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-785d2748cf56039d050c940fcbdce4e6d049fb8573b17891556628901a24461d | A sphere is cut by 20 circumferences (with the same center and radius as the sphere) such that no three of them intersect at one point. How many regions do they form? | [
"382"
] | [
"A sphere is cut by 20 circumferences (with the same center and radius as the sphere) such that no three of them intersect at one point. How many regions do they form?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be61101071f3f3213ee02f9301b50697917a4cceac614be670202e1a2ab6d0de | A sphere is cut by a plane such that the surface area of the smaller part is half of the surface area of the rest of the sphere. Find the ratio of the volumes of the two parts. | [
"\\dfrac{7}{20}"
] | [
"A sphere is cut by a plane such that the surface area of the smaller part is half of the surface area of the rest of the sphere. Find the ratio of the volumes of the two parts."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74fdca4c65d7dbb786fa9043a2ca8feacce94a5cfb7144e1ca6e788ad440c374 | A sphere is cut by parallel planes into \( n \) pieces. The total surface area of these pieces is exactly 2004 times the area of the sphere. What is the smallest possible value of \( n \)? | [
"4008"
] | [
"A sphere is cut by parallel planes into \\( n \\) pieces. The total surface area of these pieces is exactly 2004 times the area of the sphere. What is the smallest possible value of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-65ef7809468038e86e8e5a7cac0e9c93f9c7aeedcd434267cbed0f8acdcb43a6 | A sphere is cut into four congruent wedges. The circumference of the sphere is $12\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\pi$. Note: To measure the circumference, take the largest circle on the surface of the sphere. | [
"72\\pi"
] | [
"A sphere is cut into four congruent wedges. The circumference of the sphere is $12\\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\\pi$.\n\nNote: To measure the circumference, take the largest circle on the surface of the sphere."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d124193f50b0a74fdcd5a5e377277d80bbd27b22c2ebbfe71a5b13a4d3f7064 | A sphere is divided into regions by 100 planes that are passing through its center. What is the maximum possible number of regions that are created on its surface? | [
"9902"
] | [
"A sphere is divided into regions by 100 planes that are passing through its center. What is the maximum possible number of regions that are created on its surface?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-1d179e73a38ffaf8f500c61903e9677a6ecf6a32da0b27e2a3d616c3b2962a6b | A sphere is inscribed in a 6x6x6 large cube made up of unit cubes. How many unit cubes have their interiors completely inside the sphere? | [
"32"
] | [
"A sphere is inscribed in a 6x6x6 large cube made up of unit cubes. How many unit cubes have their interiors completely inside the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-9d655046331867a9e3c14870658c1e5e1bcbafd35bb871179855946294f53292 | A sphere is inscribed in a cone whose generatrices make an angle $\alpha $ with the plane of the base. Find the ratio of the volume of the sphere to that of the cone. | [
"$4\\tan^3 \\frac{\\alpha }{2}\\cot \\alpha $"
] | [
"A sphere is inscribed in a cone whose generatrices make an angle $\\alpha $ with the plane of the base. Find the ratio of the volume of the sphere to that of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa19f32e8b6971f5fa845020c091a05da2a9b3717effdd00ad528f87d710074c | A sphere is inscribed in a cone, the ratio of their volumes being equal to $k$. Find the ratio of the volumes of the spherical segments cut off from the sphere by the plane passing through the line of tangency of the sphere and cone. | [
"\\[\n\\frac{V_1}{V_2}=\\frac{z_{1,2}^2(3-z_{1,2})}{4-z_{1,2}^2(3-z_{1,2})}\n\\]\n\nwhere\n\n\\[\nz_{1,2}=\\frac{2(k+1) \\pm 2\\sqrt{k(k-2)}}{4k+1}\n\\]"
] | [
"A sphere is inscribed in a cone, the ratio of their volumes being equal to $k$. Find the ratio of the volumes of the spherical segments cut off from the sphere by the plane passing through the line of tangency of the sphere and cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2476ac49e66c20cdcc9bc2d86ec7916e98799dfc7c06e53e48270899b20b5c38 | A sphere is inscribed in a cone. Find the volume of the sphere, if the generator of the cone is equal to $l$ and is inclined to the base at an angle $\alpha $. | [
"$V=\\frac{4\\pi l^3\\cos^2 \\alpha \\tan^3\\frac{\\alpha }{2}}{3}$"
] | [
"A sphere is inscribed in a cone. Find the volume of the sphere, if the generator of the cone is equal to $l$ and is inclined to the base at an angle $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd67bdd9ce4a88daf6db63fdd0d66370cd71002613dab5e3daee9bc00c8edc10 | A sphere is inscribed in a cone. The radius of the circle along which the cone and the ball are tangent is $r$. Find the volume of the cone if the angle between the altitude and the generatrix of the cone is $\alpha $. | [
"$\\frac{\\pi r^3\\cot^3 (\\pi /4-\\alpha /2)}{3\\cos ^2 \\alpha \\sin \\alpha }$"
] | [
"A sphere is inscribed in a cone. The radius of the circle along which the cone and the ball are tangent is $r$. Find the volume of the cone if the angle between the altitude and the generatrix of the cone is $\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa4f281f5bf4d3e9fb7b2d584fc787c714a7235c201d77ce468d817c4c36ee8f | A sphere is inscribed in a cone. The ratio of the surface area of the sphere to the area of the base of the cone is $4: 3$. Find the vertex angle of the axial section of the cone. | [
"\\(\\frac{\\pi}{3}\\)"
] | [
"A sphere is inscribed in a cone. The ratio of the surface area of the sphere to the area of the base of the cone is $4: 3$. Find the vertex angle of the axial section of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d238ecf37236f293261b1f2c0154d8fb17398854b258e29e2b4f1e097847886 | A sphere is inscribed in a cone. The ratio of their volumes is equal to $n$. Find the angle of inclination of the generator to the base (calculate for $n=4$). | [
"The same as in the preceding problem. At $n=4$ we have $\\alpha _1=2\\arctan(\\cos 22^\\circ 30')\\ (\\approx 85^\\circ 28')$; $\\alpha _2=2\\arctan(\\sin 22^\\circ 30')\\ (\\approx 41^\\circ 53')$"
] | [
"A sphere is inscribed in a cone. The ratio of their volumes is equal to $n$. Find the angle of inclination of the generator to the base (calculate for $n=4$)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0415f4acf88b2e2b965430407bb286356b95dd6329c68b2f616bdb1b71379f8 | A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? | [
"8"
] | [
"A sphere is inscribed in a cube that has a surface area of $24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-400d2bf04c247af8635d5974da98aa87f855f438aa8a104b29ed4fceea8b2f20 | A sphere is inscribed in a cube with edge $a$. Determine the radius of another sphere tangent to three faces of the cube and to the first sphere. | [
"$\\frac{1}{4}a\\left(\\sqrt 3-1\\right)^2$"
] | [
"A sphere is inscribed in a cube with edge $a$. Determine the radius of another sphere tangent to three faces of the cube and to the first sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-444a60c9d547efa12e5d6a6da6a04b8dfd94a1698fe14c73ff47efa5d34647f0 | A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. | [
"81\\sqrt{3}"
] | [
"A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ee3ffaeb460c2453d7d4074a56dd107fac7aaf79b922c74d78f89203875e2cd | A sphere is inscribed in a cube, and the cube has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube? | [
"8"
] | [
"A sphere is inscribed in a cube, and the cube has a surface area of 24 square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5487512b6414ef64c473596b08f11345f96f13ea81de4e1fca87dc099ed97fcc | A sphere is inscribed in a cube. A cube is inscribed in that sphere. This process is repeated until there are 1000 cubes and spheres combined. If the surface area of the original cube is 5400, find the volume of the smallest cube. | [
"\\(\\frac{1000 \\sqrt{3}}{3^{746}}\\)"
] | [
"A sphere is inscribed in a cube. A cube is inscribed in that sphere. This process is repeated until there are 1000 cubes and spheres combined. If the surface area of the original cube is 5400, find the volume of the smallest cube."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-ef13aa4a5022521a210ecb684e52515f7681b4373fff83c6267d63becf6221e7 | A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\pi$. | [
"36\\pi"
] | [
"A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-23c1012d342448d05ebd7d8d76f8c6a021b8120774f971ce0088263b89d0c21b | A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\pi$. | [
"\\frac{\\pi}{6}"
] | [
"A sphere is inscribed in a cube. What is the ratio of the volume of the inscribed sphere to the volume of the cube? Express your answer as a common fraction in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eff24ca800c336bc344d16de003b5ac3d45d14b6bb53c721951c1e4b8c264c92 | A sphere is inscribed in a regular hexahedron, and a regular octahedron is inscribed in the sphere. What is the ratio of the volume of the octahedron to the volume of the hexahedron? | [
"\\(\\frac{1}{6}\\)"
] | [
"A sphere is inscribed in a regular hexahedron, and a regular octahedron is inscribed in the sphere. What is the ratio of the volume of the octahedron to the volume of the hexahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8771289331f9592d48fd1cc9a30055ffc5556b9cea4f80ab555d5ba147eec988 | A sphere is inscribed in a regular octahedron of side length \( s \) such that it is tangent to each of its 8 faces. Find the radius of the sphere using the volume of the regular octahedron, \(\frac{s^3 \sqrt{2}}{3}\). You may not use the space diagonal of the inscribed cube to work out the radius. | [
"\\(\\frac{s\\sqrt{6}}{6}\\)"
] | [
"A sphere is inscribed in a regular octahedron of side length \\( s \\) such that it is tangent to each of its 8 faces. Find the radius of the sphere using the volume of the regular octahedron, \\(\\frac{s^3 \\sqrt{2}}{3}\\). You may not use the space diagonal of the inscribed cube to work out the radius."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-8ad0c4fbe269302fb3de340a19b732d8adc5620105015482dcc7c8ed5afa8164 | A sphere is inscribed in a regular octahedron with edge length \(a\). Determine the volume inside the octahedron but outside the sphere. | [
"\\( a^3 \\frac{\\sqrt{2}(9 - \\pi \\sqrt{3})}{27} \\)"
] | [
"A sphere is inscribed in a regular octahedron with edge length \\(a\\). Determine the volume inside the octahedron but outside the sphere."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-5a10acc980849e885afa9352485f5b713752078b4a6f9a3ec17ba644433caf8c | A sphere is inscribed in a regular tetrahedron \( SABC \) with edge length \( a = \sqrt{19} \). Point \( M \) is the midpoint of edge \( SA \), and point \( N \) divides edge \( BC \) in the ratio \( 2:1 \), counting from \( B \). The straight line \( MN \) intersects the sphere at two points \( P \) and \( R \). Find ... | [
"\\( MN = \\frac{5\\sqrt{19}}{6} \\), \\( PR = \\frac{\\sqrt{114}}{6} \\)"
] | [
"A sphere is inscribed in a regular tetrahedron \\( SABC \\) with edge length \\( a = \\sqrt{19} \\). Point \\( M \\) is the midpoint of edge \\( SA \\), and point \\( N \\) divides edge \\( BC \\) in the ratio \\( 2:1 \\), counting from \\( B \\). The straight line \\( MN \\) intersects the sphere at two points \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e966eee40c1d38e82102973f0bdc5b64d6e9940e8a213e67ea2ea081379b6f4 | A sphere is inscribed in a regular tetrahedron with edges measuring 12 cm. What is the radius of a smaller sphere that can be inscribed touching the original sphere and three faces of the tetrahedron? | [
"\\dfrac{\\sqrt{6}}{2}"
] | [
"A sphere is inscribed in a regular tetrahedron with edges measuring 12 cm. What is the radius of a smaller sphere that can be inscribed touching the original sphere and three faces of the tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-0bb83251c1162b33b193fbd9a23da410076c2576e70c8194fa9c014a9cb927aa | A sphere is inscribed in a regular tetrahedron, and another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of the tetrahedrons. | [
"27"
] | [
"A sphere is inscribed in a regular tetrahedron, and another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of the tetrahedrons."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e5cf5468fe290a2f8a03859e6dc24a754dc0a61d852dced71538fea6c16e716 | A sphere is inscribed in a regular triangular prism. It touches the three lateral faces and both bases of the prism. Find the ratio of the surface of the sphere to the total surface area of the prism. | [
"The required ratio is $\\frac{2\\pi }{9\\sqrt 3}$"
] | [
"A sphere is inscribed in a regular triangular prism. It touches the three lateral faces and both bases of the prism. Find the ratio of the surface of the sphere to the total surface area of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92a316ebe010da6ccc7579978a19feab3c5b18e9a2bff8db9226c002eb6d5079 | A sphere is inscribed in a regular triangular pyramid with the plane vertex angle $\beta $. Find the ratio of the volumes of the sphere and the pyramid. | [
"$\\frac{4}{9}\\pi \\frac{\\cot\\frac{\\beta }{2}-\\frac{1}{\\sqrt 3}} {\\left(\\cot\\frac{\\beta }{2}+\\frac{1}{\\sqrt 3}\\right)^2}$"
] | [
"A sphere is inscribed in a regular triangular pyramid with the plane vertex angle $\\beta $. Find the ratio of the volumes of the sphere and the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3d6f2c0a925763f67316578e6d342578e1adbc66ff558a4702f50e756102734 | A sphere is inscribed in a regular triangular pyramid. Determine the angle between its lateral edge and the base if the ratio of the volume of the pyramid to the volume of the sphere is equal to $\frac{27\sqrt 3}{4\pi }$. | [
"\\(\\alpha _1=\\frac{\\pi }{6} \\mbox{and} \\alpha _2=\\arctan\\sqrt {\\frac{2}{3}}\\)"
] | [
"A sphere is inscribed in a regular triangular pyramid. Determine the angle between its lateral edge and the base if the ratio of the volume of the pyramid to the volume of the sphere is equal to $\\frac{27\\sqrt 3}{4\\pi }$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-917df8f5516c7315daf362e569f2f63ce4770449aed3aece276b566185e93010 | A sphere is inscribed in a regular triangular pyramid. Find the angle of inclination of the lateral sides of the pyramid to the plane of the base, knowing that the ratio of the volume of the pyramid to that of the sphere is $27\sqrt 3/(4\pi )$. | [
"$\\varphi _1=\\frac{\\pi }{3},\\ \\varphi _2=2\\arctan \\frac{\\sqrt 6}{3}$"
] | [
"A sphere is inscribed in a regular triangular pyramid. Find the angle of inclination of the lateral sides of the pyramid to the plane of the base, knowing that the ratio of the volume of the pyramid to that of the sphere is $27\\sqrt 3/(4\\pi )$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f3392c998d383eaeed3ac00722612246acd3f7e588f1c4b87778feeda23efe8 | A sphere is inscribed in a right circular cone. The ratio of the volumes of the cone and sphere is equal to two. Find the ratio of the total surface area of the cone to the surface area of the sphere. | [
"2"
] | [
"A sphere is inscribed in a right circular cone. The ratio of the volumes of the cone and sphere is equal to two. Find the ratio of the total surface area of the cone to the surface area of the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e1ee2e7008b499471b61b7c8e51423fae8c35094cdd0602d58f54cfd8bc328b | A sphere is inscribed in a right circular cone. The ratio of the volumes of the cone and the sphere is equal to 2. Find the ratio of the total surface of the cone to the surface of the sphere. | [
"2"
] | [
"A sphere is inscribed in a right circular cone. The ratio of the volumes of the cone and the sphere is equal to 2. Find the ratio of the total surface of the cone to the surface of the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b6f564367c66906765a3b200f6ed63242e67e2e25ae5b8673ddd837125d1b58 | A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\sqrt{c} - a$ cm. What is the value of $a + c$? [asy] import three; size(120); defaultpen(linewidth(1)); pen dashes = linetype("2 2") + linewidth(1); currentprojection = orthograph... | [
"11"
] | [
"A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\\sqrt{c} - a$ cm. What is the value of $a + c$? [asy]\nimport three; size(120); defaultpen(linewidth(1)); pen dashes = linetype(\"2 2\") + linewidth(1);\ncurrentprojection = ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c3190d42ad4e3fb4f3f2319481a7b6b3f324443ce18590c72bae76a26d470b0 | A sphere is inscribed in a right prism whose base is a right triangle. In this triangle a perpendicular of length $h$ dropped from the vertex of the right angle on the hypotenuse forms an angle $\alpha $ with a leg of the triangle. Find the volume of the prism. | [
"\\(\\frac{2h^3}{\\sin\\alpha (1+\\sin\\alpha +\\cos\\alpha )}\\)"
] | [
"A sphere is inscribed in a right prism whose base is a right triangle. In this triangle a perpendicular of length $h$ dropped from the vertex of the right angle on the hypotenuse forms an angle $\\alpha $ with a leg of the triangle. Find the volume of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28707ea578dc8366cf25dc183c9f346660f7e31691dfc22d68dda48bc1bc8669 | A sphere is inscribed in a spherical sector. The radius of the circle along which the sphere is tangent to the sector is equal to $r$. A diametral plane of the spherical sector cuts out of it a circular sector with central angle $2\varphi $. Determine the radius of the spherical sector. | [
"$2r{\\rm cosec}\\, 2\\varphi (1+\\sin \\varphi )$"
] | [
"A sphere is inscribed in a spherical sector. The radius of the circle along which the sphere is tangent to the sector is equal to $r$. A diametral plane of the spherical sector cuts out of it a circular sector with central angle $2\\varphi $. Determine the radius of the spherical sector."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f68326bceaa8fb6075c3cfccb0a62a4a569e7a171a1f315a92c52c53d2d4808d | A sphere is inscribed in a tetrahedron \(ABCD\). Its four tangent planes, which are parallel to the faces of the tetrahedron, cut four smaller tetrahedra from the tetrahedron. Denote by \(R\) the radius of the inscribed sphere and by \(h_A, h_B, h_C, h_D\) the four heights of the given tetrahedron, labeled according to... | [
"16"
] | [
"A sphere is inscribed in a tetrahedron \\(ABCD\\). Its four tangent planes, which are parallel to the faces of the tetrahedron, cut four smaller tetrahedra from the tetrahedron. Denote by \\(R\\) the radius of the inscribed sphere and by \\(h_A, h_B, h_C, h_D\\) the four heights of the given tetrahedron, labeled a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b8ccd0c7c1c2553c2c7bed622a7ce221f7c03ee09c03b9222e266a213bac83a | A sphere is inscribed in a tetrahedron, and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Let $\theta$ be the angle formed between any two lines from a point of contact to the vertices of the corresponding face. Calculate the value of $\cos(\theta)$ ... | [
"-\\dfrac{1}{2}"
] | [
"A sphere is inscribed in a tetrahedron, and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Let $\\theta$ be the angle formed between any two lines from a point of contact to the vertices of the corresponding face. Calculate the value of $\\cos(\\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6498ded384581c261583468e5be11a519af9087f0fb706f2dee630ec0f88433d | A sphere is inscribed in a truncated cone (parallel bases) of volume $V$. The sphere (of radius $R$) is tangent both to the lateral surface and to both bases. What is the angle formed by the generatrix of the cone and the greater base? | [
"$\\arccos \\sqrt {\\frac{4V-2\\pi R^3}{3V+2\\pi R^3}}$"
] | [
"A sphere is inscribed in a truncated cone (parallel bases) of volume $V$. The sphere (of radius $R$) is tangent both to the lateral surface and to both bases. What is the angle formed by the generatrix of the cone and the greater base?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5eb0e0e13b87393bbcc66ac6c46df41550f096e5c504a20b5cae2831e85923c9 | A sphere is inscribed in a truncated cone (parallel bases) whose generatrix is of length $l$ and forms with the base an angle $\alpha $. Determine the radius of the circle along which the sphere is tangent to the truncated cone. | [
"$\\frac{1}{2}l\\sin ^2 \\alpha $"
] | [
"A sphere is inscribed in a truncated cone (parallel bases) whose generatrix is of length $l$ and forms with the base an angle $\\alpha $. Determine the radius of the circle along which the sphere is tangent to the truncated cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78e66744d0acd79fdcb9ad46ec66b61c86c19f4a677195b5b180e7eb7b9a6dab | A sphere is inscribed in a truncated cone in which the radii of the upper and lower bases are $R$ and $r$. Find the radius of the second ball which touches the first sphere, the lateral surface of the truncated cone and the upper base. | [
"$\\sqrt {\\frac{R}{r}}\\left(\\sqrt {R+r}-\\sqrt R\\right)^2$"
] | [
"A sphere is inscribed in a truncated cone in which the radii of the upper and lower bases are $R$ and $r$. Find the radius of the second ball which touches the first sphere, the lateral surface of the truncated cone and the upper base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f87417301ba1e8b901b24e8be28c3c165dc8a78500a31caeb4c3f9a746de3771 | A sphere is inscribed in a truncated right circular cone. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? | [
"\\(\\frac{3+\\sqrt{5}}{2}\\)",
"\\dfrac{3 + \\sqrt{5}}{2}"
] | [
"A sphere is inscribed in a truncated right circular cone. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 94,
"genselect": 8,
"tir": 53
} | 155 | 0 | 0 |
omr-ws-2a4d82abb4b348cb1944a2222577725a052319591ecb75e8b25857c83f0c17ef | A sphere is inscribed in polyhedral \( P \). The faces of \( P \) are colored with black and white such that no two black faces share an edge. Let \( B \) be the sum of the surface areas of the black faces and \( W \) be the sum of the surface areas of the white faces. If the total surface area of \( P \) is \( 1000 \)... | [
"500"
] | [
"A sphere is inscribed in polyhedral \\( P \\). The faces of \\( P \\) are colored with black and white such that no two black faces share an edge. Let \\( B \\) be the sum of the surface areas of the black faces and \\( W \\) be the sum of the surface areas of the white faces. If the total surface area of \\( P \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-5900eafc58ec03056b29a2f02bde37d8cd64af2cef73107dd27c3140c79fc2e5 | A sphere is inscribed in tetrahedron \(ABCD\) and is tangent to faces \(BCD\), \(CAD\), \(ABD\), and \(ABC\) at points \(P\), \(Q\), \(R\), and \(S\) respectively. Segment \(PT\) is the sphere's diameter, and lines \(TA\), \(TQ\), \(TR\), and \(TS\) meet the plane \(BCD\) at points \(A'\), \(Q'\), \(R'\), and \(S'\) re... | [
"\\dfrac{5}{2}"
] | [
"A sphere is inscribed in tetrahedron \\(ABCD\\) and is tangent to faces \\(BCD\\), \\(CAD\\), \\(ABD\\), and \\(ABC\\) at points \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) respectively. Segment \\(PT\\) is the sphere's diameter, and lines \\(TA\\), \\(TQ\\), \\(TR\\), and \\(TS\\) meet the plane \\(BCD\\) at points \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-25037886e51213b7199f2cddab2a0d5f7cf090650c19db9edd09bca53ac33388 | A sphere is inscribed in tetrahedron \(ABCD\) and tangent to faces \(BCD\), \(ACD\), \(ABD\), \(ABC\) at points \(P\), \(Q\), \(R\), \(S\) respectively. Segment \(PT\) is the sphere's diameter, and lines \(TA\), \(TQ\), \(TR\), \(TS\) meet the plane \(BCD\) at points \(A'\), \(Q'\), \(R'\), \(S'\) respectively. If the ... | [
"4"
] | [
"A sphere is inscribed in tetrahedron \\(ABCD\\) and tangent to faces \\(BCD\\), \\(ACD\\), \\(ABD\\), \\(ABC\\) at points \\(P\\), \\(Q\\), \\(R\\), \\(S\\) respectively. Segment \\(PT\\) is the sphere's diameter, and lines \\(TA\\), \\(TQ\\), \\(TR\\), \\(TS\\) meet the plane \\(BCD\\) at points \\(A'\\), \\(Q'\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8686d6092c5cc93e113af9837e7b4b2c56a56341d53ea9e6a131701df44871ef | A sphere is inscribed inside a cube that is inscribed inside another sphere. The larger sphere has a volume of \( 972\pi \). Calculate the volume of the space inside the larger sphere but outside the smaller sphere. Express your answer in simplest radical form and in terms of \( \pi \). | [
"108\\pi(9 - \\sqrt{3})"
] | [
"A sphere is inscribed inside a cube that is inscribed inside another sphere. The larger sphere has a volume of \\( 972\\pi \\). Calculate the volume of the space inside the larger sphere but outside the smaller sphere. Express your answer in simplest radical form and in terms of \\( \\pi \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b165dfe48250ec07ff50031f7cd82cf033bdc4df2ff3ec5e42b90883fa8731c0 | A sphere is inscribed inside a pyramid with a square base. The height of the pyramid is \(\frac{\sqrt{15}}{2}\) times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube? | [
"\\(\\frac{25\\sqrt{3}}{6}\\)"
] | [
"A sphere is inscribed inside a pyramid with a square base. The height of the pyramid is \\(\\frac{\\sqrt{15}}{2}\\) times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-876e8c754e21bf7fb0805b5d2de6e0e3278434fabeb2bd23dfcd3d183d48da93 | A sphere is inscribed into a cone, the surface of the ball being equal to that of the base of the cone. Find the vertex angle in the axial section of the cone. | [
"$\\pi -4\\arctan \\frac{1}{2}$"
] | [
"A sphere is inscribed into a cone, the surface of the ball being equal to that of the base of the cone. Find the vertex angle in the axial section of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fe049b69f5ac3d3309ea453ee03c06f4a647079ceef0b2df81a316424dcf8d4 | A sphere is inscribed into a cone. Find the volume of the ball if the generatrix of the cone is $l$ and makes an angle a with the plane of the base. | [
"$\\frac{4}{3}\\, \\pi l^3\\, \\frac{\\sin ^3 \\alpha \\cos ^3 \\alpha }{(1+\\cos \\alpha )^3}$"
] | [
"A sphere is inscribed into a cone. Find the volume of the ball if the generatrix of the cone is $l$ and makes an angle a with the plane of the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0162a8e3d25affa0f83b9b89f7d9f177019fdde11468ac7e916454ec4783cf2f | A sphere is inscribed into a cube with edge $a$. Find the radius of another sphere which touches three faces of the cube and the first sphere. | [
"$\\frac{1}{4}a\\left(\\sqrt 3-1\\right)^2$"
] | [
"A sphere is inscribed into a cube with edge $a$. Find the radius of another sphere which touches three faces of the cube and the first sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98806d613ff996e2fa3694503fa4e24d0ab8822bc9614cc597fc84b222f9c0f1 | A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. If the lengths of segments $AA_o, BB_o, CC_o$ are $a, b, c$ respectively, and the volume of the prism is $V$, find the value of $\frac{V}{abc}$ given that $a = 2, b = 3, c = 6$. | [
"2"
] | [
"A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. If the lengths of segments $AA_o, BB_o, CC_o$ are $a, b, c$ respectively, and the volume of the prism is $V$, find the value of $\\frac{V}{abc}$ given that $a = 2, b = 3, c ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d19c13413f135e3a203dad787bb62de44483ebd8c4261cce74ff6a2de4aafa5d | A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' = \angle C_oAA'$. Find all possible values of these angles. | [
"\\(\\frac{\\pi}{3}\\)"
] | [
"A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. It is known that $\\angle A_oBB' = \\angle B_oCC' = \\angle C_oAA'$. Find all possible values of these angles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 7,
"tir": 6
} | 29 | 0 | 0 |
omr-ws-dc8104e6e63d0223c11f245489ea4de80bee0e1a83937d9a265af54d741c9f1e | A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. Let $P, Q, R$ be the projections of the incenter of the prism onto the faces $A'B', B'C', C'A'$ respectively. If the side length of the regular triangle formed by $P, Q, R$ is $s... | [
"3r^2"
] | [
"A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. Let $P, Q, R$ be the projections of the incenter of the prism onto the faces $A'B', B'C', C'A'$ respectively. If the side length of the regular triangle formed by $P, Q, R$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f5212ba8050de310638c54fa9137174b8b0077b4ca1ea384145492c81f46cf8 | A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected onto the edges of the base. Let these projections be points $A$, $B$, $C$, and $D$. If the base of the pyramid is a square with side length $s$ and the radius of the sphere is $r$, find the ra... | [
"\\dfrac{s}{2}"
] | [
"A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected onto the edges of the base. Let these projections be points $A$, $B$, $C$, and $D$. If the base of the pyramid is a square with side length $s$ and the radius of the sphere is $r$, find t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d63b352dfd4e04239fee8e751ad4b11d925ab1821bb8780dbdc5bca2959fbfa | A sphere is inscribed within a cube which has a surface area of 24 square meters. A second cube is inscribed within the sphere. What is the surface area of the inner cube? | [
"8"
] | [
"A sphere is inscribed within a cube which has a surface area of 24 square meters. A second cube is inscribed within the sphere. What is the surface area of the inner cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-b1bd6ff60ebf588db783c8b2028f1a8bb11fc8895f2072fcee3e23547cebbde5 | A sphere is inside a circumscribing cube. What fraction of the volume of the cube is inside the sphere? | [
"\\dfrac{\\pi}{6}"
] | [
"A sphere is inside a circumscribing cube. What fraction of the volume of the cube is inside the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-506de04f67dfd7a46f566ca46cd0665f89eb3686dc4670afa9cd6f9a8a42037d | A sphere is placed inside a cube-shaped box such that the length of any side of the cube is equal to the diameter of the sphere. Four polyhedra, each consisting of four 1x1x1 cubes, are glued to the corners of the box. The sphere touches each of the three corners of the polyhedra that do not come in contact with the bo... | [
"\\( 36\\pi \\)"
] | [
"A sphere is placed inside a cube-shaped box such that the length of any side of the cube is equal to the diameter of the sphere. Four polyhedra, each consisting of four 1x1x1 cubes, are glued to the corners of the box. The sphere touches each of the three corners of the polyhedra that do not come in contact with t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5b7128d7724eb62973103ef1ecfaf2c6c3b14f612b59552341af529f899aa4f5 | A sphere is rotated such that a point \( P \) moves from \( P_0 \) to \( P_1 \) and then to \( P_2 \). The point \( P \) is then rotated back to \( P_0 \) within the plane containing \( P_0, P_2, \) and the center \( O \) of the sphere. If the angle of this final rotation is \( \theta \), find \( \theta \) given that \... | [
"\\frac{1}{\\sin \\theta} \\left( \\sin \\phi, \\cos \\phi \\sin \\lambda, 0 \\right)"
] | [
"A sphere is rotated such that a point \\( P \\) moves from \\( P_0 \\) to \\( P_1 \\) and then to \\( P_2 \\). The point \\( P \\) is then rotated back to \\( P_0 \\) within the plane containing \\( P_0, P_2, \\) and the center \\( O \\) of the sphere. If the angle of this final rotation is \\( \\theta \\), find \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b01175f87e872d4b0c3b1d8d6218454c7a92145d32d6ac01ed78c1b2303e1b11 | A sphere is tangent to all lateral edges of a regular right hexagonal prism whose base lies outside the sphere. Find the ratio of that portion of the area of the lateral surface of the prism which lies inside the sphere to the portion of the surface of the sphere that lies outside the prism. | [
"$\\left(2+\\sqrt 3\\right)/4$"
] | [
"A sphere is tangent to all lateral edges of a regular right hexagonal prism whose base lies outside the sphere. Find the ratio of that portion of the area of the lateral surface of the prism which lies inside the sphere to the portion of the surface of the sphere that lies outside the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06350601260123f021601c6b2007b17de1b91838ca3d156b139d53ef272b7d3f | A sphere lies on the base of a regular triangular pyramid with the altitude $H$ and the radius of the ball inscribed in the base equal to $r$. The ball touches the base at its center. Find the radius of the sphere if the plane drawn through the vertex of the pyramid and the midpoints of two sides of the base touches th... | [
"$\\frac{Hr}{r+\\sqrt {r^2+4H^2}}$"
] | [
"A sphere lies on the base of a regular triangular pyramid with the altitude $H$ and the radius of the ball inscribed in the base equal to $r$. The ball touches the base at its center. Find the radius of the sphere if the plane drawn through the vertex of the pyramid and the midpoints of two sides of the base touch... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12a1bfcdf25d132b849c51f078fb42d7eaec1286638668bd7c093f1099d48147 | A sphere of clay with radius 13 has a cylindrical hole with radius 5 cut through its center. The remaining "bead" is reshaped into a new sphere. What is the radius of this new sphere? | [
"12"
] | [
"A sphere of clay with radius 13 has a cylindrical hole with radius 5 cut through its center. The remaining \"bead\" is reshaped into a new sphere. What is the radius of this new sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-3ee2e4321398140a713ba2f62b08abbe8c8c1bb4e2998e9261ce5340319d3b09 | A sphere of radius $1$ contains equal circles $\gamma_0$, $\gamma_1$, ..., $\gamma_n$ of radius $r$ $(n \geq 3)$. The circle $\gamma_0$ touches all the circles $\gamma_1$, ..., $\gamma_n$, and the circles $\gamma_1$ and $\gamma_2$, $\gamma_2$ and $\gamma_3$, ..., $\gamma_n$ and $\gamma_1$ touch each other. Determine th... | [
"\\dfrac{\\pi}{4}"
] | [
"A sphere of radius $1$ contains equal circles $\\gamma_0$, $\\gamma_1$, ..., $\\gamma_n$ of radius $r$ $(n \\geq 3)$. The circle $\\gamma_0$ touches all the circles $\\gamma_1$, ..., $\\gamma_n$, and the circles $\\gamma_1$ and $\\gamma_2$, $\\gamma_2$ and $\\gamma_3$, ..., $\\gamma_n$ and $\\gamma_1$ touch each o... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bb83cc3426d8191b925454cefb82724551c2c3e8f8b2acf4ef8788664a450280 | A sphere of radius $1$ is inscribed in a right circular cone whose height is $6$. Find the radius of this cone. | [
"\\(\\frac{\\sqrt{6}}{2}\\)"
] | [
"A sphere of radius $1$ is inscribed in a right circular cone whose height is $6$. Find the radius of this cone."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8734f66e4d779118af5dbce2a71ed8cf78316c7a09804c88159625f4ab60ed7d | A sphere of radius $1$ is tangent to the plane $\Pi$ at point $A$. A line $\ell$, making an angle $\phi$ with $\Pi$, intersects $\Pi$ at point $C$ and is tangent to the sphere at point $B$. Given that $AC = 2$ and $\tan \phi = 512$, the square of the length of $AB$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ a... | [
"4456449"
] | [
"A sphere of radius $1$ is tangent to the plane $\\Pi$ at point $A$. A line $\\ell$, making an angle $\\phi$ with $\\Pi$, intersects $\\Pi$ at point $C$ and is tangent to the sphere at point $B$. Given that $AC = 2$ and $\\tan \\phi = 512$, the square of the length of $AB$ can be expressed as $\\frac{a}{b}$, where ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d370e9c728125e9da296da1d29344129c37c26e0f59b2c4f29942c3f989aab9c | A sphere of radius $R$ is inscribed in a cone with vertex angle of its axial section equal to $\alpha $. Find the volume of the part of the cone above the sphere. | [
"V=\\frac{1}{3}\\pi R^3\\left[\\frac{\\cos^4 \\frac{\\alpha }{2}}{\\sin\\frac{\\alpha }{2}} -\\left(1-\\sin\\frac{\\alpha }{2}\\right)^2 \\left(2+\\sin\\frac{\\alpha }{2}\\right)\\right]."
] | [
"A sphere of radius $R$ is inscribed in a cone with vertex angle of its axial section equal to $\\alpha $. Find the volume of the part of the cone above the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de9804f7dbbed25c77c1ac7eb0a3f8f7a1e0714a8e200cd9828e4f667b28fbe9 | A sphere of radius $R$ is inscribed in a dihedral angle of $60^\circ$. Find the radius of the sphere inscribed in that angle and touching the given sphere if it is known that the straight line connecting the centers of the two spheres makes an angle of $45^\circ$ with an edge of the dihedral angle. | [
"$\\frac{2\\sqrt 2-1}{2\\sqrt 2+1}R,\\ \\frac{2\\sqrt 2+1}{2\\sqrt 2-1}R$"
] | [
"A sphere of radius $R$ is inscribed in a dihedral angle of $60^\\circ$. Find the radius of the sphere inscribed in that angle and touching the given sphere if it is known that the straight line connecting the centers of the two spheres makes an angle of $45^\\circ$ with an edge of the dihedral angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdd2fe04cfbccfede30d77984334e3a35b3e5b4c69de42bd86809e66045e7ac0 | A sphere of radius $R$ is inscribed in a pyramid whose base is a rhombus with acute angle $\alpha $. The lateral faces of the pyramid are inclined to the plane of the base at an angle $\psi $. Find the volume of the pyramid. | [
"\\(\\frac{8}{3}R^3\\cdot \\frac{\\cos^4 \\frac{\\psi }{2}} {\\sin\\alpha \\cos\\psi \\sin^2\\frac{\\psi }{2}}\\)"
] | [
"A sphere of radius $R$ is inscribed in a pyramid whose base is a rhombus with acute angle $\\alpha $. The lateral faces of the pyramid are inclined to the plane of the base at an angle $\\psi $. Find the volume of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8660270f3e87143fe19085882a3a81d2acca2d0708a81bc21d7e2d07d7b1cca | A sphere of radius $R$ is inscribed in a pyramid, whose base is a rhombus with the acute angle $\alpha $. The lateral faces of the pyramid are inclined to the base at an angle $\varphi $. Find the volume of the pyramid. | [
"$V=\\frac{4R^3\\tan\\varphi \\cot^3 \\frac{\\varphi }{2}}{3\\sin \\alpha }$"
] | [
"A sphere of radius $R$ is inscribed in a pyramid, whose base is a rhombus with the acute angle $\\alpha $. The lateral faces of the pyramid are inclined to the base at an angle $\\varphi $. Find the volume of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a26a3cd835084154cec87106bdf02fdada98f9a5dc84748fda54b0aedeb0e354 | A sphere of radius $R$ is inscribed into a pyramid whose base is a rhombus with an acute angle $\alpha $. The lateral faces of the pyramid make an angle $\varphi $ with the plane of the base. Find the volume of the pyramid. | [
"$\\frac{4}{3}\\, R^3\\, \\frac{(1+\\cos \\varphi )^3}{\\cos \\varphi \\sin ^2 \\varphi \\sin \\alpha }$"
] | [
"A sphere of radius $R$ is inscribed into a pyramid whose base is a rhombus with an acute angle $\\alpha $. The lateral faces of the pyramid make an angle $\\varphi $ with the plane of the base. Find the volume of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7dc5efb8390c06ec89f281a878f3d35e4190979ab1392678bcf62440adb081e9 | A sphere of radius $\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron? | [
"252"
] | [
"A sphere of radius $\\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-ea02d5db78504b7c6563c836d166272345565b3f98645ecb3b45a50415b84fcd | A sphere of radius $r$ is inscribed in a cone. Find the volume of the cone knowing that the distance from the vertex of the cone to the tangent plane to the sphere which is perpendicular to an element of the cone is equal to $d$. | [
"\\[\n\\frac{\\pi r^2\\left(r+\\sqrt {r^2+(d+r)^2}\\right)^3}{3(d+r)^2}\n\\]\nand\n\\[\n\\frac{\\pi r^2\\left(r+\\sqrt {r^2+(d-r)^2}\\right)^3}{3(d-r)^2}\n\\]"
] | [
"A sphere of radius $r$ is inscribed in a cone. Find the volume of the cone knowing that the distance from the vertex of the cone to the tangent plane to the sphere which is perpendicular to an element of the cone is equal to $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c652a7da63e8f78be276f40ea19134c0740c39d9f624cd9ba40d7430c210014 | A sphere of radius $r$ is inscribed in a cone. Find the volume of the cone, if it is known that a plane tangent to the sphere and perpendicular to the generator of the cone is drawn at a distance $d$ from the vertex of the cone. | [
"$V=\\frac{\\pi r^2\\left[r+\\sqrt {(d+r)^2+r^2}\\right]^3}{3(d+r)^2}$ or $V=\\frac{\\pi r^2\\left[r+\\sqrt {(d-r)^2+r^2}\\right]^3}{3(d-r)^2}$"
] | [
"A sphere of radius $r$ is inscribed in a cone. Find the volume of the cone, if it is known that a plane tangent to the sphere and perpendicular to the generator of the cone is drawn at a distance $d$ from the vertex of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af3d445eea59411110869e15e6651b311977eed0177c670fbc586207c0d157bd | A sphere of radius $r$ is inscribed in a frustum of a cone, whose generator is inclined to the plane of the base at an angle $\alpha $. Find the volume of the frustum. | [
"$V=\\frac{2\\pi r^3}{3}\\left(\\frac{4}{\\sin^2 \\alpha }-1\\right)$"
] | [
"A sphere of radius $r$ is inscribed in a frustum of a cone, whose generator is inclined to the plane of the base at an angle $\\alpha $. Find the volume of the frustum."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 3
} | 40 | 0 | 0 |
omr-ws-520297cfb2ed20255175a780e8465a8b390d91522d06e3df879dd2f573414390 | A sphere of radius $r$ is inscribed in a frustum of a cone. The generator of the cone is inclined to the base at an angle $\alpha $. Find the curved surface of the frustum. | [
"$S_{curved}=\\frac{4\\pi r^2}{\\sin^2 \\alpha }$"
] | [
"A sphere of radius $r$ is inscribed in a frustum of a cone. The generator of the cone is inclined to the base at an angle $\\alpha $. Find the curved surface of the frustum."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb89b694aa7277378caa4669bac157a12735cdfc43aaea7eac1482b9d867ed51 | A sphere of radius $r$ is inscribed into a cone. Find the volume of the cone if it is known that the plane which touches the sphere and is perpendicular to one of the generatrices of the cone is at the distance $d$ from the vertex of the cone. | [
"$\\frac{\\pi r^2\\left(r+\\sqrt {r^2+(d-r)^2}\\right)^3}{3(d-r)^2}$"
] | [
"A sphere of radius $r$ is inscribed into a cone. Find the volume of the cone if it is known that the plane which touches the sphere and is perpendicular to one of the generatrices of the cone is at the distance $d$ from the vertex of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b0281ca859b5f269e16e51c4e0bdcda7b8fce1bc6cef733858a0588ef12279e | A sphere of radius $r$ is tangent to all edges of a regular quadrangular pyramid with base side $a$. Find the volume of the pyramid. | [
"$V=a^2h/2$, where for $a/2\\le r<a/\\sqrt 2$, $h=\\frac{r\\pm \\sqrt {r^2-a^2/4}}{1-2r^2/a^2}$; for $r=a/\\sqrt 2$, $h=a/4$; for $r>a/\\sqrt 2$, $h=\\frac{r-\\sqrt {r^2-a^2/4}}{1-2r^2/a^2}$"
] | [
"A sphere of radius $r$ is tangent to all edges of a regular quadrangular pyramid with base side $a$. Find the volume of the pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad61e342c83d0960be26097ce92aa4f30848a3965366caebd7060bad57b06949 | A sphere of radius 1 is concentric with a cube of side length 4. A point is randomly chosen on a face of the cube, and a second point is randomly chosen on the opposite face. What is the probability that the line connecting the two points does not intersect the sphere? | [
"\\(\\frac{16 - 3\\pi}{8}\\)"
] | [
"A sphere of radius 1 is concentric with a cube of side length 4. A point is randomly chosen on a face of the cube, and a second point is randomly chosen on the opposite face. What is the probability that the line connecting the two points does not intersect the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-71a20866c5589d8b5a26efcd1cef0d06a35f4a7f9410c9049cc498080030fb04 | A sphere of radius 1 is in the interior of a regular triangular pyramid. At the point which bisects the altitude of the pyramid it is externally tangent to a hemisphere. The hemisphere rests on a circle inscribed in the base of the pyramid; the sphere touches the lateral faces of the pyramid. Find the lateral area of t... | [
"$S=3\\sqrt {15}\\left(\\sqrt 5+1\\right)^2,\\ \\alpha =2\\arcsin \\sqrt {\\frac{2}{5}}$"
] | [
"A sphere of radius 1 is in the interior of a regular triangular pyramid. At the point which bisects the altitude of the pyramid it is externally tangent to a hemisphere. The hemisphere rests on a circle inscribed in the base of the pyramid; the sphere touches the lateral faces of the pyramid. Find the lateral area... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15fa7a8b8df3181bb72bcb8dd10f9876abdc6ce9677a9b90ea6b0127bbd1ccd1 | A sphere of radius 4 inches is inscribed in a cone with a base of radius 6 inches. What is the height of the cone? Express your answer as a decimal to the nearest tenth. | [
"14.4"
] | [
"A sphere of radius 4 inches is inscribed in a cone with a base of radius 6 inches. What is the height of the cone? Express your answer as a decimal to the nearest tenth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-9fefe3b8b5a0762f4d2a1a0a9281141b8c1d0679a939d1ef4745013cb4e1f011 | A sphere of radius 900 is centered at the origin. Let \( A \) be a point on the sphere and \( B \) be the endpoint of the diagonal from \( A \). A cube is inscribed in the sphere such that one of its diagonals \( AC \) is \( \frac{7}{9} \) of the length of \( AB \), where \( C \) is on \( AB \). Inside this cube, there... | [
"\\( x = 0 \\)"
] | [
"A sphere of radius 900 is centered at the origin. Let \\( A \\) be a point on the sphere and \\( B \\) be the endpoint of the diagonal from \\( A \\). A cube is inscribed in the sphere such that one of its diagonals \\( AC \\) is \\( \\frac{7}{9} \\) of the length of \\( AB \\), where \\( C \\) is on \\( AB \\). I... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4dfccb87cab7dac7ecf8c017312a274f98b6f82806a5e26db83608d0f9f66e4 | A sphere of radius \( r \) is inscribed in a tetrahedron. Planes tangent to this sphere and parallel to the faces of the tetrahedron cut off four smaller tetrahedra from the original tetrahedron. These smaller tetrahedra have inscribed spheres with radii \( a, b, c, \) and \( d \). Calculate the value of \( a + b + c +... | [
"2r"
] | [
"A sphere of radius \\( r \\) is inscribed in a tetrahedron. Planes tangent to this sphere and parallel to the faces of the tetrahedron cut off four smaller tetrahedra from the original tetrahedron. These smaller tetrahedra have inscribed spheres with radii \\( a, b, c, \\) and \\( d \\). Calculate the value of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b95b5bc397b4e77844c800a7ab81248f19f9122605553206d60665aa7cb15fe9 | A sphere of radius \( r \) passes through the origin and intersects the coordinate axes at points \( A, B, \) and \( C \). Find the distance of the centroid of the triangle \( ABC \) from the origin. | [
"\\(\\frac{2r}{3}\\)"
] | [
"A sphere of radius \\( r \\) passes through the origin and intersects the coordinate axes at points \\( A, B, \\) and \\( C \\). Find the distance of the centroid of the triangle \\( ABC \\) from the origin."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-be881371020c1ad1d2b50bb4d7b844098a50eb26c4c0289a56a8e3f40dc9b622 | A sphere passes through all the vertices of one face of a cube and is tangent to all the edges of the opposite face of the cube. Find the ratio of the volume of the sphere to the volume of the cube. | [
"\\(\\frac{41\\pi\\sqrt{41}}{384}\\)"
] | [
"A sphere passes through all the vertices of one face of a cube and is tangent to all the edges of the opposite face of the cube. Find the ratio of the volume of the sphere to the volume of the cube."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-e480908d43457912147a9d44085a6255035d6532d77a4a9d44f4e3bf86e31810 | A sphere shrinks at a rate of 1 cm^3 per second. At what rate does the diameter decrease when the diameter is 10 cm? | [
"\\(\\frac{-1}{50 \\pi}\\)"
] | [
"A sphere shrinks at a rate of 1 cm^3 per second. At what rate does the diameter decrease when the diameter is 10 cm?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9ec69bb2d419ec93fb4d1aad7779735f5222abcc948cfa6a0c2dac034cace68e | A sphere touches all the edges of a tetrahedron, where two opposite edges are equal to $a$ and $b$, and all other edges are equal to each other. Find the radius of this sphere. | [
"\\(\\frac{\\sqrt{4a^2b^2 - (a^2 + b^2 - c^2)^2}}{4(a + b + 4c)}\\)"
] | [
"A sphere touches all the edges of a tetrahedron, where two opposite edges are equal to $a$ and $b$, and all other edges are equal to each other. Find the radius of this sphere."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-863139255a1f442c8c9a1a3cbd3ff4d647ffb5bec56532e4b312dc242df31358 | A sphere touches the base of a cone at its center. The surface of the sphere cuts the lateral surface of the cone along two circles one of which has a radius equal to that of the sphere and lies in the plane which is parallel to the base of the cone. The radius of the base is known to be 4/3 times as large as the radiu... | [
"$\\frac{9}{16}$"
] | [
"A sphere touches the base of a cone at its center. The surface of the sphere cuts the lateral surface of the cone along two circles one of which has a radius equal to that of the sphere and lies in the plane which is parallel to the base of the cone. The radius of the base is known to be 4/3 times as large as the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a81a3a1c4b3e392565832ac3fac68afa082ab326f1ca0b906e2a71e8fa7c1120 | A sphere touches the lateral edges of a regular right hexagonal prism whose base lies outside of the sphere. Find the ratio of the lateral area of the prism included in the sphere to the surface area of the sphere which lies outside of the prism. | [
"$\\frac{2+\\sqrt 3}{4}$"
] | [
"A sphere touches the lateral edges of a regular right hexagonal prism whose base lies outside of the sphere. Find the ratio of the lateral area of the prism included in the sphere to the surface area of the sphere which lies outside of the prism."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2631acebeab81fe5934f21e98761db4b3a4646daa12258d46c9e5d16d1571788 | A sphere whose radius is $a/\left(2\sqrt 3\right)$ is inscribed in a regular quadrangular pyramid $SABCD$ with vertex $S$ and the side of the base $a$. A plane $P$ which makes an angle of $30^\circ$ with the plane of the base touches the sphere and meets the plane of the base, without meeting the base itself, along a l... | [
"$\\frac{\\sqrt 3\\left(2-\\sqrt 3\\right)}{4}a^2$"
] | [
"A sphere whose radius is $a/\\left(2\\sqrt 3\\right)$ is inscribed in a regular quadrangular pyramid $SABCD$ with vertex $S$ and the side of the base $a$. A plane $P$ which makes an angle of $30^\\circ$ with the plane of the base touches the sphere and meets the plane of the base, without meeting the base itself, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f73a3dc550db3719c1950053c617f25a62fdc12db568a653499f2413d626f7f | A sphere with a diameter of 10 is cut with a plane. The area of intersection between the plane and the sphere is a circle with area $25\pi$. What is the absolute difference in volume between the two pieces created by the plane? | [
"0"
] | [
"A sphere with a diameter of 10 is cut with a plane. The area of intersection between the plane and the sphere is a circle with area $25\\pi$. What is the absolute difference in volume between the two pieces created by the plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fe069b70502eee5a5d6a1643ef0c5fe4f9b08fa970d065457d36559d59723e00 | A sphere with a radius of 5 is cut by a plane that is 4 units away from the center of the sphere. What is the volume of the larger of the two resulting solids? | [
"\\(162\\pi\\)"
] | [
"A sphere with a radius of 5 is cut by a plane that is 4 units away from the center of the sphere. What is the volume of the larger of the two resulting solids?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-c3a5a66d4584872da9e83156e3b41e760c36cd61dffd0ad7ef4367641d6e9daa | A sphere with a radius of 6 meters is filled with water. A hole is drilled in the bottom of the sphere causing the volume of the water to decrease at a rate of 3 cubic meters per minute. Find the rate of change of the height of the water in the sphere when the height is 4 meters. | [
"\\(-\\frac{3}{32\\pi}\\) m/min\n\nNote: The unit should be m/min, not m^3/min, as we are looking for the rate of change of the height of the water, which is a linear measurement."
] | [
"A sphere with a radius of 6 meters is filled with water. A hole is drilled in the bottom of the sphere causing the volume of the water to decrease at a rate of 3 cubic meters per minute. Find the rate of change of the height of the water in the sphere when the height is 4 meters."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9b5269219ec4733344fe50c34eb27ff923493c4c487098c518161d7f6c564180 | A sphere with center $O$ has radius $6$. A triangle with sides of length $15$, $15$, and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | [
"\\(2\\sqrt{5}\\)"
] | [
"A sphere with center $O$ has radius $6$. A triangle with sides of length $15$, $15$, and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-21e3eaae7dae26d032c3fbac0fda4c12fbad3cdd793ee885250c55fd8c213cde | A sphere with center \( O \) and radius \( r \) is cut by a horizontal plane at a distance \( \frac{1}{2}r \) above \( O \), forming a circle \( K \). The part of the sphere above the plane is removed and replaced by a right circular cone with \( K \) as its base and vertex \( V \) at a distance \( 2r \) vertically abo... | [
"r(\\sqrt{3} - 1)"
] | [
"A sphere with center \\( O \\) and radius \\( r \\) is cut by a horizontal plane at a distance \\( \\frac{1}{2}r \\) above \\( O \\), forming a circle \\( K \\). The part of the sphere above the plane is removed and replaced by a right circular cone with \\( K \\) as its base and vertex \\( V \\) at a distance \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22fff5595793ce2f4675167db20228fc8a75aee0d33363bb21b3aee67fbcdda7 | A sphere with center \( O \) has radius \( 6 \). A triangle with sides of length \( 15, 15, \) and \( 24 \) is situated in space so that each of its sides is tangent to the sphere. What is the distance between \( O \) and the plane determined by the triangle? | [
"\\( 2\\sqrt{5} \\)"
] | [
"A sphere with center \\( O \\) has radius \\( 6 \\). A triangle with sides of length \\( 15, 15, \\) and \\( 24 \\) is situated in space so that each of its sides is tangent to the sphere. What is the distance between \\( O \\) and the plane determined by the triangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c2cd3a07c031de04d1d849829e317b1552b27b556cd924ba29070749d05b64e7 | A sphere with center \(O\) has radius \(6\). A triangle with sides of length \(15\), \(15\), and \(24\) is situated in space so that each of its sides is tangent to the sphere. What is the distance between \(O\) and the plane determined by the triangle? | [
"\\(4\\)"
] | [
"A sphere with center \\(O\\) has radius \\(6\\). A triangle with sides of length \\(15\\), \\(15\\), and \\(24\\) is situated in space so that each of its sides is tangent to the sphere. What is the distance between \\(O\\) and the plane determined by the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f244bb756332a284ac2fbaa5b51dbd1e9480ffb30792dd629ae620a64a41c19 | A sphere with its center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through points $A$, $B$, and $C$, and intersects the edges $SA$, $SB$, and $SC$ again at points $A_1$, $B_1$, and $C_1$, respectively. The planes through $A_1$, $B_1$, and $C_1$ that are tangent to the sphere meet at point $O$. If $S... | [
"\\dfrac{12\\sqrt{33}}{11}"
] | [
"A sphere with its center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through points $A$, $B$, and $C$, and intersects the edges $SA$, $SB$, and $SC$ again at points $A_1$, $B_1$, and $C_1$, respectively. The planes through $A_1$, $B_1$, and $C_1$ that are tangent to the sphere meet at point $O$. ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0662dc2180a16f5c6bb3529244de84429ab132fa8fba5bff6edae5320522f89 | A sphere with radius $3$ is inscribed in a frustrum of slant height $10$. What is the volume of the frustrum? (A frustrum is the non-conical region formed by cutting a cone with a plane parallel to its base.) | [
"\\( 182 \\pi \\)"
] | [
"A sphere with radius $3$ is inscribed in a frustrum of slant height $10$. What is the volume of the frustrum? (A frustrum is the non-conical region formed by cutting a cone with a plane parallel to its base.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6dbc75a1a5a9a7f0248b83e4919b19eebe2747e6252a912d5b46dffe379dc239 | A sphere with radius $3$ is inscribed in a frustrum of slant height $10$. What is the volume of the frustrum? (A frustrum is the non-conical region formed by cutting a cone with a plane parallel to the base.) | [
"\\( 182\\pi \\)"
] | [
"A sphere with radius $3$ is inscribed in a frustrum of slant height $10$. What is the volume of the frustrum? (A frustrum is the non-conical region formed by cutting a cone with a plane parallel to the base.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-0e348289fce2c09dbb03c991063db9daa31d0dc851b93e81b19b5ab10c2d8713 | A sphere with radius $\alpha$ is initially centered at $O(x_1, y_1, z_1 + \alpha)$ and touches a point $P_m(x_1, y_1, z_1)$. The sphere then rotates such that it touches both $P_m$ and another point $P_1$. The new center of the sphere is $O'$. Given the coordinates of $P_m$ and $P_1$, determine the coordinates of the n... | [
" \\frac{P_m + P_1}{2} + \\sqrt{\\alpha^2 - \\left( \\frac{|P_1 - P_m|}{2} \\right)^2} \\cdot \\frac{(P_1 - P_m) \\times \\vec{k}}{|(P_1 - P_m) \\times \\vec{k}|} "
] | [
"A sphere with radius $\\alpha$ is initially centered at $O(x_1, y_1, z_1 + \\alpha)$ and touches a point $P_m(x_1, y_1, z_1)$. The sphere then rotates such that it touches both $P_m$ and another point $P_1$. The new center of the sphere is $O'$. Given the coordinates of $P_m$ and $P_1$, determine the coordinates o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac4cb796917acf3ad83c98a0e29f420932ef383dff5aa20944d6f7a04fdbf915 | A sphere with radius 1 is centered at (0,0,0). A point on the surface is stretched to (3,4,5), and a diametrically opposite point is stretched to (-3,-4,-5). Find the surface area and volume enclosed by the new skin. | [
"\\dfrac{1000\\sqrt{2}}{3}\\pi"
] | [
"A sphere with radius 1 is centered at (0,0,0). A point on the surface is stretched to (3,4,5), and a diametrically opposite point is stretched to (-3,-4,-5). Find the surface area and volume enclosed by the new skin."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1d3d30e562f794b5eef0f345d624341ff61da1f058af87d21cd697cd72b51e89 | A sphere with radius 1 is internally tangent to a cylinder on both a base and the side, with the sphere's center on the cylinder's axis. A series of spheres are externally tangent to the respective sphere before them, and each has a radius of 7/8 the length of the previous sphere, with their centers on the axis of the ... | [
"\\(\\frac{2}{3}\\)"
] | [
"A sphere with radius 1 is internally tangent to a cylinder on both a base and the side, with the sphere's center on the cylinder's axis. A series of spheres are externally tangent to the respective sphere before them, and each has a radius of 7/8 the length of the previous sphere, with their centers on the axis of... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f109d5e1611f9fb042188840b00eaa3b61e436aa435731b33305ca5582e7bb3 | A sphere with radius 10 has a cone inscribed within it of maximum volume. What is the volume of this cone? | [
"\\(\\frac{32000\\pi}{81}\\)"
] | [
"A sphere with radius 10 has a cone inscribed within it of maximum volume. What is the volume of this cone?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.