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-c5bfc33b501883008b7e9284ef4be5b78ce6815d143f41a7c68229119d06fd38 | A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full price tickets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-ab63cc3bb599e39a6ebe6270b0596a5b793d75cebfdc4f58e8d79ba62d9bf817 | A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets? | [
"782"
] | [
"A charity sells 140 benefit tickets for a total of 2001 dollars. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-800c39cfd59c9e561efa343eaef0440fc1568ac0bb00c3fac708c1ad23d6294f | A charity sells 140 benefit tickets for a total of \$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets? | [
"782",
"\\$782"
] | [
"A charity sells 140 benefit tickets for a total of \\$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 45
} | 77 | 0 | 0 |
omr-ws-47a134816b75621a96cf40a30efca42727afe70ffaf15cd8bf1fc762dd1d87b2 | A charmander is traveling along the lines of a regular coordinate grid with an increment of one unit from the origin (0,0) to the point (8,8). There is a lake defined by the equation $(x-4)^2+(y-4)^2=4$ that the charmander cannot cross. If the charmander can only move in the direction of the Pokécenter, how many paths ... | [
"472"
] | [
"A charmander is traveling along the lines of a regular coordinate grid with an increment of one unit from the origin (0,0) to the point (8,8). There is a lake defined by the equation $(x-4)^2+(y-4)^2=4$ that the charmander cannot cross. If the charmander can only move in the direction of the Pokécenter, how many p... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65c5f8c04a09672b3d1302e78be9610839b4efbab92ce8d6c189e1ef77593f50 | A charter plane service needs to provide flights for at least 500 first class, 1000 tourist-class, and 1500 economy class passengers. The company has two types of planes: Type A costs \$12,000 per flight and carries 50 first class, 60 tourist class, and 100 economy class passengers. Type B costs \$10,000 per flight and... | [
"17 \\text{ Type A and } 0 \\text{ Type B}"
] | [
"A charter plane service needs to provide flights for at least 500 first class, 1000 tourist-class, and 1500 economy class passengers. The company has two types of planes: Type A costs \\$12,000 per flight and carries 50 first class, 60 tourist class, and 100 economy class passengers. Type B costs \\$10,000 per fli... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-4313c099cff88f7dd271ebd065994cc78cd68ac68f5ef6df268a83c1e3785015 | A checker is at point $(1,1)$ on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger one. Determine the necessary and sufficient conditions on the lattice points such that the checker can reach them. | [
"\\(\\gcd(x, y) = 2^n\\) for some integer \\(n\\)"
] | [
"A checker is at point $(1,1)$ on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger one. Determine the necessary and sufficient conditions on the lattice points such that the checker can reach them."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 15
} | 34 | 0 | 0 |
omr-ws-3610029dcab84b3faeae5f918c3cecbbfb27b595548d0be55814b5f19261a650 | A checker is at point (1,1) on a lattice with positive integer coordinates. At each move, it can either double one coordinate or subtract the smaller coordinate from the larger. Which points can the checker reach? | [
"The checker can reach any point (m, n) where the GCD of m and n is a power of 2."
] | [
"A checker is at point (1,1) on a lattice with positive integer coordinates. At each move, it can either double one coordinate or subtract the smaller coordinate from the larger. Which points can the checker reach?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-cb29b0cbde9b7ae2cd01bf6ef8ddb99bc869965784ed15b3841cf57668d3870d | A checker is at point (1,1) on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger. Let $S$ be the set of all coordinates $(x,y)$ that the checker can reach. Find the sum of all distinct values of $x+y$ for $(x,y) \in S$ where ... | [
"20099"
] | [
"A checker is at point (1,1) on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger. Let $S$ be the set of all coordinates $(x,y)$ that the checker can reach. Find the sum of all distinct values of $x+y$ for $(x,y) \\in S$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4c8e9db2edfa9bbfef7f5f699716e97cde30986d97ad83929be247fc9d1725c9 | A checker is placed at point \((1,1)\) on a lattice where \(x\) and \(y\) are positive integers. The checker can move by either doubling one of its coordinates or subtracting the smaller coordinate from the larger coordinate. Which points on the lattice can the checker reach? | [
"\\((2^lp, 2^lq)\\) for any \\(l, p, q \\in \\mathbb{Z}^+_0\\) with \\(\\gcd(p,q) = 1\\)"
] | [
"A checker is placed at point \\((1,1)\\) on a lattice where \\(x\\) and \\(y\\) are positive integers. The checker can move by either doubling one of its coordinates or subtracting the smaller coordinate from the larger coordinate. Which points on the lattice can the checker reach?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-618b8ed0c449fea308b72d1612866fc5a2baedfcebe336c5c721242af2527261 | A checker is placed in the central cell of a $2003 \times 2003$ table. Two players move it alternately to one of the neighboring cells (cells sharing a common side). The first player can move the checker forward or turn it left. The second player can move the checker forward or turn it right. The player who cannot move... | [
"Second player has a winning strategy."
] | [
"A checker is placed in the central cell of a $2003 \\times 2003$ table. Two players move it alternately to one of the neighboring cells (cells sharing a common side). The first player can move the checker forward or turn it left. The second player can move the checker forward or turn it right. The player who canno... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 5,
"tir": 12
} | 42 | 0 | 0 |
omr-ws-acb0d74c32e01bac9f2774edfa3fae25e01bf7b7dd972c965c9b0f17fb81b565 | A checker is placed on a square of an infinite checkerboard, where each square is $1$ cm by $1$ cm. It moves according to the following rules: 1. In the first move, the checker moves $1$ square North. 2. All odd-numbered moves are North or South, and all even-numbered moves are East or West. 3. In the $n$-th move, the ... | [
"2"
] | [
"A checker is placed on a square of an infinite checkerboard, where each square is $1$ cm by $1$ cm. It moves according to the following rules:\n1. In the first move, the checker moves $1$ square North.\n2. All odd-numbered moves are North or South, and all even-numbered moves are East or West.\n3. In the $n$-th mo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-afa0421b42f3ed58ca0f7bd65e25c4f30e983c2fddae7ea1343246f42e0515fb | A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules: - In the first move, the checker moves 1 square North. - All odd-numbered moves are North or South and all even-numbered moves are East or West. - In the n-th move, the checker make... | [
"\\(\\sqrt{\\left( \\sum_{i=1, \\text{odd}}^{n} i \\right)^2 + \\left( \\sum_{i=2, \\text{even}}^{n} i \\right)^2}\\)"
] | [
"A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules:\n- In the first move, the checker moves 1 square North.\n- All odd-numbered moves are North or South and all even-numbered moves are East or West.\n- In the n-th move, the chec... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-8b11875e7e3ac40441dcb63cabd14ab217511aaaf7c7693ce1045256c31d24ef | A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules: - In the first move, the checker moves 1 square North. - All odd-numbered moves are North or South and all even-numbered moves are East or West. - In the n-th move, the checker make... | [
"\\(6\\sqrt{2}\\)"
] | [
"A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules:\n- In the first move, the checker moves 1 square North.\n- All odd-numbered moves are North or South and all even-numbered moves are East or West.\n- In the n-th move, the chec... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-203f612f2d59a0c860cc264ce32a8b49be7280f135df1319b56a9d0e54d5b3f9 | A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules: - In the first move, the checker moves 1 square North. - All odd-numbered moves are North or South, and all even-numbered moves are East or West. - In the n-th move, the checker mak... | [
"} 6\\sqrt{2} \\]"
] | [
"A checker is placed on a square of an infinite checkerboard, where each square is 1 cm by 1 cm. It moves according to the following rules:\n- In the first move, the checker moves 1 square North.\n- All odd-numbered moves are North or South, and all even-numbered moves are East or West.\n- In the n-th move, the che... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4e82eb51bb73525c1b5d4f96fb341a0ba5c7e7e8ecde7e6514ef1a93338edfa | A checker moves from the lower left corner of a $100 \times 100$ board to the upper right corner, moving one cell to the right or one cell up at each step. Let $a$ be the number of paths in which exactly 70 steps are below the diagonal from the lower left corner to the upper right corner, and let $b$ be the number of p... | [
"a = b"
] | [
"A checker moves from the lower left corner of a $100 \\times 100$ board to the upper right corner, moving one cell to the right or one cell up at each step. Let $a$ be the number of paths in which exactly 70 steps are below the diagonal from the lower left corner to the upper right corner, and let $b$ be the numbe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2f294e41120d130b39796a97749d1946c87ff90e7db4454ba263c303ff6f7bd | A checker piece of radius 1 is placed at an arbitrary location on an infinitely large checkerboard of $1 \times 1$ squares. The checker piece covers $n$ squares, either partially or completely. How many possible values for $n$ are there? | [
"4"
] | [
"A checker piece of radius 1 is placed at an arbitrary location on an infinitely large checkerboard of $1 \\times 1$ squares. The checker piece covers $n$ squares, either partially or completely. How many possible values for $n$ are there?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1c66cb756374a2263e242ad276f2393716d0285a3b9f6dc5195ae94266587c7 | A checker starts at point (1,1) on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger. Which points (x,y) on the lattice can the checker reach? | [
"\\(\\gcd(x, y) = 2^n\\) for some non-negative integer \\(n\\)."
] | [
"A checker starts at point (1,1) on a lattice with positive integer coordinates. It can move by either doubling one coordinate or subtracting the smaller coordinate from the larger. Which points (x,y) on the lattice can the checker reach?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-ef8f1d5be2e3165f5b46f94e2ca9a629f7440c3a928db1a3dbc2662527df90ed | A checker starts at the origin $(0,0)$ on an infinite lattice. The checker can only move to a point $(x,y)$ if the greatest common divisor of $x$ and $y$ is a power of $2$. Let $N$ be the number of distinct points $(x,y)$ with $0 \leq x, y \leq 100$ that the checker can reach. Find the remainder when $N$ is divided by ... | [
"622"
] | [
"A checker starts at the origin $(0,0)$ on an infinite lattice. The checker can only move to a point $(x,y)$ if the greatest common divisor of $x$ and $y$ is a power of $2$. Let $N$ be the number of distinct points $(x,y)$ with $0 \\leq x, y \\leq 100$ that the checker can reach. Find the remainder when $N$ is divi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7b24474c3f9e538c7ee3b84d6696046868a9054d0440ed69b0a0abe31c42366 | A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. What is the maximum possible value of $n$? | [
"12 or more\n\nExplanation:\nPost 2 provides a diagram and explanation showing that by placing the square card at a 45-degree angle, it can cover parts of 12 squares. This aligns with the official solution mentioned in Post 3. Therefore, the maximum possible value of \\( n \\) is 12 or more."
] | [
"A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. What is the maximum possible value of $n$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c82fd23ad5e21723bb4c9fbc622d36392576ec07c6476f8901ca9ec0226223e | A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, so that the first row is numbered $1, 2, \ldots, 17$, the second row $18, 19, \ldots, 34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1, 2, \ldots, 13$, ... | [
"555"
] | [
"A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, so that the first row is numbered $1, 2, \\ldots, 17$, the second row $18, 19, \\ldots, 34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1, 2, \\ldot... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-208482c9c71de173cb44612770f4d59f7791d39cb1df247037dff43c62ad173b | A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, so that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the second colu... | [
"555"
] | [
"A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, so that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the second... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-2a66f226565263d27e48d877c0fdf742661fca3b7d66cc99ee1985f86716f90f | A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, such that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the second co... | [
"555"
] | [
"A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper left corner, such that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the seco... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-bda8086d8ea2d6052cc6ddd6a6722f02eb1e70c492729b39c827ffad2b30b6f0 | A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper-left corner, so that the first row is numbered $1, 2, \ldots, 17$, the second row $18, 19, \ldots, 34$, and so on down the board. If the board is renumbered so that the left column, top to bottom is $1, 2, \ldots, 13$, t... | [
"555"
] | [
"A checkerboard of 13 rows and 17 columns has numbers written in each square, beginning in the upper-left corner, so that the first row is numbered $1, 2, \\ldots, 17$, the second row $18, 19, \\ldots, 34$, and so on down the board. If the board is renumbered so that the left column, top to bottom is $1, 2, \\ldots... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-5307996ad385c586484b879adedc257d1ccf49bfe89b2d65857873999cb236b8 | A checkerboard of 13 rows and 17 columns has numbers written in each square, starting from the upper left corner such that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the second co... | [
"555"
] | [
"A checkerboard of 13 rows and 17 columns has numbers written in each square, starting from the upper left corner such that the first row is numbered 1, 2, ..., 17, the second row 18, 19, ..., 34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1, 2, ..., 13, the seco... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-2b89230906d141a8d660b4925486ce87c663951a50b6089fc2e978d6d9a0b081 | A checkerboard with size \( 109 \times 21 \) will be tiled with some tiles with size \( 3 \times 1 \). What is the maximum number of tiles which can be placed on the board so that no two tiles overlap, or touch (touching on each tile's vertices are also not allowed)? | [
"297"
] | [
"A checkerboard with size \\( 109 \\times 21 \\) will be tiled with some tiles with size \\( 3 \\times 1 \\). What is the maximum number of tiles which can be placed on the board so that no two tiles overlap, or touch (touching on each tile's vertices are also not allowed)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-e2ed612e80c64f34cddf357a02ff7f476bae2d189e93b2991b41c8b6bb1e168b | A checkered grid of $6m \times 6m$ was made from fickford cord with cells one meter each. One meter of cord burns in one minute. What is the shortest time it takes for the cord to burn out with four fuses? | [
"3"
] | [
"A checkered grid of $6m \\times 6m$ was made from fickford cord with cells one meter each. One meter of cord burns in one minute. What is the shortest time it takes for the cord to burn out with four fuses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-b558821de66c5326976811f80bc2858ef5fbd39450bda9807de23b19bf20c69e | A checkered polygon \( A \) is drawn on the checkered plane. A cell of \( A \) is called internal if all 8 of its adjacent cells belong to \( A \). All other cells of \( A \) are called boundary. It is known that: 1. Each boundary cell has exactly two common sides with no boundary cells. 2. The union of all boundary ce... | [
"2"
] | [
"A checkered polygon \\( A \\) is drawn on the checkered plane. A cell of \\( A \\) is called internal if all 8 of its adjacent cells belong to \\( A \\). All other cells of \\( A \\) are called boundary. It is known that:\n1. Each boundary cell has exactly two common sides with no boundary cells.\n2. The union of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-839e2440d3e854b20208d565524692e7b56d17b8eaef586e62b29ee2c94cbaeb | A checkered square is divided into several rectangular cells, where all rectangles have the same perimeter but are not necessarily the same size. Find the smallest possible size of the square. | [
"4"
] | [
"A checkered square is divided into several rectangular cells, where all rectangles have the same perimeter but are not necessarily the same size. Find the smallest possible size of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-436e1501b5d584199618a609c1b4226d8d54464b6b8ed90549be078b25050ba0 | A cheesecake is taken out of the oven with an ideal internal temperature of 170 degrees Fahrenheit and is placed into a 45-degree Fahrenheit refrigerator. After 8 minutes, the cheesecake has cooled down to 155 degrees Fahrenheit. If we want to wait until the cheesecake is at 70 degrees Fahrenheit for internal temperatu... | [
"92.72"
] | [
"A cheesecake is taken out of the oven with an ideal internal temperature of 170 degrees Fahrenheit and is placed into a 45-degree Fahrenheit refrigerator. After 8 minutes, the cheesecake has cooled down to 155 degrees Fahrenheit. If we want to wait until the cheesecake is at 70 degrees Fahrenheit for internal temp... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3e8952e33bfed83c4db3d1506ff2786a6be284213665c8a93fe453af337e40bd | A cheetah is chasing a gazelle. A hunter shoots the cheetah every 5 seconds. The gazelle runs 200 meters per 5 seconds, and the cheetah runs 100, 200, 300, or 400 meters per 5 seconds (changing speed randomly every 5 seconds). The initial distance between the gazelle and the cheetah is 100 meters. What is the minimum h... | [
"\\(1 - (0.5)^{0.25}\\)"
] | [
"A cheetah is chasing a gazelle. A hunter shoots the cheetah every 5 seconds. The gazelle runs 200 meters per 5 seconds, and the cheetah runs 100, 200, 300, or 400 meters per 5 seconds (changing speed randomly every 5 seconds). The initial distance between the gazelle and the cheetah is 100 meters. What is the mini... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-ba638f12a6a22feea8b0730fef4f1fe85ea408ebee58d320d1d253ff00968d84 | A cheetah starts to run in a 720-mile loop at 80 miles per hour. J immediately starts to chase after it, starting at the same point, at 10 miles per hour at 12:00 PM, but realizes one hour later that it would be more wise to turn around and run in the opposite direction in the loop, so he does this. Assuming both maint... | [
"20:13 PM"
] | [
"A cheetah starts to run in a 720-mile loop at 80 miles per hour. J immediately starts to chase after it, starting at the same point, at 10 miles per hour at 12:00 PM, but realizes one hour later that it would be more wise to turn around and run in the opposite direction in the loop, so he does this. Assuming both ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c6117de165ea2c887d547b7533db4b0073fa6f5217ab68edb7c206f2f0e820cb | A chemist has 80 ml of a solution containing 20% acid. How many ml must be removed and replaced by pure acid in order to obtain a 40% solution? | [
"\\( \\frac{40}{3} \\) or approximately 13.33 ml"
] | [
"A chemist has 80 ml of a solution containing 20% acid. How many ml must be removed and replaced by pure acid in order to obtain a 40% solution?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c6b175c3dd04540fb3bfb35496da9abf506b37d9addebfc63a089e4a292ef87 | A chemist has \( m \) ounces of a solution that is \( m\% \) salt. How many ounces of salt must be added to make a solution that is \( 2m\% \) salt? | [
"\\(\\frac{m^2}{100 - 2m}\\)"
] | [
"A chemist has \\( m \\) ounces of a solution that is \\( m\\% \\) salt. How many ounces of salt must be added to make a solution that is \\( 2m\\% \\) salt?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-e1dc8d65b85c2852385cc3f0c98b98ede6099811114297539feabd3d7899383c | A chess board has $n$ squares marked. John attempts to mark all $100$ squares of the board following the rule: If each of two marked squares shares one common segment with a third square, then John can mark that third square. Find the minimum value of $n$ so that John can mark the whole board. | [
"10"
] | [
"A chess board has $n$ squares marked. John attempts to mark all $100$ squares of the board following the rule: If each of two marked squares shares one common segment with a third square, then John can mark that third square. Find the minimum value of $n$ so that John can mark the whole board."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-cb05762fb089a4899013fe93cc569bb7bec92ca9c278d1a1065e7ee82c3b1846 | A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G > B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$, and both players gain $1/2$ point when a tie occ... | [
"4"
] | [
"A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G > B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$, and both players gain $1/2$ point when a ti... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-83c6141b960c1d10cb2347d3a548af3a02b804ae873af9b71c99c7783d7849e3 | A chess competition had six participants and one two-way tie. How many different final rankings are possible? | [
"1800"
] | [
"A chess competition had six participants and one two-way tie. How many different final rankings are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-397e994223b81e5dcd4fec2b468174c9c46cbf0a27fb6898767c0a8df1680846 | A chess contest is held among 10 players in a single round (each of two players will have a match). The winner of each game earns 2 points while the loser earns none, and each of the two players will get 1 point for a draw. After the contest, none of the 10 players gets the same score, and the player of the second plac... | [
"16"
] | [
"A chess contest is held among 10 players in a single round (each of two players will have a match). The winner of each game earns 2 points while the loser earns none, and each of the two players will get 1 point for a draw. After the contest, none of the 10 players gets the same score, and the player of the second... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6beb51721236adbc9864f691a94f10532220b944a4172a650323db59714fa83e | A chess king starts from a cell and visits each cell exactly once before returning to the original position, forming a closed path. Let $d$ be the number of diagonal moves the king makes. If the chessboard is an $8 \times 8$ board, calculate the remainder when $d$ is divided by $4$. | [
"0"
] | [
"A chess king starts from a cell and visits each cell exactly once before returning to the original position, forming a closed path. Let $d$ be the number of diagonal moves the king makes. If the chessboard is an $8 \\times 8$ board, calculate the remainder when $d$ is divided by $4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-69ab61044bda480d98597d84d060f8ad5ee4e8e68a6482a5942e30803a406628 | A chess king starts on one square of an 8 × 8 chess board and travels to each square exactly once then returns to the original square without moving diagonally. Let $H$ be the number of horizontal moves and $V$ be the number of vertical moves in this tour. Find the absolute difference $|H - V|$. | [
"0"
] | [
"A chess king starts on one square of an 8 × 8 chess board and travels to each square exactly once then returns to the original square without moving diagonally. Let $H$ be the number of horizontal moves and $V$ be the number of vertical moves in this tour. Find the absolute difference $|H - V|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-f7660e9bbf8dc54deac567ee66b9036182cd43c089006839c7d7aaa1277063ef | A chess king tours an entire $8 \times 8$ chessboard by visiting each square exactly once and returning to the starting position. Let $D$ be the number of diagonal moves the king makes during this tour. Find the remainder when $D$ is divided by $2$. | [
"0"
] | [
"A chess king tours an entire $8 \\times 8$ chessboard by visiting each square exactly once and returning to the starting position. Let $D$ be the number of diagonal moves the king makes during this tour. Find the remainder when $D$ is divided by $2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26a6f2178d6a25b746842c8758bdd367bbda32aca5395eedf9e1d42c2f635399 | A chess piece can jump 8 or 9 squares either vertically or horizontally on a $15 \times 15$ board. It cannot visit the same square twice. What is the maximum number of squares this piece can visit? | [
"169"
] | [
"A chess piece can jump 8 or 9 squares either vertically or horizontally on a $15 \\times 15$ board. It cannot visit the same square twice. What is the maximum number of squares this piece can visit?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-6361c083384ac74f2407b4b8af15ce055e0249481c1dea539d10d583d4f3032b | A chess player plays a minimum of once every day for seven days with a total of $m$ matches. The maximum value of $m$ such that there are two or more consecutive days with a total of four matches is ... | [
"13"
] | [
"A chess player plays a minimum of once every day for seven days with a total of $m$ matches. The maximum value of $m$ such that there are two or more consecutive days with a total of four matches is ..."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-298c64b6f2e9bcdf4ea24e0d0e127b38d8b9aabef2214ae64d655ae43d12f37c | A chess player plays at least 1 game a day and not more than 10 games per week for 77 consecutive days. Calculate the total number of distinct periods of consecutive days during which the player plays exactly 21 games. | [
"57"
] | [
"A chess player plays at least 1 game a day and not more than 10 games per week for 77 consecutive days. Calculate the total number of distinct periods of consecutive days during which the player plays exactly 21 games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f5676a3bd9ce5b426c76c36c7f35c90cb45a29919a4d80f88e1b07e43eda18d9 | A chess rook attacks all pieces in orthogonal (horizontal or vertical) directions. I set up a rule as thus: When placed, a chess rook can attack only an odd number of rooks (exception for the first rook). An attacked rook counts as one which is the first in the line of sight. What is the maximum number of rooks possibl... | [
"31"
] | [
"A chess rook attacks all pieces in orthogonal (horizontal or vertical) directions. I set up a rule as thus: When placed, a chess rook can attack only an odd number of rooks (exception for the first rook). An attacked rook counts as one which is the first in the line of sight. What is the maximum number of rooks po... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-4c9edc49caa060e3589df37fd0a222d6f2d8c0aa417dc99787bef16a3e20875d | A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many boys are on the chess team? | [
"6"
] | [
"A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many boys are on the chess team?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5cdf5683e77cb6c6140f3751f6fc1e5dfe2769bd75b0395ddd2b8d9d438f3b5 | A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many girls are on the chess team? | [
"20"
] | [
"A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many girls are on the chess team?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5069c838f20cfc5ac03d1afe7af1f9dc0e7e3dbe8ecf460944a177b70d2832c2 | A chessboard has 64 cells painted black and white in the usual way. A bishop path is a sequence of distinct cells such that two consecutive cells have exactly one common point. At least how many bishop paths can the set of all white cells be divided into? | [
"4"
] | [
"A chessboard has 64 cells painted black and white in the usual way. A bishop path is a sequence of distinct cells such that two consecutive cells have exactly one common point. At least how many bishop paths can the set of all white cells be divided into?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-b11abd39636cb1dbfad7852c2c318a40bb8a762ece82fd5d2256160bc924c785 | A chessboard has squares of side length $1$ unit. A circle is drawn through the four corners of each black square and its interior is coloured black. What is the area of the remaining white space? | [
"\\( 59 - \\frac{27\\pi}{2} \\)"
] | [
"A chessboard has squares of side length $1$ unit. A circle is drawn through the four corners of each black square and its interior is coloured black. What is the area of the remaining white space?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf170b613cfb21c96c68f06822172d3a4cd325be96392d49d982f43c07780b7f | A chessboard is the set \( C = \{(i,j), 1 \leq i, j \leq 8\} \). We place 21 \( 3 \times 1 \) tiles on the chessboard such that no two tiles overlap. Determine which points in \( C \) cannot be covered by any tile. | [
"\\((1,1), (1,4), (1,7), (4,1), (4,4), (4,7), (7,1), (7,4), (7,7)\\) (one of these points)"
] | [
"A chessboard is the set \\( C = \\{(i,j), 1 \\leq i, j \\leq 8\\} \\). We place 21 \\( 3 \\times 1 \\) tiles on the chessboard such that no two tiles overlap. Determine which points in \\( C \\) cannot be covered by any tile."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 3,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-855dbc1f6469c5a97b98d808dd998b0899ff808fc505211be3de88532d40ecac | A chessboard polygon is defined as a polygon on a chessboard whose sides are along the grid lines. A tasteful tiling of a chessboard polygon by dominoes means that no two dominoes form a 2x2 square. If a chessboard polygon can be tiled by dominoes, find the maximum number of dominoes that can be placed in a tasteful ti... | [
"6"
] | [
"A chessboard polygon is defined as a polygon on a chessboard whose sides are along the grid lines. A tasteful tiling of a chessboard polygon by dominoes means that no two dominoes form a 2x2 square. If a chessboard polygon can be tiled by dominoes, find the maximum number of dominoes that can be placed in a tastef... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-ce8ad48dde4f3a1a8ecefc58ffb2ac42f4d79907b08c8780eaaf07f8ffdf99c5 | A chessboard polygon is said to be tastefully tiled by dominoes if no two dominoes share a long edge. Given a chessboard polygon that can be tiled by dominoes, calculate the minimum number of dominoes required to tile it tastefully, assuming the polygon consists of 20 squares. | [
"10"
] | [
"A chessboard polygon is said to be tastefully tiled by dominoes if no two dominoes share a long edge. Given a chessboard polygon that can be tiled by dominoes, calculate the minimum number of dominoes required to tile it tastefully, assuming the polygon consists of 20 squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e4a0b8bfd818dc50ade9d20bf58b9211cd3ce8c7ef0f74d1c32511d125d3c4c | A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games in total. Let $N$ be the number of different sequences of successive days on which he can play exactly 21 games. Find the minimum possible value of $N$. | [
"0"
] | [
"A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games in total. Let $N$ be the number of different sequences of successive days on which he can play exactly 21 games. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-c0c49c7df8864663829866b9a4ed348f7618bf803fd4162f808e6c813f09ec5f | A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games in total. Let $N$ be the number of different sequences of successive days on which he plays exactly 21 games. Find the minimum possible value of $N$. | [
"6"
] | [
"A chessmaster has 77 days to prepare for a tournament. He wants to play at least one game per day, but not more than 132 games in total. Let $N$ be the number of different sequences of successive days on which he plays exactly 21 games. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-855eb4555ae9af5b7a689af4cd0295b13c310c1d99af836a19ab84baa39f1542 | A chicken breeder sold 72 chickens for *679* escudos (the digits of the units and tens of thousands were illegible). What was the price per chick? | [
"511"
] | [
"A chicken breeder sold 72 chickens for *679* escudos (the digits of the units and tens of thousands were illegible). What was the price per chick?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-5a26a3525bfecfc71d6b8960a8505bea09568b7ad7647734a4528bfa1d764a6a | A chicken puff costs 5 dollars and a curry puff costs 3 dollars, while three blueberry puffs cost 1 dollar. John bought 100 puffs altogether for a total of 100 dollars. How many ways could he have bought the different puffs? He is NOT required to buy at least one of each. | [
"4"
] | [
"A chicken puff costs 5 dollars and a curry puff costs 3 dollars, while three blueberry puffs cost 1 dollar. John bought 100 puffs altogether for a total of 100 dollars. How many ways could he have bought the different puffs? He is NOT required to buy at least one of each."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a73d9ec235c7a2ac31cada36bf03b2a4e753bbd3f90617fc766a2daf1a13a89d | A child bought 9 cakes and sweets altogether. Each cake had a discount of dollars, and each sweet had a discount of 3 dollars. The child saved a total of 24 dollars. How many cakes and how many sweets did the child buy? | [
"c = 3, s = 6"
] | [
"A child bought 9 cakes and sweets altogether. Each cake had a discount of dollars, and each sweet had a discount of 3 dollars. The child saved a total of 24 dollars. How many cakes and how many sweets did the child buy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 26
} | 31 | 0 | 0 |
omr-ws-e43b9cbb8e88f470e17c8fbbde3e498c4523ec302fe429f61dc3c7ecfdd525e0 | A child enters a carousel at the point (0, 1) and the carousel revolves counter-clockwise, completing one full revolution in one minute. What are the coordinates of the child after 105 seconds? (Round your answer to three decimal places.) | [
"(1.000, 0.000)"
] | [
"A child enters a carousel at the point (0, 1) and the carousel revolves counter-clockwise, completing one full revolution in one minute. What are the coordinates of the child after 105 seconds? (Round your answer to three decimal places.)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a20e697c459a60d29a661183dc4390c25ceb5e4448f033840f8dadf8ce9874c4 | A child enters a carousel at the point (0, 1) and the carousel revolves counter-clockwise, taking one minute to complete one revolution. After how many seconds will the child have coordinates (−0.707, −0.707) if the ride lasts 5 minutes? (Round your answers to the nearest tenth of a second. Enter your answers as a comm... | [
"37.5, 97.5, 157.5, 217.5, 277.5"
] | [
"A child enters a carousel at the point (0, 1) and the carousel revolves counter-clockwise, taking one minute to complete one revolution. After how many seconds will the child have coordinates (−0.707, −0.707) if the ride lasts 5 minutes? (Round your answers to the nearest tenth of a second. Enter your answers as a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22919be2fa561f86b187c1aa2a47c690c6382f47e11bf604c1c4fa86812a1909 | A child filled the cells of a $3 \times 3$ table by the numbers $1, 2, 3, \dots, 9$ so that each number is used once. He circled the median of the three numbers in each row, and found that the median of the three circled numbers is 5. In how many different ways could he fill the table? | [
"576"
] | [
"A child filled the cells of a $3 \\times 3$ table by the numbers $1, 2, 3, \\dots, 9$ so that each number is used once. He circled the median of the three numbers in each row, and found that the median of the three circled numbers is 5. In how many different ways could he fill the table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9174246a81117d24cd3088fcff80507fd38370d3805c1696e741eef15ee3c7a9 | A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the "plastic medium red circle" in exactly two ways? | [
"29"
] | [
"A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the \"plastic medium red circle\" in exactly two ways?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a3c82f8995a4c3353e6848143d284eb53f242109ebdc7ee59ab255e931c88b9e | A child has four colors: red, yellow, blue, and green. There are 10 regions in a given picture, and each region must be colored such that no two adjacent regions share the same color. Rotations and reflections of a coloring are considered distinct. Find the remainder when the number of ways to color the regions, denote... | [
"464"
] | [
"A child has four colors: red, yellow, blue, and green. There are 10 regions in a given picture, and each region must be colored such that no two adjacent regions share the same color. Rotations and reflections of a coloring are considered distinct. Find the remainder when the number of ways to color the regions, d... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd653b197b6015cb3f1440f64fc84be8b212abe58d5862727a9b0a36a7aa99fe | A child holds two balloons in one hand by means of a pair of strings at an angle. The first balloon is oriented to the north with an angle of elevation of \(21^\circ\) and the string that supports it is \(3\sqrt{5}\) m. The second balloon is oriented to the south with an angle of elevation of \(24^\circ\) and the strin... | [
"15"
] | [
"A child holds two balloons in one hand by means of a pair of strings at an angle. The first balloon is oriented to the north with an angle of elevation of \\(21^\\circ\\) and the string that supports it is \\(3\\sqrt{5}\\) m. The second balloon is oriented to the south with an angle of elevation of \\(24^\\circ\\)... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1eae16660b26e671b8396abdd979f4e950fe4412f53970ed0ba0aced6693d7c5 | A child is standing 10 ft away from a statue mounted on a pedestal. The angle of elevation to the head of the statue is 60 degrees, and to the foot of the statue is 30 degrees. If the child is 4 ft tall, what is the total height of the statue and the pedestal? | [
"4 + 10\\sqrt{3}",
"\\( 10\\sqrt{3} + 4 \\)"
] | [
"A child is standing 10 ft away from a statue mounted on a pedestal. The angle of elevation to the head of the statue is 60 degrees, and to the foot of the statue is 30 degrees. If the child is 4 ft tall, what is the total height of the statue and the pedestal?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 24
} | 56 | 0 | 0 |
omr-ws-4b04f1758daf1d0abf22cba62e84a1e5de66d7dbe1f7fd20644084cab4a9d8b1 | A child is standing 10 ft away from a statue mounted on a pedestal. The angle of elevation to the head of the statue is 60 degrees, and to the foot of the statue is 30 degrees. What is the height of the statue? | [
"\\(\\frac{20\\sqrt{3}}{3}\\)",
"\\dfrac{20\\sqrt{3}}{3}"
] | [
"A child is standing 10 ft away from a statue mounted on a pedestal. The angle of elevation to the head of the statue is 60 degrees, and to the foot of the statue is 30 degrees. What is the height of the statue?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 29
} | 61 | 0 | 0 |
omr-ws-63f74f9d5d563671dbff58d1babc177a5a055cbd0e6d537be7dcdb7babc6fdc9 | A child receives an amount of money that can be received in three different ways: - $1$ dollar in advance and the rest in monthly installments of $39$ dollars. - $10$ dollars in advance and the rest in monthly installments of $11$ dollars. - $2$ dollars in advance and the rest in monthly installments of $7$ dollars. De... | [
"1990"
] | [
"A child receives an amount of money that can be received in three different ways:\n- $1$ dollar in advance and the rest in monthly installments of $39$ dollars.\n- $10$ dollars in advance and the rest in monthly installments of $11$ dollars.\n- $2$ dollars in advance and the rest in monthly installments of $7$ dol... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-2265f768a0501307c539406789bba5877fce38a7dc8eef44e40e0937f9ddc278 | A child throws a coin with 1 on one side and 2 on the other side. If 1 comes up, he goes 1 step, and if 2 comes up, he goes 2 steps. Find the probability that the child reaches exactly $10^9$ steps. | [
"\\(\\frac{2}{3} + \\frac{1}{2^{10^9} \\times 3}\\)"
] | [
"A child throws a coin with 1 on one side and 2 on the other side. If 1 comes up, he goes 1 step, and if 2 comes up, he goes 2 steps. Find the probability that the child reaches exactly $10^9$ steps."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 7,
"tir": 25
} | 75 | 0 | 0 |
omr-ws-ba6ef32a3b6d0ed7cbef7de810330affac39583daf671c29ec892b81c994e45d | A child twirls a weight at the end of a 2-meter string at a rate of 1 revolution per second. Find the tangential acceleration ($a_{T}$) and the normal (centripetal) acceleration ($a_{N}$). Given: - Radius ($\rho$) = 2 meters - Angular speed ($\omega$) = 1 revolution per second = $2\pi$ radians per second - Speed ($v$) ... | [
"\\( a_T = 0 \\), \\( a_N = 8\\pi^2 \\)"
] | [
"A child twirls a weight at the end of a 2-meter string at a rate of 1 revolution per second. Find the tangential acceleration ($a_{T}$) and the normal (centripetal) acceleration ($a_{N}$).\n\nGiven:\n- Radius ($\\rho$) = 2 meters\n- Angular speed ($\\omega$) = 1 revolution per second = $2\\pi$ radians per second\n... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-9c37f5bcba5c6fad7efcfede2096d4fcc8cda69e2ab6def7515a647ab95cdf13 | A child walks from the point \((0, n)\) to the point \((m, 0)\) on the Cartesian plane, taking steps of length 1 that are parallel to either the \( X \) or \( Y \) axis. For each step, the distance from the child to the axis parallel to the step is calculated. If the step causes the child to be further from the origin ... | [
"0"
] | [
"A child walks from the point \\((0, n)\\) to the point \\((m, 0)\\) on the Cartesian plane, taking steps of length 1 that are parallel to either the \\( X \\) or \\( Y \\) axis. For each step, the distance from the child to the axis parallel to the step is calculated. If the step causes the child to be further fro... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d47e5a9fe521d9087c03f56ec83a5ae71102cfb9dcc4a4f21b8075f54d9ca696 | A child watches TV for at least one hour each day for 7 weeks but never more than 11 hours in any one week. Determine the number of hours of TV the child watches in a period of consecutive days, knowing that there is exactly one such period where the total is 20 hours. Find the length of this period. | [
"20"
] | [
"A child watches TV for at least one hour each day for 7 weeks but never more than 11 hours in any one week. Determine the number of hours of TV the child watches in a period of consecutive days, knowing that there is exactly one such period where the total is 20 hours. Find the length of this period."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-135222d9be85df0996cc64060eba6463bc862fd6f8ec86b77d874890c3a03f01 | A child’s toy is made by fixing a solid right circular cone with base radius \( r \) cm and height \( h \) cm on the flat circular face of a solid hemisphere of radius \( r \) cm. The centre of the base of the cone coincides with the centre of the hemisphere. Given that \( h + 6r = 15 \), find the upper bound for the v... | [
"\\( r < 2.5 \\) \n\nThe reasoning provided is that for the equation \\( h + 6r = 15 \\), the maximum value of \\( r \\) occurs when \\( h \\) approaches 0. This results in \\( 6r < 15 \\), leading to \\( r < 2.5 \\)."
] | [
"A child’s toy is made by fixing a solid right circular cone with base radius \\( r \\) cm and height \\( h \\) cm on the flat circular face of a solid hemisphere of radius \\( r \\) cm. The centre of the base of the cone coincides with the centre of the hemisphere. Given that \\( h + 6r = 15 \\), find the upper bo... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-94a9c6fc9a162a3a855f54c1ae28b8b5b799027a1387e9a7dc8ccf9428bdf992 | A chip is placed at point \((m, n)\). Players can move the chip to \((m', n)\) or \((m, n')\) where \(0 \leq m' < m\) and \(0 \leq n' < n\). The player who makes the last move and puts the chip onto \((0, 0)\) wins. Determine the winning strategy for this game. | [
"m \\neq n"
] | [
"A chip is placed at point \\((m, n)\\). Players can move the chip to \\((m', n)\\) or \\((m, n')\\) where \\(0 \\leq m' < m\\) and \\(0 \\leq n' < n\\). The player who makes the last move and puts the chip onto \\((0, 0)\\) wins. Determine the winning strategy for this game."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-3b3fb913652eee592dd618a31467c1bb23169546b3139aa0e5ae54bd16432b6b | A chip is placed on a square of a checkered board. In one move, the chip can move either 1 square to the right, or diagonally 1 square to the left and 1 square up, or 1 square to the left and 3 squares down. The chip makes \( n \) moves and returns to the starting square. Calculate the smallest possible value of \( n \... | [
"8"
] | [
"A chip is placed on a square of a checkered board. In one move, the chip can move either 1 square to the right, or diagonally 1 square to the left and 1 square up, or 1 square to the left and 3 squares down. The chip makes \\( n \\) moves and returns to the starting square. Calculate the smallest possible value of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-249b167b16929acb6c192edaa39d7656260800f032bc7d604cd1030299a1c591 | A chocolate bar consists of \( n \) squares arranged in a rectangular pattern. You can break the bar only along the lines between the squares, and only one piece can be broken at a time. Determine the total number of breaks required to break the bar into \( n \) separate squares, and express your answer in terms of \( ... | [
"n-1"
] | [
"A chocolate bar consists of \\( n \\) squares arranged in a rectangular pattern. You can break the bar only along the lines between the squares, and only one piece can be broken at a time. Determine the total number of breaks required to break the bar into \\( n \\) separate squares, and express your answer in ter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-234015d2847379c7bf7b936453a7c2735a1e01f4f69c35f49802d08c8a54ee0b | A chocolate bar in the shape of an equilateral triangle with side length $n$ consists of triangular chips with sides of length $1$, parallel to the sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up, and passes the leftovers t... | [
"\\text{Second player wins if } n \\equiv 2 \\text{ or } 3 \\pmod{5}, \\text{ otherwise first player wins}"
] | [
"A chocolate bar in the shape of an equilateral triangle with side length $n$ consists of triangular chips with sides of length $1$, parallel to the sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up, and passes the leftov... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-c648af9522d479ec791f7239814b312453d3f56cccba09895b1e153d8b479ad8 | A chocolate bar of length $1$ is broken into $10$ pieces at random. Let $S_n$ denote the average size of the $n$th smallest piece. Calculate $S_5$. | [
"\\dfrac{1627}{25200}"
] | [
"A chocolate bar of length $1$ is broken into $10$ pieces at random. Let $S_n$ denote the average size of the $n$th smallest piece. Calculate $S_5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27120bf98909c7a9266df0387a8bf9720474fc2b80f90539e618648beeb21d5c | A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of $4$ and the group must have at least one singer. Let $N$ be the number of groups that could be selected. What is the rema... | [
"95"
] | [
"A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the numbers of tenors and basses must be a multiple of $4$ and the group must have at least one singer. Let $N$ be the number of groups that could be selected. What is the... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4c3a863ae67e71ae4b2262233bf727ce40cdeff8081797747da67a68196b2b25 | A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \( N \) be the number of groups that can be selected. What is the remainde... | [
"95"
] | [
"A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \\( N \\) be the number of groups that can be selected. What is the r... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-209e5684af822e161b1a88ebd3312e11415d787b8d6d982a6bbcff9f424e71c1 | A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \( N \) be the number of groups that could be selected. What is the remain... | [
"95"
] | [
"A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \\( N \\) be the number of groups that could be selected. What is the... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7d48d3f75097e40abe71c58dd5e1e2d891e4bb4413f80d07991f82ada6c84023 | A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \(N\) be the number of groups that could be selected. What is the remainde... | [
"95"
] | [
"A choir director must select a group of singers from among his 6 tenors and 8 basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of 4, and the group must have at least one singer. Let \\(N\\) be the number of groups that could be selected. What is the r... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1996bb0e883ab633db5e04842715e654a07300b2d31c02d73870819c01cf5fc4 | A chord $AB = a$ is drawn in a circle of radius $B$. A circle with center on line $AB$ passes through $A$ and intersects the first circle a second time at point $C$. Let $M$ be an arbitrary point on the second circle. The straight lines $MA$ and $MC$ intersect the first circle a second time at points $P$ and $Q$. Find ... | [
"\\sqrt{4B^2 - a^2}"
] | [
"A chord $AB = a$ is drawn in a circle of radius $B$. A circle with center on line $AB$ passes through $A$ and intersects the first circle a second time at point $C$. Let $M$ be an arbitrary point on the second circle. The straight lines $MA$ and $MC$ intersect the first circle a second time at points $P$ and $Q$. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-01ff1fb06a4817e6197e0dbd7017d31d74a104f5a68fef3f6af820c7cf35a6f0 | A chord $AB$ is drawn in a circle. Points $P$ and $Q$ are taken on the extensions of $AB$ beyond $A$ and $B$ respectively such that $AP = BQ$. Tangents to the circle are drawn through $P$ and $Q$, intersecting at point $M$. Determine the locus of points $M$ as $P$ and $Q$ move along the straight line. | [
"The locus of points \\( M \\) is the perpendicular bisector of the chord \\( AB \\)."
] | [
"A chord $AB$ is drawn in a circle. Points $P$ and $Q$ are taken on the extensions of $AB$ beyond $A$ and $B$ respectively such that $AP = BQ$. Tangents to the circle are drawn through $P$ and $Q$, intersecting at point $M$. Determine the locus of points $M$ as $P$ and $Q$ move along the straight line."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 6
} | 13 | 0 | 0 |
omr-ws-8baddca5e01182c4268da0351e35e838e58104e89d369464147a7e1c59405684 | A chord $AB$ is drawn in circle $O$ with radius $r$. Points $P$ and $Q$ are taken on the extension of $AB$ beyond points $A$ and $B$, respectively, such that $AP = BQ$. As $P$ and $Q$ vary along line $AB$, they determine two pairs of tangents to circle $O$. These four tangents determine four new points of intersection ... | [
"\\text{the perpendicular bisector of } AB"
] | [
"A chord $AB$ is drawn in circle $O$ with radius $r$. Points $P$ and $Q$ are taken on the extension of $AB$ beyond points $A$ and $B$, respectively, such that $AP = BQ$. As $P$ and $Q$ vary along line $AB$, they determine two pairs of tangents to circle $O$. These four tangents determine four new points of intersec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-7a980bd6610e8f2941322ca14cc7692945f5308797a346f57374cd0ce25abb84 | A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC$ and $AC$ at points $A'$ and $B'$, respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. If $AA'$, $BB'$, and $CC'$ ar... | [
"5"
] | [
"A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC$ and $AC$ at points $A'$ and $B'$, respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. If $AA'$, $BB'$, and $CC... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-c5a3fa111267e272e1aceb44ecbde7d9334655b26382d4ed10ed18005d620c98 | A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the length of $AB$ is $20$ and the length of $ST$ is $16$, calculate the measure of $\angle SPM$ in degrees for any position of $ST$. | [
"53^\\circ"
] | [
"A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the length of $AB$ is $20$ and the length of $ST$ is $16$, calculate the measure of $\\angle SPM$ in degrees for any position of $ST$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-34e166f7ed02b5e064a0ff397702a99e466faa78561e890a952af5bbb9b81816 | A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the length of $AB$ is $2r$, find the measure of angle $SPM$ in degrees for all positions of $ST$. | [
"45"
] | [
"A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the length of $AB$ is $2r$, find the measure of angle $SPM$ in degrees for all positions of $ST$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-edd4605077f0dc8c9f317b1d5695765d74ddc7e0a24ff760c4303d19d3d268c8 | A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the radius of the semicircle is $5$ and the length of $ST$ is $8$, calculate the measure of $\angle SPM$ in degrees. | [
"53"
] | [
"A chord $ST$ of constant length slides around a semicircle with diameter $AB$. Let $M$ be the midpoint of $ST$ and $P$ be the foot of the perpendicular from $S$ to $AB$. If the radius of the semicircle is $5$ and the length of $ST$ is $8$, calculate the measure of $\\angle SPM$ in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb449970f9589079b1057d787e52bb91f9845a3a7b776bff5b8bd44208e07550 | A chord $\overline{ST}$ of constant length slides around a semicircle with diameter $\overline{AB}$. Let $M$ be the midpoint of $\overline{ST}$ and $P$ be the foot of the perpendicular from $S$ to $\overline{AB}$. If the angle $SPM$ is constant for all positions of $\overline{ST}$, find the value of this constant angle... | [
"45"
] | [
"A chord $\\overline{ST}$ of constant length slides around a semicircle with diameter $\\overline{AB}$. Let $M$ be the midpoint of $\\overline{ST}$ and $P$ be the foot of the perpendicular from $S$ to $\\overline{AB}$. If the angle $SPM$ is constant for all positions of $\\overline{ST}$, find the value of this cons... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11f50e557d835d4fe383368faac5e1022d1b711dc40dfe0c95d2858888b69393 | A chord AB divides the area of circle O in the ratio 1:7. What is the measure of the arc AB? (Or, what is the measure of the angle AOB? Preferably in radians.) | [
"\\(1.76634\\) radians"
] | [
"A chord AB divides the area of circle O in the ratio 1:7. What is the measure of the arc AB? (Or, what is the measure of the angle AOB? Preferably in radians.)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-bab14dab45e6f7db94299c1d58bb74c4e4f5b0c35429b029a6b60eb2e0b4b541 | A chord \( C \) of a circle \( S \) with radius \( r \) subtends an angle of \( \frac{2\pi}{3} \) at the center of \( S \). Find the area of the region \( R \) which is closer to the chord \( C \) than to the circumference of the circle. | [
"\\(\\frac{r^2 (2\\pi + 3\\sqrt{3})}{6}\\)"
] | [
"A chord \\( C \\) of a circle \\( S \\) with radius \\( r \\) subtends an angle of \\( \\frac{2\\pi}{3} \\) at the center of \\( S \\). Find the area of the region \\( R \\) which is closer to the chord \\( C \\) than to the circumference of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2f51e5e92177e2defd5333d9afc55c2a44c9710c7f0b7b1a39f62f323896c27 | A chord \( CD = a \) moves on a semicircle with diameter \( AB = 2r \). The lines \( AD \) and \( BC \) intersect at point \( S \). If \( M \) is the midpoint of \( CD \) and \( SH \perp CD \), calculate the distance between the two fixed points through which the lines \( SH \) and \( SM \) pass. | [
"2r"
] | [
"A chord \\( CD = a \\) moves on a semicircle with diameter \\( AB = 2r \\). The lines \\( AD \\) and \\( BC \\) intersect at point \\( S \\). If \\( M \\) is the midpoint of \\( CD \\) and \\( SH \\perp CD \\), calculate the distance between the two fixed points through which the lines \\( SH \\) and \\( SM \\) pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5074dd2d5bbaa0e41399aadf407fc6d84f23e88d0a4d56cc782d4037dfe194a4 | A chord \( ST \) of constant length slides around a semicircle with diameter \( AB \). Let \( M \) be the midpoint of \( ST \) and \( P \) be the foot of the perpendicular from \( S \) to \( AB \). If the length of \( AB \) is 10 units and the length of \( ST \) is 8 units, calculate the measure of angle \( SPM \) in d... | [
"53"
] | [
"A chord \\( ST \\) of constant length slides around a semicircle with diameter \\( AB \\). Let \\( M \\) be the midpoint of \\( ST \\) and \\( P \\) be the foot of the perpendicular from \\( S \\) to \\( AB \\). If the length of \\( AB \\) is 10 units and the length of \\( ST \\) is 8 units, calculate the measure ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-89b82926438ea1d2c8dc20fa63d7eb1a932e37b7da3be4ee8a687ae5dbd53e53 | A chord \( ST \) of constant length slides around a semicircle with diameter \( AB \). Let \( M \) be the midpoint of \( ST \) and \( P \) the projection of \( S \) to \( AB \). If the angle \( \angle SPM \) is constant for all positions of \( ST \), calculate the measure of this constant angle in degrees. | [
"45"
] | [
"A chord \\( ST \\) of constant length slides around a semicircle with diameter \\( AB \\). Let \\( M \\) be the midpoint of \\( ST \\) and \\( P \\) the projection of \\( S \\) to \\( AB \\). If the angle \\( \\angle SPM \\) is constant for all positions of \\( ST \\), calculate the measure of this constant angle ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2c177e7fc24318b57e3e0738a09b5b9e792164dee791e5f631fcf14648ea13c6 | A chord \(AB\) is drawn in a circle, and the smaller of the two arcs \(AB\) corresponds to a central angle of \(120^\circ\). A tangent \(p\) to this arc is drawn. Two circles with radii \(R\) and \(r\) are constructed, each touching the smaller arc \(AB\) and the lines \(AB\) and \(p\). Find the radius of the original ... | [
"R + r + 2\\sqrt{Rr}"
] | [
"A chord \\(AB\\) is drawn in a circle, and the smaller of the two arcs \\(AB\\) corresponds to a central angle of \\(120^\\circ\\). A tangent \\(p\\) to this arc is drawn. Two circles with radii \\(R\\) and \\(r\\) are constructed, each touching the smaller arc \\(AB\\) and the lines \\(AB\\) and \\(p\\). Find the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-d854cca5787c4dc8fd383595f2dce9e7ccf01e1d60c4dfb0b2c03968c182c4e1 | A chord cuts the diameter of a circle at an angle of 45º, dividing the chord into segments \( m \) and \( n \). If the radius of the circle is \( r \), find the value of \( \frac{m^2 + n^2}{r^2} \). | [
"2"
] | [
"A chord cuts the diameter of a circle at an angle of 45º, dividing the chord into segments \\( m \\) and \\( n \\). If the radius of the circle is \\( r \\), find the value of \\( \\frac{m^2 + n^2}{r^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69405cc6d5c211b2bcbf601b5a88455173c76e0462fc4eecf2bda951ec62a4b9 | A chord is perpendicular to a diameter of a circle at a point which divides the diameter into segments with lengths in the ratio 1:3. In what ratio does the chord divide the circumference? | [
"1:3"
] | [
"A chord is perpendicular to a diameter of a circle at a point which divides the diameter into segments with lengths in the ratio 1:3. In what ratio does the chord divide the circumference?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1368e625f64887acdd7db75b31b1a75d63b1313717c60cbeb86a0792706e0603 | A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a(\pi) + b\sqrt{c}}{d(\pi) + e\sqrt{f}}$, where $a, b, c, d, e,$ and $f$ are positive integers... | [
"592"
] | [
"A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a(\\pi) + b\\sqrt{c}}{d(\\pi) + e\\sqrt{f}}$, where $a, b, c, d, e,$ and $f$ are positiv... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-37e3a0cfa3376f730429a3f7c685632107d5acff14856e05bee17a7e292217d4 | A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers... | [
"592"
] | [
"A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are positiv... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9036f4de9d8614fa8058f1d2b72d3f6a586dcb652bccc276dcb1897672264927 | A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}}$, where $a, b, c, d, e,$ and $f$ are positive integers, $a$ an... | [
"592"
] | [
"A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}}$, where $a, b, c, d, e,$ and $f$ are positive intege... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2514807ba34011572607ee287173c27ecd8c5c72e1e272e8c8d447962cbca7c2 | A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}},$ where $a, b, c, d, e,$ and $f$ are positive integers, $a$ an... | [
"592"
] | [
"A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}},$ where $a, b, c, d, e,$ and $f$ are positive... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-743eb233bd42b92db0eb48417527f0647939e0cf42c8fc1243ea6c17b784bb7e | A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form \(\frac{a\pi + b\sqrt{c}}{d\pi - e\sqrt{f}}\), where \(a, b, c, d, e,\) and \(f\) are positive intege... | [
"592"
] | [
"A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form \\(\\frac{a\\pi + b\\sqrt{c}}{d\\pi - e\\sqrt{f}}\\), where \\(a, b, c, d, e,\\) and \\(f\\) are... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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