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-b7b5bf4f2f1f4a39ffaa046566ae8451a4c65bd5ff996a09a67134505603aa78 | A regular tetrahedron is inscribed in a sphere with a volume of 864π. What is the total surface area of the tetrahedron? | [
"288\\sqrt[6]{3}"
] | [
"A regular tetrahedron is inscribed in a sphere with a volume of 864π. What is the total surface area of the tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-689284d5ed0caf84961d064282b903ebc37b43502df0d5db777dced3549c6fb0 | A regular tetrahedron is inscribed in a sphere, and another sphere is inscribed in the tetrahedron. Find the ratio of the surface areas of the spheres. | [
"9"
] | [
"A regular tetrahedron is inscribed in a sphere, and another sphere is inscribed in the tetrahedron. Find the ratio of the surface areas of the spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea26e5e3ce18aa0fd0ada8ed91226da485a7c27c0fae92fb838897ff79b46d07 | A regular tetrahedron is inside a right cone such that the tip of the cone is at one vertex of the tetrahedron and the other three vertices lie on the circle at the base of the cone. If each side of the tetrahedron has length 1, find the volume of the region outside the tetrahedron but inside the cone. | [
"\\dfrac{\\pi \\sqrt{6}}{27} - \\dfrac{\\sqrt{2}}{12}"
] | [
"A regular tetrahedron is inside a right cone such that the tip of the cone is at one vertex of the tetrahedron and the other three vertices lie on the circle at the base of the cone. If each side of the tetrahedron has length 1, find the volume of the region outside the tetrahedron but inside the cone."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30f5bfff721df6941f8e8fbe9891845efd9954d9a9872078df15fe49c6ca287b | A regular tetrahedron of height \( h \) has a smaller tetrahedron of height \( xh \) cut off by a plane parallel to the base. When the remaining frustum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. Find the value of \( x \) that satisfies the equation \( x^3 + x^2 ... | [
"0.811"
] | [
"A regular tetrahedron of height \\( h \\) has a smaller tetrahedron of height \\( xh \\) cut off by a plane parallel to the base. When the remaining frustum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. Find the value of \\( x \\) that satisfies the equation \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-6d523ba93dfca42c82815583dc6a0f0e4b7307e1b966cc12e06974b8744611b8 | A regular tetrahedron of side length \( m \) is concentric to a smaller regular tetrahedron such that the smallest distance between the two tetrahedrons is \( p \). Find the volume of the smaller tetrahedron in terms of \( m \) and \( p \). | [
"\\dfrac{(m - 2\\sqrt{6}p)^3}{6\\sqrt{2}}"
] | [
"A regular tetrahedron of side length \\( m \\) is concentric to a smaller regular tetrahedron such that the smallest distance between the two tetrahedrons is \\( p \\). Find the volume of the smaller tetrahedron in terms of \\( m \\) and \\( p \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6ec791c22612b9843bba8ec18f35f0de649c78a71cf2573f812db37fd4c6029 | A regular tetrahedron of unit volume has the midpoints of two crossing ribs. The tetrahedron was rotated $90^\circ$ around a straight line, passing through these points. Find the volume of the common part of these two tetrahedra. | [
"1/2"
] | [
"A regular tetrahedron of unit volume has the midpoints of two crossing ribs. The tetrahedron was rotated $90^\\circ$ around a straight line, passing through these points. Find the volume of the common part of these two tetrahedra."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 16
} | 28 | 0 | 0 |
omr-ws-3e538b78d899ecfe8122a11fd4408cf1504313c2560507c46b1ffe9efa01d919 | A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the upper liquid and the area of the free surface of the water. | [
"\\( h_w = 10 \\sqrt{6} \\left( \\frac{4 \\sqrt{2}}{9} \\right)^{1/3} \\), \\( A_w = \\left( \\frac{4 \\sqrt{2}}{9} \\right)^{2/3} \\cdot 225 \\sqrt{3} \\)"
] | [
"A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the upper liquid and the area of the free surface of the water."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86fdd6222c4f4599c5ebe24b193b36fee890ad863378b9ded972a8b4deed61fa | A regular tetrahedron with sides measuring four inches has each vertex cut away by a plane passing through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches? | [
"\\( 20\\sqrt{3} \\)"
] | [
"A regular tetrahedron with sides measuring four inches has each vertex cut away by a plane passing through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c2ffd2d9ab7616bf6569e366e2687d3573bfbce64a0b329decb12c8e3a32b6f | A regular tetrahedron, whose edge is equal to $a$, is cut by a plane containing one of its edges and dividing the opposite edge in the ratio $2:1$. Determine the area of the section figure and its angles. | [
"$S=\\frac{\\sqrt {19}a^2}{12};\\ \\alpha =\\arccos \\frac{3}{2\\sqrt 7};\\ \\beta =\\pi -2\\arccos \\frac{3}{2\\sqrt 7}$"
] | [
"A regular tetrahedron, whose edge is equal to $a$, is cut by a plane containing one of its edges and dividing the opposite edge in the ratio $2:1$. Determine the area of the section figure and its angles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c895ee15775c4df6319e097f50721faf6b0baf1e54c09191b72b7610d792cd5f | A regular triangle and a regular $9$-gon are drawn in the same circle. Which one has the larger sum of the squares of all the sides? | [
"The regular triangle has the larger sum of the squares of all the sides."
] | [
"A regular triangle and a regular $9$-gon are drawn in the same circle. Which one has the larger sum of the squares of all the sides?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-9f98c15828f740b6db32261b6cebf7c4c19020486638761c9d73e919326817b6 | A regular triangle is divided into $5$ isosceles triangles, of which exactly $2$ are regular. If the side length of the original regular triangle is $s$, find the sum of all possible distinct side lengths of the non-regular isosceles triangles formed in this division. | [
"s"
] | [
"A regular triangle is divided into $5$ isosceles triangles, of which exactly $2$ are regular. If the side length of the original regular triangle is $s$, find the sum of all possible distinct side lengths of the non-regular isosceles triangles formed in this division."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-ca6b717a22e393827e90cad9ba1f4776c67ae8ca020a701679fb89d53b261e09 | A regular triangle with side $a$ lies in the plane $P$. Its medians divide it into four triangles and three regular pyramids with the altitude $a$ are constructed on three of them, adjacent to the vertices as the bases (the three pyramids lie on the same side of $P$). Find the radius of the sphere which lies between th... | [
"$\\frac{a}{6}$"
] | [
"A regular triangle with side $a$ lies in the plane $P$. Its medians divide it into four triangles and three regular pyramids with the altitude $a$ are constructed on three of them, adjacent to the vertices as the bases (the three pyramids lie on the same side of $P$). Find the radius of the sphere which lies betwe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d741e089d34f505aab7e32b2da1ee193b23aa4649c5ed9177efa1623067587f6 | A regular triangular prism has an altitude \( h \) and equilateral triangle bases with side length \( a \). Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. A ray of light enters the prism through the dream-hole in the upper base, reflects once off one of the mirrors, and exit... | [
"\\(\\theta = \\arctan\\left(\\frac{h}{a} \\cdot \\left(\\frac{3}{7}\\right)^{0.5}\\right)\\), \\(\\text{Length} = \\sqrt{h^2 + \\frac{7a^2}{3}}\\)"
] | [
"A regular triangular prism has an altitude \\( h \\) and equilateral triangle bases with side length \\( a \\). Dream-holes are made in the centers of both bases, and the three lateral faces are mirrors. A ray of light enters the prism through the dream-hole in the upper base, reflects once off one of the mirrors,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79f0cce9e4458d903679ce8fc8c45b6d0a38071d68b5d0797fee75a2678d0f3c | A regular triangular prism has height \( h \) and a base of side length \( a \). Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once, and leaves through the hole in the bottom. F... | [
"\\(\\theta = \\tan^{-1}\\left(\\frac{h}{3a}\\right)\\), \\(L = \\sqrt{9a^2 + h^2}\\)"
] | [
"A regular triangular prism has height \\( h \\) and a base of side length \\( a \\). Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once, and leaves through the hole in the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-425dc5f4d55474d172184f3585f1e6f4c4efe66a87e3c4554986eba0cec776bd | A regular triangular prism has height \( h \) and a base with side length \( a \). Both bases have small holes in their centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once, and leaves through the hole in the botto... | [
"\\sqrt{\\frac{7a^2 + 3h^2}{3}}"
] | [
"A regular triangular prism has height \\( h \\) and a base with side length \\( a \\). Both bases have small holes in their centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once, and leaves through the hole in ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-d3dd4c2de32e9f6dd1209b337136787efee2283a34971af5bc5541f10a283e8d | A regular triangular prism is circumscribed about a sphere and a sphere is circumscribed about the prism. Find the ratio of the surfaces of the spheres. | [
"$5:1$"
] | [
"A regular triangular prism is circumscribed about a sphere and a sphere is circumscribed about the prism. Find the ratio of the surfaces of the spheres."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-638eb6753ee482ba8e5f4aef7d6b9441a664454bcfd0c6712ef5821b90953cc4 | A regular triangular prism is inscribed into a regular triangular pyramid with the plane angle $\alpha $ so that the lower base of the prism lies on the base of the pyramid and the upper base coincides with the section of the pyramid by a plane passing through the upper base of the prism. The length of a lateral edge o... | [
"$\\frac{6\\sin \\frac{\\alpha }{2}\\left(1-\\frac{4}{3}\\sin ^2 \\frac{\\alpha }{2}\\right)} {\\left(2\\sin \\frac{\\alpha }{2}+\\sqrt {1-\\frac{4}{3}\\sin ^2 \\frac{\\alpha }{2}}\\right)^3}$"
] | [
"A regular triangular prism is inscribed into a regular triangular pyramid with the plane angle $\\alpha $ so that the lower base of the prism lies on the base of the pyramid and the upper base coincides with the section of the pyramid by a plane passing through the upper base of the prism. The length of a lateral ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-311905b29b744c7330773352ebcfdb06ade42525404abfc3d63168499304a1c2 | A regular triangular pyramid has altitude $h$ and lateral edge $l$. Find the area of a section that is parallel to the base and distant $a$ from it. | [
"$3/4\\sqrt {3}h^2(l^2-h^2)(h-a)^{-2}$"
] | [
"A regular triangular pyramid has altitude $h$ and lateral edge $l$. Find the area of a section that is parallel to the base and distant $a$ from it."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50387f864cedec93d27c5c87de6898d7a2d6d5e240581145ed6f3287c7d44c45 | A regular triangular pyramid has base edges of 8 and a height of 6. A similar pyramid has a height of 6. a. Find the base edge of the smaller pyramid. b. Find the ratio of the total areas of the pyramids. | [
"1"
] | [
"A regular triangular pyramid has base edges of 8 and a height of 6. A similar pyramid has a height of 6.\na. Find the base edge of the smaller pyramid.\nb. Find the ratio of the total areas of the pyramids."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-673ac9c70154d82c30533fef49edf935ab1736249975bb074719c7ad81c17cf4 | A regular triangular pyramid is cut by a plane parallel to the base so that the lateral surface is cut in half. In what ratio is the altitude divided? | [
"$\\sqrt 2+1$"
] | [
"A regular triangular pyramid is cut by a plane parallel to the base so that the lateral surface is cut in half. In what ratio is the altitude divided?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c7ee578a008a621f425a55681b60cbac7aea65556f9025dc13ba0a9ebbc2eef | A regular triangular pyramid is cut by a plane perpendicular to the base and bisecting two sides of the base. Determine the volume of the cut-off pyramid, given the side $a$ of the base of the original pyramid and dihedral angle $\alpha $ at the base. | [
"$V=\\frac{a^3\\tan\\alpha }{128}$"
] | [
"A regular triangular pyramid is cut by a plane perpendicular to the base and bisecting two sides of the base. Determine the volume of the cut-off pyramid, given the side $a$ of the base of the original pyramid and dihedral angle $\\alpha $ at the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e2697304a4418f65ce97233f6c232d4694c430a2b01ea874f2455ece8d556f9 | A regular triangular pyramid is cut by a plane which is perpendicular to the base and divides the two sides of the base in half. Find the volume of the cut-off pyramid if the side of the base of the initial pyramid is $a$ and the base dihedral angle is $45^\circ$. | [
"$\\frac{a^3}{128}$"
] | [
"A regular triangular pyramid is cut by a plane which is perpendicular to the base and divides the two sides of the base in half. Find the volume of the cut-off pyramid if the side of the base of the initial pyramid is $a$ and the base dihedral angle is $45^\\circ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be5fbdb605b7a0c1136bb7626aaa7ee5bb9d7ee5ebddf2c596b61e9561389422 | A regular triangular pyramid is cut by the plane passing through a vertex of the base and the midpoints of two lateral edges. Find the ratio of the lateral area of the pyramid to the area of the base if it is known that the cutting plane is perpendicular to the lateral face opposite that vertex. | [
"\\sqrt{6}"
] | [
"A regular triangular pyramid is cut by the plane passing through a vertex of the base and the midpoints of two lateral edges. Find the ratio of the lateral area of the pyramid to the area of the base if it is known that the cutting plane is perpendicular to the lateral face opposite that vertex."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76519c89bed2b351644ee1ab27a4b126b563dd91cbdec9cc5c7a04ca9cfe3c41 | A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the fewest relatively prime dates? | [
"10"
] | [
"A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the fewest relatively prime dates?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b208c838840238d24c401a51348695137ecce299e12d1265d693c5860df87f3a | A relief organization purchased 15 blankets, 5 cots, and 10 lanterns for a total cost of $1100. The next week, at the same prices, the organization purchased 20 blankets, 10 cots, and 15 lanterns for a total cost of $1800. The third week, at the same prices, the organization purchased 10 blankets, 15 cots, and 5 lanter... | [
"50"
] | [
"A relief organization purchased 15 blankets, 5 cots, and 10 lanterns for a total cost of $1100. The next week, at the same prices, the organization purchased 20 blankets, 10 cots, and 15 lanterns for a total cost of $1800. The third week, at the same prices, the organization purchased 10 blankets, 15 cots, and 5 l... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bafbc0353c6bf8b31be36218cf0e5c8ae1f578d20001b5db6b02d4a5cfc18b74 | A remote-controlled car can only move forward in a straight line or by describing arcs of a circle with a radius of 63 cm. The car is initially located at point A, facing north. What is the minimum distance the car must travel to return to point A, but oriented towards the south? (Assume \(\pi = \frac{22}{7}\)). | [
"522"
] | [
"A remote-controlled car can only move forward in a straight line or by describing arcs of a circle with a radius of 63 cm. The car is initially located at point A, facing north. What is the minimum distance the car must travel to return to point A, but oriented towards the south? (Assume \\(\\pi = \\frac{22}{7}\\)... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96ef8f19835015edbf4d80366e1e508525cbbe19305df4ca630ad476b97e3b91 | A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. Given that one van with brake problems is detected in the first two tests, what is the probability that the remaining van is found on the third or fourth test? | [
"\\dfrac{1}{2}"
] | [
"A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. Given that one van with brake problems is detected in the first two tests, what is the probability that the remaining van is found on the third or fourth test?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-088653b6f385e94172a71a3baecd649b82b6978754e6925fcd1d6f664a20b67a | A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. What is the probability that no more than four vans need to be tested before both brake problems are detected? | [
"\\(\\frac{1}{3}\\)"
] | [
"A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. What is the probability that no more than four vans need to be tested before both brake problems are detected?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e62e7150b1fbba1bb3633a628e1d13a3d696ee113969007938ae9df9bd22f45 | A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. What is the probability that the last van with brake problems is the fourth tested? | [
"\\(\\frac{1}{5}\\)"
] | [
"A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. What is the probability that the last van with brake problems is the fourth tested?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-3ede0cca78eb2794cfb1cc6d65ebfc997d7393ba9fe47d0269c4683fa8339645 | A reptile crawls on a rope initially 1 meter long at a speed of 1 cm/s. After every 1 cm of travel, the rope stretches by an additional 1 meter. Assuming the rope can stretch infinitely and the reptile is immortal, determine if the reptile can reach the end of the rope. If it can, calculate the time required. | [
"\\( e^{100} \\) seconds"
] | [
"A reptile crawls on a rope initially 1 meter long at a speed of 1 cm/s. After every 1 cm of travel, the rope stretches by an additional 1 meter. Assuming the rope can stretch infinitely and the reptile is immortal, determine if the reptile can reach the end of the rope. If it can, calculate the time required."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 19
} | 39 | 0 | 0 |
omr-ws-7c25d52c3860b4c5d65a286d51fe50074fa745d0242bb9a55b4048badcdc9e3d | A researcher divides all the matches into 3 categories: S (matches less than 105 minutes), M (matches between 105 and 115 minutes), and L (matches more than 115 minutes). What is the probability that there will be 10S, 44M, and 10L matches in the World Cup finals tournament, given that the duration of a match is normal... | [
"0.0072"
] | [
"A researcher divides all the matches into 3 categories: S (matches less than 105 minutes), M (matches between 105 and 115 minutes), and L (matches more than 115 minutes). What is the probability that there will be 10S, 44M, and 10L matches in the World Cup finals tournament, given that the duration of a match is n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-456e214375431b3f8a922de7e6a7dcd677f1c4d60716ff06934766eb0071838e | A researcher wants to estimate the population proportion of people in Australia who only use recyclable shopping bags. The researcher found 75 out of 100 people in the sample said that they only use recyclable bags. a) Calculate a 90% confidence interval for the population proportion of people in Australia who only use... | [
"(a) (0.6789, 0.8211), (b) We are 90% confident that the true population proportion of people in Australia who only use recyclable shopping bags lies between 0.6789 and 0.8211, (c) 317"
] | [
"A researcher wants to estimate the population proportion of people in Australia who only use recyclable shopping bags. The researcher found 75 out of 100 people in the sample said that they only use recyclable bags.\na) Calculate a 90% confidence interval for the population proportion of people in Australia who on... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 13
} | 15 | 0 | 0 |
omr-ws-e8d3bd285e4e9ee4e1b399a84da8832f153409bf6427e5a90411bf0eefda4040 | A restaurant gives one of five types of coupons with each meal, each with equal probability. A customer receives a free meal after collecting one coupon of each type. How many meals does a customer expect to need to buy before getting a free meal? | [
"11.4165"
] | [
"A restaurant gives one of five types of coupons with each meal, each with equal probability. A customer receives a free meal after collecting one coupon of each type. How many meals does a customer expect to need to buy before getting a free meal?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3a21aa7491360a115314f3ba97a30ab7230a209213718ff2eea3ff054dfcb352 | A restaurant offers 10% off the price of the bill that costs at least 10 dollars, 20% off the price of a bill that costs at least 20 dollars, and so on. Everything on the menu costs an integer number of dollars. What is the most you will pay? | [
"29.5"
] | [
"A restaurant offers 10% off the price of the bill that costs at least 10 dollars, 20% off the price of a bill that costs at least 20 dollars, and so on. Everything on the menu costs an integer number of dollars. What is the most you will pay?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 3,
"tir": 9
} | 59 | 0 | 0 |
omr-ws-beb3d8b79827464cccb61a50ac0aa3979d429f329c57c60bcc36842f94078645 | A restaurant offers three desserts and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year 2003? | [
"8"
] | [
"A restaurant offers three desserts and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year 2003?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 14
} | 33 | 0 | 0 |
omr-ws-3374ed7d1ac2d586031d1db7e4d39776c38d95556ed1c4aad5da102842cda1b8 | A restaurant serves 6 different flavors of ice cream. There are 150 regular customers, each ordering the same flavor every day. If 130 customers have already been served, there will always be 6 different flavors ordered. What is the minimum number of customers that must order to guarantee that there will be 4 distinct ... | [
"88"
] | [
"A restaurant serves 6 different flavors of ice cream. There are 150 regular customers, each ordering the same flavor every day. If 130 customers have already been served, there will always be 6 different flavors ordered. What is the minimum number of customers that must order to guarantee that there will be 4 dist... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-9ec3551a194a98c2e024619a003692e948edb0676c41cd2ba4e01266673561c7 | A retail jeweler sells a pair of gold earrings for $213.90. The jeweler uses a 130% markup based on cost. Based on this information, what was the wholesale cost of the earrings? | [
"93"
] | [
"A retail jeweler sells a pair of gold earrings for $213.90. The jeweler uses a 130% markup based on cost. Based on this information, what was the wholesale cost of the earrings?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c212afa414537898a325a76530a939c9a41e4e9aa59fe6ba26916ce97ca8a591 | A retailer buys an article at a discount of 15% on the printed price from a wholesaler. He marks up the price by 10%. Due to competition in the market, he allows a discount of 5% to a buyer. If the buyer pays 451.44 for the article inclusive of 8% sales tax, find: i) The printed price of the article. ii) The profit per... | [
"i) The printed price of the article is approximately 568. ii) The profit percentage of the retailer is -14.2%."
] | [
"A retailer buys an article at a discount of 15% on the printed price from a wholesaler. He marks up the price by 10%. Due to competition in the market, he allows a discount of 5% to a buyer. If the buyer pays 451.44 for the article inclusive of 8% sales tax, find:\ni) The printed price of the article.\nii) The pro... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1d5832288e4a58d88e75c74628998a70357ff744f205751c812b9e148bdf2cf | A retired person has $70000$ coins to invest and two types of bonds are available. The first type of bond yields an annual income of $8\%$ and the second type yields $10\%$ per annum. The person must invest at least $10000$ coins in the first type and no more than $30000$ coins in the second type. How should the person... | [
"40000 coins in bond A and 30000 coins in bond B"
] | [
"A retired person has $70000$ coins to invest and two types of bonds are available. The first type of bond yields an annual income of $8\\%$ and the second type yields $10\\%$ per annum. The person must invest at least $10000$ coins in the first type and no more than $30000$ coins in the second type. How should the... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 25
} | 34 | 0 | 0 |
omr-ws-ce1557dc27431e8f52e7a6930aa5f87f8d33cdc6e0fe226b758cfc5d18870cce | A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater, whereas, had he served $b$ years more ($b \neq a$), his pension would have been $q$ dollars greater than the original annua... | [
"\\(\\frac{aq^2 - bp^2}{2(bp - aq)}\\)"
] | [
"A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater, whereas, had he served $b$ years more ($b \\neq a$), his pension would have been $q$ dollars greater than the original... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-162ace66eae99296612fd571272357e86f6f8714f7533af112b8d6e9c55a0fef | A retiring employee receives an annual pension proportional to the square root of the number of years of service. Had he served \(a\) years more, his pension would have been \(p\) dollars greater, whereas, had he served \(b\) years more (with \(a\) and \(b\) being distinct), his pension would have been \(q\) dollars gr... | [
"\\(\\frac{q^2a - p^2b}{2(pb - qa)}\\)"
] | [
"A retiring employee receives an annual pension proportional to the square root of the number of years of service. Had he served \\(a\\) years more, his pension would have been \\(p\\) dollars greater, whereas, had he served \\(b\\) years more (with \\(a\\) and \\(b\\) being distinct), his pension would have been \... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-3c86b7d6967f0e52d1eb189a44c13263f15a66052123907e0b5c5cb91ec4b260 | A revolving beacon of light sits 2 miles from a straight shoreline, revolving at 1 revolution per minute. Find the speed of the spot of light on the shore in miles per hour when it is 1 mile from the point closest to the beacon. | [
"\\( 300\\pi \\)"
] | [
"A revolving beacon of light sits 2 miles from a straight shoreline, revolving at 1 revolution per minute. Find the speed of the spot of light on the shore in miles per hour when it is 1 mile from the point closest to the beacon."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-88db3ca36808581d29baacf4a952bfecb024f73fae22541faa8a2c0d4c221cf6 | A rhino moves on the coordinate plane in the following way: It begins at the origin and moves from any lattice point $(x,y)$ to $(x,y+1)$, $(x+1,y)$, or $(x+1,y+1)$. Compute the number of possible paths that are possible for the rhino to move from $(0,0)$ to $(3,3)$. | [
"63"
] | [
"A rhino moves on the coordinate plane in the following way: It begins at the origin and moves from any lattice point $(x,y)$ to $(x,y+1)$, $(x+1,y)$, or $(x+1,y+1)$. Compute the number of possible paths that are possible for the rhino to move from $(0,0)$ to $(3,3)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-7d5f41d36f6c70dc596490f973a2ee875e85a320e0668d6d58a382719386b6bf | A rhomblet is formed by joining the sides of two equilateral triangles with side length $1$. How many ways can you tile a regular hexagon of side length $3$ using rhomblets? | [
"980"
] | [
"A rhomblet is formed by joining the sides of two equilateral triangles with side length $1$. How many ways can you tile a regular hexagon of side length $3$ using rhomblets?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85779ce538ec2fbc78221742e9de0c5fd899a37e929da7c8e5dd0eac22436c6f | A rhombus $ABCD$ has side length $10$. A circle centered at $A$ passes through $C$, and a circle centered at $B$ passes through $D$. If the two circles are tangent to each other, what is the area of the rhombus? | [
"75"
] | [
"A rhombus $ABCD$ has side length $10$. A circle centered at $A$ passes through $C$, and a circle centered at $B$ passes through $D$. If the two circles are tangent to each other, what is the area of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-fe0eebf078f3089be2094c0cb2b669bec25ff6cc0696c3af206b4403f6b2426d | A rhombus $ABCD$ is given with $\angle BAD = 60^\circ$. Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, and $CP = 3$. Determine the length of the segment $AP$. | [
"\\(\\sqrt{7}\\)"
] | [
"A rhombus $ABCD$ is given with $\\angle BAD = 60^\\circ$. Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, and $CP = 3$. Determine the length of the segment $AP$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2749e0daa09fb9fd3a0830fb41f4984f4a004af3464847f671d7b7a6892b1434 | A rhombus $ABCD$ with side $1+\sqrt 5$ and an acute angle of $60^\circ$ contains a circle which is inscribed in a triangle $ABD$. A tangent, drawn from the point $C$ to the circle, cuts the side $AB$ at a point $E$. Find the length of the segment $AE$. | [
"2"
] | [
"A rhombus $ABCD$ with side $1+\\sqrt 5$ and an acute angle of $60^\\circ$ contains a circle which is inscribed in a triangle $ABD$. A tangent, drawn from the point $C$ to the circle, cuts the side $AB$ at a point $E$. Find the length of the segment $AE$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6081803a897d6218bd612caa336c48bce83257a5be9faab1af9dca2499de85c1 | A rhombus $ABCD$ with side length $6$ is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is $\sin(\angle ABC)$? | [
"\\(\\frac{\\pi}{9}\\)"
] | [
"A rhombus $ABCD$ with side length $6$ is rolled to form a cylinder of volume $6$ by taping $\\overline{AB}$ to $\\overline{DC}$. What is $\\sin(\\angle ABC)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8849f42f25b201c2b15a397ddd950a4a66e9d89c958bae6f1f7f7fc0b91fd909 | A rhombus ABCD has a perimeter of 52 meters and a diagonal AC of 24 meters. What is the area of the rhombus in square meters? | [
"120"
] | [
"A rhombus ABCD has a perimeter of 52 meters and a diagonal AC of 24 meters. What is the area of the rhombus in square meters?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-271fbbd4a24994c13794dfb7b3e822155ea2052104ce97d0b7dae1e3d3b954a5 | A rhombus has a height of 15 and diagonals with lengths in a ratio of $3:4$. What is the length of the longer diagonal of the rhombus? | [
"25"
] | [
"A rhombus has a height of 15 and diagonals with lengths in a ratio of $3:4$. What is the length of the longer diagonal of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-dbaa810921eda9cda8ab88581a4887cdc06e2bb304af52500819837cb65babad | A rhombus has a perimeter of 64 and one diagonal of 12. Find the other diagonal, the area, and the altitude. | [
"\\dfrac{3\\sqrt{55}}{2}"
] | [
"A rhombus has a perimeter of 64 and one diagonal of 12. Find the other diagonal, the area, and the altitude."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2aebd4704128936f5513c624d34e729c30faca7eba7fcdb3876186bffe531b8c | A rhombus has an area of 108 square units. The lengths of its diagonals have a ratio of 3 to 2. What is the length of the longest diagonal, in units? | [
"18"
] | [
"A rhombus has an area of 108 square units. The lengths of its diagonals have a ratio of 3 to 2. What is the length of the longest diagonal, in units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-011edbf5833156d0e2126a468b109ee1975a9012673c528424758e5bf336d5fb | A rhombus has an area of 540 square cm. One of its diagonals is 45 cm. What is the distance from the point of intersection of the diagonals to a side of the rhombus? | [
"\\(\\frac{180}{17}\\)"
] | [
"A rhombus has an area of 540 square cm. One of its diagonals is 45 cm. What is the distance from the point of intersection of the diagonals to a side of the rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4384379029e8da5af958d323b8355dd075808f2310364d4b3dc6deac6f2e56e2 | A rhombus has half the area of a square with the same side-length. What is the ratio of the long diagonal to the short diagonal? | [
"\\( 2 + \\sqrt{3} \\)"
] | [
"A rhombus has half the area of a square with the same side-length. What is the ratio of the long diagonal to the short diagonal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bd9450b6c82452bd68e360c62e9b74199e5983f6c0032eae971c079d7f3519e0 | A rhombus has interior angles of 30, 150, 30, and 150 degrees, and an area of 3. Find its perimeter. | [
"\\(4\\sqrt{6}\\)"
] | [
"A rhombus has interior angles of 30, 150, 30, and 150 degrees, and an area of 3. Find its perimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9758b5a3d12afae166d3918053fd0a9325f2c9239bc6c1f3d6fb62f75da51613 | A rhombus has one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches. | [
"\\(\\frac{\\sqrt{5K}}{2}\\)"
] | [
"A rhombus has one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e78273f07f3237147570aa98c3135d6ac2d5879442cbff3dc93369c2bd77eec1 | A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal? | [
"90"
] | [
"A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91924624b8143b322f6076c9b6bc6077be998a8ca920e31adede782fec0fb231 | A rhombus is circumscribed around a square with side length 1997. Find the lengths of its diagonals if they are known to be different integers. | [
"3990006"
] | [
"A rhombus is circumscribed around a square with side length 1997. Find the lengths of its diagonals if they are known to be different integers."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c498542cd586484e8ca650dae073bf0449485ebb43b50cf4725276dcae187268 | A rhombus is circumscribed around a square with side length 7, and the diagonals of the rhombus are parallel to the sides of the square. Find the lengths of the diagonals of the rhombus, given that they are integers. | [
"14 and 14"
] | [
"A rhombus is circumscribed around a square with side length 7, and the diagonals of the rhombus are parallel to the sides of the square. Find the lengths of the diagonals of the rhombus, given that they are integers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-2a0475e5d35e0f53bff52cae04ace72980a4889a495ac1ce90960fe545c845e5 | A rhombus is formed by two chords and two radii of a circle with radius \( R \) meters. What is the area of the rhombus in square meters, expressed as a common fraction in terms of \( R \)? | [
"\\(\\frac{R^{2}\\sqrt{3}}{2}\\)"
] | [
"A rhombus is formed by two chords and two radii of a circle with radius \\( R \\) meters. What is the area of the rhombus in square meters, expressed as a common fraction in terms of \\( R \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-02d79b869d5b7aee81e49919f41908e660865b3a1fefb7af5a9e6d6bcecad918 | A rhombus is formed by two radii and two chords of a circle whose radius is 16 feet. What is the area of the rhombus in square feet? | [
"\\( 128\\sqrt{3} \\)"
] | [
"A rhombus is formed by two radii and two chords of a circle whose radius is 16 feet. What is the area of the rhombus in square feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6b87f0dfec63e528908f40d1961f5e47944381bd4be6da749253e52f3ae4bad5 | A rhombus is inscribed in a circle, and the length of one diagonal is 8x. What is the length of the other diagonal? | [
"\\(8x\\)"
] | [
"A rhombus is inscribed in a circle, and the length of one diagonal is 8x. What is the length of the other diagonal?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5704e2011ad776af70758681b04ffb14622e5f89a11a6a0d25568746fabfed1d | A rhombus is inscribed in a convex quadrilateral such that its sides are parallel to the diagonals of the quadrilateral. If the lengths of the diagonals are $d_1$ and $d_2$, calculate the length of the side of the rhombus in terms of $d_1$ and $d_2$. | [
"\\(\\frac{\\sqrt{d_1^2 + d_2^2}}{2}\\)"
] | [
"A rhombus is inscribed in a convex quadrilateral such that its sides are parallel to the diagonals of the quadrilateral. If the lengths of the diagonals are $d_1$ and $d_2$, calculate the length of the side of the rhombus in terms of $d_1$ and $d_2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33ffecc51abbf960502046f3537786b2e05bd9487d4f30fdae7a25d3e0247797 | A rhombus is inscribed in triangle \( \triangle ABC \) with vertex \( A \) as one of its vertices. Two sides of the rhombus lie along \( AB \) and \( AC \), where \( AC = 6 \), \( AB = 12 \), and \( BC = 8 \). Find the length of a side of the rhombus. | [
"4"
] | [
"A rhombus is inscribed in triangle \\( \\triangle ABC \\) with vertex \\( A \\) as one of its vertices. Two sides of the rhombus lie along \\( AB \\) and \\( AC \\), where \\( AC = 6 \\), \\( AB = 12 \\), and \\( BC = 8 \\). Find the length of a side of the rhombus."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-68f744bcb83a8273aaf552b77a4febd8a1ed8ebb880420c12587958a682cc5ff | A rhombus is inscribed in triangle \(ABC\) such that one of its vertices is \(A\) and two of its sides lie along \(AB\) and \(AC\). Given that \(\overline{AC} = 6\) inches, \(\overline{AB} = 12\) inches, and \(\overline{BC} = 8\) inches, find the side length of the rhombus in inches. | [
"4"
] | [
"A rhombus is inscribed in triangle \\(ABC\\) such that one of its vertices is \\(A\\) and two of its sides lie along \\(AB\\) and \\(AC\\). Given that \\(\\overline{AC} = 6\\) inches, \\(\\overline{AB} = 12\\) inches, and \\(\\overline{BC} = 8\\) inches, find the side length of the rhombus in inches."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-30edfe2162ff0560d0be95c31191d153fe1093a3526a39cd9eb28407278ab99d | A rhombus of side length \( s \) has the property that there is a point on its longest diagonal such that the distances from that point to the vertices are 1, 1, 1, and \( s \). What is the value of \( s \)? | [
"\\(\\frac{\\sqrt{5} + 1}{2}\\)"
] | [
"A rhombus of side length \\( s \\) has the property that there is a point on its longest diagonal such that the distances from that point to the vertices are 1, 1, 1, and \\( s \\). What is the value of \\( s \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d347ff5894419b2f7454d5da7bdb1cd3e0ca4313d5d1177c2b86fd390c68f9e6 | A rhombus with diagonals $AC = a$ and $BD = b$ is the base of a pyramid $SABCD$. The lateral edge $SA$ of length $q$ is perpendicular to the base. Through the point $A$ and the midpoint $K$ of the edge $SC$ a plane parallel to the diagonal $BD$ of the base is drawn. Determine the area of the plane section thus obtained... | [
"\\(\\frac{b}{6}\\sqrt{q^2 + a^2}\\)"
] | [
"A rhombus with diagonals $AC = a$ and $BD = b$ is the base of a pyramid $SABCD$. The lateral edge $SA$ of length $q$ is perpendicular to the base. Through the point $A$ and the midpoint $K$ of the edge $SC$ a plane parallel to the diagonal $BD$ of the base is drawn. Determine the area of the plane section thus obt... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32a51e27c81c388dbaf8ee4df6dae0bf51ce6f4b73435342bc86060bd0d63870 | A rhombus with its incircle is given. At each vertex of the rhombus, a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii \( r_1 \) and \( r_2 \), while the incircle has radius \( r \). Given that \( r_1 \) and \( r_2 \) are natural numbers and that \( r_1 r_2 = r \),... | [
"\\( r_1 = 1 \\), \\( r_2 = 1 \\), \\( r = 1 \\)"
] | [
"A rhombus with its incircle is given. At each vertex of the rhombus, a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii \\( r_1 \\) and \\( r_2 \\), while the incircle has radius \\( r \\). Given that \\( r_1 \\) and \\( r_2 \\) are natural numbers and that \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3138a23562dbc8a395431a1e01df2249147ea22526c9952abb0913f22b7d1bf5 | A rhombus with the acute angle $\alpha $ and side $a$ is divided into three equal parts by straight lines emanating from the vertex of this angle. Determine the lengths of the line-segments. | [
"$AF=AE=\\frac{a}{3}\\sqrt {13+12\\cos \\alpha }$"
] | [
"A rhombus with the acute angle $\\alpha $ and side $a$ is divided into three equal parts by straight lines emanating from the vertex of this angle. Determine the lengths of the line-segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34dc3854c1b7ddca91f5f38d9dcd262028ed70cf4904491e49285b2e5662fdd4 | A rhombus with the larger diagonal$ $d and acute angle $\gamma $ rotates about an axis passing outside it through its vertex and perpendicular to its larger diagonal. Determine the volume of the solid thus obtained. | [
"$V=\\frac{\\pi d^3\\tan\\frac{\\gamma }{2}}{2}$"
] | [
"A rhombus with the larger diagonal$ $d and acute angle $\\gamma $ rotates about an axis passing outside it through its vertex and perpendicular to its larger diagonal. Determine the volume of the solid thus obtained."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-202ec35b7c03f9dfaa24d3ed39598f2fb92550156aa78c80779f21d8118c16d3 | A rhombus-shaped shed with diagonals of length 8 and 10 feet is surrounded by grass. A goat is tied to a vertex of the shed with a leash of length 8 feet. The goat eats 5 square feet of grass a day and tramples 1 square foot of grass a day. If the goat is tied to the shed for 12 days and eats all the grass within its r... | [
"1"
] | [
"A rhombus-shaped shed with diagonals of length 8 and 10 feet is surrounded by grass. A goat is tied to a vertex of the shed with a leash of length 8 feet. The goat eats 5 square feet of grass a day and tramples 1 square foot of grass a day. If the goat is tied to the shed for 12 days and eats all the grass within ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-a340f76ef548fe3dbadb2857123b8012408b82dc4437fd7798f5b06dcf7a9037 | A ribbon 56 centimeters long is cut into two pieces. One piece is three times longer than the other. Find the lengths of both pieces of ribbon in centimeters. | [
"14 cm and 42 cm"
] | [
"A ribbon 56 centimeters long is cut into two pieces. One piece is three times longer than the other. Find the lengths of both pieces of ribbon in centimeters."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-1372a6cd85205b841249ec4ccbc8d218072013a98ea56a1667bed6feda239bda | A ribbon is tied around a box with volume 6 such that it goes along an edge, then along the diagonal of the top side, down the opposite edge, and along the diagonal of the bottom side. Minimize the length of the ribbon. | [
"\\(6 \\cdot 6^{1/3}\\)"
] | [
"A ribbon is tied around a box with volume 6 such that it goes along an edge, then along the diagonal of the top side, down the opposite edge, and along the diagonal of the bottom side. Minimize the length of the ribbon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-344b73790d2dfb19ee042cdb8824806fcb14062346af7e552d02244a339e7bd8 | A ribbon is wrapped around a cube of side length $1$ such that it cuts eight of the cube’s edges in the same $x: 1-x$ ratio. Given that the length of the ribbon on each of the six faces of the cube is the same, compute the total length of the ribbon. | [
"\\( 6\\sqrt{2}(\\sqrt{3} - 1) \\)"
] | [
"A ribbon is wrapped around a cube of side length $1$ such that it cuts eight of the cube’s edges in the same $x: 1-x$ ratio. Given that the length of the ribbon on each of the six faces of the cube is the same, compute the total length of the ribbon."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa906fe2881d6e167352b03fdb354691b58d2b29d6a95d291af23170dc93d829 | A rich knight has a chest and every day he puts into the chest some quantity of coins from the numbers $1, 2, \ldots, 100$. Let $N$ be the maximum number of days the knight can continue putting coins into the chest without stopping. Calculate the value of $N$. | [
"7"
] | [
"A rich knight has a chest and every day he puts into the chest some quantity of coins from the numbers $1, 2, \\ldots, 100$. Let $N$ be the maximum number of days the knight can continue putting coins into the chest without stopping. Calculate the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-b461bc37ef14bbd294940d579b10f0d514ffa8cbeaf2431e6215038d8a92450a | A rich man deposits $500,000 into a bank that pays an annual interest of 9.5%, compounded daily. Each day, he collects the day's interest and then deposits an additional $40,000 into the bank. By the end of the year, how much profit would he make from the interest? | [
"739100"
] | [
"A rich man deposits $500,000 into a bank that pays an annual interest of 9.5%, compounded daily. Each day, he collects the day's interest and then deposits an additional $40,000 into the bank. By the end of the year, how much profit would he make from the interest?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-c57026575a0ad6915f35c667386efefba74b86e4c909830b8432f2d173a1be92 | A ride at an amusement park consists of a small circle of radius 20 metres rolling inside a large circle of radius 100 metres. The large circle is tangent to the ground at its lowest point and remains fixed. When the center of the smaller circle has rotated $120^\circ$ around the center of the larger circle, how far of... | [
"\\(80 + 40\\sqrt{3}\\)"
] | [
"A ride at an amusement park consists of a small circle of radius 20 metres rolling inside a large circle of radius 100 metres. The large circle is tangent to the ground at its lowest point and remains fixed. When the center of the smaller circle has rotated $120^\\circ$ around the center of the larger circle, how ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95cd52649671455f53d40c2a5a9402c8fee8c14c2850dafb02d37fc1295a18b9 | A rider is pedaling a bicycle at an angular velocity of $\omega_1$ rpm, and the pedals are attached to a chain ring with radius $r_1$ feet. The chain ring is connected by a bicycle chain to a back sprocket with radius $r_2$ feet, which is turning at $\omega_2$ rpm. The back sprocket is attached to the back wheel of the... | [
"\\dfrac{\\pi \\omega_1 r_1 R}{30 r_2}"
] | [
"A rider is pedaling a bicycle at an angular velocity of $\\omega_1$ rpm, and the pedals are attached to a chain ring with radius $r_1$ feet. The chain ring is connected by a bicycle chain to a back sprocket with radius $r_2$ feet, which is turning at $\\omega_2$ rpm. The back sprocket is attached to the back wheel... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5700a897dff7a25e5ba9b60a4d320dab1fcefb75a19ae103c0c1a9026a455af7 | A rigged coin has the property that when it is flipped 2005 times, the probability of getting heads 589 times is equal to the probability of getting heads 590 times. If \(\frac{m}{n}\) is the probability of getting two heads in a row, where \(m\) and \(n\) are relatively prime, find \(m+n\). | [
"314"
] | [
"A rigged coin has the property that when it is flipped 2005 times, the probability of getting heads 589 times is equal to the probability of getting heads 590 times. If \\(\\frac{m}{n}\\) is the probability of getting two heads in a row, where \\(m\\) and \\(n\\) are relatively prime, find \\(m+n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2dac5ab97577467568176b9143d5edad4cebd1455c32d09b15f8a462309c5399 | A right angle has its vertex at (3,4) and the equation of the angle bisector is \(2x - y - 2 = 0\). Find the equations of the sides of the angle. | [
"x - 3y + 9 = 0"
] | [
"A right angle has its vertex at (3,4) and the equation of the angle bisector is \\(2x - y - 2 = 0\\). Find the equations of the sides of the angle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-76d74f941464c365a56f5db84251210b2f42c8833fc137361e85519f6dbb796f | A right angle has its vertex at the centroid of an equilateral triangle with a side length of 1 unit. Find the maximum possible area that is interior to both the angle and the triangle. | [
"\\(\\frac{\\sqrt{3} - 1}{6}\\)"
] | [
"A right angle has its vertex at the centroid of an equilateral triangle with a side length of 1 unit. Find the maximum possible area that is interior to both the angle and the triangle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6fd2c23c556fac85e3d0a310b9fc3f0de0d6c02e6d0a860ad2dfe64abbe739c | A right angle has its vertex moving on a fixed circle, and one of its sides passes through a fixed point. Determine the curve to which the other side of the angle is always tangent. | [
"hyperbola"
] | [
"A right angle has its vertex moving on a fixed circle, and one of its sides passes through a fixed point. Determine the curve to which the other side of the angle is always tangent."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-1ad09fd0e100138a72dbca68c61a6a0a7b3e91e6de96e69ee7f93bb24177dd04 | A right angle is given on a plane. A circle with center lying in the interior of the angle touches one of its sides, cuts the other side at points $A$ and $B$ and cuts the bisector of the angle at points $C$ and $D$. The length of the chord $CD$ is $\sqrt 7$ and that of the chord $A B$ is $\sqrt 6$. Find the radius of ... | [
"$R=\\sqrt 2$ cm. %"
] | [
"A right angle is given on a plane. A circle with center lying in the interior of the angle touches one of its sides, cuts the other side at points $A$ and $B$ and cuts the bisector of the angle at points $C$ and $D$. The length of the chord $CD$ is $\\sqrt 7$ and that of the chord $A B$ is $\\sqrt 6$. Find the rad... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1410f77636b77e4e73864491cbbe756e783b6eaa5ba160a398a36989381c09f | A right angle is rotated about a fixed vertex inside a circle. The legs of the right angle meet the circle at two points. Draw two tangent lines to the circle at these two points. Find the locus of the intersection of the two tangent lines. | [
"a circle"
] | [
"A right angle is rotated about a fixed vertex inside a circle. The legs of the right angle meet the circle at two points. Draw two tangent lines to the circle at these two points. Find the locus of the intersection of the two tangent lines."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-365797051f00cb4e7f0c80f9f1a4e3f325d1f376499085d74baf67330684415f | A right angle moves so that its sides touch the curve $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ all the time. Find the locus of the vertices of the angle. | [
"The circle $x^2+y^2=a^2+b^2$"
] | [
"A right angle moves so that its sides touch the curve $$\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$$ all the time. Find the locus of the vertices of the angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dd2bc43905f08299c3751f3979aa90ce2f58c7990418e55bc2e383498091d7a | A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points of the sides of the angle and a given circle. | [
"\\left(x - \\frac{h}{2}\\right)^2 + \\left(y - \\frac{k}{2}\\right)^2 = \\frac{h^2 + k^2}{4}"
] | [
"A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points of the sides of the angle and a given circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-d6e288ec315fcfb9c6a848d9b514f2fdbbd86874a69db0036658d34930e6c066 | A right circular cone and a right cylinder have the same height of 20 units and their bases are coplanar with the same center. The radius of the cone's base is 20 units, and the radius of the cylinder's base is 10 units. What is the ratio of the volume of the part of the cone inside the cylinder to the volume of the pa... | [
"1"
] | [
"A right circular cone and a right cylinder have the same height of 20 units and their bases are coplanar with the same center. The radius of the cone's base is 20 units, and the radius of the cylinder's base is 10 units. What is the ratio of the volume of the part of the cone inside the cylinder to the volume of t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-c2fb639e7572889a8b381cc41f4eb32b878bd11205b13b3cc58860b5671e007a | A right circular cone and a right cylinder with the same height \( 20 \) do not have the same circular base but the circles are coplanar and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are \( 20 \) and \( 10 \), respectively, what is the ratio of the v... | [
"1"
] | [
"A right circular cone and a right cylinder with the same height \\( 20 \\) do not have the same circular base but the circles are coplanar and their centers are the same. If the cone and the cylinder are on the same side of the plane and their base radii are \\( 20 \\) and \\( 10 \\), respectively, what is the rat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 8
} | 51 | 0 | 0 |
omr-ws-30c8840b379fa557858228b83ce46a04e485d04b8ecda8491d1739ae0bf16532 | A right circular cone has a base of radius $1$ and a height of $3$. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the length $s$ of an edge of the cube? | [
"\\dfrac{9\\sqrt{2} - 6}{7}"
] | [
"A right circular cone has a base of radius $1$ and a height of $3$. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the length $s$ of an edge of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-afd688191295733fa9a1ff972a6df2995c1e2ef43a47f9b5215f44e0fa8caed5 | A right circular cone has a base of radius $1$ and height $3$. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side length of the cube? | [
"\\dfrac{9\\sqrt{2} - 6}{7}"
] | [
"A right circular cone has a base of radius $1$ and height $3$. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side length of the cube?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d9d492d5889cc2dd98eac2f9d15d4c182ef26849a1ddfc7b8235b42b32f5f05b | A right circular cone has a base of radius 1 and height 3. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side-length of the cube? | [
"\\(\\frac{3}{4}\\)"
] | [
"A right circular cone has a base of radius 1 and height 3. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side-length of the cube?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7023d1fd82c2992683e10481af4acc203feb6768776512467fb8c1d9eb0406fd | A right circular cone has a base radius of $1$ and a slant height of $3$. For a point $P$ on the circumference of the base, the shortest path from $P$ around the cone and back to $P$ is drawn. Find the minimum distance from the vertex $V$ of the cone to this path. | [
"\\(\\frac{3}{2}\\)"
] | [
"A right circular cone has a base radius of $1$ and a slant height of $3$. For a point $P$ on the circumference of the base, the shortest path from $P$ around the cone and back to $P$ is drawn. Find the minimum distance from the vertex $V$ of the cone to this path."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-2ffcd92187891907cd4b7d2e1e0b55c20539ad41450cce7c98274f842d92c0e6 | A right circular cone has a base radius of 1 and a height of 3. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side-length of the cube? | [
"\\(\\frac{6}{3\\sqrt{2}+2}\\)"
] | [
"A right circular cone has a base radius of 1 and a height of 3. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is the side-length of the cube?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-90c81bd25b8c0a20bf5ce377eae568fef81f0f6f9c79bc76136aa56f57c5e175 | A right circular cone has a base radius of 1 cm and a slant height of 3 cm. Point \( P \) is on the circumference of the base. The shortest path from \( P \) around the cone and back to \( P \) is drawn. What is the minimum distance from the vertex \( V \) to this path? | [
"\\(\\frac{3}{2}\\)"
] | [
"A right circular cone has a base radius of 1 cm and a slant height of 3 cm. Point \\( P \\) is on the circumference of the base. The shortest path from \\( P \\) around the cone and back to \\( P \\) is drawn. What is the minimum distance from the vertex \\( V \\) to this path?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-f39d65771778c200e642dafc5dddc1b1849650bc21d35a5338c1b91414f32bda | A right circular cone has a base radius of 1 cm and a slant height of 3 cm. Point \( P \) is on the circumference of the base. What is the minimum distance from the vertex \( V \) to the shortest path around the cone starting from \( P \)? | [
"\\dfrac{3}{2}"
] | [
"A right circular cone has a base radius of 1 cm and a slant height of 3 cm. Point \\( P \\) is on the circumference of the base. What is the minimum distance from the vertex \\( V \\) to the shortest path around the cone starting from \\( P \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8991c2019805212bd70856ea838c39bbc8b9a7c686290cfea1611f2a1f327230 | A right circular cone has a base radius of 17 and a slant height of 34. A point \( P \) is on the circumference of the base. The shortest path from \( P \) around the cone and back to \( P \) is drawn. If the length of this path is \( m\sqrt{n} \) in simplest form, find \( m + n \). | [
"70"
] | [
"A right circular cone has a base radius of 17 and a slant height of 34. A point \\( P \\) is on the circumference of the base. The shortest path from \\( P \\) around the cone and back to \\( P \\) is drawn. If the length of this path is \\( m\\sqrt{n} \\) in simplest form, find \\( m + n \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f01979b3e5e06483c921aa43cd6f308f40a5a838409593222ca69f0449f76537 | A right circular cone has a base with radius $600$ and height $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}.$... | [
"625"
] | [
"A right circular cone has a base with radius $600$ and height $200\\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 25
} | 57 | 0 | 0 |
omr-ws-b66b671acdb7fa5fb08b5c9bdee713df78043e7ddeb1e462a846b610cc69ec51 | A right circular cone has a base with radius 600 and height $200\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}$. Fin... | [
"625"
] | [
"A right circular cone has a base with radius 600 and height $200\\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\\sqrt{2... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-9ebe5413f89fbb52e1689eda3ff8bb09e90a86972d2346645a3de6c73ddc6f6a | A right circular cone has a base with radius 600 and height \(200\sqrt{7}\). A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is \(375\sqrt{2}\).... | [
"625"
] | [
"A right circular cone has a base with radius 600 and height \\(200\\sqrt{7}\\). A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is \\(375\\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 20
} | 37 | 0 | 0 |
omr-ws-af7b1aa65e0eee70b3bcc51fd22ba5eb60b800e2456248acf712c30357e39a2c | A right circular cone has a height equal to its radius. What is the ratio of its volume to that of a cube inscribed inside it, with the base of the cube lying on the base of the cone? Simplify your answer, using the binomial theorem if relevant. | [
"\\(\\frac{8 \\pi}{3}\\)"
] | [
"A right circular cone has a height equal to its radius. What is the ratio of its volume to that of a cube inscribed inside it, with the base of the cube lying on the base of the cone? Simplify your answer, using the binomial theorem if relevant."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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