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-7babc065702e7eca27a7f45d7793aff70306edfdcc6e3bfcb473e4f86ce193a2 | A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$, where $m$ and... | [
"409"
] | [
"A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 3 and the other with three sides each of length 5, has length equal to $m/n$, where $m... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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omr-ws-d5f1ef3b2dc9c248a05db2bc21e25760b8ca27f9e7cd8559cd98e469ca1d3e5b | A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides of length 5. The chord of the circle that divides the hexagon into two trapezoids with three sides of lengths 3 and 5, respectively, has length equal to \( \frac{m}{n} \), where \( m \) and \( n \) are relatively pr... | [
"409"
] | [
"A hexagon inscribed in a circle has three consecutive sides each of length 3 and three consecutive sides of length 5. The chord of the circle that divides the hexagon into two trapezoids with three sides of lengths 3 and 5, respectively, has length equal to \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are r... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
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"tir": 11
} | 61 | 0 | 0 |
omr-ws-c8e8d4f7111c0dfd948de7251e7eeaea0f408af3d934fdb48db5ca9deb989772 | A hexagon inscribed in a circle has three consecutive sides of length $a$ and three consecutive sides of length $b$. Find the radius of the circle. | [
"\\( r = \\sqrt{\\frac{a^2 + b^2 + ab}{3}} \\)"
] | [
"A hexagon inscribed in a circle has three consecutive sides of length $a$ and three consecutive sides of length $b$. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
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omr-ws-c93d7a4e9a253900549bc276c63cdcec8e34f2937c75b1580a1a28f20fcb3b7d | A hexagon inscribed in a circle has three consecutive sides of length 3 and three consecutive sides of length 5. The chord of the circle that divides the hexagon into 2 trapezoids (one with 3 sides of length 3 and one with 3 sides of length 5) has length equal to \( \frac{m}{n} \), where \( m \) and \( n \) are relativ... | [
"409"
] | [
"A hexagon inscribed in a circle has three consecutive sides of length 3 and three consecutive sides of length 5. The chord of the circle that divides the hexagon into 2 trapezoids (one with 3 sides of length 3 and one with 3 sides of length 5) has length equal to \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 14
} | 53 | 0 | 0 |
omr-ws-e7efa36b0beea2c6d369d1f77f08a32e1dba2e403d45ccc0b562b4cd76eeae54 | A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of length 3, and the other with three sides, each of length 5, has length equal to $m/n$, where $... | [
"409"
] | [
"A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into two trapezoids, one with three sides, each of length 3, and the other with three sides, each of length 5, has length equal to $m/n$, wh... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-adfcbc36463495bc9cf49992e80afcc088048e91e257ee609a152070b5d4e45e | A hexagon is inscribed in a circle of radius $r$. If two sides of the hexagon are $7$ units long and the other four sides are $20$ units long, find the radius $r$. | [
"16"
] | [
"A hexagon is inscribed in a circle of radius $r$. If two sides of the hexagon are $7$ units long and the other four sides are $20$ units long, find the radius $r$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
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} | 30 | 0 | 0 |
omr-ws-2f429d5377319869b13c1254a5f1df8048c60573d3aa3c4c3552af34f685e137 | A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$, and two have length $3$. If $r$ is a root of the polynomial $2r^3 - 7r - 3 = 0$, find the value of $r^2 + 2r + 1$. | [
"4"
] | [
"A hexagon is inscribed in a circle of radius $r$. Two of the sides of the hexagon have length $1$, two have length $2$, and two have length $3$. If $r$ is a root of the polynomial $2r^3 - 7r - 3 = 0$, find the value of $r^2 + 2r + 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-37858dc9fd60d13f417485f6bc026554b0731c9a264dfd059d3c9e1ad8f142cc | A hexagon is inscribed in a circle of radius \( r \). If two sides of the hexagon are 7 units long and the other four sides are 20 units long, find \( r \). | [
"16"
] | [
"A hexagon is inscribed in a circle of radius \\( r \\). If two sides of the hexagon are 7 units long and the other four sides are 20 units long, find \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9007d3821ab6ab26194130a97a050708110fe7aaf0caebf5f8d6f3e73c03c18d | A hexagon is inscribed in a circle of radius \( r \). If two sides of the hexagon are 7 units long, while the other four sides are 20 units long, find \( r \). | [
"16"
] | [
"A hexagon is inscribed in a circle of radius \\( r \\). If two sides of the hexagon are 7 units long, while the other four sides are 20 units long, find \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 16,
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} | 31 | 0 | 0 |
omr-ws-6e69316e7bd9439759ebf8c2458f9272bb7b7ab7036a111acb190fc3008366f2 | A hexagon is inscribed in a circle of radius \( r \). Two sides of the hexagon are 7 units long, while the other four sides are 20 units long. Find \( r \). | [
"16"
] | [
"A hexagon is inscribed in a circle of radius \\( r \\). Two sides of the hexagon are 7 units long, while the other four sides are 20 units long. Find \\( r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9cc5af5d18c244e0bc90a7bb1ad66602e20f09cfbd493b5d689cae2e2c2380af | A hexagon is inscribed in a circle with 3 sides of length 10 and 3 sides of length 22. If the radius of the circle is expressed as $\sqrt{n}$, what is $n$? | [
"67"
] | [
"A hexagon is inscribed in a circle with 3 sides of length 10 and 3 sides of length 22. If the radius of the circle is expressed as $\\sqrt{n}$, what is $n$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-611af2a766ee772029c9e028c5aa2d442840ecac1c9a4618a5277345a5b2266c | A hexagon is inscribed in a circle with a radius of 14 cm. Three of its sides are each 22 cm long, and the other three sides are equal. Find the length of each of the other three sides. | [
"4"
] | [
"A hexagon is inscribed in a circle with a radius of 14 cm. Three of its sides are each 22 cm long, and the other three sides are equal. Find the length of each of the other three sides."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5425560cedd86c604aa38a9b4bb828b02994822bba91edecb534aea12eff984f | A hexagon is inscribed in a circle with radius \( r \). Two of its sides have length 1, two have length 2, and the last two have length 3. Calculate the value of \( r^3 \) if \( r \) is a root of the equation \( 2r^3 - 7r - 3 = 0 \). | [
"\\dfrac{7r + 3}{2}"
] | [
"A hexagon is inscribed in a circle with radius \\( r \\). Two of its sides have length 1, two have length 2, and the last two have length 3. Calculate the value of \\( r^3 \\) if \\( r \\) is a root of the equation \\( 2r^3 - 7r - 3 = 0 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f55e3241c8970d4c9388a3fcae7b0e5ac126465c5b8ed9c8c0b8b0d8fc9bdd05 | A hexagon is inscribed in a circle, and an equilateral triangle is inscribed inside the hexagon such that three of its sides are parallel to the hexagon's sides. What is the ratio of the area of the equilateral triangle to the area of the circle? | [
"\\(\\frac{3\\sqrt{3}}{8\\pi}\\)"
] | [
"A hexagon is inscribed in a circle, and an equilateral triangle is inscribed inside the hexagon such that three of its sides are parallel to the hexagon's sides. What is the ratio of the area of the equilateral triangle to the area of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87c0d82a1e9ffa6389ae58f1fcab1e75bc5f0d8a490473341a1df45d8996a926 | A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth side, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from vertex $A$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth side, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from vertex $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-63a764bbfb4c42710b848263dade424158462ca4f1316e2c4da5bfd021fb57e4 | A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-130c9d0d2ba4c5b3906874ed4d03419c781fc4303d2582a50f36a50b8b86cac0 | A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$."
] | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 7,
"tir": 20
} | 74 | 0 | 0 |
omr-ws-0fae3fbcd7d2a7a7fdf5c423971ed5ead79015d0c0ee53dea7c36afe681da441 | A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth side, denoted by $AB$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth side, denoted by $AB$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-4de133fe0dc5fe24289ef5dc812ac2c82b25406a0fb1f646d1822a896376d246 | A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth side, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from vertex $A$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth side, denoted by $\\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from vertex $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-49efe881e5b0cabd2a2760188f8e8477f3b476a16801c5d3e12c4c5ccf853786 | A hexagon is inscribed in a circle. Five of the sides of the hexagon have a length of $81$ and the $6th$, denoted by $\overline{AB}$, has a length of $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five of the sides of the hexagon have a length of $81$ and the $6th$, denoted by $\\overline{AB}$, has a length of $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-87e6d6361fd4bc4eb4cbd75150c80b723df4f62fe1a9be79c966d1714f4febd9 | A hexagon is inscribed in a circle. Five sides have length 81, and the other side has length 31. Find the sum of the three diagonals from a vertex on the short side. | [
"384"
] | [
"A hexagon is inscribed in a circle. Five sides have length 81, and the other side has length 31. Find the sum of the three diagonals from a vertex on the short side."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d9d60ff0abcaf5301030e736a0dc737d075aa49fb1999e1741642d7e5e13b3b | A hexagon is inscribed in a circle. The hexagon has three sides of length 6 and three sides of length 10. Compute the radius of the circle. | [
"\\(\\frac{14 \\sqrt{3}}{3}\\)"
] | [
"A hexagon is inscribed in a circle. The hexagon has three sides of length 6 and three sides of length 10. Compute the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-df2d35567709ef6bfb902bae08e86af45b09ef6fa3ae6601d2d46978757965cf | A hexagon is inscribed in a circle. The lengths of three consecutive sides are $a$, and the lengths of the other three sides are $b$. Express the radius $r$ of the circle in terms of $a$ and $b$. | [
"\\(\\sqrt{\\frac{a^2 + ab + b^2}{3}}\\)"
] | [
"A hexagon is inscribed in a circle. The lengths of three consecutive sides are $a$, and the lengths of the other three sides are $b$. Express the radius $r$ of the circle in terms of $a$ and $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-3d8594607bd6b57aeec7dd1bfd23126ab87f982d8ddff4ea7546bd66ed397aa3 | A hexagon is inscribed in a cube of side \( a \) with the vertices of the hexagon being the centers of the corresponding edges of the cube. Find the area of the hexagon. | [
"\\(\\frac{3\\sqrt{3}}{4}a^{2}\\)"
] | [
"A hexagon is inscribed in a cube of side \\( a \\) with the vertices of the hexagon being the centers of the corresponding edges of the cube. Find the area of the hexagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-271caf695072027a45e0d213ce6348162ab2b046d198e89d007fa44465dce4da | A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\sqrt{2}+c\sqrt{5}$, where $a$, $b$ and $c$ are whole numbers. Find $a+b+c$. | [
"6"
] | [
"A hexagon is obtained by joining, in order, the points $(0,1)$, $(1,2)$, $(2,2)$, $(2,1)$, $(3,1)$, $(2,0)$, and $(0,1)$. The perimeter of the hexagon can be written in the form $a+b\\sqrt{2}+c\\sqrt{5}$, where $a$, $b$ and $c$ are whole numbers. Find $a+b+c$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a22f9f06898b7b697001bd29c2123ebf7c39f5133fe4c0e8759b048d48ac9c04 | A hexagon is placed on a unit square such that two pairs of opposite vertices of the hexagon are on the sides of the square. What is the minimum area of the hexagon if it is placed on top of at least one pair of opposite sides of the square? | [
"\\dfrac{1}{2}"
] | [
"A hexagon is placed on a unit square such that two pairs of opposite vertices of the hexagon are on the sides of the square. What is the minimum area of the hexagon if it is placed on top of at least one pair of opposite sides of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-7f510d1c0f055c880f8e4763c55a8b88584e15a0475896e2def0c9766a36f371 | A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$. | [
"272"
] | [
"A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 10
} | 57 | 0 | 0 |
omr-ws-aa066bed8d5340c12273e14610b08bac424fb65027c292cbf90531b4990f3f81 | A hexagon with area \( H \) is constructed from two congruent equilateral triangles such that two pairs of parallel lines have distances of 2 and 10. Compute \( H^2 \). | [
"1728"
] | [
"A hexagon with area \\( H \\) is constructed from two congruent equilateral triangles such that two pairs of parallel lines have distances of 2 and 10. Compute \\( H^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d31b2c3388f7d372dd8a2ddd84ba31a23d12b4394dd1f485d7b8fa4f311d9fd | A hexagon with consecutive sides of lengths $2$, $2$, $7$, $7$, $11$, and $11$ is inscribed in a circle. Find the radius of the circle. | [
"7"
] | [
"A hexagon with consecutive sides of lengths $2$, $2$, $7$, $7$, $11$, and $11$ is inscribed in a circle. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-63a55e5ad01497888ed8c4561ba367a3b79643b528f59882f8a9ad92c5f38973 | A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. Find the radius of the circle. | [
"7"
] | [
"A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. Find the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b048b38b46d914c9c720f6c4cf575001c61e447e5dc96becb1f857df1268a0b0 | A hexagon with side lengths (in order) of $2, 2, 7, 7, 11, 11$ is inscribed in a circle. Find the radius $r$ of this circle. | [
"7"
] | [
"A hexagon with side lengths (in order) of $2, 2, 7, 7, 11, 11$ is inscribed in a circle. Find the radius $r$ of this circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43c5c79e831c4c033ea9dd64cb7be0ebe40742f6564203e702dc4e66f3882252 | A hexagon with side lengths 15, 16, 15, 15, 16, 15 (in that order) is inscribed in a circle. If the radius of the circle can be written as $\dfrac{\sqrt{a}}{b}+c$, find $a+b+c$. | [
"520"
] | [
"A hexagon with side lengths 15, 16, 15, 15, 16, 15 (in that order) is inscribed in a circle. If the radius of the circle can be written as $\\dfrac{\\sqrt{a}}{b}+c$, find $a+b+c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6ecaa55c3b295f970f2ac79961c54442221dbd6af7b1629daf3b10edc9f0a948 | A hexagon with sides of lengths $2, 2, 7, 7, 11,$ and $11$ is inscribed in a circle. Find the diameter of the circle. | [
"14"
] | [
"A hexagon with sides of lengths $2, 2, 7, 7, 11,$ and $11$ is inscribed in a circle. Find the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a9f13c2ec7d50c57cf81532e87885e3073b88df748b82419722476e9e673da4a | A hexagon with two side lengths of $7$ and four side lengths of $20$ is inscribed in a circle of radius $r$. Find the radius of the circle. | [
"16"
] | [
"A hexagon with two side lengths of $7$ and four side lengths of $20$ is inscribed in a circle of radius $r$. Find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8371cf02e5b91886ea045bceb1a50f84505bb3136fd9dc9698bf9c22d3200e36 | A hexagonal doghouse has sides of 10 feet each. A dog is leashed to one corner of the doghouse with a leash that is 40 feet long. What is the area that the dog can access? | [
"\\(\\frac{4700}{3}\\pi - 400\\cos^{-1}\\left(\\frac{\\sqrt{3}}{4}\\right) + 25\\sqrt{3}(\\sqrt{13} - 1)\\)"
] | [
"A hexagonal doghouse has sides of 10 feet each. A dog is leashed to one corner of the doghouse with a leash that is 40 feet long. What is the area that the dog can access?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da68c40a12542d45efde94df0e0a5e4e086d59758f4abf97ddec6c85ce7d561a | A hexagonal pencil has each side of the hexagon measuring 5 mm and a total length of 15 cm. One end is cut straight, revealing a hexagonal cross-section with a central circular lead. The other end is sharpened to form a cone with a vertex angle of 20 degrees. Calculate the volume of the pencil. | [
"\\(\\left(\\frac{75\\sqrt{3}}{2}\\right)\\left(150 - \\frac{5}{\\tan10^\\circ}\\right) + \\frac{1}{3}(25\\pi)\\left(\\frac{5}{\\tan10^\\circ}\\right)\\)"
] | [
"A hexagonal pencil has each side of the hexagon measuring 5 mm and a total length of 15 cm. One end is cut straight, revealing a hexagonal cross-section with a central circular lead. The other end is sharpened to form a cone with a vertex angle of 20 degrees. Calculate the volume of the pencil."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-8dd73cab05b4a587e8f685c183aaea7f0c00c9663b14bf23c5a58ade59330e0c | A hexagonal room is covered with 19 numbered rugs of four different colors. Two rugs with a common segment along their borders must have different colors. Only two colors are used in the middle layer, and at most one rug in the outer layer has the same color as the central rug. Two coloring patterns are different if th... | [
"2064"
] | [
"A hexagonal room is covered with 19 numbered rugs of four different colors. Two rugs with a common segment along their borders must have different colors. Only two colors are used in the middle layer, and at most one rug in the outer layer has the same color as the central rug. Two coloring patterns are different ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e438122d8687b18a445a3ff3d87b121a802c593ba61a5b1390de1edc381b3e2a | A hexagonal table has 2012 columns. Each odd column contains 2012 hexagons, and each even column contains 2013 hexagons. The number $i$ is written in each hexagon of the $i$-th column. We can change the numbers by selecting three adjacent hexagons, rotating the numbers, and adjusting them by ±1 depending on the directi... | [
"\\(2012 \\times 2012 + 1006 \\times 2013 - 1\\) \n\nThis expression represents the total number of hexagons minus one. However, for simplicity, we can state the result as:"
] | [
"A hexagonal table has 2012 columns. Each odd column contains 2012 hexagons, and each even column contains 2013 hexagons. The number $i$ is written in each hexagon of the $i$-th column. We can change the numbers by selecting three adjacent hexagons, rotating the numbers, and adjusting them by ±1 depending on the di... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 4
} | 8 | 0 | 0 |
omr-ws-8a4a8d08d8b9988b2fda1902c3a08ed3fedd3a9ae6a1e405ac0ad48a162417c3 | A hexagonal table is given, which has 2012 columns. There are 2012 hexagons in each of the odd columns, and there are 2013 hexagons in each of the even columns. The number $i$ is written in each hexagon from the $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select thre... | [
"4049148"
] | [
"A hexagonal table is given, which has 2012 columns. There are 2012 hexagons in each of the odd columns, and there are 2013 hexagons in each of the even columns. The number $i$ is written in each hexagon from the $i$-th column. Changing the numbers in the table is allowed in the following way: We arbitrarily select... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-ddfa1cb4892ea2ca9b3efe817f9422e3552ff98f17d2618ab538c60af0a5e821 | A hexagram (a six-pointed star) is comprised of $2$ overlapping equilateral triangles. These triangles form a regular hexagon surrounded by $6$ congruent equilateral triangles. The radius of the circle that contains the centers of those $6$ small triangles can be written as $a\sqrt{3}$. If the side of one of the $2$ la... | [
"212"
] | [
"A hexagram (a six-pointed star) is comprised of $2$ overlapping equilateral triangles. These triangles form a regular hexagon surrounded by $6$ congruent equilateral triangles. The radius of the circle that contains the centers of those $6$ small triangles can be written as $a\\sqrt{3}$. If the side of one of the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-7f237b3c87600308c53e08cd4fcdda475f036922f9827c55a8205958ce6ddeb1 | A hexahedron consists of six triangular faces and five vertices. Two opposite vertices are the meeting points of three edges, and the other three vertices are the meeting points of four edges. Andrew wants to write an integer at each vertex so that the sum of the numbers at the three vertices of each face is the same. ... | [
"11"
] | [
"A hexahedron consists of six triangular faces and five vertices. Two opposite vertices are the meeting points of three edges, and the other three vertices are the meeting points of four edges. Andrew wants to write an integer at each vertex so that the sum of the numbers at the three vertices of each face is the s... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1ff8151edd63c3ac3fa48f3218fb908d80ffbe4769985d6e42a3f90c092f7038 | A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arit... | [
"34"
] | [
"A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 80,
"tir": 46
} | 126 | 0 | 0 |
omr-ws-826501c7e01a515cd3a3661b92577193265da4b6855d617a68a267dcd8dc78e9 | A high school is forming a band consisting of 10 students, with 5 boys and 5 girls. There are 33 boys and 22 girls auditioning. If you are a girl who attended the audition, what is the probability that you will be selected to play in the school band? | [
"\\(\\frac{5}{22}\\)"
] | [
"A high school is forming a band consisting of 10 students, with 5 boys and 5 girls. There are 33 boys and 22 girls auditioning. If you are a girl who attended the audition, what is the probability that you will be selected to play in the school band?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 30
} | 47 | 0 | 0 |
omr-ws-152134253038abc5407d41f3507132acf6ef2ed126a6ebddb7523c626f4325f5 | A high school marching band can be arranged in a rectangular formation with exactly three boys in each row and exactly five girls in each column. What is the sum of all possible sizes of the marching band for which this is possible? | [
"312"
] | [
"A high school marching band can be arranged in a rectangular formation with exactly three boys in each row and exactly five girls in each column. What is the sum of all possible sizes of the marching band for which this is possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-a69093d02b2b82f18868ca6e2a150c675c15a5d37255dc465384a829d5d0b134 | A high school math club has 99 members. They sent 60, 65, and 70 students to international math tournaments A, B, and C, respectively. 40 students participated in both A and B, 45 students participated in both B and C, and 50 students participated in both C and A. Let M and m be the maximum and minimum possible number ... | [
"M = 39, m = 0"
] | [
"A high school math club has 99 members. They sent 60, 65, and 70 students to international math tournaments A, B, and C, respectively. 40 students participated in both A and B, 45 students participated in both B and C, and 50 students participated in both C and A. Let M and m be the maximum and minimum possible nu... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5e2ca80e1761c57f691ac0254f7166d3157423765467b4b17d0870772fbeaaf | A high school's football stadium has 11 rows of 199 seats each. Several groups, each not exceeding 39 people, attend a game. If people from the same group sit in the same row, what is the maximum number of tickets that can be sold so that everyone is guaranteed a seat? | [
"2145"
] | [
"A high school's football stadium has 11 rows of 199 seats each. Several groups, each not exceeding 39 people, attend a game. If people from the same group sit in the same row, what is the maximum number of tickets that can be sold so that everyone is guaranteed a seat?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-80c3d8ecda2d7a633b44d55800dcfd485e20caf7906b17e03be45c052a9c0e19 | A highway patrol plane flies 3 miles above a level, straight road at a constant rate of 120 mph. At a certain instant, the line-of-sight distance from the plane to an oncoming car is 5 miles and is decreasing at a rate of 160 mph. Find the car's speed along the highway. | [
"200 mph"
] | [
"A highway patrol plane flies 3 miles above a level, straight road at a constant rate of 120 mph. At a certain instant, the line-of-sight distance from the plane to an oncoming car is 5 miles and is decreasing at a rate of 160 mph. Find the car's speed along the highway."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-abf705777af433e0a7f42593b3ed61b3afefbe4e7765e400b244fc53d430fcf6 | A highway with round-the-clock bus service crosses a railway. Every hour two trains run along the railway and approach the level crossing exactly at $n$ hours and at $n$ hours 38 minutes respectively where $n$ assumes the values from 0 to 23. When a train passes the crossing the lifting gate stops the road traffic for ... | [
"The possible values of $T$ (expressed in minutes) are 20, 15, 12, $7\\frac{1}{2}$ and $5\\frac{5}{11}$"
] | [
"A highway with round-the-clock bus service crosses a railway. Every hour two trains run along the railway and approach the level crossing exactly at $n$ hours and at $n$ hours 38 minutes respectively where $n$ assumes the values from 0 to 23. When a train passes the crossing the lifting gate stops the road traffic... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e095c40c49d52ef3026e71771d6e79ce7427e45d60872f65f283eb2ff5ffb36 | A hiker is at the point $(1, 1, 995)$ on a mountain whose height is given by $z = 1000 - 3x^2 - 2y^2$. In what direction (in 3-dimensional space) must she travel in order to ascend as rapidly as possible? | [
"\\((-6, -4, 2\\sqrt{13})\\)"
] | [
"A hiker is at the point $(1, 1, 995)$ on a mountain whose height is given by $z = 1000 - 3x^2 - 2y^2$. In what direction (in 3-dimensional space) must she travel in order to ascend as rapidly as possible?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-9d994aea05bc4592c130aae0812c0cbf396eeda3870b6c8eb911a46fc86f90d4 | A hiker starts out from point $A$ in the direction of $B$; $a$ hours later a cyclist starts out from $B$ to meet the hiker and meets him $b$ hours after the start. How long will it take the cyclist and the hiker to cover the whole distance between $A$ and $B$, if the cyclist requires $c$ hours less than does the hiker? | [
"It takes the cyclist $\\frac{a+2b-c+\\sqrt {(a+2b-c)^2+4bc}}{2}$ hours and the hiker $$\\frac{a\\!+\\!2b\\!+\\!c\\!+\\!\\sqrt {(a+2b-c)^2+4bc}}{2} \\!=\\!\\frac{(a+2b+c)+\\sqrt {(a+2b+c)^2-4(a+b)c}}{2}$$ hours to cover the whole distance $AB$"
] | [
"A hiker starts out from point $A$ in the direction of $B$; $a$ hours later a cyclist starts out from $B$ to meet the hiker and meets him $b$ hours after the start. How long will it take the cyclist and the hiker to cover the whole distance between $A$ and $B$, if the cyclist requires $c$ hours less than does the h... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5eb1e62b207f409ea545b76a22be2e3e57306b26d0af16f901eea0bb25ca9e94 | A hiker walks 8000 m on a course of South 81 degrees East. They then change direction and hike 5000 m on a course of North 32 degrees East. How far are they from their starting point? | [
"6047.5 m"
] | [
"A hiker walks 8000 m on a course of South 81 degrees East. They then change direction and hike 5000 m on a course of North 32 degrees East. How far are they from their starting point?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7bd7ba0b5eeca9bfc63d25db3b18b9bf0fab1c424453b61ebf5234793fb415d4 | A hiker walks 8000 m on a course of South 81 degrees East. They then change direction and hike 5000 m on a course of North 32 degrees East. On what course must they travel to return to the starting point? | [
"S74^\\circ W"
] | [
"A hiker walks 8000 m on a course of South 81 degrees East. They then change direction and hike 5000 m on a course of North 32 degrees East. On what course must they travel to return to the starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-78b9695c8e36c883d2b3bb47d3d0d738e2f3ce13258ea7536d3480764ce052e4 | A hill is 300 feet tall and makes an angle of depression of 26° from the top. How long does the barrier need to be to cover the hill's slope? | [
"684"
] | [
"A hill is 300 feet tall and makes an angle of depression of 26° from the top. How long does the barrier need to be to cover the hill's slope?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88dab577e4e52d9bd77a2db25c486f8707eb9e8576861211f58998e99e31015c | A hilly island has 2023 lookouts. Each lookout is in line of sight with at least 42 other lookouts. For any two distinct lookouts \(X\) and \(Y\), there is a sequence of lookouts \(A_1, A_2, \dots, A_{n+1}\) such that \(A_1 = X\), \(A_{n+1} = Y\), and each \(A_i\) is in line of sight with \(A_{i+1}\). The smallest such... | [
"140"
] | [
"A hilly island has 2023 lookouts. Each lookout is in line of sight with at least 42 other lookouts. For any two distinct lookouts \\(X\\) and \\(Y\\), there is a sequence of lookouts \\(A_1, A_2, \\dots, A_{n+1}\\) such that \\(A_1 = X\\), \\(A_{n+1} = Y\\), and each \\(A_i\\) is in line of sight with \\(A_{i+1}\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5eb55f41c2babbf612f1ff33300ac10b57b728aa6a4813b30ea402abd4d308dc | A hoax number \(k \in \mathbb{N}\) is one for which the sum of its digits (in base \(10\)) equals the sum of the digits of its distinct prime factors (in base \(10\)). Find the second hoax number. | [
"58"
] | [
"A hoax number \\(k \\in \\mathbb{N}\\) is one for which the sum of its digits (in base \\(10\\)) equals the sum of the digits of its distinct prime factors (in base \\(10\\)). Find the second hoax number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c53c0cf0787314208b8f7579e81a17b51be209c70a11ecb8edef2c6b907409c | A hockey player has a 25% chance of scoring a goal. If 8 shots are attempted, find the probability that the player scores 5 goals in a row. | [
"\\(\\frac{27}{16384}\\)"
] | [
"A hockey player has a 25% chance of scoring a goal. If 8 shots are attempted, find the probability that the player scores 5 goals in a row."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97c3c091ced0450469990d4c2a0d4612239afa4b6543818c4078459182da8a56 | A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders, and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielder, and 1 forward. Given a maximum of three substitutions can be made, 11 players end the game, and the team formation is never changed (i.e., a midfielder can only be replace... | [
"118"
] | [
"A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders, and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielder, and 1 forward. Given a maximum of three substitutions can be made, 11 players end the game, and the team formation is never changed (i.e., a midfielder can only be re... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-85b29c8a9029f4c5af2c95edcc5c656ed78f80fe06ba96904c506cbf36f857e8 | A hockey team of 15 players went to a pizzeria after their game. Every player had 3 different slices of pizza from a variety of types. No two players had exactly the same set of three slices. At least how many different types of pizza were available? | [
"6"
] | [
"A hockey team of 15 players went to a pizzeria after their game. Every player had 3 different slices of pizza from a variety of types. No two players had exactly the same set of three slices. At least how many different types of pizza were available?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f508430240371687bf8f411a4f9b688fa620e9fab0f3ed8e4680500897469b54 | A hockey team receives 2 points for a win and 1 point for a tie. In one season, the team won a championship with 58 points. They won 11 more games than they tied. How many wins and how many ties did the team have? | [
"23 wins and 12 ties."
] | [
"A hockey team receives 2 points for a win and 1 point for a tie. In one season, the team won a championship with 58 points. They won 11 more games than they tied. How many wins and how many ties did the team have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 19
} | 21 | 0 | 0 |
omr-ws-afd60ed1cac165b89a94eb0ad27cadf444d4b71ad91a9f46cc3b9a5a4bfc9301 | A hole 6 inches deep is drilled all the way through the center of a sphere. Find the volume of the resulting object after removing the "spherical caps" and the cylindrical section that is 6 inches in height. | [
"36\\pi"
] | [
"A hole 6 inches deep is drilled all the way through the center of a sphere. Find the volume of the resulting object after removing the \"spherical caps\" and the cylindrical section that is 6 inches in height."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1fc7d089528905fed348276db9876639afe40beefaa1ebfd09f1240c7139e51 | A hole is cut in a square sheet of plywood with side equal to 10 units of length. The hole is in the form of a rectangle whose diagonal is equal to 5 units of length. The opening is bordered with a fine wire frame. A unit of area (that is, the area of a square with side equal to unit length) of the plywood weighs 2 gra... | [
"3 and 4"
] | [
"A hole is cut in a square sheet of plywood with side equal to 10 units of length. The hole is in the form of a rectangle whose diagonal is equal to 5 units of length. The opening is bordered with a fine wire frame. A unit of area (that is, the area of a square with side equal to unit length) of the plywood weighs ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5333b5747b9f2ad2b8d074efcab0fcd1e40891b1b12ead0ab9f44f4e3534ef18 | A hole is drilled through the center of a ball of radius 4, leaving a solid with a hollow cylindrical core of height \( h \). Calculate the volume of the remaining solid when \( h = 6 \). | [
"36\\pi"
] | [
"A hole is drilled through the center of a ball of radius 4, leaving a solid with a hollow cylindrical core of height \\( h \\). Calculate the volume of the remaining solid when \\( h = 6 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-c0e39220dfdb9f366a2067a2d8a8c5ea13b86b9421e711c76a39f9ebe975d622 | A hole is drilled through the center of a ball of radius \( r \), leaving a solid with a hollow cylindrical core of height \( h \). Calculate the volume of the solid in terms of \( h \) only, assuming \( h < 2r \). | [
"\\dfrac{\\pi h^3}{6}"
] | [
"A hole is drilled through the center of a ball of radius \\( r \\), leaving a solid with a hollow cylindrical core of height \\( h \\). Calculate the volume of the solid in terms of \\( h \\) only, assuming \\( h < 2r \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-26395c9cb4a2535f81ee035236c21460bc02a4b93660442cbb49f79e18fec7cc | A hole of area 1 is covered by 5 pieces of cloth, each with an area of at least 0.5. Determine the minimum possible value of the maximum overlapping area between any two pieces of cloth. | [
"\\dfrac{3}{8}"
] | [
"A hole of area 1 is covered by 5 pieces of cloth, each with an area of at least 0.5. Determine the minimum possible value of the maximum overlapping area between any two pieces of cloth."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-9c818c91cca6fa8563e66aabd18a67c58e2c998ddfaf5dc20f8fce807bd38979 | A hole with radius $r$ is bored through the center of a sphere with radius $R$, such that the volume of the region left after boring out the hole is exactly half the volume of the original sphere. Express $r$ in terms of $R$. | [
"\\( r = \\sqrt{\\frac{\\sqrt[3]{4} - 1}{\\sqrt[3]{2}}}R \\)"
] | [
"A hole with radius $r$ is bored through the center of a sphere with radius $R$, such that the volume of the region left after boring out the hole is exactly half the volume of the original sphere. Express $r$ in terms of $R$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 15
} | 33 | 0 | 0 |
omr-ws-7edccebfd2e0192390516d89d0e538e98bbe079e1b866338816ce279cd3f5afc | A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box? | [
"216"
] | [
"A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-44ccf8f85e57f7138e7fa0d8f735f47246e2c2ae48dfe7824f59b8ee25365090 | A hollow cone is held vertex down. A sphere of radius $2$ is dropped in so that it is tangent to the surface of the cone. A sphere of radius $3$ is dropped in so that it is tangent to the surface of the cone and the first sphere. A third sphere of radius $5$ is dropped in, and a final sphere of radius $1$ is dropped in... | [
"620"
] | [
"A hollow cone is held vertex down. A sphere of radius $2$ is dropped in so that it is tangent to the surface of the cone. A sphere of radius $3$ is dropped in so that it is tangent to the surface of the cone and the first sphere. A third sphere of radius $5$ is dropped in, and a final sphere of radius $1$ is dropp... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-d51af9b7616c56993b57d936047c0a27ad507197cfec5c9a87cf8ff6dcfc2408 | A hollow cone is held with the vertex down. A sphere of radius $2$ is dropped into the cone so that it is tangent to the surface of the cone. A second sphere of radius $3$ is dropped in so that it is tangent to the cone's surface and the sphere of radius $2$. A third sphere of radius $5$ is dropped in so that it is tan... | [
"620"
] | [
"A hollow cone is held with the vertex down. A sphere of radius $2$ is dropped into the cone so that it is tangent to the surface of the cone. A second sphere of radius $3$ is dropped in so that it is tangent to the cone's surface and the sphere of radius $2$. A third sphere of radius $5$ is dropped in so that it i... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2608cbb1bb65d7273a9b3ba4f544dfb508a6c019ce343c7613f317d3bf00aad9 | A hollow ice cream cone has a slant height of 63 and its base has a circumference of \(42\pi\). An beetle is on the base of the cone and wishes to travel to the diametrically opposite point on the base. What is the length of the shortest path that the beetle can take to reach its destination? | [
"\\(63\\sqrt{3}\\)"
] | [
"A hollow ice cream cone has a slant height of 63 and its base has a circumference of \\(42\\pi\\). An beetle is on the base of the cone and wishes to travel to the diametrically opposite point on the base. What is the length of the shortest path that the beetle can take to reach its destination?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e79599544b41b854ebfe97da170dd10fd37d67f90c771bfd47de7beec267b20 | A hollow inverted right circular cone has height \(6 + 2\sqrt{2}\). A sphere of radius 2 is resting at the bottom inside the cone. A sphere of radius 3 is tangent to the first sphere, and to the lateral and top faces of the cone. How many more spheres of radius 3 can fit inside the cone? | [
"1"
] | [
"A hollow inverted right circular cone has height \\(6 + 2\\sqrt{2}\\). A sphere of radius 2 is resting at the bottom inside the cone. A sphere of radius 3 is tangent to the first sphere, and to the lateral and top faces of the cone. How many more spheres of radius 3 can fit inside the cone?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-0cb3c7c2be847d9b7dad12d2b3dadddae1983efd55a0dd46762136a2c0b89528 | A hollow rubber ball has an outside diameter of 5.6 cm and a thickness of 0.3 cm. Determine the amount of rubber used to construct the ball. | [
"\\( 8.44 \\pi \\)"
] | [
"A hollow rubber ball has an outside diameter of 5.6 cm and a thickness of 0.3 cm. Determine the amount of rubber used to construct the ball."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 25
} | 28 | 0 | 0 |
omr-ws-516b17347ce21849724276bcbeacc5e93466d9873a8186737108f92bee9ef21d | A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the maximum amount that this buyer can afford to borr... | [
"\\$89035"
] | [
"A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the maximum amount that this buyer can afford to... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-2ddc09ec2b7e40a67fc07de1315ce80fbb9beca8aaae4bf24ed5adb023bbbee8 | A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the total interest paid during the term of the mortga... | [
"$102,965"
] | [
"A home buyer can afford to spend no more than $800/month on mortgage payments. Suppose that the interest rate is 9% and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. Determine the total interest paid during the term of the m... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-7c53a5cd176605e1782b75041b6253cfb4b2de34a095c1728e33eb7fc0e5090e | A home buyer can afford to spend no more than \$N per month on mortgage payments. Suppose that the interest rate is $r$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and the payments are made continuously. Determine the maximum amount that this buyer can afford to borrow... | [
"\\(\\frac{12N}{r}\\left(1-e^{-20r}\\right)\\)"
] | [
"A home buyer can afford to spend no more than \\$N per month on mortgage payments. Suppose that the interest rate is $r$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and the payments are made continuously. Determine the maximum amount that this buyer can afford to ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-997616d3741e60c31c1fe1763ee68bcb9e28f1dbd480ac5819e8df6cb4a8bbf3 | A homeowner wants to increase the size of a rectangular deck that measures 15 feet by 20 feet. The local building code states that the deck cannot exceed 900 square feet. If the length and width are to be increased by the same amount, \( x \), find, to the nearest tenth, the maximum number of feet by which the length o... | [
"12.6"
] | [
"A homeowner wants to increase the size of a rectangular deck that measures 15 feet by 20 feet. The local building code states that the deck cannot exceed 900 square feet. If the length and width are to be increased by the same amount, \\( x \\), find, to the nearest tenth, the maximum number of feet by which the l... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ec83e2d08a0625e0eb02f3c71e793375e9ef6417faa6bbb10899c6c86dcb826a | A hook consists of three segments of length 1 forming two right angles. We have a square of side length \( n \) divided into \( n^2 \) squares of side length 1 by lines parallel to its sides. Hooks are placed on this square such that each segment of the hook covers one side of a small square, and no two segments of a h... | [
"All integers \\( n \\)."
] | [
"A hook consists of three segments of length 1 forming two right angles. We have a square of side length \\( n \\) divided into \\( n^2 \\) squares of side length 1 by lines parallel to its sides. Hooks are placed on this square such that each segment of the hook covers one side of a small square, and no two segmen... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b04b34c27a9b8f2726bcc9f10a8df606ac24ab5c61a4e90cdaed40b252688ffe | A hoop is resting vertically at a staircase with $AB = 12 \text{ cm}$ and $BC = 8 \text{ cm}$. What is the radius of the hoop? | [
"13"
] | [
"A hoop is resting vertically at a staircase with $AB = 12 \\text{ cm}$ and $BC = 8 \\text{ cm}$. What is the radius of the hoop?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0668a3d751098d23dc5e543aa046ca9f1d5c1897925138cd96b45872ef212089 | A horizontal bridge is in the shape of a parabolic arch. The center height from the bridge to the water is 10 feet, and the arch bridge is 20 feet across. What is the height \( h \) of the arch 2 feet from the shore? | [
"9.6"
] | [
"A horizontal bridge is in the shape of a parabolic arch. The center height from the bridge to the water is 10 feet, and the arch bridge is 20 feet across. What is the height \\( h \\) of the arch 2 feet from the shore?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-52943539def592c5a85a83390ed6a1039773fa5c3d2bc9bc7db9086931b73d5a | A horizontal cylindrical vessel has a length $l$ and a base radius $R$. The vessel contains a liquid with a height $h$. Calculate the volume $V$ of the liquid in the vessel using the formula $$V=2l\int_{0}^{h}\sqrt{R^2-(x-R)^2}dx$$ and find $V$ when $l=10$, $R=5$, and $h=3$. | [
"198.15"
] | [
"A horizontal cylindrical vessel has a length $l$ and a base radius $R$. The vessel contains a liquid with a height $h$. Calculate the volume $V$ of the liquid in the vessel using the formula\n$$V=2l\\int_{0}^{h}\\sqrt{R^2-(x-R)^2}dx$$\nand find $V$ when $l=10$, $R=5$, and $h=3$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-d976fa45c392f631a481b157f8b6977916de351a208eeabc0a2156404c2d7354 | A horizontal flat square board with a width of 6 inches is ruled with a grid of fine lines, 2 inches apart, and has a vertical rim around the edge. If a coin with a diameter of 2/3 inches is tossed onto the board, what is the probability that it rests without crossing a line? | [
"0.5625"
] | [
"A horizontal flat square board with a width of 6 inches is ruled with a grid of fine lines, 2 inches apart, and has a vertical rim around the edge. If a coin with a diameter of 2/3 inches is tossed onto the board, what is the probability that it rests without crossing a line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 7,
"tir": 13
} | 39 | 0 | 0 |
omr-ws-cbff7d75447b05e10ba5795e6045a97fcface78d52b0669614fe62edf579cf41 | A horse 24 feet from the center of a merry-go-round makes 32 revolutions. In order to travel the same distance, how many revolutions would a horse 8 feet from the center have to make? | [
"96"
] | [
"A horse 24 feet from the center of a merry-go-round makes 32 revolutions. In order to travel the same distance, how many revolutions would a horse 8 feet from the center have to make?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61bc83a2c5297bbf1ce2a29cb9e861cb4e0fa9ed55ce8031169d5b1122fe485b | A horse is tied outside an equilateral triangular garden at one of the vertices. The side length of the triangular fence is 8 m, and the rope is 10 m long. Find the area over which the horse can graze outside the fence. | [
"\\( 86\\pi \\)"
] | [
"A horse is tied outside an equilateral triangular garden at one of the vertices. The side length of the triangular fence is 8 m, and the rope is 10 m long. Find the area over which the horse can graze outside the fence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-70b17e36a4ef777c9ebd48b6f658182e86c1fdb5fd877179f65950e1f5b8ce3a | A horse is tied outside an equilateral triangular garden at one of the vertices. The side length of the triangular fence is 8 meters, and the rope is 10 meters long. Find the area over which the horse can graze outside the fence. | [
"\\( 86\\pi \\)"
] | [
"A horse is tied outside an equilateral triangular garden at one of the vertices. The side length of the triangular fence is 8 meters, and the rope is 10 meters long. Find the area over which the horse can graze outside the fence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-60a9389fc2ce328b0a22004913e7e5f0a629f0e0b3137f9c0b13c03eef67b6c3 | A horse is tied to the edge of a circular pasture with a radius of 10 meters using a rope that is 15 meters long. How much area can the horse graze? | [
"233.012"
] | [
"A horse is tied to the edge of a circular pasture with a radius of 10 meters using a rope that is 15 meters long. How much area can the horse graze?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-8ea8bf60b83900d0b051120d7ccfe930ca54955ada9f41123a623c38c68c7102 | A horseshoe shaped as a semi-circle with a diameter of $1$ is thrown randomly onto a square grid where each square has a side length of $1$. What is the expected number of intersections between the horseshoe and the grid lines? | [
"2"
] | [
"A horseshoe shaped as a semi-circle with a diameter of $1$ is thrown randomly onto a square grid where each square has a side length of $1$. What is the expected number of intersections between the horseshoe and the grid lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5904efae695ec41014baeb1f42869cd95a9f9e8bcc8f676a7077e049a9d93e92 | A hospital receives $X$ patients in a day, where $X$ follows a Poisson distribution with parameter $100$. Each patient is sent for a blood test with probability $0.25$, independently of the others. Let $Y$ be the number of patients who are sent for a blood test in a day. What is the probability distribution of $Y$? | [
"\\( Y \\sim \\operatorname{Pois}(25) \\) or \\( P(Y = y) = \\frac{e^{-25} \\cdot 25^y}{y!} \\)"
] | [
"A hospital receives $X$ patients in a day, where $X$ follows a Poisson distribution with parameter $100$. Each patient is sent for a blood test with probability $0.25$, independently of the others. Let $Y$ be the number of patients who are sent for a blood test in a day. What is the probability distribution of $Y$... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3dcfc7af8c826788513b0e3217e1b70066496d93f9f3dc02aa99c0c74c8ac26e | A host invites \( n \) couples to a party. She wants to ask a subset of the \( 2n \) guests to give a speech, but she does not want to ask both members of any couple to give speeches. If she wants at least one guest to speak, in how many ways can she proceed? | [
"\\(3^n - 1\\)"
] | [
"A host invites \\( n \\) couples to a party. She wants to ask a subset of the \\( 2n \\) guests to give a speech, but she does not want to ask both members of any couple to give speeches. If she wants at least one guest to speak, in how many ways can she proceed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58c0f78d7ffa6ce9cb9816e03af667841cc6330d973652a4ded5f45d4d9db9a5 | A host invites \( n \) couples to a party. She wants to ask a subset of the \( 2n \) guests to give a speech, but she does not want to ask both members of any couple to give speeches. In how many ways can she proceed? | [
"\\( 3^n \\)"
] | [
"A host invites \\( n \\) couples to a party. She wants to ask a subset of the \\( 2n \\) guests to give a speech, but she does not want to ask both members of any couple to give speeches. In how many ways can she proceed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0fd3a71b46dcef1d597ea00e4e44d3a3f251e73a7f76f1d1d5eebbbad764cbfb | A hostel needs to buy 200 tables, 300 desks, and 500 chairs. Company "abc" sells a package containing 20 tables, 18 desks, and 25 chairs for 20,000 rupees. Company "xyz" sells a package containing 10 tables, 24 desks, and 50 chairs for 30,000 rupees. What is the most cost-effective combination of packages from "abc" an... | [
"350000"
] | [
"A hostel needs to buy 200 tables, 300 desks, and 500 chairs. Company \"abc\" sells a package containing 20 tables, 18 desks, and 25 chairs for 20,000 rupees. Company \"xyz\" sells a package containing 10 tables, 24 desks, and 50 chairs for 30,000 rupees. What is the most cost-effective combination of packages from... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 25
} | 32 | 0 | 0 |
omr-ws-385771d7d0528ea268498f4165682d318a9c09bec5262839c9bec28f72a4f962 | A hot beverage initially at 198 degrees Fahrenheit cools to 174 degrees Fahrenheit in 5 minutes in a room at a constant 74 degrees Fahrenheit. How long will it take for the beverage to cool to 105 degrees Fahrenheit? | [
"32.2"
] | [
"A hot beverage initially at 198 degrees Fahrenheit cools to 174 degrees Fahrenheit in 5 minutes in a room at a constant 74 degrees Fahrenheit. How long will it take for the beverage to cool to 105 degrees Fahrenheit?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b644407ca3a2ca407fe951c8fffd8f16ebbbd21ddf780046ce9d3c3a2aa68237 | A hotel can sell out its 250 rooms if it charges $175 per room. For every $10 increase in price, the hotel rents 3 fewer rooms. What price should the hotel charge to maximize its revenue? | [
"504.17"
] | [
"A hotel can sell out its 250 rooms if it charges $175 per room. For every $10 increase in price, the hotel rents 3 fewer rooms. What price should the hotel charge to maximize its revenue?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b4a04b8ea908ca671dcf8e6c31e9569755473b7b1ae70249149ceb3b13cadaf7 | A hotel charges $20 for the use of its dining room and $2.50 per plate for each dinner. An association charges $3 per plate but invites four nonpaying guests. If each person has one plate, how many paying persons must attend for the association to collect the exact amount needed to pay the hotel? | [
"60"
] | [
"A hotel charges $20 for the use of its dining room and $2.50 per plate for each dinner. An association charges $3 per plate but invites four nonpaying guests. If each person has one plate, how many paying persons must attend for the association to collect the exact amount needed to pay the hotel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0391f5db57014459bde614602dfbe6ab2fc4bfa9ecb1a8ee7ed8253a9114e6c0 | A hotel has 10 rooms on each side of a corridor. 7 rooms are to be booked so that no 2 rooms on the same side of the corridor are adjacent. How many ways can this be done? | [
"4352"
] | [
"A hotel has 10 rooms on each side of a corridor. 7 rooms are to be booked so that no 2 rooms on the same side of the corridor are adjacent. How many ways can this be done?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbed5e1fbbbdd3e1693b9400e419caf60ff108f22df849429823109a2533f3cd | A hotel has 100 rooms where the $i$-th room can hold a maximum of $100 + i$ ants for all $i = 1, 2, \dots, 100$. The total number of ants in the hotel is $n$. The queen ant requires a room of her own. Determine the maximum value of $n$ such that the manager can always choose two rooms $A$ and $B$, move all guests from ... | [
"14949"
] | [
"A hotel has 100 rooms where the $i$-th room can hold a maximum of $100 + i$ ants for all $i = 1, 2, \\dots, 100$. The total number of ants in the hotel is $n$. The queen ant requires a room of her own. Determine the maximum value of $n$ such that the manager can always choose two rooms $A$ and $B$, move all guests... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-2ba41c5b0bcd49a2b226274bad1d9dee261e41427e8d914e96264b1862312bdf | A hotel has 113 floors, with the lobby listed as the first floor. How many total digits are there on the elevator buttons combined? | [
"229"
] | [
"A hotel has 113 floors, with the lobby listed as the first floor. How many total digits are there on the elevator buttons combined?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d47b52f09ba3dde8a6fd2ab8116bf6b354427b5f886879088b99b134243aaa09 | A hotel has rooms that can accommodate up to two people. Couples can share a room, but otherwise men will share only with men and women only with women. How many rooms are needed to guarantee that any group of 100 people can be accommodated? | [
"51"
] | [
"A hotel has rooms that can accommodate up to two people. Couples can share a room, but otherwise men will share only with men and women only with women. How many rooms are needed to guarantee that any group of 100 people can be accommodated?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-eb2635ef8a18becd997383cdc96343e36b72b39e7888163204e1d1982e6a0ae6 | A hotel has ten rooms along each side of a corridor. An olympiad team leader wishes to book seven rooms on the corridor so that no two reserved rooms on the same side of the corridor are adjacent. In how many ways can this be done? | [
"4352"
] | [
"A hotel has ten rooms along each side of a corridor. An olympiad team leader wishes to book seven rooms on the corridor so that no two reserved rooms on the same side of the corridor are adjacent. In how many ways can this be done?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 13
} | 61 | 0 | 0 |
omr-ws-ae31cd4ee1c8cbf5ee8347250fa765d3ca38d6dc7c0afe936ce752ad03872c8f | A hotel packed breakfast for each of three guests, with each breakfast consisting of one nut roll, one cheese roll, and one fruit roll. The rolls were wrapped and became indistinguishable. The preparer randomly placed three rolls in a bag for each guest. Given that the probability each guest got one roll of each type i... | [
"79"
] | [
"A hotel packed breakfast for each of three guests, with each breakfast consisting of one nut roll, one cheese roll, and one fruit roll. The rolls were wrapped and became indistinguishable. The preparer randomly placed three rolls in a bag for each guest. Given that the probability each guest got one roll of each t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-5c46d803a226034f24c26d79fae98aaa82f586d43d996c68d9c49e8651f249b5 | A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each ... | [
"79"
] | [
"A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1d5b5dc342c2d497dedd208179b7127168309f86d8b3a2a842b4326f4015fa4 | A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each ... | [
"079"
] | [
"A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
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