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-d8bedda830fedb0581db9ab77a8c6de09ded97e4b8d502799f74dfa04eb8334a | 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? | [
"n-3"
] | [
"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"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-f936ecd1df5063627d9080059f8628a243c1064306f3d70b20d93759b697570e | A regular n-gon is in the same position after a rotation of 35 degrees about its center. What is the smallest possible value of n? | [
"72"
] | [
"A regular n-gon is in the same position after a rotation of 35 degrees about its center. What is the smallest possible value of n?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
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omr-ws-399ad1961f24635afe2ebd8d2021549e3e0bd160b0e33675796bb5187256c999 | A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\bigtriangleup ADG$. Express your answer in simplest radical form. | [
"4+3\\sqrt{2}"
] | [
"A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\\bigtriangleup ADG$. Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f21c893b6176a97047cf864c3e4d1fa98453b2c00c3a10af5f140320b2236fd4 | A regular octagon \( ABCDEFGH \) has an area of one square unit. What is the area of the rectangle \( ABEF \)? | [
"\\(\\frac{1}{2}\\)"
] | [
"A regular octagon \\( ABCDEFGH \\) has an area of one square unit. What is the area of the rectangle \\( ABEF \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4eb3d6231a06402bb3a270c97ddae5189598a34fe4ae34f5444ea065ada17a30 | A regular octagon \(A_1A_2A_3A_4A_5A_6A_7A_8\) is inscribed in a circle of area 1. Point \(P\) lies inside the circle such that the region bounded by \(\overline{PA_1}, \overline{PA_2}\), and the minor arc \(\widehat{A_1A_2}\) has area \(\frac{1}{7}\), and the region bounded by \(\overline{PA_3}, \overline{PA_4}\), and... | [
"504"
] | [
"A regular octagon \\(A_1A_2A_3A_4A_5A_6A_7A_8\\) is inscribed in a circle of area 1. Point \\(P\\) lies inside the circle such that the region bounded by \\(\\overline{PA_1}, \\overline{PA_2}\\), and the minor arc \\(\\widehat{A_1A_2}\\) has area \\(\\frac{1}{7}\\), and the region bounded by \\(\\overline{PA_3}, \... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-c37fb9ff6469c2344d9a8111de3124418e9b1786922a1a9bbec98048540fe5f9 | A regular octagon and a circle lie on the plane. The octagon can be rotated around its center and reflected symmetrically about any of its sides. Let $N$ be the minimum number of operations (rotations and reflections) required to ensure that the center of the octagon lies inside the given circle, starting from any init... | [
"7"
] | [
"A regular octagon and a circle lie on the plane. The octagon can be rotated around its center and reflected symmetrically about any of its sides. Let $N$ be the minimum number of operations (rotations and reflections) required to ensure that the center of the octagon lies inside the given circle, starting from any... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-a4373fe5aacfb711064fe05b282652b5dc88b6a6a911d56643c2a0fdb8e3656e | A regular octagon and a circle lie on the plane. The octagon can be rotated around the center and reflected symmetrically about any of its sides. Let $d$ be the distance from the center of the octagon to the center of the circle. If the radius of the circle is $r$, find the smallest integer $n$ such that after performi... | [
"3"
] | [
"A regular octagon and a circle lie on the plane. The octagon can be rotated around the center and reflected symmetrically about any of its sides. Let $d$ be the distance from the center of the octagon to the center of the circle. If the radius of the circle is $r$, find the smallest integer $n$ such that after per... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7d534511f00100ebcc8a7e7eb7d6be4e1025005767a0c691081c45aa54da699 | A regular octagon and its circumcircle are drawn. The lines tangent to its circumcircle at each vertex of the octagon are drawn, and points are created where two or more lines intersect. Lines connect these points, and the ratios of the side lengths of the four octagons, from smallest to largest, can be expressed as \(... | [
"\\( 24 - 16\\sqrt{2} \\)"
] | [
"A regular octagon and its circumcircle are drawn. The lines tangent to its circumcircle at each vertex of the octagon are drawn, and points are created where two or more lines intersect. Lines connect these points, and the ratios of the side lengths of the four octagons, from smallest to largest, can be expressed ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-899524003715f8f753691ccbf54b3572f4370ebd934efce82304ba64c4d7f773 | A regular octagon has side one. All points whose distance from at least one vertex is less than one are deleted. Find the remaining area. | [
"2(1 + \\sqrt{2}) - 3\\pi"
] | [
"A regular octagon has side one. All points whose distance from at least one vertex is less than one are deleted. Find the remaining area."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-bd466654cc1fcf7b421fac79080d48f8fac4f697078e9807a23e168adf3e199d | A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length 2000. What is the length of each side of the octagon? | [
"\\( 2000(\\sqrt{2} - 1) \\)"
] | [
"A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length 2000. What is the length of each side of the octagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-671aa355b2ccdc6e7e2a15f8b0f8e8cadf583de2ab03a1bbbb3954a9131400a0 | A regular octagon is formed by cutting off four equal isosceles right-angled triangles from the corners of a square with side length 1. What is the area of the octagon? | [
"\\( 2(\\sqrt{2}-1) \\)"
] | [
"A regular octagon is formed by cutting off four equal isosceles right-angled triangles from the corners of a square with side length 1. What is the area of the octagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d3c29f424cd5e762019c95e9964b0b08dbf51ce1ccd6bb651c56b2ac95170e73 | A regular octagon is inscribed in a circle with a radius of 15.8 cm. Find the perimeter of the octagon. | [
"96.7423717"
] | [
"A regular octagon is inscribed in a circle with a radius of 15.8 cm. Find the perimeter of the octagon."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 28
} | 35 | 0 | 0 |
omr-ws-0a38d259a831dfee18be9c40c54092da90311b64a3811168d1c66645432f6252 | A regular octagon is inscribed in a regular octagon of area $1$ square unit. Find the smallest area the inscribed regular octagon could have. | [
"\\dfrac{2 + \\sqrt{2}}{4}"
] | [
"A regular octagon is inscribed in a regular octagon of area $1$ square unit. Find the smallest area the inscribed regular octagon could have."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eab6d94dd79e345bce5f8858f07888e460ba6a604125056266750865e5e5fdeb | A regular octagon is inscribed within a circle and has a side measure of 6. Find the area of this octagon. | [
"\\( 72(1 + \\sqrt{2}) \\)"
] | [
"A regular octagon is inscribed within a circle and has a side measure of 6. Find the area of this octagon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4e27afc180732ec4bbb2961f4a79e7746adaf695389ae9b7a8f92b1ff14a2679 | A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square with sides of one unit. What is the length of the leg of each of the triangles? | [
"\\(\\frac{2 - \\sqrt{2}}{2}\\)"
] | [
"A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square with sides of one unit. What is the length of the leg of each of the triangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-22fd819e0407feb34d9d9be7322ae8009b330ffeaef72600647fac78b085b06f | A regular octagon with center \( O \) has 1000 equally spaced lighthouses on it, with 8 of them on the vertices. A circular road is constructed around the octagon such that the circle and the octagon share the same center and do not intersect. Mia drives once around the circular track and stares at the closest lighthou... | [
"3500"
] | [
"A regular octagon with center \\( O \\) has 1000 equally spaced lighthouses on it, with 8 of them on the vertices. A circular road is constructed around the octagon such that the circle and the octagon share the same center and do not intersect. Mia drives once around the circular track and stares at the closest l... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d141d2e715a9faf81ae2e3801b2e4a0c1bdf8f18d34385ebe6ad4e231bf26f2f | A regular octagon with side length $a$ and a regular hexagon with side length $b$ have the same area. Find the square of the ratio of their side lengths, $a^2/b^2$. | [
"\\(\\frac{81 - 54\\sqrt{2}}{16}\\)"
] | [
"A regular octagon with side length $a$ and a regular hexagon with side length $b$ have the same area. Find the square of the ratio of their side lengths, $a^2/b^2$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df304c5ede5b8ef7d0cc154f9a5672c39394580fa233215e70c53926aff05940 | A regular octagon with side length 26 is partitioned into parallelograms. For each partition $ P_i $, let $ A_i $ denote the sum of the areas of all the rectangles formed. Find the maximum value of $ A_i $ over all possible partitions $ P_i $. | [
"\\( 2 \\cdot 26^2 \\)"
] | [
"A regular octagon with side length 26 is partitioned into parallelograms. For each partition $ P_i $, let $ A_i $ denote the sum of the areas of all the rectangles formed. Find the maximum value of $ A_i $ over all possible partitions $ P_i $."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-3eb8f0e82e50f7ade1a7e47c94d56309e021f49b524df8fcfa8c8ecd21963018 | A regular octahedron $ABCDEF$ has points $B$, $C$, $D$, and $E$ coplanar and forming the vertices of a square. The octahedron is divided along a plane $\mathcal{P}$, parallel to line $BC$, into two polyhedra of equal volume. The cosine of the acute angle that plane $\mathcal{P}$ makes with plane $BCDE$ is $\frac{1}{3}$... | [
"300"
] | [
"A regular octahedron $ABCDEF$ has points $B$, $C$, $D$, and $E$ coplanar and forming the vertices of a square. The octahedron is divided along a plane $\\mathcal{P}$, parallel to line $BC$, into two polyhedra of equal volume. The cosine of the acute angle that plane $\\mathcal{P}$ makes with plane $BCDE$ is $\\fra... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-31abdbde7c6e16d039c31570d1fd34bdb5fe29777305c32fe7371c133efa5a96 | A regular octahedron ABCDEF has a volume of 360 cubic units. What is the volume of the portion of the octahedron that consists of all points closer to vertex A than to any other vertex? | [
"60"
] | [
"A regular octahedron ABCDEF has a volume of 360 cubic units. What is the volume of the portion of the octahedron that consists of all points closer to vertex A than to any other vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-66f4736214d829ea0aa16ce490dd15997a7d8ec7e5ef00e2f07fbd62b05c305f | A regular octahedron has a side length of 1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has an area of \( \frac{a\sqrt{b}}{c} \), where \( a, b, \) and \( c \) are positive integers, \( a \) and \( c... | [
"14"
] | [
"A regular octahedron has a side length of 1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has an area of \\( \\frac{a\\sqrt{b}}{c} \\), where \\( a, b, \\) and \\( c \\) are positive integers, \\(... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-039a50f76c151486d87a30909cafb5b0a618273ce80beb8008d11a0dfaa9ff4d | A regular octahedron has a volume of $120 \text{ cm}^3$. A portion of this octahedron consists of the part that is closer to the top vertex than to any other vertex. This portion extends down to the center of the octahedron. What is the volume, in cubic centimeters, of this portion? | [
"15"
] | [
"A regular octahedron has a volume of $120 \\text{ cm}^3$. A portion of this octahedron consists of the part that is closer to the top vertex than to any other vertex. This portion extends down to the center of the octahedron. What is the volume, in cubic centimeters, of this portion?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7820b1f7a5ccac9c37fbdc7f897eb5e61aec1a205a082985cc4b936104baa91 | A regular octahedron has a volume of $120$. It is oriented such that the two end vertices are facing exactly up and down, and the other four vertices form a square parallel to the ground. Let the vertex at the very top be $A$. Determine the volume of the portion of the octahedron consisting of points that are closer to... | [
"60"
] | [
"A regular octahedron has a volume of $120$. It is oriented such that the two end vertices are facing exactly up and down, and the other four vertices form a square parallel to the ground. Let the vertex at the very top be $A$. Determine the volume of the portion of the octahedron consisting of points that are clos... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4896764273893fb16b54ce49bad1323e9c889b165a018730fac36cecd84f9e39 | A regular octahedron has a volume of 120. Let \( A \) be one of its vertices, and let \( S \) be the set of all points on the interior of the octahedron that are closer to \( A \) than to any other vertex. What is the volume of \( S \)? | [
"20"
] | [
"A regular octahedron has a volume of 120. Let \\( A \\) be one of its vertices, and let \\( S \\) be the set of all points on the interior of the octahedron that are closer to \\( A \\) than to any other vertex. What is the volume of \\( S \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-d2a76161cf80189010504f478f4927b4bb20a098876c90d2caa3480f9f75f95c | A regular octahedron has edges of length 6 cm. If \( d \) cm is the shortest distance from the center of one face to the center of the opposite face measured around the surface of the octahedron, what is the value of \( d^2 \)? | [
"72"
] | [
"A regular octahedron has edges of length 6 cm. If \\( d \\) cm is the shortest distance from the center of one face to the center of the opposite face measured around the surface of the octahedron, what is the value of \\( d^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-6469e5d05cd8967ad1473d97d6c4ad9bb15702f0b8f7d00bee82826e5e50483e | A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, a... | [
"14"
] | [
"A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\\frac{a\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-1cd99b25700894d954880d919d14a1b788d3cae5d89761677ed8eb6097fe0ba5 | A regular octahedron has side length 1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and... | [
"14"
] | [
"A regular octahedron has side length 1. A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\\frac{a\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively pri... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-f9ac6c009bcfe54289d9a98b36dea5ce69e3605401193074c271b62bfc68f94d | A regular octahedron has side length \(1\). A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are positive integers, \(a\) and \(c\) are relat... | [
"14"
] | [
"A regular octahedron has side length \\(1\\). A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area \\(\\frac{a\\sqrt{b}}{c}\\), where \\(a\\), \\(b\\), and \\(c\\) are positive integers, \\(a\\) ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 19,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-7d39138203d2fd6bc630a8d56a2657c61ac609b24adc1da3b04940ca866997de | A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. Let \(T\) be a regular octahedron of edge length \(28\). What is the total surface area of \(T\), rounded to the nearest integer? | [
"2718"
] | [
"A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. Let \\(T\\) be a regular octahedron of edge length \\(28\\). What is the total surface area of \\(T\\), rounded to the nearest integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-54d53b5ffc735949081543243d593aecd16b0baa457d1918e03b7675de00aae3 | A regular octahedron is formed by connecting the centers of the faces of a $6 \times 6 \times 6$ cube. What is the volume of the octahedron? | [
"36"
] | [
"A regular octahedron is formed by connecting the centers of the faces of a $6 \\times 6 \\times 6$ cube. What is the volume of the octahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b8e92625b64df4555bd4d20c585a259c0fc8fca269f779e3dc01f17d0bf1d6e | A regular octahedron is formed by setting its vertices at the centers of the faces of a cube. Another regular octahedron is formed around the cube by making the center of each triangle of the octahedron hit at a vertex of the cube. What is the ratio of the volume of the larger octahedron to that of the smaller octahedr... | [
"27"
] | [
"A regular octahedron is formed by setting its vertices at the centers of the faces of a cube. Another regular octahedron is formed around the cube by making the center of each triangle of the octahedron hit at a vertex of the cube. What is the ratio of the volume of the larger octahedron to that of the smaller oct... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-47099e32537029ba298393e7a4008651e4b54fc434096ed0b004501336017ce5 | A regular pentagon $ABCDE$ is inscribed in a circle with center $O$. Points $P$ and $R$ are inside the pentagon. Perpendiculars $PJ, PK, PL, PM, PN$ are dropped from $P$ to the sides or their extensions, and perpendiculars $RT, RU, RV, RW, RX$ are dropped from $R$ to the sides or their extensions. If the perpendicular ... | [
"5OH"
] | [
"A regular pentagon $ABCDE$ is inscribed in a circle with center $O$. Points $P$ and $R$ are inside the pentagon. Perpendiculars $PJ, PK, PL, PM, PN$ are dropped from $P$ to the sides or their extensions, and perpendiculars $RT, RU, RV, RW, RX$ are dropped from $R$ to the sides or their extensions. If the perpendic... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-4193f734be7f7d2b4a2bb0632de2e2f386d2ca8a5c8ff4f69ee10f603b167fd6 | A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each vertex $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two additional spheres $K_1$ and $K_2$, each of radius $\frac{s}{2}$, that touch all five spheres $K_i$. Determine whether $K_1$ and $K_2$ intersect each other, touc... | [
"\\text{They have no common points.}"
] | [
"A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each vertex $A_i$, a sphere $K_i$ of radius $\\frac{s}{2}$ is constructed. There are two additional spheres $K_1$ and $K_2$, each of radius $\\frac{s}{2}$, that touch all five spheres $K_i$. Determine whether $K_1$ and $K_2$ intersect each othe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-49d9ae4af01ed96a7f35b5654a38eba3969fd8de90804e5424479576191c18d2 | A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$, for $1 \leq i \leq 5$ and $1 \leq j \leq 4$. Find the maximum of $f(P, S)$ over all pairs of shapes. | [
"\\dfrac{\\pi}{20}"
] | [
"A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs $A_iB_j$, for $1 \\leq i \\leq 5$ and $1 \\leq j \\leq 4$. Find the maximum of $f(P, S)$ over all pairs of s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-be8bbc7096531ef845cb363bf62cda6aa938b440d20221181cbec996744d05d9 | A regular pentagon \( MNPQR \) is inscribed in a circle, and \( F \) is a point on the arc \( MQ \). Calculate the value of \( \overline{FM} + \overline{FQ} - \overline{FN} - \overline{FP} - \overline{FR} \). | [
"0"
] | [
"A regular pentagon \\( MNPQR \\) is inscribed in a circle, and \\( F \\) is a point on the arc \\( MQ \\). Calculate the value of \\( \\overline{FM} + \\overline{FQ} - \\overline{FN} - \\overline{FP} - \\overline{FR} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02bfba5bf395c445eb2bd2d4c50597de81729d775acc00932f0505e89c69828a | A regular pentagon has a perimeter of 50 inches. What is its area? | [
"172.05"
] | [
"A regular pentagon has a perimeter of 50 inches. What is its area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-68a8ddaa4779046bb64f17690cc7b255ea55d0290095ef5fe92d978e5a0469d0 | A regular pentagon has an area of $1+\sqrt{5}$. Each vertex is folded over so that it coincides with the center of the pentagon, forming a smaller pentagon. What is the area of the new pentagon? | [
"$\\sqrt{5} - 1$"
] | [
"A regular pentagon has an area of $1+\\sqrt{5}$. Each vertex is folded over so that it coincides with the center of the pentagon, forming a smaller pentagon. What is the area of the new pentagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 26
} | 76 | 0 | 0 |
omr-ws-67343d7711225e3d753db9ed79083a4bb604826e40b0bdb7e85fbafec9b3cc94 | A regular pentagon has its side extended to form a five-pointed star. If the ratio of the area of the pentagon to the area of the star is $\sin(\theta)$ with $0<\theta <90$, find $\theta$. | [
"18"
] | [
"A regular pentagon has its side extended to form a five-pointed star. If the ratio of the area of the pentagon to the area of the star is $\\sin(\\theta)$ with $0<\\theta <90$, find $\\theta$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 2,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-bae1420487c65ac30677672dc7b998c2185f962ff71a89e58eb9888da0e89ef1 | A regular pentagon has side length $x$. Calculate the expression for the radius $R$ of the circumcircle in terms of $x$. Simplify your answer to the form $R = \frac{x}{a}$, where $a$ is a simplified trigonometric expression. Find $a$. | [
"2\\sin 36^\\circ"
] | [
"A regular pentagon has side length $x$. Calculate the expression for the radius $R$ of the circumcircle in terms of $x$. Simplify your answer to the form $R = \\frac{x}{a}$, where $a$ is a simplified trigonometric expression. Find $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0aeec163fc88bac6ada7da3d1fa3330e4cadc4f29fbb4761caca267784ff2232 | A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the re... | [
"50"
] | [
"A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-fc696ea07267a8309fecf2ab96234ab1503fb1b30f37d0034653d46b3ac1db13 | A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the re... | [
"200"
] | [
"A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f046e960c46a89317bc25a2d2e38961216815ca2491ca3ee8bea68cf7f81f33 | A regular pentagon is drawn in the plane, along with all its diagonals. All segments are extended infinitely in both directions, dividing the plane into many infinite and finite areas. Bob the banana starts in the center region of the pentagon and every minute crosses a side of the region he is in. If Bob will explode ... | [
"200"
] | [
"A regular pentagon is drawn in the plane, along with all its diagonals. All segments are extended infinitely in both directions, dividing the plane into many infinite and finite areas. Bob the banana starts in the center region of the pentagon and every minute crosses a side of the region he is in. If Bob will exp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-469b05344500fc2ca82fb473178f0499f1504f16486d7afd497ef825a4bf0b17 | A regular pentagon is inscribed inside a regular hexagon with area \( A \) such that they share a vertex. If \( A \) satisfies \( 2 \sqrt{A - \sqrt{A}} = \sqrt{5} + 3 \), then the area of the pentagon can be expressed as \( \frac{\sqrt{m + n \sqrt{p}}}{q} \), where \( m, n, p \) are positive integers with \( p \) squar... | [
"27"
] | [
"A regular pentagon is inscribed inside a regular hexagon with area \\( A \\) such that they share a vertex. If \\( A \\) satisfies \\( 2 \\sqrt{A - \\sqrt{A}} = \\sqrt{5} + 3 \\), then the area of the pentagon can be expressed as \\( \\frac{\\sqrt{m + n \\sqrt{p}}}{q} \\), where \\( m, n, p \\) are positive intege... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-805f49ea8fc2d311df47c3d25f5cd8e1bcc883e3b31f85c2c5bce9b521130596 | A regular pentagon is rotated $36^{\circ}$ around its center to produce a second pentagon. What fraction of the area of the original pentagon is the area of the intersection of the two pentagons, to the nearest whole percent? | [
"89"
] | [
"A regular pentagon is rotated $36^{\\circ}$ around its center to produce a second pentagon. What fraction of the area of the original pentagon is the area of the intersection of the two pentagons, to the nearest whole percent?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-26d816f3dc72e3a8c9163a78fb78366c0b513e4f877430659b9bcc074562b58b | A regular pentagon with area $1$ has its non-adjacent vertices connected with line segments, dividing the pentagon into ten triangular regions and one smaller pentagonal region. All obtuse triangles are colored. This process is repeated using the smaller pentagon, and continued infinitely. What is the total area that i... | [
"\\dfrac{\\sqrt{5}-1}{2}"
] | [
"A regular pentagon with area $1$ has its non-adjacent vertices connected with line segments, dividing the pentagon into ten triangular regions and one smaller pentagonal region. All obtuse triangles are colored. This process is repeated using the smaller pentagon, and continued infinitely. What is the total area t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-bbbf76986102e7c33c8a1b452301b99d866cfcf03de30cc4c1715137d104a3ab | A regular pentagon with area $1+\sqrt{5}$ has all of its corners folded into the center to create a smaller regular pentagon. What is the area of this smaller pentagon? | [
"\\sqrt{5} - 1"
] | [
"A regular pentagon with area $1+\\sqrt{5}$ has all of its corners folded into the center to create a smaller regular pentagon. What is the area of this smaller pentagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-e95087b41200a36c3bdf01bc9c9e734a5ef1e304b86529e0e6ebcb845903c7cd | A regular pentagon with area $1+\sqrt{5}$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? | [
"\\sqrt{5} - 1"
] | [
"A regular pentagon with area $1+\\sqrt{5}$ is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-21130e5b40ec52df624e80710e4fd4b4c2e6d7d3d24cdc38988b9823548d1ca1 | A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? | [
"\\(\\sqrt{5} - 1\\)"
] | [
"A regular pentagon with area $\\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 3,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-6ae1de85695a1f7397908d1a2e0a8a20364d7cc9d61ffb23591665d340d513f2 | A regular pentagon with side length 1 has a regular pentagon constructed externally on each of its sides. The figure is then folded such that the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained containe... | [
"\\(\\frac{5^{\\frac{5}{4}}}{2(1+\\sqrt{5})} \\cot 36^\\circ (1 + \\cos 108^\\circ \\tan 36^\\circ)^2\\)"
] | [
"A regular pentagon with side length 1 has a regular pentagon constructed externally on each of its sides. The figure is then folded such that the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained con... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-568c283b0774426b6499c817dc06832d1679098137e9cc1a0093a9ffc4ede3fd | A regular pentagon with side length \( p \) is modified such that three consecutive sides are held rigid, while the other two are bent inward, maintaining the side length \( p \). If the area of this new pentagon is \( \frac{49}{2} \), what was the area of the original pentagon? | [
"\\( 54.78 \\) (approximate value)"
] | [
"A regular pentagon with side length \\( p \\) is modified such that three consecutive sides are held rigid, while the other two are bent inward, maintaining the side length \\( p \\). If the area of this new pentagon is \\( \\frac{49}{2} \\), what was the area of the original pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-2825ca4a831b90ef63a33ff88ff354ec0e5b422475c127e37a5671192d5a22b1 | A regular pentagonal pyramid $SABCDE$ is intersected by the plane passing through the vertices $A$ and $C$ of the base and the midpoints of the lateral edges $DS$ and $ES$. Find the area of the section if the length of the side of the base is equal to $q$ and the length of the lateral edge is equal to $b$. | [
"\\(\\frac{q}{16}\\left(2+\\sqrt{5}\\right)\\sqrt{4b^2+3q^2}\\)"
] | [
"A regular pentagonal pyramid $SABCDE$ is intersected by the plane passing through the vertices $A$ and $C$ of the base and the midpoints of the lateral edges $DS$ and $ES$. Find the area of the section if the length of the side of the base is equal to $q$ and the length of the lateral edge is equal to $b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdaf11531891cf75e14a55c430585ac52c2ea12b92f11cb33382a51e828781cb | A regular polygon has \(6n + 1\) vertices, and these vertices are colored in two colors (say red and blue). Suppose there are \(k\) blue vertices. Calculate the total number of isosceles monochromatic triangles formed, expressed as a function of \(n\) and \(k\). | [
"no_valid_answer_found"
] | [
"A regular polygon has \\(6n + 1\\) vertices, and these vertices are colored in two colors (say red and blue). Suppose there are \\(k\\) blue vertices. Calculate the total number of isosceles monochromatic triangles formed, expressed as a function of \\(n\\) and \\(k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f73f0f69708e2fa1ea7bae4639837aae4aedc35e693b86542edca0ebfe86adf | A regular polygon has an even number of sides. If the product of the length of its side and the distance between two opposite sides is (1/4)th of its area, find the number of sides it has. | [
"16"
] | [
"A regular polygon has an even number of sides. If the product of the length of its side and the distance between two opposite sides is (1/4)th of its area, find the number of sides it has."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d191e8c1024525ff65aff8f3b6102a4b6cf0d3d5337fbb7e2fa693888d1c2573 | A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have? | [
"24"
] | [
"A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e974bef870cde02f01c9c37c351ec79c1f466eb12d9280ab7c2f0a09900a7ad | A regular polygon has exterior angles each measuring 15 degrees. How many sides does the polygon have? | [
"24"
] | [
"A regular polygon has exterior angles each measuring 15 degrees. How many sides does the polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f779c21d5ed218268a759efa6f65de8beb048d645dded7db71f49a0cb663007c | A regular polygon has interior angles of 120 degrees. How many sides does the polygon have? | [
"6"
] | [
"A regular polygon has interior angles of 120 degrees. How many sides does the polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1340c392b1dfbfe69586278b1d576cbbd7d9c3874aaf6fc8f9f167e174cbafb | A regular polygon has interior angles of 162 degrees. How many sides does the polygon have? | [
"20"
] | [
"A regular polygon has interior angles of 162 degrees. How many sides does the polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-793adf1de1582bca6477f4d973fe1ff9df37a300180a957d337e0c2d5f401ec7 | A regular polygon has perimeter 108 cm and each side has length 12 cm. How many sides does this polygon have? | [
"9"
] | [
"A regular polygon has perimeter 108 cm and each side has length 12 cm. How many sides does this polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f69ea3ff368e24f4e490c7c21bcd77d273c64cb8e81872be21dd3095cbdad2eb | A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units? | [
"15"
] | [
"A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e265fb528f0004b8487bff9f6ab1845bb6fbcd66ed6fc57616102baee2eac712 | A regular polygon of \( m \) sides is exactly enclosed (no overlaps, no gaps) by \( m \) regular polygons of \( n \) sides each. If \( m = 10 \), what is the value of \( n \)? | [
"5"
] | [
"A regular polygon of \\( m \\) sides is exactly enclosed (no overlaps, no gaps) by \\( m \\) regular polygons of \\( n \\) sides each. If \\( m = 10 \\), what is the value of \\( n \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74f338d4055dd9ee7d31ca66265932d7ad343bad86d8869f6b982b8025393eb3 | A regular polygon with $2n$ sides and side length $1$ is dissected into rhombuses with side length $1$. If $n=10$, calculate the total number of rhombuses required for this dissection. | [
"90"
] | [
"A regular polygon with $2n$ sides and side length $1$ is dissected into rhombuses with side length $1$. If $n=10$, calculate the total number of rhombuses required for this dissection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f6de4eeb1696d6634d7a2ddebab25a5f3fba68dcb588954b89f49e85ada4e5cf | A regular polygon with $2n$ sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 600. Compute the value of the integer $n$. | [
"6"
] | [
"A regular polygon with $2n$ sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 600. Compute the value of the integer $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-4de8156f5660481f3aaeffa37c4ba1459c2f04983c9c6bd32fd98676f8ed0496 | A regular polygon with \( f \) faces, \( v \) vertices, and \( e \) edges is made by cutting a regular tetrahedron with an edge length of 1 unit from each of the four corners of a tetrahedron with an edge length of 3 units. Compute \( f + v + e \). | [
"38"
] | [
"A regular polygon with \\( f \\) faces, \\( v \\) vertices, and \\( e \\) edges is made by cutting a regular tetrahedron with an edge length of 1 unit from each of the four corners of a tetrahedron with an edge length of 3 units. Compute \\( f + v + e \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed8da37aac4f77a169df29e52884ad6c17dfce61ebabb45f88a7d1c8e5dc4338 | A regular polygon with \( n \) sides is divided into \( n \) isosceles triangles by segments joining its center to the vertices. Initially, \( n + 1 \) frogs are placed inside the triangles. At every second, two frogs in some common triangle jump into the interiors of the two neighboring triangles (one frog into each n... | [
"5"
] | [
"A regular polygon with \\( n \\) sides is divided into \\( n \\) isosceles triangles by segments joining its center to the vertices. Initially, \\( n + 1 \\) frogs are placed inside the triangles. At every second, two frogs in some common triangle jump into the interiors of the two neighboring triangles (one frog ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8dd2e120b3c491ee573b75abe1f06dcf448c004c6c86b47dc1c573d27f3873cd | A regular polygon with \( n \) sides is inscribed in a unit circle. A cut of length 1 is made through the midpoint of one of the polygon's sides. Let \( A(n) \) be the area of the smallest piece that can be removed by such a cut. Find \( A(6) \). | [
"\\dfrac{1}{2}"
] | [
"A regular polygon with \\( n \\) sides is inscribed in a unit circle. A cut of length 1 is made through the midpoint of one of the polygon's sides. Let \\( A(n) \\) be the area of the smallest piece that can be removed by such a cut. Find \\( A(6) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b60a09960eb2f8e0cc52c5cce122e6d0f31393a480a734f9efb47aa807eb9b77 | A regular polygon with \(2n\) sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 1200. Compute the value of the integer \(n\). | [
"12"
] | [
"A regular polygon with \\(2n\\) sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 1200. Compute the value of the integer \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3b5c48170d66cc599f8a6ddc2bd854685998803d781153f3831f0fb828ff326 | A regular polygon with \(2n\) sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 600. Compute the value of the integer \(n\). | [
"6"
] | [
"A regular polygon with \\(2n\\) sides is inscribed in a circle of radius 5. The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 600. Compute the value of the integer \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c83fa41239c2d3695d3af5ff47aae7ea13dae0cded9ad133fd51043ca01d6c74 | A regular polygon with exactly 20 diagonals is inscribed in a circle. The area of the polygon is $144\sqrt{2}$. Find the area of the circle. | [
"\\( 72\\pi \\)"
] | [
"A regular polygon with exactly 20 diagonals is inscribed in a circle. The area of the polygon is $144\\sqrt{2}$. Find the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 27
} | 46 | 0 | 0 |
omr-ws-72630ca212554b6e4c0c27aaaece65184dc552ce9241ad24885ee1129e13d520 | A regular polygon with exactly 20 diagonals is inscribed in a circle. The area of the polygon is \(144\sqrt{2}\). Find the area of the circle. | [
"\\(72\\pi\\)"
] | [
"A regular polygon with exactly 20 diagonals is inscribed in a circle. The area of the polygon is \\(144\\sqrt{2}\\). Find the area of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-6678c02edb6a7de08d144d582a5e93e736c32585d20c7e269ef3044c3ddf487f | A regular polyhedron has 9 edges. Each vertex is colored either white or black. Calculate the minimum number of triangles with vertices of the same color that must exist in this polyhedron. | [
"0"
] | [
"A regular polyhedron has 9 edges. Each vertex is colored either white or black. Calculate the minimum number of triangles with vertices of the same color that must exist in this polyhedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-1a74e5a2d48e2978b50765658028e1d174a698abfafa010ac9df855a47d95285 | A regular polyhedron is created by cutting a regular tetrahedron with edge length one from each of the four corners of a tetrahedron. Compute the sum of the number of faces (f), vertices (v), and edges (e) of the resulting polyhedron. | [
"26"
] | [
"A regular polyhedron is created by cutting a regular tetrahedron with edge length one from each of the four corners of a tetrahedron. Compute the sum of the number of faces (f), vertices (v), and edges (e) of the resulting polyhedron."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-3974d12075ebd5151fa0ffaed21a76dec09b266b7a5d2ca7821151245cacc215 | A regular pyramid is placed on a cube \(ABCDA_1B_1C_1D_1\) such that its base coincides with the square \(A_1B_1C_1D_1\), and its vertex \(M\) is outside the cube. If the length of all edges of the pyramid is \(2 - \sqrt{2}\), find the distance from point \(A\) to the line \(MC_1\). | [
"1"
] | [
"A regular pyramid is placed on a cube \\(ABCDA_1B_1C_1D_1\\) such that its base coincides with the square \\(A_1B_1C_1D_1\\), and its vertex \\(M\\) is outside the cube. If the length of all edges of the pyramid is \\(2 - \\sqrt{2}\\), find the distance from point \\(A\\) to the line \\(MC_1\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 4
} | 52 | 0 | 0 |
omr-ws-4292bb1a6d0493a7e288df80a3878c0d726f4291177691a43f0a6986df1c7dc9 | A regular pyramid whose base is a square with side $a$ is rotated about a straight line passing through the vertex of the pyramid parallel to one of the sides of the base. Compute the volume of the solid of revolution if the plane angle at the vertex of the pyramid is equal to $\alpha $. | [
"$1/12\\pi a^3[1+2\\cot^2 (\\alpha /2)]$"
] | [
"A regular pyramid whose base is a square with side $a$ is rotated about a straight line passing through the vertex of the pyramid parallel to one of the sides of the base. Compute the volume of the solid of revolution if the plane angle at the vertex of the pyramid is equal to $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50e94414aef3de25e9dfdf81199da6c91bc9de77c1b5e7217ff188cd5797a386 | A regular pyramid-shaped container with a square base holds 120 cm\(^3\) of water. When placed with the base on the ground, the water covers up to 5 cm in height, and when placed upside down, the water covers 10 cm in height. Determine the side of the base and the height of the container. | [
"15"
] | [
"A regular pyramid-shaped container with a square base holds 120 cm\\(^3\\) of water. When placed with the base on the ground, the water covers up to 5 cm in height, and when placed upside down, the water covers 10 cm in height. Determine the side of the base and the height of the container."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-8fa78b4b6536a1932410e2194838749ca6616f6942673e45a132c16ac8379cab | A regular quadrangular is inscribed in a sphere of radius $R$. Find the volume of the pyramid if the radius of the circle circumscribed about its base is $r$. | [
"$\\frac{1}{6}rR\\left(R\\pm \\sqrt {R^2-r^2}\\right)$"
] | [
"A regular quadrangular is inscribed in a sphere of radius $R$. Find the volume of the pyramid if the radius of the circle circumscribed about its base is $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80b5a935aa99294e207cb876f6c8bddf367a515af121efa1c1e428938fff3294 | A regular quadrangular pyramid contains a sphere of radius 2. The ball touches the lateral faces of the pyramid and is externally tangent to a hemisphere resting on the sphere inscribed in the base of the pyramid. The point of tangency of the ball and the hemisphere is at the distance from the base of the pyramid which... | [
"$V=4\\left(\\sqrt {10}+1\\right)^3,\\ \\alpha =2\\arcsin \\sqrt {\\frac{11}{20}}$"
] | [
"A regular quadrangular pyramid contains a sphere of radius 2. The ball touches the lateral faces of the pyramid and is externally tangent to a hemisphere resting on the sphere inscribed in the base of the pyramid. The point of tangency of the ball and the hemisphere is at the distance from the base of the pyramid ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42216364102b8329846ef87924ea28fbaff5637f9c6f91d36abb590cebf32923 | A regular quadrangular pyramid is cut by a plane parallel to the base. In what ratio is the volume of the pyramid divided if the area of the section is one third the area of the base? | [
"$\\left(3\\sqrt 3+1\\right)/26$"
] | [
"A regular quadrangular pyramid is cut by a plane parallel to the base. In what ratio is the volume of the pyramid divided if the area of the section is one third the area of the base?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2aa36a1423b6fc621bb9de797dbeff46d959912aa746787572c50e633a1976c3 | A regular quadrangular pyramid is inscribed in a sphere. The radius of the sphere is $R$, the edge angle is $\alpha $. Find the lateral surface of the pyramid. | [
"$4R^2\\sin 2\\alpha $"
] | [
"A regular quadrangular pyramid is inscribed in a sphere. The radius of the sphere is $R$, the edge angle is $\\alpha $. Find the lateral surface of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edd9166cfb41b6d0283b598aff4744679cb94c7534a0fd30bd115bec0b9779db | A regular quadrangular pyramid with base side a and dihedral angle at the base equal to $2\alpha $ is cut by a plane that bisects the dihedral angle at the base. Find the area of the section. | [
"$a^2\\sin 2\\alpha \\cos \\alpha {\\rm cosec}^2 3\\alpha $"
] | [
"A regular quadrangular pyramid with base side a and dihedral angle at the base equal to $2\\alpha $ is cut by a plane that bisects the dihedral angle at the base. Find the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dc29305d4e25a430641036c99c7136503a751c85afa87aba23bab2753acb464 | A regular quadrangular pyramid with side of its base equal to $a$ is cut by a plane bisecting its dihedral angle at the base which is equal to $2\alpha $. Find the area of the section. | [
"a^2\\, \\frac{\\sin^2 2\\alpha \\cos\\alpha } {\\sin^2 3\\alpha }"
] | [
"A regular quadrangular pyramid with side of its base equal to $a$ is cut by a plane bisecting its dihedral angle at the base which is equal to $2\\alpha $. Find the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b34933420fcd69e5ece6e04fb9bb1cffa73764aa3428bfc9ea2e2a37375b89d | A regular quadrilateral pyramid is cut by a plane passing through one of the vertices of the base and is perpendicular to the opposite lateral edge. The area of the intersection is half the area of the base. Determine the angle between a lateral edge and the base. | [
"\\arctan\\left( \\sqrt{\\dfrac{9 + \\sqrt{33}}{6}} \\right)"
] | [
"A regular quadrilateral pyramid is cut by a plane passing through one of the vertices of the base and is perpendicular to the opposite lateral edge. The area of the intersection is half the area of the base. Determine the angle between a lateral edge and the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-504d680888df15ab627d7abee7348fabb3b6db37ad68c2bd2479e56c020100df | A regular tetrahedron $ABCD$ has a side length of $6$. Let $\ell$ be the line passing through the midpoints of $\overline{AB}$ and $\overline{CD}$. The tetrahedron is rotated about $\ell$, and let $\mathcal{S}$ be the intersection of all the rotated tetrahedra. The square of the volume of $\mathcal{S}$ can be expressed... | [
"89"
] | [
"A regular tetrahedron $ABCD$ has a side length of $6$. Let $\\ell$ be the line passing through the midpoints of $\\overline{AB}$ and $\\overline{CD}$. The tetrahedron is rotated about $\\ell$, and let $\\mathcal{S}$ be the intersection of all the rotated tetrahedra. The square of the volume of $\\mathcal{S}$ can b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-560da5a34285e3b73c8010f2d198813bbf254f0b6852b304343d2aed09e995f5 | A regular tetrahedron $ABCD$ has edge length 1. Find the shortest distance between edge $AB$ and edge $CD$. | [
"$\\frac{\\sqrt{2}}{2}$"
] | [
"A regular tetrahedron $ABCD$ has edge length 1. Find the shortest distance between edge $AB$ and edge $CD$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e7fd73246bde80e08cb7ec087a609e5dd9b9a4db789f073b25532e0b12f71318 | A regular tetrahedron $ABCD$ has three white faces and one black face opposite vertex $D$. The tetrahedron is placed on a plane $S$ with the black face down. The tetrahedron can be rolled on the plane by rotating about its edges. After a sequence of rolls that returns the tetrahedron to its original position, calculate... | [
"3"
] | [
"A regular tetrahedron $ABCD$ has three white faces and one black face opposite vertex $D$. The tetrahedron is placed on a plane $S$ with the black face down. The tetrahedron can be rolled on the plane by rotating about its edges. After a sequence of rolls that returns the tetrahedron to its original position, calc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce663626d7790904277e5659fdd1891f783020232a40d93ca8223976e2af0622 | A regular tetrahedron ABCD has a side length of 4. A plane cuts through the midpoints of edges AB, AC, CD, and BD. Find the area of the intersection of the plane and the tetrahedron. | [
"4"
] | [
"A regular tetrahedron ABCD has a side length of 4. A plane cuts through the midpoints of edges AB, AC, CD, and BD. Find the area of the intersection of the plane and the tetrahedron."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-549c2131986dda995bbf7d2517cc9522544798d404824205afe14c94e0736202 | A regular tetrahedron ABCD has all sides of length four. A plane intersects edges AC, AD, BC, and BD at points W, X, Y, and Z, respectively. Given AW=2, AX=3, and BZ=1, determine the length of BY. | [
"\\dfrac{2}{5}"
] | [
"A regular tetrahedron ABCD has all sides of length four. A plane intersects edges AC, AD, BC, and BD at points W, X, Y, and Z, respectively. Given AW=2, AX=3, and BZ=1, determine the length of BY."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-534a19c6d89b82778693bd8db965f4eb3a6e9d063282b40a9e6ea8552ae8959f | A regular tetrahedron ABCD has an inscribed sphere O with radius 1. Find the side length of the tetrahedron. | [
"\\( \\frac{\\sqrt{11} + 1}{\\sqrt{\\frac{5}{6}}} \\)"
] | [
"A regular tetrahedron ABCD has an inscribed sphere O with radius 1. Find the side length of the tetrahedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87d7b018fcb6eb7e6a479bfc7e20ac4f4fd4ef22953185ec8baa5c20262d0c53 | A regular tetrahedron ABCD has an inscribed sphere O with radius 1. Four other spheres, W, X, Y, and Z, are loaded into the tetrahedron such that they are tangent to sphere O and three of the four faces. This process is repeated indefinitely with four more spheres tangent to W, X, Y, and Z, and three of the four faces.... | [
"\\(\\frac{2\\left(\\sqrt{3} + \\sqrt{33}\\right)^3 \\pi}{3 \\left(36 \\sqrt{110} - 123 \\sqrt{33} + 236 \\sqrt{10} - 213\\sqrt{3}\\right)}\\)"
] | [
"A regular tetrahedron ABCD has an inscribed sphere O with radius 1. Four other spheres, W, X, Y, and Z, are loaded into the tetrahedron such that they are tangent to sphere O and three of the four faces. This process is repeated indefinitely with four more spheres tangent to W, X, Y, and Z, and three of the four f... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43ead6d624c05449030151de3596105d8f0b99e3380b520933b175b3618ab328 | A regular tetrahedron ABCD has side length four. A plane intersects edges AC, AD, BC, and BD at points W, X, Y, and Z, respectively. Given that AW=2, AX=3, and BZ=1, determine the length of BY. | [
"\\(\\frac{2}{5}\\)"
] | [
"A regular tetrahedron ABCD has side length four. A plane intersects edges AC, AD, BC, and BD at points W, X, Y, and Z, respectively. Given that AW=2, AX=3, and BZ=1, determine the length of BY."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 10
} | 58 | 0 | 0 |
omr-ws-d9d40e1f1e607d0fb72f87d72ef6b089cb757167fc4594d6d4b169f493eb649f | A regular tetrahedron \(ABCD\) with edge length \(b\) is inscribed in a cone with a vertex angle of 90° such that edge \(AB\) lies on a generator of the cone. Find the distance from the vertex of the cone to the straight line \(CD\). | [
"\\dfrac{b\\sqrt{34}}{8}"
] | [
"A regular tetrahedron \\(ABCD\\) with edge length \\(b\\) is inscribed in a cone with a vertex angle of 90° such that edge \\(AB\\) lies on a generator of the cone. Find the distance from the vertex of the cone to the straight line \\(CD\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3dd280007e1ae826e87b91cd4f2252b7fda6b6ca641d92f73e6f6684b9a6c14c | A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube? | [
"\\sqrt{3}"
] | [
"A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a329a5a0449ef4f869ab22f8c16b4c3b2703d03e04243fc3e4387f7f29bcc47 | A regular tetrahedron and a regular octahedron have equal edges. Find the ratio of their volumes. | [
"\\(\\frac{1}{4}\\)"
] | [
"A regular tetrahedron and a regular octahedron have equal edges. Find the ratio of their volumes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-11bcbd207a6ace5ff7eef3aec85faeaeeaf8d98894ccf60936cc0ff62c9de2e7 | A regular tetrahedron and a regular quadrangular pyramid, where the side face of the pyramid is equal to the face of the tetrahedron, are folded together along equal faces. How many faces does the resulting polyhedron have? | [
"7"
] | [
"A regular tetrahedron and a regular quadrangular pyramid, where the side face of the pyramid is equal to the face of the tetrahedron, are folded together along equal faces. How many faces does the resulting polyhedron have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-f52647f2bd5a08a993ad61a184ba18942fbb4a99b9e13684c632a54b8fa58cdb | A regular tetrahedron has faces $\Delta_1, \Delta_2, \Delta_3, \Delta_4$. Spheres $\Gamma$ and $\gamma$ satisfy the property that for all $i \in \{1,2,3,4\}$, the circumcircle of $\Delta_i$ lies on the surface of $\Gamma$ and the incircle of $\Delta_i$ lies on the surface of $\gamma$. What is the ratio of the volume of... | [
"\\(3\\sqrt{3}\\)"
] | [
"A regular tetrahedron has faces $\\Delta_1, \\Delta_2, \\Delta_3, \\Delta_4$. Spheres $\\Gamma$ and $\\gamma$ satisfy the property that for all $i \\in \\{1,2,3,4\\}$, the circumcircle of $\\Delta_i$ lies on the surface of $\\Gamma$ and the incircle of $\\Delta_i$ lies on the surface of $\\gamma$. What is the rati... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-a23188ed2a1bb595d39e26928ccefeaa56ac11cd90811daa40154da6f7b7a787 | A regular tetrahedron has two vertices at \( (0, 0, 0) \) and \( (18, 8, 14) \). Let the minimum possible \( x \) coordinate of one of the other two vertices be \( a - \sqrt{b} \), where \( a \) and \( b \) are both positive integers. Find \( a + b \). | [
"204"
] | [
"A regular tetrahedron has two vertices at \\( (0, 0, 0) \\) and \\( (18, 8, 14) \\). Let the minimum possible \\( x \\) coordinate of one of the other two vertices be \\( a - \\sqrt{b} \\), where \\( a \\) and \\( b \\) are both positive integers. Find \\( a + b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fcab87b2effafa5a4f803d72037b15158f426f6fdd8f13908c171cfdb96739e | A regular tetrahedron has two vertices on the body diagonal of a cube with side length 12. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron. | [
"4\\sqrt{3}"
] | [
"A regular tetrahedron has two vertices on the body diagonal of a cube with side length 12. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-391376e0bdee92c02fee60226efbd32bfb0f531dd59cf31c86b554bf1d7f4328 | A regular tetrahedron is a three-dimensional figure whose faces are four equilateral triangles. A regular tetrahedron is on Johnny's desk. Inscribed inside it is a smaller, upside-down regular tetrahedron. What is the ratio of the volume of the larger tetrahedron to the upside-down tetrahedron? | [
"27"
] | [
"A regular tetrahedron is a three-dimensional figure whose faces are four equilateral triangles. A regular tetrahedron is on Johnny's desk. Inscribed inside it is a smaller, upside-down regular tetrahedron. What is the ratio of the volume of the larger tetrahedron to the upside-down tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffd73d597258752fd5fe621ed4170e8ccb23663144b2e060a4272601d2174bc2 | A regular tetrahedron is cut off each vertex of a cube with edge length 1 in such a way that each original face of the cube becomes a regular octagon. Find the length of an edge of the new 14-sided solid. | [
"\\( \\sqrt{2} - 1 \\)"
] | [
"A regular tetrahedron is cut off each vertex of a cube with edge length 1 in such a way that each original face of the cube becomes a regular octagon. Find the length of an edge of the new 14-sided solid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 30
} | 51 | 0 | 0 |
omr-ws-fbf55cb01290edae0d0f1a06689838722b176f6fb9572cf9894812b73156e2bf | A regular tetrahedron is cut simultaneously from six different planes, each cutting exactly halfway through a corner and bisecting the opposite edge. How many pieces are there? | [
"8"
] | [
"A regular tetrahedron is cut simultaneously from six different planes, each cutting exactly halfway through a corner and bisecting the opposite edge. How many pieces are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-d48bbb0ee567997b26b1a9c16db37cddadde52276fb5be8dfe2237d0881708bf | A regular tetrahedron is inscribed in a hemisphere of radius $1$ with maximal volume. Its surface area can be written in the form $\frac{a\sqrt{b}}{c}$ such that $a$ and $c$ are relatively prime and $b$ is not the square of any prime. Find $a+b+c$. | [
"14"
] | [
"A regular tetrahedron is inscribed in a hemisphere of radius $1$ with maximal volume. Its surface area can be written in the form $\\frac{a\\sqrt{b}}{c}$ such that $a$ and $c$ are relatively prime and $b$ is not the square of any prime. Find $a+b+c$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
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