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-27d4b92d56b998a597b91dde4444831313eb1a66c826d59405e5413476778aaa | A regular $n$-gon is inscribed in a unit circle. Compute the product of the distances from a fixed vertex to all the other vertices. | [
"n"
] | [
"A regular $n$-gon is inscribed in a unit circle. Compute the product of the distances from a fixed vertex to all the other vertices."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61578717c634e76bdc770b1d1770dee96171fdd60c73b383f46161339756aa19 | A regular $n$-gon with vertices $P_1, P_2, \ldots, P_n$ is inscribed in a unit circle. What is the value of the sum $$\sum_{k=2}^{n} P_1P_k$$ where $P_1P_k$ denotes the distance between vertices $P_1$ and $P_k$? | [
"\\(2 \\cot \\left(\\frac{\\pi}{2n}\\right)\\)"
] | [
"A regular $n$-gon with vertices $P_1, P_2, \\ldots, P_n$ is inscribed in a unit circle. What is the value of the sum\n$$\\sum_{k=2}^{n} P_1P_k$$\nwhere $P_1P_k$ denotes the distance between vertices $P_1$ and $P_k$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-2f4cc1905884db551871e8ea51558be3f5ba3e019cfe85c263fe27796abf7adf | A regular $n$-sided polygon can form exactly $2023$ distinct squares from its vertices. What is the minimum possible value of $n$? | [
"8092"
] | [
"A regular $n$-sided polygon can form exactly $2023$ distinct squares from its vertices. What is the minimum possible value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-14aa841b5d74304d9015e4839bb94be251384cf3269f2c07a71b2bb08da38195 | A regular (5 × 5)-array of lights is defective such that toggling the switch for one light causes each adjacent light in the same row and column, as well as the light itself, to change state. Initially, all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the po... | [
"9"
] | [
"A regular (5 × 5)-array of lights is defective such that toggling the switch for one light causes each adjacent light in the same row and column, as well as the light itself, to change state. Initially, all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-c2163c58894bf83857df9ca47cbafedd110d20fc0a43a2feaba45c3c94bf3e27 | A regular (n+2)-gon is inscribed in a circle. Let \( T_{n} \) denote the number of ways to join its vertices in pairs so that the resulting segments do not intersect one another. Given that \( T_{0}=1 \), calculate \( T_{5} \). | [
"42"
] | [
"A regular (n+2)-gon is inscribed in a circle. Let \\( T_{n} \\) denote the number of ways to join its vertices in pairs so that the resulting segments do not intersect one another. Given that \\( T_{0}=1 \\), calculate \\( T_{5} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-c6b588014ddd38405a30a61b487ca4c6c99e5739f90e850e8907667420d9038c | A regular 10-gon must be completely covered by triangles that do not go outside the 10-gon. The triangles themselves can overlap. Determine: a) how to do it with the help of 8 triangles, b) What is the smallest number of triangles that can do this? | [
"8"
] | [
"A regular 10-gon must be completely covered by triangles that do not go outside the 10-gon. The triangles themselves can overlap. Determine:\na) how to do it with the help of 8 triangles,\nb) What is the smallest number of triangles that can do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 10
} | 12 | 0 | 0 |
omr-ws-7bba24490ed8891c44571d8feda34c472f05d29726c6d4123e3008c9ead816a8 | A regular 11-gon is inscribed in a circle. How many triangles are there whose three vertices are all vertices of the 11-gon and whose interiors contain the center of the circle? | [
"55"
] | [
"A regular 11-gon is inscribed in a circle. How many triangles are there whose three vertices are all vertices of the 11-gon and whose interiors contain the center of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-0d3ec67f513dc28820c0ec3f4b076b2e2bc26c7856adf51464a546b37e85eff3 | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$. | [
"720"
] | [
"A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers. Find $a + b + c + d^{}_{}$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-d803fd80bced4f5e31baec044acaabc26462403d50947c222ce7c0335c2742d8 | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \( a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6} \), where \( a \), \( b \), \( c \), and \( d \) are positive integers. Find \( a + b + c + d \). | [
"720"
] | [
"A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \\( a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6} \\), where \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are positive integers. Find \\( a + b + c + d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-38efabe17b456be68baef0bd1ea5f184d490b5e746c54423cfc6a6a3fe5d1516 | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \( a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6} \) where \( a \), \( b \), \( c \), and \( d \) are positive integers. Find \( a + b + c + d \). | [
"720"
] | [
"A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \\( a + b\\sqrt{2} + c\\sqrt{3} + d\\sqrt{6} \\) where \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are positive integers. Find \\( a + b + c + d \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5ef4ea5a4f8d99173b3d7fee06ffbbf3f8c567a83abd3685fa6f9499a37cfcd | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \(a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}\), where \(a\), \(b\), \(c\), and \(d\) are positive integers. Find \(a + b + c + d\). | [
"720"
] | [
"A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \\(a + b\\sqrt{2} + c\\sqrt{3} + d\\sqrt{6}\\), where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are positive integers. Find \\(a + b + c + d\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-28c92febc867f3a14c2342fe905be6f98ff5ec4c30bc242fbe9bde7b71647f65 | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form \[a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. | [
"720"
] | [
"A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form\n\\[a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},\\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbce9a9e6695d26a7df7e253bf0a4f1bcc10b4406996457f5c5ac3329d675f61 | A regular 12-gon is inscribed in a circle of radius 5. What is the sum of the squares of the distances from any one fixed vertex to all the others? | [
"600"
] | [
"A regular 12-gon is inscribed in a circle of radius 5. What is the sum of the squares of the distances from any one fixed vertex to all the others?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-13d6806e92dbb408a5b2c147722615bca33773c262cdd350c6e5d95258bab85f | A regular 12-sided die is rolled $n$ times where $11$ does not divide $n$. Let the probability that the sum of the numbers rolled is a multiple of $11$ be $\frac{p}{q}$ where $(p,q)=1$. Find $|11p-q|$. | [
"1"
] | [
"A regular 12-sided die is rolled $n$ times where $11$ does not divide $n$. Let the probability that the sum of the numbers rolled is a multiple of $11$ be $\\frac{p}{q}$ where $(p,q)=1$. Find $|11p-q|$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f09b544ca8f7347a8b9de77123dc22af118a569920f0932755e412b10e9dd852 | A regular 1985-gon is given in the plane. If a straight line is passed through each side of the polygon, determine the number of parts into which these lines divide the plane. | [
"1968081"
] | [
"A regular 1985-gon is given in the plane. If a straight line is passed through each side of the polygon, determine the number of parts into which these lines divide the plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3acf2f28d6b99c8b5c2299b1b6148bbdb87f01fc303af0aebcd038ef001ea9c | A regular 1997-gon is decomposed into triangles using non-intersecting diagonals. How many of these triangles are acute? | [
"1"
] | [
"A regular 1997-gon is decomposed into triangles using non-intersecting diagonals. How many of these triangles are acute?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-13517eb928deb04ee49b7c822db322d9a32c5b007358a9977fb0a3252c0f819c | A regular 2001-gon is inscribed in a circle. Consider a regular 667-gon whose vertices are at vertices of the 2001-gon. Let $A$ be the area of the part of the 2001-gon lying outside the 667-gon. If $A = k\cdot\sin^3\frac{\pi}{2001}\cdot\cos^3\frac{\pi}{2001}$, find the value of $k$. | [
"10672"
] | [
"A regular 2001-gon is inscribed in a circle. Consider a regular 667-gon whose vertices are at vertices of the 2001-gon. Let $A$ be the area of the part of the 2001-gon lying outside the 667-gon. If $A = k\\cdot\\sin^3\\frac{\\pi}{2001}\\cdot\\cos^3\\frac{\\pi}{2001}$, find the value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3f14744df2e5ebe43ec672367cbdb53589f927be885b33e643d2b9ad0c3aaa0 | A regular 2004-gon is divided into 2002 triangles. Find the maximal number of obtuse triangles in this dissection. | [
"2000"
] | [
"A regular 2004-gon is divided into 2002 triangles. Find the maximal number of obtuse triangles in this dissection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-afe60cd9ab7bf9bb5c0cc64db369a0edfb665502e721363f2af0818142ce6dd1 | A regular 2004-gon is divided into 2003 triangles. Find the maximal number of obtuse triangles in this dissection. | [
"2002"
] | [
"A regular 2004-gon is divided into 2003 triangles. Find the maximal number of obtuse triangles in this dissection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-5b3afa3ab8ced590e42044e7c7383dd5cf29a0e314735f79ff7206160101f341 | A regular 201-sided polygon is inscribed in a circle with center \( C \). Triangles are formed by connecting any three of the 201 vertices of the polygon. Find the number of triangles that have the point \( C \) lying inside the triangle, and then find the sum of the digits of this number. | [
"22"
] | [
"A regular 201-sided polygon is inscribed in a circle with center \\( C \\). Triangles are formed by connecting any three of the 201 vertices of the polygon. Find the number of triangles that have the point \\( C \\) lying inside the triangle, and then find the sum of the digits of this number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-ebee9f8cfeb64f94d889112f7188deb546b03fdbb8f2d7fc931da19274e853d5 | A regular 2012-gon has one vertex joined with all the vertices that are not adjacent to it. This divides the 2012-gon into triangles. How many of these triangles are obtuse? | [
"1004"
] | [
"A regular 2012-gon has one vertex joined with all the vertices that are not adjacent to it. This divides the 2012-gon into triangles. How many of these triangles are obtuse?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-110dd5f8858574a5514b1973c151eb18816f537ae14be7f0867151fea17cb1a7 | A regular 2013-gon and all its diagonals are drawn. They divide the polygon into many smaller polygons that do not overlap. How many of these smaller polygons have 2013 sides? | [
"1"
] | [
"A regular 2013-gon and all its diagonals are drawn. They divide the polygon into many smaller polygons that do not overlap. How many of these smaller polygons have 2013 sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ffb2a6445ad58cdcd8ccfee82f373c2f72a393b14d75936478842ce8ade89383 | A regular 2689-gon is cut by 2686 diagonals into 2687 triangles. What is the maximum number of different triangles that could be formed? | [
"1344"
] | [
"A regular 2689-gon is cut by 2686 diagonals into 2687 triangles. What is the maximum number of different triangles that could be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-d78ed5d31bffca38ee9de5fc439e416c440e8a670d6605522bbc42f2539b8bf7 | A regular 2n-gon with center A is inscribed in a circle. After rotating the 2n-gon about A, the rays from A to the vertices of the 2n-gon intersect the circle at 2n new points. Let \(P\) be the sum of the x-coordinates and \(Q\) be the sum of the y-coordinates of the original 2n vertices. Calculate the value of \(P + Q... | [
"0"
] | [
"A regular 2n-gon with center A is inscribed in a circle. After rotating the 2n-gon about A, the rays from A to the vertices of the 2n-gon intersect the circle at 2n new points. Let \\(P\\) be the sum of the x-coordinates and \\(Q\\) be the sum of the y-coordinates of the original 2n vertices. Calculate the value o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f056363f2201b5337575073d6c99609efd503ef8a2c4d426b095b7651e5b7402 | A regular 6-faced die is thrown three times. Calculate the probability that the sum of the dice points on all three throws is 12. | [
"\\dfrac{25}{216}"
] | [
"A regular 6-faced die is thrown three times. Calculate the probability that the sum of the dice points on all three throws is 12."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8448d68e05d3f74352db61d9697996feb83f6f4d60ebebc214a43b9dad9389af | A regular 6-sided die is rolled three times. Given that one 6 was rolled, what is the probability that a 6 was rolled at least twice? | [
"\\(\\frac{16}{91}\\)"
] | [
"A regular 6-sided die is rolled three times. Given that one 6 was rolled, what is the probability that a 6 was rolled at least twice?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0b067b856717392b969008b5cee01fc61024b08467f9cd41d74e31dbbc617c4c | A regular 61-gon with vertices \( A_1, A_2, A_3, \ldots, A_{61} \) is inscribed in a circle with radius \( r \). Suppose \( (\overline{A_1A_2})(\overline{A_1A_3})(\overline{A_1A_4}) \cdots (\overline{A_1A_{61}}) = r \). If \( r^{2006} = \frac{p}{q} \) where \( p \) and \( q \) are coprime positive integers, find the re... | [
"642"
] | [
"A regular 61-gon with vertices \\( A_1, A_2, A_3, \\ldots, A_{61} \\) is inscribed in a circle with radius \\( r \\). Suppose \\( (\\overline{A_1A_2})(\\overline{A_1A_3})(\\overline{A_1A_4}) \\cdots (\\overline{A_1A_{61}}) = r \\). If \\( r^{2006} = \\frac{p}{q} \\) where \\( p \\) and \\( q \\) are coprime positi... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-f8d9eb23427a6d0936110d3709d2eb6d4b823548bcf25afb623c48ae0cf239bf | A regular 9-gon has vertices labeled \( P_1 \) through \( P_9 \). Given \( P_1P_2 = 1 \) and \( P_1P_3 = x \), let the intersection of \( P_1P_2 \) and \( P_4P_5 \) be \( Q \). The area of \( \triangle P_1QP_5 \) is given by \( A \cdot x^2 + B \cdot x + C \). Find the ordered triple \( (A, B, C) \). | [
"\\left(0, \\frac{1}{4}, 0\\right)"
] | [
"A regular 9-gon has vertices labeled \\( P_1 \\) through \\( P_9 \\). Given \\( P_1P_2 = 1 \\) and \\( P_1P_3 = x \\), let the intersection of \\( P_1P_2 \\) and \\( P_4P_5 \\) be \\( Q \\). The area of \\( \\triangle P_1QP_5 \\) is given by \\( A \\cdot x^2 + B \\cdot x + C \\). Find the ordered triple \\( (A, B,... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9872e95c597402a86adba7d78d359c37617ef5e35046e5a7a641735a9e9741a9 | A regular \( 5 \times 5 \)-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggle... | [
"(3,3)"
] | [
"A regular \\( 5 \\times 5 \\)-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-0fac1a813b05e8011c418bbf855149896c4fd2c4973b2a89921a7d3fdd1fbb55 | A regular \( n \)-gon is inscribed in a circle of radius 1. Let \( a_1, \cdots, a_{n-1} \) be the distances of one of the vertices of the polygon to all the other vertices. Calculate the value of \( (5 - a_1^2) \cdots (5 - a_{n-1}^2) \) for \( n = 10 \). | [
"3025"
] | [
"A regular \\( n \\)-gon is inscribed in a circle of radius 1. Let \\( a_1, \\cdots, a_{n-1} \\) be the distances of one of the vertices of the polygon to all the other vertices. Calculate the value of \\( (5 - a_1^2) \\cdots (5 - a_{n-1}^2) \\) for \\( n = 10 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91e0f04419375e4ae3670b556f7dca905b2d6fecf4bd94c54c9984665379fdcc | A regular \(4n\)-gon is decomposed into parallelograms. Let \(S\) be the sum of the areas of all rectangles from the decomposition. If the side length of the \(4n\)-gon is 1, find the value of \(S\) when \(n=3\). | [
"6"
] | [
"A regular \\(4n\\)-gon is decomposed into parallelograms. Let \\(S\\) be the sum of the areas of all rectangles from the decomposition. If the side length of the \\(4n\\)-gon is 1, find the value of \\(S\\) when \\(n=3\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-f87e3a66d699d23b5aaf9d832c92a3cd3d97b55851e70c06145ec900d535b738 | A regular decagon (10-sided polygon) $ABCDEFGHIJ$ has an area of $2017$ square units. Determine the area (in square units) of the rectangle $CDHI$. | [
"806.8"
] | [
"A regular decagon (10-sided polygon) $ABCDEFGHIJ$ has an area of $2017$ square units. Determine the area (in square units) of the rectangle $CDHI$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-89e9cad8653e952662c60becb623cc874ae79d6ae0c6e480466a2304d3b2c354 | A regular decagon is formed. Two triangles are drawn by choosing three vertices at random each time. What is the probability that over half of the decagon is shaded if both triangles are shaded? | [
"\\dfrac{1}{2}"
] | [
"A regular decagon is formed. Two triangles are drawn by choosing three vertices at random each time. What is the probability that over half of the decagon is shaded if both triangles are shaded?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-914f94ec7062847b860fdde8510cd38a3de9db78c5fac37604c4a84cf2bb834a | A regular decagon is inscribed in a circle. What is the probability that the length of a chord drawn by connecting any two non-consecutive vertices is greater than the radius of the circle? | [
"1"
] | [
"A regular decagon is inscribed in a circle. What is the probability that the length of a chord drawn by connecting any two non-consecutive vertices is greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-b2cfcd1fc204ff00a99bdeaaba0116198b8959e56873c3a65b439944872432c7 | A regular decagon-shaped shield has three lines drawn on it that delimit a small triangle with an area of 21 cm². What is the surface area of the entire shield, rounded to the nearest whole number, given that \(\cos 36^\circ = 0.809\)? | [
"2209 cm²"
] | [
"A regular decagon-shaped shield has three lines drawn on it that delimit a small triangle with an area of 21 cm². What is the surface area of the entire shield, rounded to the nearest whole number, given that \\(\\cos 36^\\circ = 0.809\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-996aabe8114061b1ca85449046ed5df1841592fceb18e2c062329f4af806a5cb | A regular dodecagon ($12$-sided polygon) is inscribed within a circle. To the nearest integer, what percent of the area of the circle is outside of the dodecagon? | [
"5"
] | [
"A regular dodecagon ($12$-sided polygon) is inscribed within a circle. To the nearest integer, what percent of the area of the circle is outside of the dodecagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1adef31601df0e1ef77e8e29cb772a7ce9844f4a64c8fc64d4698c262b4bb38 | A regular dodecagon \(ABCDEF\cdots L\) is inscribed in a square with an area of 24. Points \(A\), \(D\), \(G\), and \(J\) are at the midpoints of each side of the square. Find the area of the dodecagon. | [
"18"
] | [
"A regular dodecagon \\(ABCDEF\\cdots L\\) is inscribed in a square with an area of 24. Points \\(A\\), \\(D\\), \\(G\\), and \\(J\\) are at the midpoints of each side of the square. Find the area of the dodecagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 14
} | 27 | 0 | 0 |
omr-ws-8724d14d468af2bc1d2e4199e803a54bff0bb2d7a3995f284275fa9686d154c0 | A regular dodecagon \(A_1A_2\ldots A_{12}\) is inscribed in a circle with a diameter of 20 cm. Calculate the perimeter of the pentagon \(A_1A_3A_6A_8A_{11}\). | [
"\\(30 + 20\\sqrt{2}\\) cm"
] | [
"A regular dodecagon \\(A_1A_2\\ldots A_{12}\\) is inscribed in a circle with a diameter of 20 cm. Calculate the perimeter of the pentagon \\(A_1A_3A_6A_8A_{11}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-492df3f428a6da6ab6c4cee79adb94b25a8918a600082c45b827319a5b630bfe | A regular dodecagon has side length $2$. The area of the smallest rectangle whose vertices are vertices of the dodecagon can be expressed in the form $a + b\sqrt{3}$, where $a$ and $b$ are natural numbers. Compute $a + b$. | [
"8\n\nGiven the forum discussions, the first post provides a clear and concise solution to the problem. It states that the area of the smallest rectangle whose vertices are vertices of the dodecagon is $4 + 4\\sqrt{3}$, leading to $a + b = 8$. The second post seems to have a different approach and arrives at a diff... | [
"A regular dodecagon has side length $2$. The area of the smallest rectangle whose vertices are vertices of the dodecagon can be expressed in the form $a + b\\sqrt{3}$, where $a$ and $b$ are natural numbers. Compute $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e7bd345cd52643dd2122ae37134005b10e1a38d998e8f0b92255f9356697437 | A regular dodecahedron is a figure with 12 identical pentagons for each of its faces. Let $x$ be the number of ways to color the faces of the dodecahedron with 12 different colors, where two colorings are identical if one can be rotated to obtain the other. Compute $\frac{x}{12!}$. | [
"\\dfrac{1}{60}"
] | [
"A regular dodecahedron is a figure with 12 identical pentagons for each of its faces. Let $x$ be the number of ways to color the faces of the dodecahedron with 12 different colors, where two colorings are identical if one can be rotated to obtain the other. Compute $\\frac{x}{12!}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42e0f20173a6bfebd82a9a92ca0daa443a8c3f25b0008259a58bdd63500a941e | A regular dodecahedron is inscribed in a unit sphere. Vectors are drawn from the center of the sphere to each of the vertices. What is the sum of these vectors? | [
"0"
] | [
"A regular dodecahedron is inscribed in a unit sphere. Vectors are drawn from the center of the sphere to each of the vertices. What is the sum of these vectors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-b236601b7a586807eda5de1824488d46798528e90cce5a60450690b7babde610 | A regular heptagon \(A_1A_2 \ldots A_7\) is inscribed in a circle \(C\). Point \(P\) is taken on the shorter arc \(A_7A_1\). Calculate the value of \(PA_1 + PA_3 + PA_5 + PA_7 - (PA_2 + PA_4 + PA_6)\). | [
"0"
] | [
"A regular heptagon \\(A_1A_2 \\ldots A_7\\) is inscribed in a circle \\(C\\). Point \\(P\\) is taken on the shorter arc \\(A_7A_1\\). Calculate the value of \\(PA_1 + PA_3 + PA_5 + PA_7 - (PA_2 + PA_4 + PA_6)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-c5936ef29ffe5634d457f79f12e10558653509f00fb48f8a09cae406cde0c37b | A regular heptagon \(A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}A_{7}\) is inscribed in a unit circle \(\odot O\). Point \(P\) is on the extension line of \(OA_1\) such that \(OP = 2\). Find the value of \(|PA_1||PA_2| \cdots |PA_7|\). | [
"127"
] | [
"A regular heptagon \\(A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}A_{7}\\) is inscribed in a unit circle \\(\\odot O\\). Point \\(P\\) is on the extension line of \\(OA_1\\) such that \\(OP = 2\\). Find the value of \\(|PA_1||PA_2| \\cdots |PA_7|\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7ab228e1ed5e148b76fec82a9b87551145b2d4d34b1939a8415ee7842e200659 | A regular heptagon circumscribes a circular cookie. The apothem of the heptagon is 13, and the distance from the center of the circle to a vertex of the heptagon is 19. Compute the area outside the circle and inside the heptagon. | [
"\\( 728\\sqrt{3} - 169\\pi \\)"
] | [
"A regular heptagon circumscribes a circular cookie. The apothem of the heptagon is 13, and the distance from the center of the circle to a vertex of the heptagon is 19. Compute the area outside the circle and inside the heptagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-8ef0d187ada551f584c3499c5e973fda64c8757619a20d76cbbfa8dce5b3b5da | A regular heptagon is circumscribed about a circle. Given that the area of the region between the heptagon and the circle is 1, find the radius of the circle. | [
"\\( r = \\sqrt{\\frac{2}{2\\pi - 7\\sin\\frac{\\pi}{7}}} \\)"
] | [
"A regular heptagon is circumscribed about a circle. Given that the area of the region between the heptagon and the circle is 1, find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0435bf6805c959cfeab503e660290bc5916f8a692c6f0d068066cd5b42a809b | A regular heptagon is triangulated by joining all its vertices. Each side of the resulting triangles is colored either red or blue. Let $N$ be the number of monochromatic triangles (triangles with all sides the same color). Find the minimum possible value of $N$. | [
"5"
] | [
"A regular heptagon is triangulated by joining all its vertices. Each side of the resulting triangles is colored either red or blue. Let $N$ be the number of monochromatic triangles (triangles with all sides the same color). Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-8a11a8eda684faa25b67b676c320960dbb57eb2653cbc8a630ccf35275a76e4c | A regular hexagon $ABCDEF$ and a regular triangle $APQ$ have no common interior points. It is known that $PB < QB$, and the point $M$ is the midpoint of the segment $PB$. Find the angle between lines $AM$ and $QF$. | [
"60^\\circ"
] | [
"A regular hexagon $ABCDEF$ and a regular triangle $APQ$ have no common interior points. It is known that $PB < QB$, and the point $M$ is the midpoint of the segment $PB$. Find the angle between lines $AM$ and $QF$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-d18c497c051d87ac918d1e10a177f1243a8a5e504f1cced139249952b6095485 | A regular hexagon $ABCDEF$ has an area of $12$ cm$^2$. Drawing segments from a vertex, the hexagon is divided into $4$ triangles. Calculate the areas of these triangles. | [
"$2, 2, 4, 4$"
] | [
"A regular hexagon $ABCDEF$ has an area of $12$ cm$^2$. Drawing segments from a vertex, the hexagon is divided into $4$ triangles. Calculate the areas of these triangles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-18d01617d09968f27c5dca45cc1611e8ef665a03dcd66e1d4966902306e1dcd1 | A regular hexagon $ABCDEF$ has sides of length $2\sqrt{3}$. A circle passes through points $A$ and $B$ and is tangent to sides $CD$ and $EF$. Find the length of the circle's radius. | [
"\\(6 - 2\\sqrt{2}\\)"
] | [
"A regular hexagon $ABCDEF$ has sides of length $2\\sqrt{3}$. A circle passes through points $A$ and $B$ and is tangent to sides $CD$ and $EF$. Find the length of the circle's radius."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-51f615c9a277c6c45d52572a63981556c050003a14cf68e24a25d30efa22da0b | A regular hexagon $ABCDEF$ is inscribed in a rectangle $PQRS$ with $AB = 1,$ and $A$ and $B$ on side $PQ,$ $C$ on side $QR,$ $D$ and $E$ on side $RS,$ and $F$ on side $SP$. What is the area of $PQRS$? | [
"\\(3\\sqrt{3}\\)"
] | [
"A regular hexagon $ABCDEF$ is inscribed in a rectangle $PQRS$ with $AB = 1,$ and $A$ and $B$ on side $PQ,$ $C$ on side $QR,$ $D$ and $E$ on side $RS,$ and $F$ on side $SP$. What is the area of $PQRS$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-127d4dbc20304817fdd6303229baedbe17e8e4d8c44963020f9e8a9f562eddbd | A regular hexagon \( ABCDEF \) has an area of 1. The sides \( AB \), \( CD \), and \( EF \) are extended to form a larger polygon with \( ABCDEF \) in the interior. Find the area of this larger polygon. | [
"1.5"
] | [
"A regular hexagon \\( ABCDEF \\) has an area of 1. The sides \\( AB \\), \\( CD \\), and \\( EF \\) are extended to form a larger polygon with \\( ABCDEF \\) in the interior. Find the area of this larger polygon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8bb595b5440d8ad35d2b4f509d7b6fcac4543735b4e1dcc5f38d9f190e92b4b6 | A regular hexagon \( H \) with side length 20 is divided completely into equilateral triangles with side length 1. How many regular hexagons with sides parallel to the sides of \( H \) are formed by lines in the grid? | [
"8000"
] | [
"A regular hexagon \\( H \\) with side length 20 is divided completely into equilateral triangles with side length 1. How many regular hexagons with sides parallel to the sides of \\( H \\) are formed by lines in the grid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-f5c7e43982bba74f7ee894d177ef29d6bfe26753b527e4a0f671b8df20e6ed48 | A regular hexagon \(ABCDEF\) has vertices \(A\) and \(C\) at \((0,0)\) and \((7,1)\), respectively. What is the area of the hexagon? | [
"\\( 25\\sqrt{3} \\)"
] | [
"A regular hexagon \\(ABCDEF\\) has vertices \\(A\\) and \\(C\\) at \\((0,0)\\) and \\((7,1)\\), respectively. What is the area of the hexagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 60
} | 92 | 0 | 0 |
omr-ws-a61d7a025e60e83e4d808a285cdddb7fbe0ded9a5cbd149707a495c5b8a2e2f6 | A regular hexagon \(ABCDEF\) is inscribed in a circle \(w_1\). Another circle \(w_2\) is tangent to \(AB\) at its midpoint and internally tangent to \(w_1\) at the midpoint of arc \(AB\). If the radius of \(w_2\) is 6, what is the area of \(ABCDEF\)? | [
"\\(10368 + 6048\\sqrt{3}\\)"
] | [
"A regular hexagon \\(ABCDEF\\) is inscribed in a circle \\(w_1\\). Another circle \\(w_2\\) is tangent to \\(AB\\) at its midpoint and internally tangent to \\(w_1\\) at the midpoint of arc \\(AB\\). If the radius of \\(w_2\\) is 6, what is the area of \\(ABCDEF\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-81a2374cdf2d384d9a68018e68b7f3001c9ea0874681d40e780e8f5e700634a6 | A regular hexagon \(ABCDEF\) is rotated 180 degrees clockwise around the axis \(AD\). If \(AD = 8\), what is the volume of the solid generated by the rotation? | [
"64\\pi"
] | [
"A regular hexagon \\(ABCDEF\\) is rotated 180 degrees clockwise around the axis \\(AD\\). If \\(AD = 8\\), what is the volume of the solid generated by the rotation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 8
} | 12 | 0 | 0 |
omr-ws-80d0ccea26f483959856460454a15dbea2852589d9a67fa19cc8f2d6b4955e07 | A regular hexagon and a regular pentagon have a common edge as shown. Find the measure of the angle \(BAC\). | [
"24"
] | [
"A regular hexagon and a regular pentagon have a common edge as shown. Find the measure of the angle \\(BAC\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa9b7cc6191592db4ce58d008549b63cc72c54a704764f54f1c0b81d6e97738f | A regular hexagon and a square are inscribed in a given circle. An altitude drawn from the center of the circle to a side of the hexagon is 1 unit longer than an altitude drawn from the center of the circle to a side of the square. Find the length of the radius of the circle. | [
"\\(2(\\sqrt{3} + \\sqrt{2})\\)"
] | [
"A regular hexagon and a square are inscribed in a given circle. An altitude drawn from the center of the circle to a side of the hexagon is 1 unit longer than an altitude drawn from the center of the circle to a side of the square. Find the length of the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-b8f2b155727433380ecfbc5db951f3c36862655af7241c9fe914068809d1da1f | A regular hexagon and an equilateral triangle are inscribed in a unit circle such that for each side of the triangle, there are two sides of the hexagon that are parallel to it. The area of the region that lies within the hexagon but outside the triangle can be expressed as $\frac{a\sqrt{b}}{c} - d$, where $a$, $b$, $c... | [
"13"
] | [
"A regular hexagon and an equilateral triangle are inscribed in a unit circle such that for each side of the triangle, there are two sides of the hexagon that are parallel to it. The area of the region that lies within the hexagon but outside the triangle can be expressed as $\\frac{a\\sqrt{b}}{c} - d$, where $a$, ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-cabcee315bb65d989b810e5b4d8058bf43d5112b20ec059d88ac8783bf8dff54 | A regular hexagon and an equilateral triangle have the same perimeter. If the area of the triangle is $\sqrt{3}$, then the area of the hexagon is ... | [
"\\dfrac{3\\sqrt{3}}{2}"
] | [
"A regular hexagon and an equilateral triangle have the same perimeter. If the area of the triangle is $\\sqrt{3}$, then the area of the hexagon is ..."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd9e366bb31d8678152799a91708a3b01f10ec2d41bf157a4c4225e1e21fab51 | A regular hexagon has a positive integer side length. A laser is emitted from one of the hexagon’s corners and is reflected off the edges of the hexagon until it hits another corner. Let \( a \) be the distance that the laser travels. What is the smallest possible value of \( a^2 \) such that \( a > 2019 \)? | [
"\\( 2020^2 \\)"
] | [
"A regular hexagon has a positive integer side length. A laser is emitted from one of the hexagon’s corners and is reflected off the edges of the hexagon until it hits another corner. Let \\( a \\) be the distance that the laser travels. What is the smallest possible value of \\( a^2 \\) such that \\( a > 2019 \\)?... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-1e035ee40bd49471fcf50ea3d54601d68f44eae73291b731aba463763c60dc65 | A regular hexagon has a side length of 6. Congruent arcs with a radius of 3 are drawn with the center at each of the vertices, creating circular sectors. What is the area of the region inside the hexagon but outside the sectors? | [
"\\( 54\\sqrt{3} - 18\\pi \\)"
] | [
"A regular hexagon has a side length of 6. Congruent arcs with a radius of 3 are drawn with the center at each of the vertices, creating circular sectors. What is the area of the region inside the hexagon but outside the sectors?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 7,
"tir": 29
} | 80 | 0 | 0 |
omr-ws-94677fd2d1a8fa7b6bcea7b3ccf1151fc38239a6d35fceb565d1b2bbe543549e | A regular hexagon has an inscribed circle with an area of 25π. Find the sum of the lengths of the diagonals. | [
"\\(60\\sqrt{3}\\)"
] | [
"A regular hexagon has an inscribed circle with an area of 25π. Find the sum of the lengths of the diagonals."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-019d9bb4eb624cde06bb298277682d4a26236fe8d5a2b70d4ce250bd085b6448 | A regular hexagon has one vertex at the origin, an adjacent vertex on the line $y=-2$, and another adjacent vertex on the line $y=-1$. Let $S_x$ be the sum of the $x$-coordinates of the vertices, and $S_y$ be the sum of the $y$-coordinates. Express $|S_x| + |S_y|$ in the form $a\sqrt{b} + c$, where $a$, $b$, $c$ are po... | [
"19"
] | [
"A regular hexagon has one vertex at the origin, an adjacent vertex on the line $y=-2$, and another adjacent vertex on the line $y=-1$. Let $S_x$ be the sum of the $x$-coordinates of the vertices, and $S_y$ be the sum of the $y$-coordinates. Express $|S_x| + |S_y|$ in the form $a\\sqrt{b} + c$, where $a$, $b$, $c$ ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-84487604a857535db7947dc71f86810e2fd25e42d2bb0fabf299c5e254ec027f | A regular hexagon is covered with $N$ parallelograms of the same area. If the area of each parallelogram is $A$, find the sum of all possible values of $N$ when the total area of the hexagon is $27A$. | [
"27"
] | [
"A regular hexagon is covered with $N$ parallelograms of the same area. If the area of each parallelogram is $A$, find the sum of all possible values of $N$ when the total area of the hexagon is $27A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8c0a580a3b85b5628da7881880ecb1e6182aa2a09346a756f6b230cf7da322c | A regular hexagon is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure, there are exactly 37 points which are vertices of at least one of those triangles. These points are enumerated in an arbitrary way. A triangle is called "clocky" if, running in a clockwise direction from... | [
"18"
] | [
"A regular hexagon is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure, there are exactly 37 points which are vertices of at least one of those triangles. These points are enumerated in an arbitrary way. A triangle is called \"clocky\" if, running in a clockwise directi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
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omr-ws-035dc2861eee644c2c885dca16f9956e3da6b350b9f968e80b7af92c67850ba8 | A regular hexagon is dissected into six triangles, each with a distinct area that is a digit from $1$ to $9$. Three students are given unique digits from the remaining three digits and are asked to deduce the six areas. None of the students alone can determine the areas. What is the product of the six areas? | [
"8064"
] | [
"A regular hexagon is dissected into six triangles, each with a distinct area that is a digit from $1$ to $9$. Three students are given unique digits from the remaining three digits and are asked to deduce the six areas. None of the students alone can determine the areas. What is the product of the six areas?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5695dbd90bf6537cab26b509533472ae83f0c672da0b4824938f418ad0ef74aa | A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On these triangles, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). If we color the triangles black and white such that no two adjacent tri... | [
"0"
] | [
"A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On these triangles, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). If we color the triangles black and white such that no two adjacen... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-d8d56545a9ac4c32fa8b311b4b2ec707ff58ec6ab2b99ef3d529467add92809e | A regular hexagon is divided into equal rhombuses, with sides parallel to the hexagon's sides. Directions are set on three non-adjacent sides of the hexagon in the order of traversing the hexagon counterclockwise. On each side of the rhombuses, arrows are directed in the same way as the parallel sides of the hexagon. C... | [
"2"
] | [
"A regular hexagon is divided into equal rhombuses, with sides parallel to the hexagon's sides. Directions are set on three non-adjacent sides of the hexagon in the order of traversing the hexagon counterclockwise. On each side of the rhombuses, arrows are directed in the same way as the parallel sides of the hexag... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-4dae67e74179a596225a72a494750d55219f70791be03bbb494fcdd9177b1741 | A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction. | [
"\\frac{4}{3}"
] | [
"A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6356d604c875b6f82515daa0712aaac9153201f94070c46ec74d4c9f139222a | A regular hexagon is inscribed in a circle of radius 2 units. In square units, what is the area of the hexagon? Express your answer in simplest radical form. | [
"6\\sqrt{3}"
] | [
"A regular hexagon is inscribed in a circle of radius 2 units. In square units, what is the area of the hexagon? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d408de80a5c968473af450f48ec13e04c4cd2ad83c3c86b4ab30bbd0980225c1 | A regular hexagon is inscribed in a circle, and a second regular hexagon is circumscribed about the same circle. If the sum of the areas of the two regular hexagons is $350\sqrt{3}$, find the radius of the circle. | [
"10"
] | [
"A regular hexagon is inscribed in a circle, and a second regular hexagon is circumscribed about the same circle. If the sum of the areas of the two regular hexagons is $350\\sqrt{3}$, find the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f64fcb4d437f0416f80c66eb00efc41588d9a4922ac320dc946662814897107 | A regular hexagon is inscribed in a circle, and another regular hexagon circumscribes the same circle. If the area of the smaller hexagon is 24, what is the area of the larger hexagon? | [
"32"
] | [
"A regular hexagon is inscribed in a circle, and another regular hexagon circumscribes the same circle. If the area of the smaller hexagon is 24, what is the area of the larger hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-f859f81593f964ac1a42aa44a3e3f7cd13700030ecb159eee6919b6b5c20a822 | A regular hexagon is inscribed in a rectangle. If the area of the shaded triangle is 20 and the area of the shaded quadrilateral is 23, find the area of the hexagon. | [
"222"
] | [
"A regular hexagon is inscribed in a rectangle. If the area of the shaded triangle is 20 and the area of the shaded quadrilateral is 23, find the area of the hexagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5d6c292269487c486f022a12c429eba38d978022c0f3bc83a1ed3fcf2ced80e | A regular hexagon is inscribed inside a circle of radius \( R \). The circle is inscribed inside a square, and the square is inscribed inside a larger regular hexagon. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon? | [
"\\(\\frac{3}{2}(2-\\sqrt{3})\\)"
] | [
"A regular hexagon is inscribed inside a circle of radius \\( R \\). The circle is inscribed inside a square, and the square is inscribed inside a larger regular hexagon. What is the ratio of the area of the smaller hexagon to the area of the larger hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 20
} | 32 | 0 | 0 |
omr-ws-3f99e8aac963e3cbb4d0962ca0707c9e13a45396866974c8256efc08a0c463a3 | A regular hexagon of area \( H \) is inscribed in a convex polygon of area \( P \). Given that the hexagon's side length is \( s \), find the maximum possible value of \( \frac{P}{H} \) and express your answer as a fraction in simplest form. | [
"\\dfrac{3}{2}"
] | [
"A regular hexagon of area \\( H \\) is inscribed in a convex polygon of area \\( P \\). Given that the hexagon's side length is \\( s \\), find the maximum possible value of \\( \\frac{P}{H} \\) and express your answer as a fraction in simplest form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-172cd5bfea3fc09f690c7dce7370c89b48fe4f62703db92f10bfad42a77dfe9e | A regular hexagon of side 1 is inscribed at the intersection of two identical parabolas that are oriented in opposite directions, with one being symmetrical to the other with respect to the line passing through their points of intersection. Find the area between the parabolas but outside the hexagon. | [
"\\(\\frac{8}{3} - \\frac{3\\sqrt{3}}{2}\\)"
] | [
"A regular hexagon of side 1 is inscribed at the intersection of two identical parabolas that are oriented in opposite directions, with one being symmetrical to the other with respect to the line passing through their points of intersection. Find the area between the parabolas but outside the hexagon."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cfc2044dcc1834ea1b8ed297f1ddfab287f2b6c4f5467696627a4daed05e40f6 | A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? | [
"2\\pi - 3\\sqrt{3}",
"\\(3\\sqrt{3} - \\pi\\)",
"\\pi"
] | [
"A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 7
} | 27 | 0 | 0 |
omr-ws-dd1a31570a79a53126c8fc0ca852ec6bdd03215d72f29b2b51b4a53587e4aef6 | A regular hexagon of side length 1 covers the plane, and a beetle crawls along the sides of the hexagon. The shortest path length from the starting point to the end point is 1000. If the beetle's path includes segments parallel to a specific direction, calculate the sum of the lengths of all such segments. | [
"1000"
] | [
"A regular hexagon of side length 1 covers the plane, and a beetle crawls along the sides of the hexagon. The shortest path length from the starting point to the end point is 1000. If the beetle's path includes segments parallel to a specific direction, calculate the sum of the lengths of all such segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-9d667659c2242f5feebbf1fd94c7ab3309697ec40813297dcaba30b8c07e8868 | A regular hexagon with a perimeter of $12\sqrt{2}$ units is centered at the origin of the Cartesian plane, with two vertices on the $x$-axis and the midpoints of two sides on the $y$-axis. The portion of the hexagon in Quadrant I is revolved around the $x$-axis to form a solid with volume $m$ cubic units, and around th... | [
"52"
] | [
"A regular hexagon with a perimeter of $12\\sqrt{2}$ units is centered at the origin of the Cartesian plane, with two vertices on the $x$-axis and the midpoints of two sides on the $y$-axis. The portion of the hexagon in Quadrant I is revolved around the $x$-axis to form a solid with volume $m$ cubic units, and aro... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-261a6fd88a2e3a711386942513d726e02a9dd053c17acf29e4eadfd5ac67bf47 | A regular hexagon with a perimeter of \( 12\sqrt{2} \) units is centered at the origin with two vertices on the x-axis and the midpoints of two sides on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \( m \) cubic units and around the y-axis to form a soli... | [
"97"
] | [
"A regular hexagon with a perimeter of \\( 12\\sqrt{2} \\) units is centered at the origin with two vertices on the x-axis and the midpoints of two sides on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \\( m \\) cubic units and around the y-axis to f... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a4fb077b765528b3125913517cc6097fb995cdeafbe6b81f1519c8c478355722 | A regular hexagon with a perimeter of \( 12\sqrt{2} \) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two of its sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \( m... | [
"2"
] | [
"A regular hexagon with a perimeter of \\( 12\\sqrt{2} \\) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two of its sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with vol... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6eaf529b33dc82b8eed857958c8826e87aa9834c1d6172d56d2d5dec588b5f2d | A regular hexagon with a perimeter of \( 12\sqrt{2} \) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \( m \) cub... | [
"97"
] | [
"A regular hexagon with a perimeter of \\( 12\\sqrt{2} \\) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-08e047e6a9bc401d1d40364e141035e92ab2e7edaf209301ecaed1d525e3dbb4 | A regular hexagon with a perimeter of \( 12\sqrt{2} \) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \( m \) cub... | [
"97"
] | [
"A regular hexagon with a perimeter of \\( 12\\sqrt{2} \\) units lies in the Cartesian plane such that its center is at the origin, two vertices lie on the x-axis, and the midpoints of two sides lie on the y-axis. The portion of the hexagon in Quadrant I is revolved around the x-axis to form a solid with volume \\(... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-de8b120aee95b18c7f34a07ac5ca745c2fb2680aff41869e6c35f63f8bf37452 | A regular hexagon with a side length of 50 is divided into equilateral triangles with unit side lengths, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice that define a segment of length 2. After several such operations, exactly one node remains. How many ways is thi... | [
"2551"
] | [
"A regular hexagon with a side length of 50 is divided into equilateral triangles with unit side lengths, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice that define a segment of length 2. After several such operations, exactly one node remains. How many ways i... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-3456d6bbe0e2c99dcc2262f8b4451147181d811e59626c1d92b360cec5ed0ff5 | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart, with one pair of sides parallel to the imaginary axis. Let \( R \) be the region outside the hexagon, and let \( S = \left\{ \frac{1}{z} \mid z \in R \right\} \). The area of \( S \) has the form \( a\pi + \sqrt... | [
"29"
] | [
"A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart, with one pair of sides parallel to the imaginary axis. Let \\( R \\) be the region outside the hexagon, and let \\( S = \\left\\{ \\frac{1}{z} \\mid z \\in R \\right\\} \\). The area of \\( S \\) has the fo... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 21
} | 21 | 0 | 0 |
omr-ws-ba028b4e4774c46b2c49c1d37815ee05365251f4ae6d06dcbe0a2e68e81c49ba | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \ | \ z \in R\right\rbrace$. Find the area of $S.$ | [
"3 \\sqrt{3} + 2 \\pi"
] | [
"A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z} \\ | \\ z \\in R\\right\\rbrace$. Find the area of $S.$"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-83f3d8cee8a41d5066163050834d300dfb4c08fff180f29dee4ad62b9b7f6a9f | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \mid z \in R\right\rbrace$. Then the area of $S$ has the form $a\pi + \sqrt{b}... | [
"29"
] | [
"A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z} \\mid z \\in R\\right\\rbrace$. Then the area of $S$ has the form $a\\... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5ba4070786d569aa94a53ddda85b7d138643097fb9fa88ebfe986b19aa3a2a01 | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace$. Then the area of $S$ has the form $a\pi + \sqrt{b}$, wh... | [
"029"
] | [
"A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \\left\\lbrace\\frac{1}{z}|z \\in R\\right\\rbrace$. Then the area of $S$ has the form $a\\p... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-fdde2f82d61936328fd053257647bc3be0945fdb1c7274009053e1fa8a10aa05 | A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let \( R \) be the region outside the hexagon, and let \( S = \left\lbrace \frac{1}{z} \mid z \in R \right\rbrace \). Then the area of \( S \) has the form \(... | [
"11"
] | [
"A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let \\( R \\) be the region outside the hexagon, and let \\( S = \\left\\lbrace \\frac{1}{z} \\mid z \\in R \\right\\rbrace \\). Then the area of \\( S \... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d99a97cfc80261213ab376b703bea578969e17a4b4e5721697438c7fe02afab6 | A regular hexagon with side length $1$ is given. There are $m$ points in its interior such that no $3$ are collinear. The hexagon is divided into triangles (triangulated), such that every point of the $m$ given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior poin... | [
"\\dfrac{3\\sqrt{3}}{4}"
] | [
"A regular hexagon with side length $1$ is given. There are $m$ points in its interior such that no $3$ are collinear. The hexagon is divided into triangles (triangulated), such that every point of the $m$ given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-645e4ddce96b4e912bb55900bc11af3ce96ef975914168e9661af02d8d0bf484 | A regular hexagon with side length 16 cm is divided into two equal parts. One part is used to construct a square, and the other part is used to construct an isosceles triangle with a base of 18 cm. What is the sum of the areas, in cm², of the newly created square and triangle? | [
"252"
] | [
"A regular hexagon with side length 16 cm is divided into two equal parts. One part is used to construct a square, and the other part is used to construct an isosceles triangle with a base of 18 cm. What is the sum of the areas, in cm², of the newly created square and triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-e37afd2a060148afc9a0bb2470421686f29d93d557c61696aa71133ae37b1df1 | A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? | [
"\\( 36\\sqrt{21} \\)"
] | [
"A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-92eb8be6e6967e595250a95b7941fcbd4b5e6ff5d96666d3b6f4a813e26885c0 | A regular hexagonal prism is circumscribed about a sphere of radius $R$. Find its surface area. | [
"$12R^2\\sqrt 3$"
] | [
"A regular hexagonal prism is circumscribed about a sphere of radius $R$. Find its surface area."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea0d4b6c0820d22058a12a08b58333ae176047016797d52f7e8c8f40fdbace21 | A regular hexagonal pyramid \( SABCDEF \) is given. Construct a cross section by a plane passing through the diagonal of the base \( BE \) and parallel to the face \( ASF \). How many times the area of the constructed section is greater than the area of the side face? | [
"\\(\\frac{5}{4}\\)"
] | [
"A regular hexagonal pyramid \\( SABCDEF \\) is given. Construct a cross section by a plane passing through the diagonal of the base \\( BE \\) and parallel to the face \\( ASF \\). How many times the area of the constructed section is greater than the area of the side face?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-a74e346813f9910735c5320950687f3c9829cc6cc16be18eb3b063e91a1c0936 | A regular hexagonal pyramid with face angle at the vertex equal to $\alpha $ is intersected by a plane passing at an angle $\beta $ to the base through its longest diagonal. Find the ratio of the area of the plane section to the area of the base. | [
"\\(\\frac{1}{6\\cos\\beta }\\lambda (\\lambda +2)\\), where \\(\\lambda =\\displaystyle\\frac{\\sqrt {\\cos^2 \\displaystyle\\frac{\\alpha }{2}-3}} {\\sqrt {\\cot^2 \\displaystyle\\frac{\\alpha }{2}-3}+\\sqrt 3\\tan\\beta }\\)"
] | [
"A regular hexagonal pyramid with face angle at the vertex equal to $\\alpha $ is intersected by a plane passing at an angle $\\beta $ to the base through its longest diagonal. Find the ratio of the area of the plane section to the area of the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-09fb30387c27516fc7c585cadfb0733f5a436ba16df3de5a70ca027da16f9291 | A regular hexahedral pyramid is intersected by a plane parallel to its lateral face and passing through the center of the base. Find the ratio of the area of the plane section to the area of the lateral face. | [
"\\frac{5}{4}"
] | [
"A regular hexahedral pyramid is intersected by a plane parallel to its lateral face and passing through the center of the base. Find the ratio of the area of the plane section to the area of the lateral face."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d4e0849c0377e8e67fb176a50c29f130d73789e6f6b99550d55872ab7b800a4 | A regular icosahedron is a $20$ -faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane... | [
"810"
] | [
"A regular icosahedron is a $20$ -faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-6972027848d01709d8caa49d87b08b32908df94398e1fac6afd326f5ebf523b2 | A regular n-gon $A_1A_2...A_N$ is inscribed in a circle with center $O$. If $\angle A_1OA_{16} = 135^\circ$, find all possible values of $n$. | [
"64"
] | [
"A regular n-gon $A_1A_2...A_N$ is inscribed in a circle with center $O$. If $\\angle A_1OA_{16} = 135^\\circ$, find all possible values of $n$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29e4466977e4422fe0075c44409b7bcc623d482df61b328a673868661c468408 | A regular n-gon has its sides and diagonals colored with \( k \) colors such that: 1) No triangle with vertices among the vertices of the n-gon is colored with more than two colors. 2) Given a segment \( AB \), either \( AB \) is colored red or there exists another vertex \( C \) such that both \( AC \) and \( BC \) ar... | [
"5"
] | [
"A regular n-gon has its sides and diagonals colored with \\( k \\) colors such that:\n1) No triangle with vertices among the vertices of the n-gon is colored with more than two colors.\n2) Given a segment \\( AB \\), either \\( AB \\) is colored red or there exists another vertex \\( C \\) such that both \\( AC \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-3107a676119514a9bd4d355696020b2c6e72e17096eafd2a4e8104259d2bdf30 | A regular n-gon is divided by its $n-3$ diagonals into $n-2$ triangles such that no two of these diagonals intersect inside the n-gon. What is the maximum number of noncongruent triangles formed by this triangulation? | [
"2"
] | [
"A regular n-gon is divided by its $n-3$ diagonals into $n-2$ triangles such that no two of these diagonals intersect inside the n-gon. What is the maximum number of noncongruent triangles formed by this triangulation?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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