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-d00ab99b4fe4853c5d9ab5a751deaf2153512a294f041e3519f0f3d08f1f9dc9 | What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$? (Here $[x]$ denotes the greatest integer less than or equal to $x$.) | [
"92"
] | [
"What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$? (Here $[x]$ denotes the greatest integer less than or equal to $x$.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-13b41993792fd96ce470ab3f315b56a7a9631d29db2f7801a61e33063f450fa5 | What is the sum of the squares of the roots of the equation $x^2 - 7\lfloor x \rfloor + 5 = 0$? | [
"92"
] | [
"What is the sum of the squares of the roots of the equation $x^2 - 7\\lfloor x \\rfloor + 5 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-99b6887b1ada64de378f98e08ca1d959f3d6e849e73c4f81275085ab5b087d94 | What is the sum of the squares of the roots of the equation \( x^2 + 7[x] + 5 = 0 \)? (Here \([x]\) denotes the greatest integer less than or equal to \( x \).) | [
"95"
] | [
"What is the sum of the squares of the roots of the equation \\( x^2 + 7[x] + 5 = 0 \\)? (Here \\([x]\\) denotes the greatest integer less than or equal to \\( x \\).)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 22 | 0 | 0 |
omr-ws-6ef664f32cdda0a70d8b096b00ca4ec827d980ff74292deb2f40f28666f02c1f | What is the sum of the squares of the roots of the equation \( x^2 - 7[x] + 5 = 0 \), where \([x]\) denotes the greatest integer function? | [
"92"
] | [
"What is the sum of the squares of the roots of the equation \\( x^2 - 7[x] + 5 = 0 \\), where \\([x]\\) denotes the greatest integer function?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-9c370594cef94f60978907699c1b58a9d4369284b80d816d5be9fc3d222552e1 | What is the sum of the tens digit and the ones digit of the integer form of $(2+3)^{23}$? | [
"7"
] | [
"What is the sum of the tens digit and the ones digit of the integer form of $(2+3)^{23}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be086581af5026e8f6c5cde3d47fdf37f6fdbd87ea0d882b3f1d63d5966702cc | What is the sum of the tens digit and the units digit in the decimal representation of $9^{2004}$? | [
"7"
] | [
"What is the sum of the tens digit and the units digit in the decimal representation of $9^{2004}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17988f352831704ea4ecfdeeb1b06146c5a30c4d635c996ce3bd4b1f1da18843 | What is the sum of the the roots of the equation $4x^3 + 5x^2 - 8x = 0$? Express your answer as a decimal to the nearest hundredth. | [
"-1.25"
] | [
"What is the sum of the the roots of the equation $4x^3 + 5x^2 - 8x = 0$? Express your answer as a decimal to the nearest hundredth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-545b31ff52a0bcec00b152241405f54909880a554c893ab3ed555884712e3b65 | What is the sum of the three last digits of \(2^{2001}\)? | [
"14"
] | [
"What is the sum of the three last digits of \\(2^{2001}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 42
} | 74 | 0 | 0 |
omr-ws-80c97b17cf1aec53d75b62f1c5c9dbbfea35b02f9d44a182b8e93e0d36e98eb6 | What is the sum of the three-digit numbers that are not werpa numbers? (Note: A number is a werpa number if it has at least two of the same digits and the sum of its digits is even.) | [
"427555"
] | [
"What is the sum of the three-digit numbers that are not werpa numbers? \n\n(Note: A number is a werpa number if it has at least two of the same digits and the sum of its digits is even.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8f120ae88acd1d222a40ba32144efec79020a108bb8cb397664e9b8a0a2ce89b | What is the sum of the two least natural numbers that each have exactly six different positive factors? | [
"30"
] | [
"What is the sum of the two least natural numbers that each have exactly six different positive factors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-20cb60ee6c0f09ce54e4ab1a8eea69d443df386b4bfbefee8e295e308e515d41 | What is the sum of the two solutions to the equation $54-15x-x^2=0$? | [
"-15"
] | [
"What is the sum of the two solutions to the equation $54-15x-x^2=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d9087d984c8b2dbbafb5e7be1fad3eef4b67c7820c5a85cd5a3f846099c7f14 | What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$? | [
"7"
] | [
"What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e31041f643c47b557e19cd8946d9cc2e2015b38ecebdd46b002ac1be5dfa1a8 | What is the sum of the values of $x$ such that $(x^2-9x+14)^{x^2-25x+46}=0$? | [
"7"
] | [
"What is the sum of the values of $x$ such that $(x^2-9x+14)^{x^2-25x+46}=0$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f29e2513717b4961984fada70d94fb40efb6751624099976668ac3b25cb0d20 | What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | [
"5"
] | [
"What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff3c1132aa6b469b39e675552974494fe235b70d5cc3a0473c1ac6d26c654535 | What is the sum of the whole-number factors of 24? | [
"60"
] | [
"What is the sum of the whole-number factors of 24?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-478f06b7f3c53861e63f2a1d55d8a09c3f0e05add6d13fea74968f501e3a603f | What is the sum of three consecutive even integers if the sum of the first and third integers is $128$? | [
"192"
] | [
"What is the sum of three consecutive even integers if the sum of the first and third integers is $128$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e797450cb9f96ae7d46f01bfe6071b9b3004b7f77c4d57f59ddfca648fe7397 | What is the sum of zero and all of the solutions of the equation \(\log_2 x \log_4 x \log_6 x = \log_2 x \log_4 x + \log_2 x \log_6 x + \log_4 x \log_6 x\)? | [
"49"
] | [
"What is the sum of zero and all of the solutions of the equation \\(\\log_2 x \\log_4 x \\log_6 x = \\log_2 x \\log_4 x + \\log_2 x \\log_6 x + \\log_4 x \\log_6 x\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-16281fcbf3b5e5b01729adfdf111dc60f1ec614f364ed318a06d4e4434d159ca | What is the sum to infinity of the series $\frac{1}{2}+\frac{1}{4}+\frac{2}{8}+\frac{3}{16}+\frac{5}{32}+\cdots$? | [
"2"
] | [
"What is the sum to infinity of the series $\\frac{1}{2}+\\frac{1}{4}+\\frac{2}{8}+\\frac{3}{16}+\\frac{5}{32}+\\cdots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-309e8d6792260114a727cd52cce502a937b5231c6101616f49f4ac1b9b57501f | What is the surface area generated by rotating the portion of the curve given by the parametric equations $x = a(\cos t)^{3}$ and $y = a(\sin t)^{3}$, for $0 \leq t \leq \frac{\pi}{2}$, about the x-axis? | [
"\\(\\frac{6\\pi a^2}{5}\\)"
] | [
"What is the surface area generated by rotating the portion of the curve given by the parametric equations $x = a(\\cos t)^{3}$ and $y = a(\\sin t)^{3}$, for $0 \\leq t \\leq \\frac{\\pi}{2}$, about the x-axis?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-edee25f4c896275e4d5800c21e6802eafc617b04cf78485cd7cb409a7724d7c4 | What is the surface area in square meters of a right rectangular pyramid with a height of 8 meters, a base width of 12 meters, and a base length of 30 meters? | [
"864"
] | [
"What is the surface area in square meters of a right rectangular pyramid with a height of 8 meters, a base width of 12 meters, and a base length of 30 meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-be81b9308db8f2cb25966119a7dead42e28c5c414a19d1754640cfdbb21e5c1d | What is the surface area of a $10 \times 10 \times 10$ cube if cubes of side length $2$ are sliced off of each corner, and each is replaced by a single unit cube? | [
"600"
] | [
"What is the surface area of a $10 \\times 10 \\times 10$ cube if cubes of side length $2$ are sliced off of each corner, and each is replaced by a single unit cube?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 6
} | 24 | 0 | 0 |
omr-ws-738b2be62b50f41460e7f95af3498a3a287319230fbbc4416073a174e81e049f | What is the surface area of a 3-dimensional model formed by vertically cutting a cone in half and attaching the apexes of each half, if the uncut cone had a radius of 4 cm and a height of 3 cm? | [
"\\( 24 + 56\\pi \\)"
] | [
"What is the surface area of a 3-dimensional model formed by vertically cutting a cone in half and attaching the apexes of each half, if the uncut cone had a radius of 4 cm and a height of 3 cm?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-117140be24c83e2f65904ae631a8e8296d3a43c1f92eb8da4767a2da826880c0 | What is the surface area of a sphere inscribed in a tetrahedron with side length eight? | [
"\\(\\frac{64\\pi}{3}\\)"
] | [
"What is the surface area of a sphere inscribed in a tetrahedron with side length eight?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2e8077ddfb959904e25b02be6193779ee564a32400803d44e07e732569d47bc | What is the surface area of a spherical segment (the part of a sphere between two parallel planes) cut from a sphere at latitudes 23.5 degrees north and 23.5 degrees south, excluding the areas of the flat circular bases? | [
"\\( 4 \\pi r^2 \\left(1 - \\cos\\left(\\frac{23.5 \\pi}{180}\\right)\\right) \\)"
] | [
"What is the surface area of a spherical segment (the part of a sphere between two parallel planes) cut from a sphere at latitudes 23.5 degrees north and 23.5 degrees south, excluding the areas of the flat circular bases?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-ee9863295e37a9b03f690dc760257aaa32172ac8a2ec8c3689fb019af93197d3 | What is the surface area of the Earth between 23.5 degrees north (Tropic of Cancer) and 23.5 degrees south (Tropic of Capricorn) latitude, if Earth is a perfect sphere with radius 3963 miles? | [
"78700000"
] | [
"What is the surface area of the Earth between 23.5 degrees north (Tropic of Cancer) and 23.5 degrees south (Tropic of Capricorn) latitude, if Earth is a perfect sphere with radius 3963 miles?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 5
} | 31 | 0 | 0 |
omr-ws-3ee54a61a8ccb4227e2ea8d6fb21d1c10b082a58fa7f82d23b5ba044ce5d7458 | What is the surface area of the portion of the cylinder $x^2 + y^2 = 9$ that lies on or above the $xy$ plane and on or below the plane $x + 2z = 8$? | [
"\\( 24\\pi \\)"
] | [
"What is the surface area of the portion of the cylinder $x^2 + y^2 = 9$ that lies on or above the $xy$ plane and on or below the plane $x + 2z = 8$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7c075ad868de11700a90960deecb0f2abc354630d214a3f8c332d98655344741 | What is the surface area of the solid generated when the curve $3y=\sqrt{x}(3-x)$, from $(0,0)$ to $(3,0)$, is rotated about the y-axis? | [
"\\(\\frac{28\\pi\\sqrt{3}}{5}\\)"
] | [
"What is the surface area of the solid generated when the curve $3y=\\sqrt{x}(3-x)$, from $(0,0)$ to $(3,0)$, is rotated about the y-axis?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-cac0e4df76a5f1c44eceee6244a8c032e8a86c2dc27527b9e9df01cdd2b948b9 | What is the surface defined by the equation $z = xy$? | [
"hyperbolic paraboloid"
] | [
"What is the surface defined by the equation $z = xy$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-1eb9de2bb142da1b048427da00bb78350811b0cd48133756379f5438efeb1b5c | What is the symmetric group of a four-dimensional hypercube? | [
"B_4"
] | [
"What is the symmetric group of a four-dimensional hypercube?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-1f2f849d6622158aa159a0b8e456945f130192fc8752849978390c53cefbcfef | What is the tens digit in the sum \(7! + 8! + 9! + \cdots + 2006!\)? | [
"4"
] | [
"What is the tens digit in the sum \\(7! + 8! + 9! + \\cdots + 2006!\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-7caded368fd50d823ac7006ccebe68f851e65b80b86c4a4b65299d3d0c282727 | What is the tens digit in the sum \(7! + 8! + 9! + \ldots + 2006!\)? | [
"4"
] | [
"What is the tens digit in the sum \\(7! + 8! + 9! + \\ldots + 2006!\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d283b17e87f0b37f2feb2e5fdd959d212df51375e7591c5863debac2fc7abc9b | What is the tens digit of $27^{27}$? (No calculators). | [
"0"
] | [
"What is the tens digit of $27^{27}$? (No calculators)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7942617dc8d9b58c12e912f7573d4b7b83dcacff58ad01b8ba9ac01f4bee9f7 | What is the tens digit of \(27^{27}\)? | [
"0"
] | [
"What is the tens digit of \\(27^{27}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-73a4d64fa0f0c1e83e89815af5df40bb07ef128cef1d958033c42b95970c3f04 | What is the tens digit of \(4^7 + 12^2 + 9^4\)? | [
"8"
] | [
"What is the tens digit of \\(4^7 + 12^2 + 9^4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-9d02bcbc5dd3ecd0a1a10145d7ec92466962b94b07b4f00a8d629b98ffc9f317 | What is the tens digit of \(6^{315} + 3^{11}\)? | [
"2"
] | [
"What is the tens digit of \\(6^{315} + 3^{11}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b182f08dbb217f53680b643793b300d6559a4183bc9b335410633a5023da2c0 | What is the tens digit of the product of all the odd numbers up to $1000$? | [
"2"
] | [
"What is the tens digit of the product of all the odd numbers up to $1000$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5fe05b6a6cf4e7741f62eab4f721c3f0a220446db9122fb26561023d0a006031 | What is the tens' digit of the product of all prime numbers between 1 and 100? | [
"7"
] | [
"What is the tens' digit of the product of all prime numbers between 1 and 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-c4f0ac94fe928847b070650d406a1583bace03103c7b7b5ec32880d78b725f42 | What is the tenth positive integer that is both odd and a multiple of 3? | [
"57"
] | [
"What is the tenth positive integer that is both odd and a multiple of 3?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5578d6623b7cb3df8e939ed04990b0f2f2fcc1f79e4432b67937bd13c614fb7e | What is the tenth term in the geometric sequence $9,3,1,\frac 13, \ldots$? | [
"\\frac{1}{2187}"
] | [
"What is the tenth term in the geometric sequence $9,3,1,\\frac 13, \\ldots$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6122d790c895a6d115e11131f40490255641f4327131691d4c524bcbd87492e2 | What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that each hemi-sphere's disc must not be visible within a 52.72077938642-degree cone of light extending from the central point of the disc? | [
"14"
] | [
"What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that each hemi-sphere's disc must not be visible within a 52.72077938642-degree cone of light extending from the central point of the disc?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b988422de9187153ce95f34c9a260ff71971a112777f9f342d8bc1287bc1a130 | What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that each hemi-sphere's disc must not be visible within a tubular beacon of light extending from the entire disc? | [
"31"
] | [
"What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that each hemi-sphere's disc must not be visible within a tubular beacon of light extending from the entire disc?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-8d3038c6a6270b463717aadc6db3a9a9b08471aa2bcb457886b95baf98cef5e8 | What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that the flat side (disc) of each hemi-sphere must not be visible from the central point of any other hemi-sphere's disc? | [
"4"
] | [
"What is the theoretical maximum number of 1m diameter hemi-spheres that can be crammed into a 5.5m diameter sphere, given that the flat side (disc) of each hemi-sphere must not be visible from the central point of any other hemi-sphere's disc?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-932c105191e70f1f615d59420a3435aeddf9b6e49fc3383125c81356a0453d8b | What is the third digit after the decimal point of the decimal representation of $\sqrt[3]{2+\sqrt 5} + \sqrt[3]{2-\sqrt 5}$? | [
"0"
] | [
"What is the third digit after the decimal point of the decimal representation of $\\sqrt[3]{2+\\sqrt 5} + \\sqrt[3]{2-\\sqrt 5}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f66296028d4e0279fd810d279b03ed09d3dc5a58d2ad73dcbea5dfc3b1733f5e | What is the third term of $x$ in the expansion of $(x-2y)^9$? | [
"\\(4x^7y^2\\)"
] | [
"What is the third term of $x$ in the expansion of $(x-2y)^9$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a25f8d16981a5034250e35635b3e216fadf6f807eee3f2aff107b05a11ce8c1 | What is the third year after 2007 where each digit is prime and the year itself is also prime? | [
"2333"
] | [
"What is the third year after 2007 where each digit is prime and the year itself is also prime?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-101731719c72b499930551a4e5406bed18d9c1ba92745f88bd11e14ab4a1b3db | What is the thousands' digit of \(101^{29}\)? | [
"2"
] | [
"What is the thousands' digit of \\(101^{29}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6e26381b2e9ea303b6c460a39acccb6bf17af967aa0aa3341cf948e04ddbaa6d | What is the thousandth digit after the decimal point in the square root of the number consisting of 1998 digits, all of which are 1? | [
"1"
] | [
"What is the thousandth digit after the decimal point in the square root of the number consisting of 1998 digits, all of which are 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-89d79adac0dc7699b2ebb329d20584fa1e9b2e8234ee0b504add752ed422c100 | What is the thousandth digit after the decimal point of the square root of the number consisting of 1998 consecutive 1s? | [
"1"
] | [
"What is the thousandth digit after the decimal point of the square root of the number consisting of 1998 consecutive 1s?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-396c2be3d3fcdc4f650bab34b8bb784c36a58b0bf370011c669c47aec2e2c325 | What is the thousandth digit of $(2+\sqrt{5})^{2012}$ after the decimal place? | [
"9"
] | [
"What is the thousandth digit of $(2+\\sqrt{5})^{2012}$ after the decimal place?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-d36408e0f705f65de1698f3050b46c6a3e2e6fd586000a8b45c1b37b52b6fb9a | What is the three-digit number that has the maximum number of factors? | [
"840"
] | [
"What is the three-digit number that has the maximum number of factors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-4bd0ad89e88e045bfa38e26e7c3eb9aec97fe0d42f9d8e350f2d8840ef44c401 | What is the time complexity of the series $\log(x) + 2\log(x/2) + 4\log(x/4) + 8\log(x/8) + \ldots$ until one of the terms becomes negative? | [
"\\( O(x \\log(x)) \\)"
] | [
"What is the time complexity of the series $\\log(x) + 2\\log(x/2) + 4\\log(x/4) + 8\\log(x/8) + \\ldots$ until one of the terms becomes negative?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-141798915831cfa0c24c002ac162fdf386a67e6d1777418fa23031d8d6bb104b | What is the total amount of water supplied per hour inside a circle of radius 13 feet, given that the sprinkler distributes water to a depth of \( e^{-r} \) feet per hour at a distance of \( r \) feet from the sprinkler? | [
"\\(-2\\pi (12 e^{-13} + 1)\\)"
] | [
"What is the total amount of water supplied per hour inside a circle of radius 13 feet, given that the sprinkler distributes water to a depth of \\( e^{-r} \\) feet per hour at a distance of \\( r \\) feet from the sprinkler?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-c87b3d03e1b960e08b4f217f10e89ce8e3b233da8ea548ee186b7460e2355276 | What is the total area between the curves \( y = \cos^{-1}(\sin x) \) and \( y = \sin^{-1}(\cos x) \) on the interval \([-9\pi, 9\pi]\)? | [
"\\( 9\\pi^2 \\)"
] | [
"What is the total area between the curves \\( y = \\cos^{-1}(\\sin x) \\) and \\( y = \\sin^{-1}(\\cos x) \\) on the interval \\([-9\\pi, 9\\pi]\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-a1a373664be30dce318268e592629a83ce96cafab3d7109cf80557bba92c3997 | What is the total area of figure 3, given that each figure has one more square than the previous figure and the side of the added square is equal to the diagonal of the largest square in the previous figure, with the square in figure 1 having an area of $2$ cm$^2$? | [
"11"
] | [
"What is the total area of figure 3, given that each figure has one more square than the previous figure and the side of the added square is equal to the diagonal of the largest square in the previous figure, with the square in figure 1 having an area of $2$ cm$^2$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-690f375a1304bc31815725b58aa4ba2b36fedac8853ff466ee048f8764437068 | What is the total area of figure 6, given that each figure has one more square than the previous figure and the side of the added square is equal to the diagonal of the largest square in the previous figure, with the square in figure 1 having an area of $2$ cm$^2$? | [
"17"
] | [
"What is the total area of figure 6, given that each figure has one more square than the previous figure and the side of the added square is equal to the diagonal of the largest square in the previous figure, with the square in figure 1 having an area of $2$ cm$^2$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-249203db46c872fee872b2cdf4d6aed766ececfbbb66b0433b6ca554fc2e2b41 | What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 6 units and lateral edges measuring 5 units? | [
"48"
] | [
"What is the total area, in square units, of the four triangular faces of a right, square-based pyramid that has base edges measuring 6 units and lateral edges measuring 5 units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c35241ca6d497e38e16284c09e8978acaa66afe5745514adeccf437b968fa54e | What is the total displacement of a student who walks 3 blocks east, 2 blocks north, 1 block west, and then 2 blocks south? | [
"2 blocks east"
] | [
"What is the total displacement of a student who walks 3 blocks east, 2 blocks north, 1 block west, and then 2 blocks south?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-07c3b5a3ee8c3c0d59fe1c4d488c838075b3c3cf0bdc6bac29c3595d409b86af | What is the total distance covered if awards are given to 15 players standing in a line with one meter between each, using the longest possible route? | [
"111"
] | [
"What is the total distance covered if awards are given to 15 players standing in a line with one meter between each, using the longest possible route?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 8,
"tir": 4
} | 30 | 0 | 0 |
omr-ws-80ea018c445a6c12ade09307f5a0a709a0d9739d5853a15ac72872285cab6518 | What is the total distance that Particle 2 travels in the interval $0 \leq t \leq \frac{1}{4}$? | [
"\\(1 - e^{-\\frac{1}{4}}\\)"
] | [
"What is the total distance that Particle 2 travels in the interval $0 \\leq t \\leq \\frac{1}{4}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d8a6db837737ce3ef2c31a88ed228d38d2e4bdce7eb2b758cbb7719b074d0d4 | What is the total distance traveled by a particle on a number line from \( t = 2 \) to \( t = 4 \), given its position is defined by the equation \( x = -t^2 + 7t + 19 \)? | [
"2.5"
] | [
"What is the total distance traveled by a particle on a number line from \\( t = 2 \\) to \\( t = 4 \\), given its position is defined by the equation \\( x = -t^2 + 7t + 19 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-48a29709534b5539603057eb672577d9bffe14e8767261418b3921866948b3a4 | What is the total gold-plated area in a unit square where the diagonal is covered by small golden squares with sides parallel to the unit square, and each neighbor's side length is either half or double of the previous square, with midpoints on the diagonal at distances $\dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \ldots... | [
"\\(\\frac{5}{27}\\)"
] | [
"What is the total gold-plated area in a unit square where the diagonal is covered by small golden squares with sides parallel to the unit square, and each neighbor's side length is either half or double of the previous square, with midpoints on the diagonal at distances $\\dfrac{1}{2}, \\dfrac{1}{4}, \\dfrac{1}{8}... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-1546596a8e38e6ac99d460a5ee2ca3f675ab4202249082dd7698fda0995bf904 | What is the total number of "rising numbers"? A "rising number" is defined as a positive integer with multiple digits such that each digit is larger than the one to its left. | [
"502"
] | [
"What is the total number of \"rising numbers\"? A \"rising number\" is defined as a positive integer with multiple digits such that each digit is larger than the one to its left."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 1
} | 33 | 0 | 0 |
omr-ws-cab1ea3d1663459baaa256738b351e0b5c46bdc2859891514b7c99267870eab8 | What is the total number of 4-digit numbers that can be formed using at most 2 distinct digits? | [
"576"
] | [
"What is the total number of 4-digit numbers that can be formed using at most 2 distinct digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-f74daa16039dea34109e7ad26bd10a3ba832e615f7259af419fc8ce836fe49c5 | What is the total number of Sylow 3-subgroups in a group of order 75? | [
"25"
] | [
"What is the total number of Sylow 3-subgroups in a group of order 75?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-48c37b1f52705f5561606955dc7736c3c5b391961637604a52d400a9541f0375 | What is the total number of digits used when the first 2002 positive even integers are written? | [
"7456"
] | [
"What is the total number of digits used when the first 2002 positive even integers are written?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b1b643541d6e2f020dc8a6b9bd055ca219ed907665be83d47788c34850a75f5 | What is the total number of glasses needed to build a pyramid where the top level has 1 × 2 glasses, the second level has 2 × 3 glasses, the third level has 3 × 4 glasses, and so on, up to the lower level which has 2001 × 2002 glasses? | [
"2,674,674,002"
] | [
"What is the total number of glasses needed to build a pyramid where the top level has 1 × 2 glasses, the second level has 2 × 3 glasses, the third level has 3 × 4 glasses, and so on, up to the lower level which has 2001 × 2002 glasses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c3491fd2a50ce2819b241d0bc5c044859c157dc6a0cc664f7d7426769b407d9b | What is the total number of integral solutions for $|x| + |y| + |z| \leq 20$? | [
"11521"
] | [
"What is the total number of integral solutions for $|x| + |y| + |z| \\leq 20$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-85e44745b601809094a99e49e23a8438704f3804e91518ce83f5e6733e140b5f | What is the total number of integral solutions to the equation \(abc = 24\)? | [
"120"
] | [
"What is the total number of integral solutions to the equation \\(abc = 24\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-c41499de6ff55a68fb635ed13d7ca36636496e0408806e84068184e4363d74cd | What is the total number of natural numbers divisible by 9, with the number of digits not exceeding 2008, and at least two of the digits being 9s? | [
"\\(\\frac{10^{2008} - 9^{2008} - 2008 \\cdot 9^{2007} + 8}{9}\\)"
] | [
"What is the total number of natural numbers divisible by 9, with the number of digits not exceeding 2008, and at least two of the digits being 9s?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-f470962eec98a29a60d908527cfa857901c9ad9a9cb48cbaca46ab88930807bf | What is the total number of one-digit factors for all of the integers from 1 to 100, inclusive? | [
"281"
] | [
"What is the total number of one-digit factors for all of the integers from 1 to 100, inclusive?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e15bc34f8023aa9a5959705967b55a5a70aac01aa40ca76666ebd439b507e84c | What is the total number of outcomes when $n$ identical coins, each with 3 possible states (head, tail, or standing straight), are flipped? | [
"$\\binom{n+2}{n}$"
] | [
"What is the total number of outcomes when $n$ identical coins, each with 3 possible states (head, tail, or standing straight), are flipped?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-de6840341cf73d269a527cbf9180c5ccef9b05b0b83e80edf679ad3c444db744 | What is the total number of people that should be present in a crowd to ensure that at least 6 people have the same birthday? | [
"1831"
] | [
"What is the total number of people that should be present in a crowd to ensure that at least 6 people have the same birthday?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-cd6f296c070ed6d9f2b6789ddc2e78ca36c1df6f43fdacca3012be75bbdd23d9 | What is the total number of permutations of \( n \) different things taken not more than \( r \) at a time, where each thing may be repeated any number of times? | [
"\\dfrac{n(n^{r} - 1)}{n - 1}"
] | [
"What is the total number of permutations of \\( n \\) different things taken not more than \\( r \\) at a time, where each thing may be repeated any number of times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-302f66b23ec8ae6d52ae598d2b228613340f11fe727ca5de1c641fcb0c305ab2 | What is the total number of possible outcomes in a 2x2 Rubik's cube after any two moves? | [
"114"
] | [
"What is the total number of possible outcomes in a 2x2 Rubik's cube after any two moves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-ffa13384bd344f843c74fb3580012f458188e2cee5a1584583122d40d6a66f84 | What is the total number of possible ways to make a total of 144 using only the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 in that order, with addition signs? | [
"4"
] | [
"What is the total number of possible ways to make a total of 144 using only the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 in that order, with addition signs?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-c13073cb4a6291eb333c6fbae7bbd410ac9fbf7a3db9704e7b261251ea1b8dda | What is the total number of real roots of the equation $(x^2+3x+4)^2+3(x^2+3x+4)+4=x$? | [
"0"
] | [
"What is the total number of real roots of the equation $(x^2+3x+4)^2+3(x^2+3x+4)+4=x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b25908b12006b6235ee4d67eaebfcfd210f4b2b8b691d2504ec5d192a74cc59 | What is the total number of rectangles that can be drawn on an \( m \times n \) chessboard? | [
"\\((m+1)(n+1)\\)"
] | [
"What is the total number of rectangles that can be drawn on an \\( m \\times n \\) chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-352cf1697227a7496a340d799710ca0d2998da6737ef1520e809874ea3e282c0 | What is the total number of rectangles that can be drawn on an \( n \times n \) chessboard? | [
"\\((n+1)^2\\)"
] | [
"What is the total number of rectangles that can be drawn on an \\( n \\times n \\) chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e53131808f84999e67b1afb191d4eb50668ecc1f82b7bd035666e55df08ffa1 | What is the total number of right angles formed by the edges of a cube? | [
"48"
] | [
"What is the total number of right angles formed by the edges of a cube?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e492afc1ff617f2d08c32bbb692cac03c2be9bb9d341bd55f47051580a3bf7cf | What is the total number of squares that can be drawn on an \( m \times n \) chessboard? | [
"\\dfrac{\\min(m, n) \\cdot (\\min(m, n) + 1) \\cdot (3 \\cdot \\max(m, n) - \\min(m, n) + 1)}{6}"
] | [
"What is the total number of squares that can be drawn on an \\( m \\times n \\) chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-a101121a0ca2385451de407a24b650f139b0fc07829ba67c0167ba074ea1b8e5 | What is the total number of subgroups of $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$? | [
"16"
] | [
"What is the total number of subgroups of $\\mathbb{Z}_2 \\times \\mathbb{Z}_2 \\times \\mathbb{Z}_2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-71cb6c052ae6f2ead413735271740e491ea7d2cce7fba1d024aa4d11b7fdc114 | What is the total number of subgroups of $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}$? | [
"16"
] | [
"What is the total number of subgroups of $\\mathbb{Z}_{2} \\times \\mathbb{Z}_{2} \\times \\mathbb{Z}_{2}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e99b599b48d22fe8f7e73471fbebb0b62918b7b9573dff8a0ba36630bfa4c25b | What is the total number of terms in the progression 86, 48, 24, 12, ..., 3/16? | [
"10"
] | [
"What is the total number of terms in the progression 86, 48, 24, 12, ..., 3/16?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 6
} | 54 | 0 | 0 |
omr-ws-9541c8ba91c717a6c0afa13055322385ceaf0d81d6830b3a2105be5e0c03516b | What is the total number of ways to distribute 20 identical biscuits among 8 students such that everyone gets at least one biscuit and exactly two of them get at least two biscuits? | [
"308"
] | [
"What is the total number of ways to distribute 20 identical biscuits among 8 students such that everyone gets at least one biscuit and exactly two of them get at least two biscuits?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c69c130d1df1c29f1f460aed8c83f1ddc7d8e2ace08fb3321b86750b351402f | What is the total number of ways to place $K$ bishops on an $N \times N$ chessboard such that no two bishops are in attacking positions? | [
"\\sum_{k=0}^{K} \\binom{N}{k}^2 \\binom{N-1}{K-k}^2 k! (K - k)!"
] | [
"What is the total number of ways to place $K$ bishops on an $N \\times N$ chessboard such that no two bishops are in attacking positions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-cbb988bb8bd5777cf51997a18adedd5b81ab88a13fb9508a9d07c85955d37b4d | What is the total number of ways to place $K$ knights on an $N \times N$ chessboard such that no two knights are in attacking positions? | [
"\\dbinom{\\left\\lceil \\frac{N^2}{2} \\right\\rceil}{K} + \\dbinom{\\left\\lfloor \\frac{N^2}{2} \\right\\rfloor}{K}"
] | [
"What is the total number of ways to place $K$ knights on an $N \\times N$ chessboard such that no two knights are in attacking positions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d4302700b69e8bf1a4a99d910dbd322c328b91b55c23265ec1caded99fe7d368 | What is the total radian measure of a circle with a 12-inch diameter? What is the formula used to find this measure? | [
"\\(2\\pi\\)"
] | [
"What is the total radian measure of a circle with a 12-inch diameter? What is the formula used to find this measure?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-2ed3ff243616fc4629da9d204e31b8cecf0b07178295c68ce156cc697bc38de3 | What is the total surface area of a cube with volume $27\sqrt{2}$? | [
"\\( 54 \\sqrt[3]{2} \\)"
] | [
"What is the total surface area of a cube with volume $27\\sqrt{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-599bfd3dab58a177278e735cbf596972d9f62e90ad614b8124d3f3aafdebb891 | What is the total value of an investment after 10 years if the initial investment is 500 dollars, annual deposits are 200 dollars, and the interest is compounded annually at 6%? | [
"3331.58"
] | [
"What is the total value of an investment after 10 years if the initial investment is 500 dollars, annual deposits are 200 dollars, and the interest is compounded annually at 6%?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc874921c91d10d568047faa11969db1d2e4938f0cb2c1b7e775757d9489387c | What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet? | [
"192"
] | [
"What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-734f4fd6bd1d992747b154b020f599d5f6995178c6ad398055c72f4f244a0dc8 | What is the triangle with the greatest area that can be inscribed in a Lemniscate of Bernoulli such that the sides of the triangle do not intersect the lemniscate and the triangle lies entirely on one side of the lemniscate? | [
"\\dfrac{a^2}{2}"
] | [
"What is the triangle with the greatest area that can be inscribed in a Lemniscate of Bernoulli such that the sides of the triangle do not intersect the lemniscate and the triangle lies entirely on one side of the lemniscate?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-154e271440fe350f2315b38b983b1e9df17beb26c508658e8de7d5bd4ad74c1e | What is the twenty-ninth term of the arithmetic sequence with \( a_1 = 13 \) and \( d = -\frac{5}{2} \)? | [
"-57"
] | [
"What is the twenty-ninth term of the arithmetic sequence with \\( a_1 = 13 \\) and \\( d = -\\frac{5}{2} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-ea55c39b269ab842176dd863b69e075f3ed7ad43c5dd154922b0dd265dc905df | What is the union of the closed intervals \([1/n, 2 - 1/n]\) as \(n\) ranges over the positive integers? | [
"\\((0, 2)\\)"
] | [
"What is the union of the closed intervals \\([1/n, 2 - 1/n]\\) as \\(n\\) ranges over the positive integers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-78ccfe896f9d14ddfbd0ed6967a505943461d78b350a5c7a2e9b7e4c9f722406 | What is the union of the closed intervals \([1/n, 2 - 1/n]\) for all positive integers \(n\)? | [
"(0, 2)"
] | [
"What is the union of the closed intervals \\([1/n, 2 - 1/n]\\) for all positive integers \\(n\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-67fc167bba2e8f968bd0d76c57078f414c8a907956dbd77716dc761e67ee68b3 | What is the union of the set \( A = \{1, 2, \{3, 4\}, \{5, 6\}\} \)? | [
"\\{1, 2, 3, 4, 5, 6\\}"
] | [
"What is the union of the set \\( A = \\{1, 2, \\{3, 4\\}, \\{5, 6\\}\\} \\)?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ae1cecab1a327cd7f686f322521b3cf01b60fab905c1b99fc24ce324720d50f3 | What is the unit digit of $\left\lfloor \frac{10^{20000}}{10^{100} + 3} \right\rfloor$? | [
"3"
] | [
"What is the unit digit of $\\left\\lfloor \\frac{10^{20000}}{10^{100} + 3} \\right\\rfloor$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-42ccf562ace8dc8ca7a9f2517fee3f319c0ef854fceddac97732b33d04e66d7b | What is the unit digit of $\left\lfloor \frac{10^{20000}}{10^{100}+3} \right\rfloor$? Where $\left\lfloor x \right\rfloor$ denotes the floor of $x$. | [
"3"
] | [
"What is the unit digit of $\\left\\lfloor \\frac{10^{20000}}{10^{100}+3} \\right\\rfloor$? Where $\\left\\lfloor x \\right\\rfloor$ denotes the floor of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ebaf5e37dab5289043f45c85daa11d4efd00da7807f7eb9a67831e7a88dc609 | What is the unit digit of the sum of the divisors (including $1$ and $2^{2022}$) of $2^{2022}$? | [
"7"
] | [
"What is the unit digit of the sum of the divisors (including $1$ and $2^{2022}$) of $2^{2022}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06ba95ceb7847022dd0273fa927575aba902f3db6b1d06e01e420bb299f7abcb | What is the unit's digit of $2^{100} + 3^{100}$? | [
"7"
] | [
"What is the unit's digit of $2^{100} + 3^{100}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-913d1d7ff27498a31cf208b8dfa158c89a3ca6ebef3f4ce49e39a1a89f4fe7cc | What is the unit's digit of \(1^1 + 2^2 + 3^3 + \ldots + 100^{100}\)? | [
"0"
] | [
"What is the unit's digit of \\(1^1 + 2^2 + 3^3 + \\ldots + 100^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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