dc005c | An action with a probability of success of \( p = 0.22 \) is repeated 100 times. What is the probability of getting 19, 20, 21, or 22 successes? | [
"0.349"
] | [
"An action with a probability of success of \\( p = 0.22 \\) is repeated 100 times. What is the probability of getting 19, 20, 21, or 22 successes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-ec6100965930053a99713418d35aa6c46922e48695e4a7f6797c58edabafe8f6 | An acute isosceles triangle \(ABC\) is inscribed in a circle. Tangents to the circle at points \(B\) and \(C\) meet at point \(D\). If \(\angle ABC = \angle ACB = 2\angle D\), find the measure of \(\angle A\). | [
"\\(\\frac{540}{7}\\)"
] | [
"An acute isosceles triangle \\(ABC\\) is inscribed in a circle. Tangents to the circle at points \\(B\\) and \\(C\\) meet at point \\(D\\). If \\(\\angle ABC = \\angle ACB = 2\\angle D\\), find the measure of \\(\\angle A\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-c07e17bc49093ba5c94304db068850ccb3a01c98d427aa3dbe561e894511dd7d | An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 2 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. [asy] import graph; unitsize(2 cm); pair O, A, B, C, D; O = (0,0); A = dir(90); B = dir... | [
"3/7"
] | [
"An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 2 \\angle D$ and $\\angle BAC = k \\pi$ in radians, then find $k$.\n\n[asy]\nimport graph;\n\nunitsize(2 cm);\n\npair O, A, B, C, D;\n\nO = (0... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89aebd07c73151aa503bda13eaea9971a0cc55261a2dd2f52f5ec98a7ae93a4c | An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Let $P$ be the point of concurrency of the circumcircles of $\triangle{BFK}, \triangle{CEL}$, and $\tri... | [
"10"
] | [
"An acute scalene triangle $\\triangle{ABC}$ with altitudes $\\overline{AD}, \\overline{BE},$ and $\\overline{CF}$ is inscribed in circle $\\Gamma$. Medians from $B$ and $C$ meet $\\Gamma$ again at $K$ and $L$ respectively. Let $P$ be the point of concurrency of the circumcircles of $\\triangle{BFK}, \\triangle{CEL... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aaa89a12be1dcc8abe7762b36b5cc3955daac718633444388531d34bd6a3b5f7 | An acute triangle $ABC$ has integer side lengths. It is known that $AC = BD$ where $D$ is a point on the line $BC$ such that $AD$ is perpendicular to $BC$. Find the smallest possible length of side $BC$. | [
"6"
] | [
"An acute triangle $ABC$ has integer side lengths. It is known that $AC = BD$ where $D$ is a point on the line $BC$ such that $AD$ is perpendicular to $BC$. Find the smallest possible length of side $BC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de4cc6de5f2fb95d49f51ce07e87f54811ec9327aa6597a13cb47109b4f82234 | An acute triangle \(ABC\) (\(AB \ne AC\)) is inscribed in a circle \(\omega\) with center \(O\). The point \(M\) is the midpoint of side \(BC\). The tangent line to \(\omega\) at point \(A\) intersects the line \(BC\) at point \(D\). A circle with center at point \(M\) and radius \(MA\) intersects the extensions of sid... | [
"(0.5, 0.5)"
] | [
"An acute triangle \\(ABC\\) (\\(AB \\ne AC\\)) is inscribed in a circle \\(\\omega\\) with center \\(O\\). The point \\(M\\) is the midpoint of side \\(BC\\). The tangent line to \\(\\omega\\) at point \\(A\\) intersects the line \\(BC\\) at point \\(D\\). A circle with center at point \\(M\\) and radius \\(MA\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-a64d96b8ba26c0e3e48c075b94bc8fc6b571dff94b8a059c418f8513d6ec80c0 | An acute triangle \(ABC\) has side lengths \(a\), \(b\), and \(c\) such that \(a\), \(b\), and \(c\) form an arithmetic sequence. Given that the area of triangle \(ABC\) is an integer, what is the smallest value of its perimeter? | [
"42"
] | [
"An acute triangle \\(ABC\\) has side lengths \\(a\\), \\(b\\), and \\(c\\) such that \\(a\\), \\(b\\), and \\(c\\) form an arithmetic sequence. Given that the area of triangle \\(ABC\\) is an integer, what is the smallest value of its perimeter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 11
} | 20 | 0 | 0 |
omr-ws-bde2528cb5590c43a497460c5a1029e330e89a59bd2317e04d9d3bd3cab32bf0 | An acute triangle \(ABC\) is inscribed in a circle \(\omega\) centered at a point \(O\). Extensions of its altitudes from vertices \(A\) and \(C\) intersect the circle \(\omega\) again at points \(A_0\) and \(C_0\) respectively. The line \(A_0C_0\) intersects sides \(AB\) and \(BC\) at points \(A_1\) and \(C_1\) respec... | [
"5"
] | [
"An acute triangle \\(ABC\\) is inscribed in a circle \\(\\omega\\) centered at a point \\(O\\). Extensions of its altitudes from vertices \\(A\\) and \\(C\\) intersect the circle \\(\\omega\\) again at points \\(A_0\\) and \\(C_0\\) respectively. The line \\(A_0C_0\\) intersects sides \\(AB\\) and \\(BC\\) at poin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
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