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-25255b9809129872d238cf5d3b74ae8d1e0805513a2c4307a2cef8be2cef61ed | What is the value of the sum \(1^2 + 2^2 + 4^2 + 5^2 + \ldots + 8^2 + 10^2 + \ldots + 100^2\), where we are summing the squares from 1 to 100 but leaving out the squares of the powers of 3? | [
"330970"
] | [
"What is the value of the sum \\(1^2 + 2^2 + 4^2 + 5^2 + \\ldots + 8^2 + 10^2 + \\ldots + 100^2\\), where we are summing the squares from 1 to 100 but leaving out the squares of the powers of 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-513f0d0a0b360c0f75efed91f9a292b6f72eaea6f2fff43e786f89953f58cfeb | What is the value of the sum \(\frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{1}{2 \cdot 3 \cdot 4 \cdot 5} + \frac{1}{3 \cdot 4 \cdot 5 \cdot 6} + \cdots + \frac{1}{28 \cdot 29 \cdot 30 \cdot 31}\)? | [
"\\(\\frac{749}{13485}\\)"
] | [
"What is the value of the sum \\(\\frac{1}{1 \\cdot 2 \\cdot 3 \\cdot 4} + \\frac{1}{2 \\cdot 3 \\cdot 4 \\cdot 5} + \\frac{1}{3 \\cdot 4 \\cdot 5 \\cdot 6} + \\cdots + \\frac{1}{28 \\cdot 29 \\cdot 30 \\cdot 31}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-e50f4ac7b3fb9412ef06eb1c6ce544f4b38c697af5605c8c8efbc7cf14d9fd98 | What is the value of the sum \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{2018}\)? Please show a good method. | [
"7.187"
] | [
"What is the value of the sum \\(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots + \\frac{1}{2018}\\)? Please show a good method."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-f83910ea338068f6c9931e1a25edd18da92b71146d0e970a096e27a289258b48 | What is the value of the sum \(\sum_{i=0}^{m} k^i \cdot \binom{m}{i}^2\) for a general \(k\)? The cases \(k = 0\) and \(k = 1\) are known, but a general form is sought. | [
"\\(_2F_1(-m, -m; 1; k)\\)"
] | [
"What is the value of the sum \\(\\sum_{i=0}^{m} k^i \\cdot \\binom{m}{i}^2\\) for a general \\(k\\)? The cases \\(k = 0\\) and \\(k = 1\\) are known, but a general form is sought."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 5
} | 11 | 0 | 0 |
omr-ws-00323aac11490e560fcfde4de83ce70395615297e76be39456c7ec2cf9361f80 | What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$? | [
"\\frac{49}{4}"
] | [
"What is the value of the sum\n\\[\n \\sum_z \\frac{1}{{\\left|1 - z\\right|}^2} \\, ,\n\\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-9d10cc5938abb96e59f1a12a6db0209556be4ec9283828c77d775e68f0559542 | What is the value of the sum of the distances from any point on an ellipse to its two foci (F and G)? | [
"2a"
] | [
"What is the value of the sum of the distances from any point on an ellipse to its two foci (F and G)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d69ce7d71b2d4c8767222123e029ffadec2273a46ce674a38e7dcddcd18ca68d | What is the value of the summation \(\sum_{n=1}^{\infty} \frac{n(n+1)}{2^n}\)? | [
"8"
] | [
"What is the value of the summation \\(\\sum_{n=1}^{\\infty} \\frac{n(n+1)}{2^n}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-16ddaba3635adacbb94553e02a5119ddd0f8eeac28de67e63ed50481ba32d115 | What is the value of the units digit of the 131st term in the sequence where the nth term is given by \(2 \times 4^{n-1}\)? | [
"2"
] | [
"What is the value of the units digit of the 131st term in the sequence where the nth term is given by \\(2 \\times 4^{n-1}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd6bf63abd0349e4a780b896d1ead4f3154193f6f393ab264f56ca973b4dd3bc | What is the vector projection of vector AP onto vector AB, given points A (0,2), B (5,0), and P (3,7)? | [
"\\(\\frac{5}{\\sqrt{29}}\\)"
] | [
"What is the vector projection of vector AP onto vector AB, given points A (0,2), B (5,0), and P (3,7)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b2cc583446d28ae3ec58b21e9a564dcd723d360aca2b876122f40a74e8512d6 | What is the vertex of the graph given by the equation \( y = x^2 + 4x - 5247842957273465746 \)? | [
"\\((-2, -5247842957273465750)\\)"
] | [
"What is the vertex of the graph given by the equation \\( y = x^2 + 4x - 5247842957273465746 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6afe0e1d0901cd5d0e4812ebf03938202845f10f0c0064b232e984b983db12d0 | What is the volume between the elliptical paraboloid \( x = \frac{y^2}{2p} + \frac{z^2}{q} \) and the plane \( x = a \)? | [
"\\(\\frac{\\pi \\sqrt{2pq} a^2}{2}\\)"
] | [
"What is the volume between the elliptical paraboloid \\( x = \\frac{y^2}{2p} + \\frac{z^2}{q} \\) and the plane \\( x = a \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-bef10f69456c993b7d84ffa225a4b381ec600bab4510583fb7ef67872645efc6 | What is the volume between the sphere \(x^2 + y^2 + z^2 = 4\) and the cylinder \(x^2 + y^2 - 2x = 0\) where \(y \geq 0\)? | [
"\\dfrac{24\\pi + 32}{9}"
] | [
"What is the volume between the sphere \\(x^2 + y^2 + z^2 = 4\\) and the cylinder \\(x^2 + y^2 - 2x = 0\\) where \\(y \\geq 0\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-1dc6f9fde3313ed3eb48d66f2678db3ff682dd2d7ab012c19d9bedabdf42d3a5 | What is the volume generated when the region between the line \( y = x \) and the parabola \( y = x^2 \) is rotated about the line \( y = x \)? | [
"\\( \\frac{\\pi \\sqrt{2}}{60} \\)"
] | [
"What is the volume generated when the region between the line \\( y = x \\) and the parabola \\( y = x^2 \\) is rotated about the line \\( y = x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-8c9b52493078162c219b973cdfe68d2382adfbfb2eef615334b2cacca8a3c571 | What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively? | [
"30"
] | [
"What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42907bf8c1891dcc68312186906d79a7b970e55e7a9fe62ad3aba61cf8a620c0 | What is the volume in litres and gallons of a plastic tube that is 50 meters long and has a circumference of 100 mm when filled with water? | [
"500 litres, 132.086 gallons"
] | [
"What is the volume in litres and gallons of a plastic tube that is 50 meters long and has a circumference of 100 mm when filled with water?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b43bdc45dbc478fffabbaf5b4afd85bc09172384a6bffc85cff6552eb1da624b | What is the volume of a cube formed by the centers of the faces of a regular octahedron with edge length \( s \)? | [
"\\(\\frac{2s^3\\sqrt{2}}{27}\\)"
] | [
"What is the volume of a cube formed by the centers of the faces of a regular octahedron with edge length \\( s \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-17086033ae4f4c3bcb581168f9b4705a6780d7177d0a8af59984fa5582889478 | What is the volume of a regular octahedron formed by joining the centers of adjacent faces of unit cubes? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the volume of a regular octahedron formed by joining the centers of adjacent faces of unit cubes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6b54a229244779f8af6fe6442cc1f955fe3ce10c0ab9fa0590c191c60a153359 | What is the volume of a spherical shell with an outer diameter of 8 cm and an inner diameter of 6 cm? | [
"\\(\\frac{14\\pi}{3}\\) cubic centimeters."
] | [
"What is the volume of a spherical shell with an outer diameter of 8 cm and an inner diameter of 6 cm?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-95343cb8238223312736c1028e8c679ca6bb4790d3ef5d4c1c366419d4555920 | What is the volume of a tetrahedron that has two faces which are equilateral triangles of unit side length and two faces which are right isosceles triangles? | [
"\\(\\frac{\\sqrt{2}}{12}\\)"
] | [
"What is the volume of a tetrahedron that has two faces which are equilateral triangles of unit side length and two faces which are right isosceles triangles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-19bf16c5903480dad3fce554d957349f52c1a5dbec86f0cf5bcaef6826c61b0c | What is the volume of a triangular prism if a sphere of unit radius is tangent to all its edges? | [
"\\( 6\\sqrt{3} \\)"
] | [
"What is the volume of a triangular prism if a sphere of unit radius is tangent to all its edges?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3195d652053fed5c31e214b570d6a6338db70a41c44db927f0f6ed8959aa2a5 | What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$? | [
"4"
] | [
"What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \\sqrt{13}$, $BD = 2\\sqrt{5}$, and $CD = 5$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-96e5d4422fc7e401912281236ea914c57a6ec4ad77ee7084290b726edefecfec | What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? | [
"4"
] | [
"What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\\sqrt{13}, BD=2\\sqrt{5},$ and $CD=5$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c9d9a400d7a9b5c7d9c14a47839740f295ee26ae46bd53b53ae7ff355739855b | What is the volume of the Euclidean sphere \( S^{n-1} \)? | [
"\\frac{2 \\pi^{n/2}}{\\Gamma\\left(\\frac{n}{2}\\right)}"
] | [
"What is the volume of the Euclidean sphere \\( S^{n-1} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8af9198faa50e79b7a1ee1cddf9b48bad49bdf7c460c6ca8bd43f234eee0bc29 | What is the volume of the intersection of the sphere \(x^2 + y^2 + z^2 \leq r^2\) and the cylinder \(x^2 + y^2 \leq rx\)? | [
"\\(\\frac{2}{3} \\pi r^3\\)"
] | [
"What is the volume of the intersection of the sphere \\(x^2 + y^2 + z^2 \\leq r^2\\) and the cylinder \\(x^2 + y^2 \\leq rx\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-b1fb9a9bc7871824786a1e6ad7f201f93721ce65302783600cb6f92b6d152d9b | What is the volume of the intersection of two spheres, each with radius $r$, when their centers intersect? | [
"\\(\\frac{5 \\pi r^3}{12}\\)"
] | [
"What is the volume of the intersection of two spheres, each with radius $r$, when their centers intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-45cc568499427cc59b5db403cb06a81925ec1e6f0cfe3f298583a982a66f041e | What is the volume of the largest right rectangular prism that can be inscribed in the ellipsoid given by the equation \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}= 1? \] | [
"\\(\\frac{8abc}{3\\sqrt{3}}\\)"
] | [
"What is the volume of the largest right rectangular prism that can be inscribed in the ellipsoid given by the equation\n\n\\[\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}+\\frac{z^{2}}{c^{2}}= 1?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c456fc4c1c128896adef2d210cfee6af58b57231522896e88df3d50eb076e74d | What is the volume of the largest sphere that fits inside a regular tetrahedron with a side length of 6? | [
"\\pi \\sqrt{6}"
] | [
"What is the volume of the largest sphere that fits inside a regular tetrahedron with a side length of 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd9c2f843235224f6d72f183d5d444e6dcaed15bf5706cfc4b3920fc966edf88 | What is the volume of the octahedron formed by joining the centers of the faces of a right rectangular prism with dimensions 3, 4, and 5? | [
"10"
] | [
"What is the volume of the octahedron formed by joining the centers of the faces of a right rectangular prism with dimensions 3, 4, and 5?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-30a9ade97c99d0c9b49541f10c9be5edbffa05a53a434e5691ae9a9a32c6b38d | What is the volume of the portion of the sphere given by the equation \((x-1)^{2}+(y-1)^{2}+(z-1)^{2}=121\) where \(z > 0\)? | [
"\\(1008\\pi\\)"
] | [
"What is the volume of the portion of the sphere given by the equation \\((x-1)^{2}+(y-1)^{2}+(z-1)^{2}=121\\) where \\(z > 0\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-746f13b85de78577b412df776e1121432ac73a1b25a1e7da90bb3d184d64dd2f | What is the volume of the region bounded by \( y = x^2 \), \( y = 1 \), and \( y = 0 \) when rotated around the y-axis? | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"What is the volume of the region bounded by \\( y = x^2 \\), \\( y = 1 \\), and \\( y = 0 \\) when rotated around the y-axis?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-9c79f45b5eb06156f7130fe7b8002e12c894abee8007a5035becaae284f259d3 | What is the volume of the region defined by $s=\{(x,y,z):|x|+|y|+|z|\le 1\}$? | [
"\\(\\frac{4}{3}\\)"
] | [
"What is the volume of the region defined by $s=\\{(x,y,z):|x|+|y|+|z|\\le 1\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7dd534af2bf5f4d41b7b8dc60868d9f8bc28749d9fccf14ea1474d648743ca52 | What is the volume of the region enclosed by the graphs of $x^{2022} + y^{2022} + z^{2022} = 1$ and $\left(x - \frac{1}{2}\right)^{2022} + y^{2022} + (z + 1)^{2022} = 2$? Round your answer to the nearest integer. | [
"3"
] | [
"What is the volume of the region enclosed by the graphs of $x^{2022} + y^{2022} + z^{2022} = 1$ and $\\left(x - \\frac{1}{2}\\right)^{2022} + y^{2022} + (z + 1)^{2022} = 2$? Round your answer to the nearest integer."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4837015470f29567ef7772e79c04f01c712f38c4846e9c05e324cb6fdff74f70 | What is the volume of the region in three-dimensional space defined by the inequalities $x^2+y^2+z^2 \le 1$ and $x^2+y^2+(z-1)^2 \le 1$? | [
"$\\frac{5\\pi}{12}$"
] | [
"What is the volume of the region in three-dimensional space defined by the inequalities $x^2+y^2+z^2 \\le 1$ and $x^2+y^2+(z-1)^2 \\le 1$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b63d722d595c676c5f4b22c1e568e4e38019abbdbfbaeda59766316e22a41216 | What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$? | [
"\\(\\frac{1}{6}\\)",
"\\frac{1}{6}"
] | [
"What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\\le1$ and $|x|+|y|+|z-1|\\le1$?"
] | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 28
} | 60 | 0 | 0 |
omr-ws-6b629135fc89afad6602177921497eb89cbb462e915eb12b8f9e3d21a0a3f789 | What is the volume of the region in three-dimensional space defined by the inequalities \( |x| + |y| + |z| \le 1 \) and \( |x| + |y| + |z-1| \le 1 \)? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the volume of the region in three-dimensional space defined by the inequalities \\( |x| + |y| + |z| \\le 1 \\) and \\( |x| + |y| + |z-1| \\le 1 \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-d15d301298c92d0dc5fb8b5a34f320ea4c8bc73642711e2fe7bce0079703334c | What is the volume of the region within a rectangular prism with sides 3, 4, and 5 that is within 1/2 unit of the prism's surface? | [
"36"
] | [
"What is the volume of the region within a rectangular prism with sides 3, 4, and 5 that is within 1/2 unit of the prism's surface?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-110c2a2cfa2d82b93a81ff09537a3340f53797fd63ea5f07d3115ab6ada2b007 | What is the volume of the right circular cone of minimum volume that can be confined within a hemisphere of radius \( R \) and supported on the diametral plane? | [
"\\(\\frac{1}{3} \\pi R^3\\)"
] | [
"What is the volume of the right circular cone of minimum volume that can be confined within a hemisphere of radius \\( R \\) and supported on the diametral plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-33af00a8b53d581acddcc0fe6d15614ee71f09aa3e8f2b3c63bc4e5a02c9ce7b | What is the volume of the set $S=\{(x,y,z):|x|+|y|+|z|\le 1\}$? | [
"\\(\\frac{4}{3}\\)"
] | [
"What is the volume of the set $S=\\{(x,y,z):|x|+|y|+|z|\\le 1\\}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-db2dc0721b6348e3498bc08b6e491cd2d58546815a146df8a35179558937bf76 | What is the volume of the smallest box that can contain four perfectly spherical steaks, each with a radius of 1 inch? | [
"32"
] | [
"What is the volume of the smallest box that can contain four perfectly spherical steaks, each with a radius of 1 inch?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-ce4f4d4aa48b52005d104d1790fb71cacbd3b223f2e05527e33257e39f55c607 | What is the volume of the smallest box that can contain four perfectly spherical steaks, each with a radius of 1 inch? Note: The problem implies that the steaks are packed in a way that minimizes the volume of the box. | [
"32"
] | [
"What is the volume of the smallest box that can contain four perfectly spherical steaks, each with a radius of 1 inch?\n\nNote: The problem implies that the steaks are packed in a way that minimizes the volume of the box."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae810b09f2afc0c39ed78b197e0196ea383c506b71f0dc303c4cc74247f08c0f | What is the volume of the solid bounded back and front by \( x = 1 \) and \( x = 2 \), above by \( z = x + 1 \), below by \( z = 0 \), and on the sides by the surfaces \( y = \frac{1}{x} \) and \( y = -\frac{1}{x} \)? | [
"2 + 2\\ln 2"
] | [
"What is the volume of the solid bounded back and front by \\( x = 1 \\) and \\( x = 2 \\), above by \\( z = x + 1 \\), below by \\( z = 0 \\), and on the sides by the surfaces \\( y = \\frac{1}{x} \\) and \\( y = -\\frac{1}{x} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b3f8ca47aefaa49c571a848e32cac9254ddf1643c62cd52246202bc0a35b517 | What is the volume of the solid bounded by eight spheres of radius 1 whose centers are located at the corners of a unit cube? | [
"8 \\left(1 - \\frac{\\sqrt{3}}{2}\\right)^3"
] | [
"What is the volume of the solid bounded by eight spheres of radius 1 whose centers are located at the corners of a unit cube?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e447c8fd477f7368d70992b5eb59e250762c15a394024dd0b8f1a2af78b4a79 | What is the volume of the solid bounded by the curves \( y = e^{-x^2} \) and \( y = e^{-x} \) when revolved about the y-axis? | [
"\\(6\\pi e^{-1}\\)"
] | [
"What is the volume of the solid bounded by the curves \\( y = e^{-x^2} \\) and \\( y = e^{-x} \\) when revolved about the y-axis?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0126de07845e00953fd46a0bbab9c8c54c84af0c8d29b7226f383923bde6e96 | What is the volume of the solid created when the area bound by the curves \( y = (x-1)^2 \) and \( y = \ln x \) is rotated around the line \( x = 1 \)? | [
"2\\pi \\int_{1}^{a} (x - 1)(\\ln x - (x - 1)^2) \\, dx"
] | [
"What is the volume of the solid created when the area bound by the curves \\( y = (x-1)^2 \\) and \\( y = \\ln x \\) is rotated around the line \\( x = 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-0ebd9f6df2972703d65f73994637a22198b8536093eceaa49a8352ff29d2e22a | What is the volume of the solid formed by rotating quadrilateral $ABCD$ with vertices $A(0,0)$, $B(3,-1)$, $C(6,2)$, and $D(2,3)$ about the x-axis? | [
"\\dfrac{82}{3}\\pi"
] | [
"What is the volume of the solid formed by rotating quadrilateral $ABCD$ with vertices $A(0,0)$, $B(3,-1)$, $C(6,2)$, and $D(2,3)$ about the x-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f81c6dcedbd1371d0f6850f8582038b259d59f5611e86af8113d1535d1871fdb | What is the volume of the solid generated when the region bounded by the graphs \( y = 2x \) and \( y = xe^x \) is revolved around the x-axis? | [
"\\pi \\left( \\frac{4}{3} (\\ln 2)^3 - 2 (\\ln 2)^2 + 2 \\ln 2 - \\frac{3}{4} \\right)"
] | [
"What is the volume of the solid generated when the region bounded by the graphs \\( y = 2x \\) and \\( y = xe^x \\) is revolved around the x-axis?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45ef8e864f8b4c3ebfafc96ff6bc610a4d6610f2f4ac794aef20490407d847f3 | What is the volume of the solid that lies inside both of the spheres given by the equations \(x^2 + y^2 + z^2 + 4x - 2y + 4z + 5 = 0\) and \(x^2 + y^2 + z^2 = 0\)? | [
"0"
] | [
"What is the volume of the solid that lies inside both of the spheres given by the equations \\(x^2 + y^2 + z^2 + 4x - 2y + 4z + 5 = 0\\) and \\(x^2 + y^2 + z^2 = 0\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-978a41d1986e254a895c8787576fe2a0f033626c6e53092ce6ff5c68ed814036 | What is the volume of the sphere given by the equation $x^2 + y^2 - 8y + z^2 + 7 = 0$ that lies below the plane $5x + y + 3z = 0$? | [
"\\dfrac{\\pi \\left(3 - \\dfrac{4}{\\sqrt{35}}\\right)^2 \\left(6 + \\dfrac{4}{\\sqrt{35}}\\right)}{3}"
] | [
"What is the volume of the sphere given by the equation $x^2 + y^2 - 8y + z^2 + 7 = 0$ that lies below the plane $5x + y + 3z = 0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-09477e6e07477256a66208ccf95b7074175df1301b301fee2dc0a603ea6ecd44 | What is the volume of the tetrahedron $ABCD$ with edge lengths $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\frac{12}{5}\sqrt{2}$? | [
"\\dfrac{24}{5}"
] | [
"What is the volume of the tetrahedron $ABCD$ with edge lengths $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\\frac{12}{5}\\sqrt{2}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-a4d43ef2dede1ad55a4a3ddebde61f1dcc4bd61e12303424b8aad7abadef810f | What is the volume of the union of two regular tetrahedrons $A$ and $B$ formed using the 8 vertices of a unit cube? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the volume of the union of two regular tetrahedrons $A$ and $B$ formed using the 8 vertices of a unit cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-7ec602149ed520ee9fe95d36c278f6ab46537a0c8d4b4d45a93dcf706ce96e5c | What is the volume swept out when a cube with side length 1 is rotated about its body diagonal? | [
"$\\frac{\\pi}{\\sqrt{3}}$"
] | [
"What is the volume swept out when a cube with side length 1 is rotated about its body diagonal?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-7fed0ba2ec4de1319324ec9151f279cb435f1968c748889c1c41f3d00e5f6790 | What is the volume, in cubic centimeters, of a cone without a base that was made from a semicircle of radius 12 cm? | [
"\\( 72\\pi\\sqrt{3} \\)"
] | [
"What is the volume, in cubic centimeters, of a cone without a base that was made from a semicircle of radius 12 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d30db4ebcfcdd8895518fddc0acdc00372186fd0a2876d294e169e54a4c8399e | What is the volume, in cubic centimetres, of the portion of a regular octahedron that is closer to the top vertex than to any other vertex, if the octahedron has a volume of $120 \text{ cm}^3$? | [
"20"
] | [
"What is the volume, in cubic centimetres, of the portion of a regular octahedron that is closer to the top vertex than to any other vertex, if the octahedron has a volume of $120 \\text{ cm}^3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-89dab3419b396b8f57629bd17754ce2061affcf42d4b031c16a9758d92501e84 | What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $24$ square inches, $16$ square inches and $6$ square inches? | [
"48"
] | [
"What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $24$ square inches, $16$ square inches and $6$ square inches?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc8bad8c688a8ace86e92f867bf8e8145a3c4e015828ef5d46bdfeaa19a586f8 | What is the volume, in cubic units, of a cube whose surface area is 600 square units? | [
"1000"
] | [
"What is the volume, in cubic units, of a cube whose surface area is 600 square units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a06a97fd203e9bdccc66f8fbb15fd28a8a59863a2233715d1373e5f946ce44d7 | What is the weakest topology $\tau$ on $\mathbb{R}$ such that the $\sigma$-algebra generated by $\tau$ is the power set $\mathcal{P}(\mathbb{R})$? | [
"\\text{The discrete topology}"
] | [
"What is the weakest topology $\\tau$ on $\\mathbb{R}$ such that the $\\sigma$-algebra generated by $\\tau$ is the power set $\\mathcal{P}(\\mathbb{R})$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6b9062caa0e01baebfd963a866b00c249ff8de22d7c37f3ce19c5fdbb17962d7 | What is the weakest topology $\tau$ on a nonempty set $X$ such that the $\sigma$-algebra generated by $\tau$ is the power set $\mathcal{P}(X)$? | [
"The discrete topology on $X$."
] | [
"What is the weakest topology $\\tau$ on a nonempty set $X$ such that the $\\sigma$-algebra generated by $\\tau$ is the power set $\\mathcal{P}(X)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-a46a80099ae5ce5836349918e7865c85e90e8fa90918e11d2030a98693fc21fb | What is the weight of an apple in the 80th percentile, given that the weight of a medium apple follows a normal distribution with a mean of 200 grams and a standard deviation of 12 grams? | [
"210.1"
] | [
"What is the weight of an apple in the 80th percentile, given that the weight of a medium apple follows a normal distribution with a mean of 200 grams and a standard deviation of 12 grams?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d133c3c5f1f4cee7773488786a4866da58d5bfbfe766add3047531323c5870ee | What is the weighted average of the Stirling numbers \( S(n, k) \) for a fixed \( n \) as \( k \) goes from 0 to \( n \)? | [
"\\( H_n \\) or \\( 1 + \\frac{1}{2} + \\frac{1}{3} + \\ldots + \\frac{1}{n} \\)"
] | [
"What is the weighted average of the Stirling numbers \\( S(n, k) \\) for a fixed \\( n \\) as \\( k \\) goes from 0 to \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-225532b94385cd4be54f3d6004935069e503d4225861f5c771022642d281fa94 | What is the width of the petals of the rose defined by $r=\cos2\theta$, given that the petals have a length of $1$? | [
"\\(\\frac{2\\sqrt6}{9}\\)"
] | [
"What is the width of the petals of the rose defined by $r=\\cos2\\theta$, given that the petals have a length of $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-0d779d54997a1b5f03422353be5b7198d97a2ba623976f440781439ffa1ec178 | What is the y-intercept of the function \(3\sin(2x+1)\)? | [
"2.5245"
] | [
"What is the y-intercept of the function \\(3\\sin(2x+1)\\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-f6305fd94b518c57871d325de682ffa6e82a85641db12b2fa3d8f289c46e9223 | What is, on average, the terrestrial distance between two points chosen completely at random on Earth's surface, assuming the Earth is a perfect sphere of radius $R$? The terrestrial distance is defined as the minimum length of an arc joining these two points traced on the Earth's surface (arc of a great circle). | [
"\\(\\frac{\\pi R}{2}\\)"
] | [
"What is, on average, the terrestrial distance between two points chosen completely at random on Earth's surface, assuming the Earth is a perfect sphere of radius $R$? The terrestrial distance is defined as the minimum length of an arc joining these two points traced on the Earth's surface (arc of a great circle)."... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c979f8836401fd2d8c4b1dbd532cf4be58fa5da7cf019e80adc5ea66b958d9ab | What kind of binary operation * can be applied to points in the Cartesian plane such that the plane is closed under *? | [
"(a, b) * (c, d) = (ac - bd, ad + bc)"
] | [
"What kind of binary operation * can be applied to points in the Cartesian plane such that the plane is closed under *?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-f1879fd24d43201ec5fed3a7d7cbf453023b7e7d1b3b5e80b3622f0b9e7503b3 | What kind of graph is produced by the equation $y = \sum^{\infty}_{n=1}\frac{\sin(nx)}{n}$? | [
"\\( y = \\begin{cases}\n 0 & \\text{for }x=2\\pi k\\\\\n \\frac{\\pi}{2} - \\frac{1}{2}x & \\text{for }2\\pi k < x < 2\\pi (k+1)\\\\\n\\end{cases} \\)"
] | [
"What kind of graph is produced by the equation $y = \\sum^{\\infty}_{n=1}\\frac{\\sin(nx)}{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-dbf28879c8913ee841fab7555e63fa5ea580dc3b40a95df0d73ce4c8496bb156 | What kind of singularity does the function \( f(z) = \sin\left(\frac{1}{\cos\left(\frac{1}{z}\right)}\right) \) have? | [
"non-isolated essential singularity"
] | [
"What kind of singularity does the function \\( f(z) = \\sin\\left(\\frac{1}{\\cos\\left(\\frac{1}{z}\\right)}\\right) \\) have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-74d4756fb2696f6577701f9cf64d563f24cf8527042ce8a687a80ba2046e3af6 | What kind of triangle \(ABC\) satisfies the condition \[ \frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = 1 + \frac{1}{\sqrt{2}} \] and the triangle \(ABC\) is not acute? | [
"\\text{right-angled isosceles}"
] | [
"What kind of triangle \\(ABC\\) satisfies the condition\n\\[\n\\frac{\\sin A + \\sin B + \\sin C}{\\cos A + \\cos B + \\cos C} = 1 + \\frac{1}{\\sqrt{2}}\n\\]\nand the triangle \\(ABC\\) is not acute?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-392b7245346580601470bee7617fedac0f05d87b7b424a5873c99dba35195e4c | What length \( D \) maximizes the probability that a randomly chosen line segment of length \( D \) on an infinite checkerboard grid (with unit side length squares) crosses exactly one line, and what is this maximal probability? | [
"\\( D = 1 \\) and the maximal probability is \\( 1 \\)."
] | [
"What length \\( D \\) maximizes the probability that a randomly chosen line segment of length \\( D \\) on an infinite checkerboard grid (with unit side length squares) crosses exactly one line, and what is this maximal probability?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c7315e8c58e0243ff58d81e9fb5bef25cfb42d78f2b621b5ed022713237544c3 | What length of rope is needed to tie a goat at the corner of a garden so that the goat can graze exactly half of the garden? | [
"a \\sqrt{\\dfrac{2}{\\pi}}"
] | [
"What length of rope is needed to tie a goat at the corner of a garden so that the goat can graze exactly half of the garden?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2d797a5a4251601773a2eb603422f876e5045dda6768eaf72e02c2ddacbc4ac6 | What length of the second path will be illuminated by light A, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m? | [
"11"
] | [
"What length of the second path will be illuminated by light A, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-3ef84c80b41108cf5c0f161e08eff10419ac96dcdfc972687d0ecb83cd445da0 | What length of the second path will be illuminated by lights A and B, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m? | [
"7"
] | [
"What length of the second path will be illuminated by lights A and B, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-758cad94e96d2b399b78bff90a2d42107b44071bfe0687ca483c1fa2cd58dce5 | What length of the second path will be illuminated by lights A, B, and C, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m? | [
"3"
] | [
"What length of the second path will be illuminated by lights A, B, and C, given that the walkways meet at an angle of 60 degrees, the lights are placed 5 m apart on one walkway, and each light has an illumination radius of 7 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-63f400fc3ef633c7e3a2ee31e56a0f4f42d914927f8556989a33b5d9de5496b3 | What line do the centers of all circles which touch the line \( y + 4 = 0 \) and the circle \( x^2 + y^2 = 4 \) describe? | [
"Two parabolas described by \\( y = \\frac{1}{12}x^2 - 3 \\) and \\( y = \\frac{1}{4}x^2 - 1 \\)."
] | [
"What line do the centers of all circles which touch the line \\( y + 4 = 0 \\) and the circle \\( x^2 + y^2 = 4 \\) describe?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-59da17c1759802c1a50e5c9b3fa59cb96dabeaf71eb3f8c78a3bc68c30042eb4 | What may be the last digit of a positive integer \( n \) if the sum \( n + n^2 + n^3 + \cdots + n^{(2n-3)} - 4 \) is a prime number? | [
"5"
] | [
"What may be the last digit of a positive integer \\( n \\) if the sum \\( n + n^2 + n^3 + \\cdots + n^{(2n-3)} - 4 \\) is a prime number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1c20c40826395a6bd476b468ca219924ee9fe9e8fadec4538741d45415f6dbf | What may be the last digit of a positive integer \( n \) if the sum \( n + n^2 + n^3 + \cdots + n^{2n-3} - 4 \) is a prime number? | [
"5"
] | [
"What may be the last digit of a positive integer \\( n \\) if the sum \\( n + n^2 + n^3 + \\cdots + n^{2n-3} - 4 \\) is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b403534681aa5ca4798cc6c04bf4d5dc4cfb5e7883de4cc2e59597d0e13f6bcc | What may be the sum of the angles of triangle $ABC$ on the surface of a cube, where each side of the triangle is the shortest polyline connecting the vertices and none of the vertices lie on the edge of the cube? | [
"270^\\circ"
] | [
"What may be the sum of the angles of triangle $ABC$ on the surface of a cube, where each side of the triangle is the shortest polyline connecting the vertices and none of the vertices lie on the edge of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d96c54f34a7fa756e27a1bf85489146be07aa0fd65ff2d01baa260e73c03bc57 | What monthly payment is needed to pay off a loan of \$500 amortized at 12% compounded monthly for 2 years? | [
"23.64"
] | [
"What monthly payment is needed to pay off a loan of \\$500 amortized at 12% compounded monthly for 2 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74d5dfe779ed07f755299d4e92149c1632360e2c6d201b26b28cac8e4f3df7a5 | What multiple of 15 is closest to 2009? | [
"2010"
] | [
"What multiple of 15 is closest to 2009?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51a31f4c23c4afa84e8895e373b2e0fd8343cead84648dadff0d6414dbf5e50a | What must be the criteria for an N-gon (i.e., number of sides, parity) such that a bumblebee can walk across each diagonal exactly once without having to fly? | [
"\\( N \\ge 5 \\) and \\( N \\equiv 1 \\pmod{2} \\)"
] | [
"What must be the criteria for an N-gon (i.e., number of sides, parity) such that a bumblebee can walk across each diagonal exactly once without having to fly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-025a4b7998045ffc6eebfe5debab3ab140b44fce1ded264dc6d12343b2e4a6e8 | What must be the dimensions of the second isosceles triangle EFG (EF=FG=?, and EG=?) that has the same circumscribed and inscribed circle as the isosceles triangle ABC with sides AB=BC=5 and AC=6? | [
"EF = FG = 5, EG = 6"
] | [
"What must be the dimensions of the second isosceles triangle EFG (EF=FG=?, and EG=?) that has the same circumscribed and inscribed circle as the isosceles triangle ABC with sides AB=BC=5 and AC=6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-780d72510bb025b698f50001a198a871208ffd8a64a7970b985381515476bdc2 | What must be the minimum value of $\omega$ so that the minimum value of the function $y=2\sin(\omega x)$ appears at least 20 times in the open interval $(0,1)$? | [
"\\dfrac{79\\pi}{2}"
] | [
"What must be the minimum value of $\\omega$ so that the minimum value of the function $y=2\\sin(\\omega x)$ appears at least 20 times in the open interval $(0,1)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ecec8da9073fcf0e4e50c3138aadb916491779148acf8fd3b7e1cccb0f2de2e5 | What must be the minimum value of $n$ so that there are at least two identically colored triangles in a $K_{n}$ graph when its vertices are painted with $r$ different colors? | [
"r + 3"
] | [
"What must be the minimum value of $n$ so that there are at least two identically colored triangles in a $K_{n}$ graph when its vertices are painted with $r$ different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-581f2d4107203032ea98f809198bebfadc12e98c7d5a3644d1d8e21da582431a | What must be the minimum value of $n$ so that there is at least one triangle with all vertices of the same color in a $K_{n}$ graph when its vertices are painted with $r$ different colors? | [
"\\( 2r + 1 \\)"
] | [
"What must be the minimum value of $n$ so that there is at least one triangle with all vertices of the same color in a $K_{n}$ graph when its vertices are painted with $r$ different colors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 5,
"tir": 2
} | 54 | 0 | 0 |
omr-ws-4fd5c49a9d607941ba919a635cab92c201f8a2e833a686a8d569534bfc3619e2 | What must be the shape of a bowl in a clepsydra (ancient water clock) so that the water level falls at a constant rate? | [
"\\( A(h) = a\\sqrt{1 + \\frac{2gh}{c^2}} \\)"
] | [
"What must be the shape of a bowl in a clepsydra (ancient water clock) so that the water level falls at a constant rate?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29cbf719734d6200cdc8f24bb0e9be248fc66453a637769cdbb84e3ddefde080 | What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$? | [
"-37"
] | [
"What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b84f42c996248a33dab9b6e6bd5b6f4533c41458f18fdb1d55505b3265dd492 | What must be true of a quadrilateral ABCD so that when the midpoints of its sides E, F, G, and H are connected, the resulting quadrilateral EFGH is a rhombus? | [
"The diagonals AC and BD of quadrilateral ABCD must be equal in length."
] | [
"What must be true of a quadrilateral ABCD so that when the midpoints of its sides E, F, G, and H are connected, the resulting quadrilateral EFGH is a rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-99e6e9528395e2c6b5dd7bbed74e4cb5bfb84a2e418b3b6aa4da3e5f18608608 | What non-zero real value for \( x \) satisfies \( (7x)^{14} = (14x)^7 \)? | [
"\\(\\frac{2}{7}\\)"
] | [
"What non-zero real value for \\( x \\) satisfies \\( (7x)^{14} = (14x)^7 \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-59b52ad9c363e36bc3e253c570d1e298364ad463037aedd0e3a55716a22bd52e | What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? Express your answer as a common fraction. | [
"\\frac{8}{5}"
] | [
"What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3725d632cf5b837fa374e32305d49a2e05b4fb872e96647f91d960d74c1826b5 | What norms exist on the algebra $\mathrm M_n(K)$, where $K = \mathbb R$ or $\mathbb C$, that make it a normed algebra with $||I|| = 1$? | [
"\\text{All operator norms induced by vector norms on } K^n"
] | [
"What norms exist on the algebra $\\mathrm M_n(K)$, where $K = \\mathbb R$ or $\\mathbb C$, that make it a normed algebra with $||I|| = 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-2e0e38aa9c030208b313c2ddb5cf0cb0c96ad49823aa83e6805ad6031b4239df | What number can be added to both the numerator and denominator of $\frac{3}{5}$ so that the resulting fraction will be equivalent to $\frac{5}{6}$? | [
"7"
] | [
"What number can be added to both the numerator and denominator of $\\frac{3}{5}$ so that the resulting fraction will be equivalent to $\\frac{5}{6}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29a38754c475b66d83a9febcb8e7627716e65565c5c0b64ca0b9fefe5e021f18 | What number can be obtained in the remainder when the hundredth power of a whole number is divided by 125? | [
"Take into account that every integer not divisible by 5 can be written in the form $5k\\pm 1$ or $5k\\pm 2$. {\\it Answer:} 0 or 1"
] | [
"What number can be obtained in the remainder when the hundredth power of a whole number is divided by 125?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d177cd115a9e249393a3c3696054357ea2400632af3401abd93f6b7ebf1acf89 | What number is a multiple of every integer? | [
"0"
] | [
"What number is a multiple of every integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91555f25afd790af32649629d1ea4754325a81751ced8f77c755c04b74b6f2d4 | What number less than 100000 has more factors than any other number less than 100000? | [
"83160"
] | [
"What number less than 100000 has more factors than any other number less than 100000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-5aa032f1fdfd111e3d0a1450e595a97d67e016c9e3d2a5e538d028e65fe6ea2c | What number less than 1000000 has more factors than any other number less than 1000000? | [
"720720"
] | [
"What number less than 1000000 has more factors than any other number less than 1000000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-b18702f389b12b8da2638660e5857c6dfb6ad2d15171a747142fca15ab9f8ba2 | What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$ | [
"-1+12i"
] | [
"What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bed249bf6517316b7de53e64eb37cc3a1627e3d6dca3ea4e2ead20bc42bed25a | What number needs to be put in place of * so that the problem "There are n straight lines on the plane, intersecting at * points. Find n." has a unique solution? | [
"1"
] | [
"What number needs to be put in place of * so that the problem \"There are n straight lines on the plane, intersecting at * points. Find n.\" has a unique solution?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4da08d140f1966e7fda701edec4b6f73746523c9f0f12eb28795544cc60cdf6 | What number results when 50 is increased by $120\%$? | [
"110"
] | [
"What number results when 50 is increased by $120\\%$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8118f941954978315f9b9c02607b375e715e5942494ebd76ca30dd675a5df0b7 | What number should be removed from the sequence $1!, 2!, 3!, \ldots, 200!$ such that the product of the remaining 199 numbers is a perfect square? | [
"99 or 100"
] | [
"What number should be removed from the sequence $1!, 2!, 3!, \\ldots, 200!$ such that the product of the remaining 199 numbers is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-77135a94d24cdb476bc1deeb09d08a3060970bf9daa5ba358d036d8928edbf8f | What number shows up most often when you roll 10 dice? | [
"35"
] | [
"What number shows up most often when you roll 10 dice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e2a1d739da189b154a4a0088ffba30898231753c2af9f1732f79595fc062192 | What number, when divided by 2, gives the same result as when 2 is subtracted from it? | [
"4"
] | [
"What number, when divided by 2, gives the same result as when 2 is subtracted from it?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31b5209eed91784caa0a906b96998814cdab7c2103ef8dec28f7a7027a096b0b | What numbers \( x \) and \( y \) come closest to satisfying the equations \( x - y = 1 \), \( 2x + y = -1 \), and \( x + 2y = 1 \)? Square and add the errors, \((x - y - 1)^2 + (2x + y + 1)^2 + (x + 2y - 1)^2\). Then minimize this expression. | [
"(0, 0)"
] | [
"What numbers \\( x \\) and \\( y \\) come closest to satisfying the equations \\( x - y = 1 \\), \\( 2x + y = -1 \\), and \\( x + 2y = 1 \\)? Square and add the errors, \\((x - y - 1)^2 + (2x + y + 1)^2 + (x + 2y - 1)^2\\). Then minimize this expression."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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