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-a08b9306984f1b61145712d07dc0099abdc7172ee242e3352bc84aa95e01ee1e | What is the greatest three-digit number which is a multiple of 13? | [
"988"
] | [
"What is the greatest three-digit number which is a multiple of 13?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50e8808ac470ed9934b88b62ac18f613784aa6638aff345af707bc130c13a01b | What is the greatest three-digit positive integer \( n \) for which the sum of the first \( n \) positive integers is not a divisor of the product of the first \( n \) positive integers? | [
"996"
] | [
"What is the greatest three-digit positive integer \\( n \\) for which the sum of the first \\( n \\) positive integers is not a divisor of the product of the first \\( n \\) positive integers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 5
} | 38 | 0 | 0 |
omr-ws-8addb96b8637dbcb8e8f89714946d50a06ff3dd66f800505355d0b618e9eca96 | What is the greatest two-digit multiple of 13? | [
"91"
] | [
"What is the greatest two-digit multiple of 13?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92de10dc87e1b6fbb0f18ec84f35f04f1753bc882a3d4db7e05cb046cb2155db | What is the greatest two-digit prime factor of $\binom{100}{50}$? | [
"97"
] | [
"What is the greatest two-digit prime factor of $\\binom{100}{50}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1ea0c1a54f21218504b087e385d0da62b20f5fc359d2d91f16cbffe310add42d | What is the greatest two-digit whole number, the product of whose digits is 8? | [
"81"
] | [
"What is the greatest two-digit whole number, the product of whose digits is 8?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a838e0502c384b8e535d3ac02cc3babb68eea2be672979f923e14d4f8d584340 | What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x, y, z$ are real numbers? | [
"\\(\\frac{3}{2}\\)"
] | [
"What is the greatest value of $\\sin x \\cos y + \\sin y \\cos z + \\sin z \\cos x$, where $x, y, z$ are real numbers?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8a75dc4742b1984dc760fd42a92aa4b5d3f866a610adecaf33f9909ede21d9e6 | What is the greatest value of \( k \) such that the inequality \[ |a+b| + |b+c| + |c+a| \geq k(|a| + |b| + |c|) \] holds for all complex numbers \( a, b, c \)? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the greatest value of \\( k \\) such that the inequality\n\\[ |a+b| + |b+c| + |c+a| \\geq k(|a| + |b| + |c|) \\]\nholds for all complex numbers \\( a, b, c \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-da258509024ed13e92bb4af71e0131e8892c915b819caa42c3c827b5f5d06edb | What is the greatest value of \( n \) such that \( 58!_{12} \) is divisible by \( 16^n \)? | [
"16"
] | [
"What is the greatest value of \\( n \\) such that \\( 58!_{12} \\) is divisible by \\( 16^n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c157c6bb6cb385c1970bcf39b809ea2b5b12b9dc4e388d22df87fb00ce7308ed | What is the greatest value of \((\sin x)^8 + (\cos x)^8\)? | [
"1"
] | [
"What is the greatest value of \\((\\sin x)^8 + (\\cos x)^8\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-53e54ee8be822342b4aebcaf8db02d537ffd63aaf50ebedffe0df7f41b994251 | What is the greatest value of the expression for \( x \in \mathbb{N^{*}} \)? $$\frac{(x^2+5x+12)(x^2+5x-12)(x^2-5x+12)(-x^2+5x+12)}{x^4}$$ | [
"576"
] | [
"What is the greatest value of the expression for \\( x \\in \\mathbb{N^{*}} \\)?\n$$\\frac{(x^2+5x+12)(x^2+5x-12)(x^2-5x+12)(-x^2+5x+12)}{x^4}$$"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-b0a7efd71ae8992a06002702c26bc81ec6162201bbbed4a2c48ba58d96df7d24 | What is the greatest value of the function \( f(x) = (3 \sin(10x + 11))^2 - 7 \) as \( x \) varies over all real values? | [
"-40"
] | [
"What is the greatest value of the function \\( f(x) = (3 \\sin(10x + 11))^2 - 7 \\) as \\( x \\) varies over all real values?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-639f549970668ebe68be86b01ffcd6c98964f4c0ee6336889159a2609e7caba5 | What is the greatest value of the function \( f(x) = 3 \sin \left( (10x + 11)^2 - 7 \right)^2 \) as \( x \) varies over all real values? | [
"49"
] | [
"What is the greatest value of the function \\( f(x) = 3 \\sin \\left( (10x + 11)^2 - 7 \\right)^2 \\) as \\( x \\) varies over all real values?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a62a8bacee646adb69acd1940de2a0e434901c87aca119bbcf4825a9d259842 | What is the greatest value that cannot be generated by the expression $al + bm + cn$, where $a$, $b$, and $c$ are coefficients and $l$, $m$, and $n$ are variables? The conjectured formula is $lmn - lm - mn - nl + l + m + n$. | [
"lmn - lm - mn - nl + l + m + n"
] | [
"What is the greatest value that cannot be generated by the expression $al + bm + cn$, where $a$, $b$, and $c$ are coefficients and $l$, $m$, and $n$ are variables? The conjectured formula is $lmn - lm - mn - nl + l + m + n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5c760467c8cb0fc6791aa297787c6af841d436912f968de4a9af75828fd771e | What is the greatest value that cannot be obtained by adding the point values of five darts, where the possible point values are 1, 2, 4, and 6? | [
"27"
] | [
"What is the greatest value that cannot be obtained by adding the point values of five darts, where the possible point values are 1, 2, 4, and 6?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3329f59056e8fc8e9c3a6c8ebc87ae1b2eae34fde6f76ca716b8555de98f3f4 | What is the greatest value that the determinant of a 4x4 matrix, whose entries are only 0 and 1, can take? | [
"3"
] | [
"What is the greatest value that the determinant of a 4x4 matrix, whose entries are only 0 and 1, can take?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-fa319c988646b9b097143477a776000d05c6e5a63782f148a5bfb8a793d40552 | What is the greatest value that the ratio of the area of triangle $A_1B_1C_1$ to the area of triangle $ABC$ can take, where $A_1B_1C_1$ is formed by reflecting the vertices of the acute-angled triangle $ABC$ with respect to the opposite sides? | [
"4"
] | [
"What is the greatest value that the ratio of the area of triangle $A_1B_1C_1$ to the area of triangle $ABC$ can take, where $A_1B_1C_1$ is formed by reflecting the vertices of the acute-angled triangle $ABC$ with respect to the opposite sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9db34e1c0cbf1fb4cec6dbbeed96bedbf1e0056ab76889653aaf2517baf60a88 | What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers? | [
"6"
] | [
"What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-568929575f3e53cb49a206ece50bfafbb96d48b54508fc8d302f94ff7444a7e9 | What is the group associated with the polynomial $t^3 - 3t + 1$? | [
"A_3"
] | [
"What is the group associated with the polynomial $t^3 - 3t + 1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84174b8b77afe4144acf1c94b308a62ce9080254a7f279b6c404a40d946c5655 | What is the group generated by the matrices \( \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \) and \( \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix} \)? | [
"\\( SL_2(\\mathbb{Z}) \\)"
] | [
"What is the group generated by the matrices \\( \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix} \\) and \\( \\begin{bmatrix} 1 & -1 \\\\ 1 & 0 \\end{bmatrix} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-f6f755db9286773d6d66761e03e947684a1f8ad60b5d8dc8b37c4c2c1e7c1174 | What is the harmonic mean of 7, 8, and 3? | [
"\\(\\frac{504}{101}\\)"
] | [
"What is the harmonic mean of 7, 8, and 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-81f44d78a970c229abbfd332aa0a30313cf987a7ef2ecf2584bec08cbec32e43 | What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. | [
"49"
] | [
"What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8d63c8d96a02f5293aa04aec6c74b2dac6cc828a58bf788c39f0b45072f2b83 | What is the height of a regular pentagon with side length 1? | [
"\\(\\frac{\\sqrt{5+2\\sqrt{5}}}{2}\\)"
] | [
"What is the height of a regular pentagon with side length 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-438c4caaf497a08413e96239ba3a9e63ea90fb51b51c97dd0bd938830032d9a7 | What is the height of a right circular cone enclosed in a sphere of radius \( a \) that has the maximum volume? | [
"\\(\\frac{4a}{3}\\)"
] | [
"What is the height of a right circular cone enclosed in a sphere of radius \\( a \\) that has the maximum volume?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-0df228d9e76feb85f9b26fbab17832d98f7f91445441cc6009a15c93b910aad7 | What is the height of a right circular cone, cut from a solid sphere of radius \( a \), when its volume is maximum? The vertex and the circumference of the base of the cone are on the surface of the sphere. | [
"\\( \\frac{4a}{3} \\)"
] | [
"What is the height of a right circular cone, cut from a solid sphere of radius \\( a \\), when its volume is maximum? The vertex and the circumference of the base of the cone are on the surface of the sphere."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-23dad8a6406180e6121264633bc1a21fd28925c3953305ea49bea47f1c6c1b0d | What is the height of an upright circular cone with minimum volume that surrounds a sphere of radius 8? | [
"32"
] | [
"What is the height of an upright circular cone with minimum volume that surrounds a sphere of radius 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-69e372adf70bf4354c89881aeab427e53547b631a8a634476c74e1fc292e365b | What is the height of the tallest cylinder with a diameter of 9 that can fit inside a rectangular prism with interior dimensions $13 \times 45 \times 70$? | [
"70"
] | [
"What is the height of the tallest cylinder with a diameter of 9 that can fit inside a rectangular prism with interior dimensions $13 \\times 45 \\times 70$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3f74d2c9b97ad4ee643f2896b8a2cad934fbdf8cfadd515003b4c3d8d3685c15 | What is the hexadecimal representation of $ \frac{1}{5} $? | [
"\\( 0.3333333_{16} \\)"
] | [
"What is the hexadecimal representation of $ \\frac{1}{5} $?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-1f91b85a7a17c5b01cdd49cac7b77a53aefd1ef753202bfb350ed7024bd3e347 | What is the highest possible area of the triangle whose vertices are the midpoints of the inscribed circles of triangles \( APB \), \( BQC \), and \( ARC \), where \( APB \), \( BQC \), and \( ARC \) are constructed outside an equilateral triangle with area \( 1 \) such that each of these triangles has an angle of \( 6... | [
"1"
] | [
"What is the highest possible area of the triangle whose vertices are the midpoints of the inscribed circles of triangles \\( APB \\), \\( BQC \\), and \\( ARC \\), where \\( APB \\), \\( BQC \\), and \\( ARC \\) are constructed outside an equilateral triangle with area \\( 1 \\) such that each of these triangles h... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a4dc768bc3d68c64fbcb53758ac8962e8e2bab6596597c5c9a6dc779d6406791 | What is the highest possible area of triangle \( PQR \), where \( P \), \( Q \), and \( R \) are the vertices of triangles \( APB \), \( BQC \), and \( ARC \) respectively, constructed outside an equilateral triangle with area \( 1 \) such that each of these triangles has an angle of \( 60^\circ \) opposite to the side... | [
"4"
] | [
"What is the highest possible area of triangle \\( PQR \\), where \\( P \\), \\( Q \\), and \\( R \\) are the vertices of triangles \\( APB \\), \\( BQC \\), and \\( ARC \\) respectively, constructed outside an equilateral triangle with area \\( 1 \\) such that each of these triangles has an angle of \\( 60^\\circ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-392e7cc0c5a1119b1093ed45144fb5d52a45b7f274ecc0e53a41928cc6129f54 | What is the highest possible probability of getting 12 out of 20 multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one? The first question has 1 choice, the second question has 2 choices, and so on. | [
"\\dfrac{1}{798336000}"
] | [
"What is the highest possible probability of getting 12 out of 20 multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one? The first question has 1 choice, the second question has 2 choices, and so on."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-ec939165970c726555c33e016cac9f7413f30b1f3646bb1dbcbbfe389e97f907 | What is the highest possible value for the function \( f(x, y) = 6 \sin x \cos y + 2 \sin x \sin y + 3 \cos x \)? | [
"7"
] | [
"What is the highest possible value for the function \\( f(x, y) = 6 \\sin x \\cos y + 2 \\sin x \\sin y + 3 \\cos x \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-101a5d323a77a675cdcc2f6810f6d9fc3089050841079851044faa2fdfaccac3 | What is the highest power of $503$ that divides $\left(2013^{503^{2012}}-1\right)$? | [
"2013"
] | [
"What is the highest power of $503$ that divides $\\left(2013^{503^{2012}}-1\\right)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-85f580e952a56c154a50c8ea95e1ff1f43a690e992fed5698f410bfee605c096 | What is the highest power of $9$ that divides evenly into $36900!$? | [
"9222"
] | [
"What is the highest power of $9$ that divides evenly into $36900!$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 32
} | 51 | 0 | 0 |
omr-ws-55eb96c5745f5a3e04d9b3717f40dcad1440412e94e0ddb11c1735d4a460dd43 | What is the highest power of 14 that divides into \(500!\)? | [
"82"
] | [
"What is the highest power of 14 that divides into \\(500!\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83dc190e9bde28c9a6e68198b67acaef2b0b77d20730f2236eaf520912879563 | What is the highest power of 3 that will divide \(13^{204} - 1\)? | [
"2"
] | [
"What is the highest power of 3 that will divide \\(13^{204} - 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-06aadb64649643ba2c7863dfd4a6e26d39e1b4cfe1e88306a45a1f7bdeb64d9c | What is the highest value of $a + b + c$ if $a^2 + 4b^2 + 9c^2 - 2a - 12b + 6c + 2 = 0$? | [
"\\(\\frac{17}{3}\\)"
] | [
"What is the highest value of $a + b + c$ if $a^2 + 4b^2 + 9c^2 - 2a - 12b + 6c + 2 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-3b989258b55e3a599c949e42f5015a6adb3bc3f10ce4a54ea2630c9b36c8cde6 | What is the highest value that the side length \( x \) of an equilateral triangle can assume if a right triangle with sides 3, 4, and 5 is inscribed within it? The right triangle can be inscribed in any orientation, with its vertices on the sides of the equilateral triangle. | [
"5"
] | [
"What is the highest value that the side length \\( x \\) of an equilateral triangle can assume if a right triangle with sides 3, 4, and 5 is inscribed within it? The right triangle can be inscribed in any orientation, with its vertices on the sides of the equilateral triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-6791135d93751de034a489fc56dad616063952744353fb4b67ee06fa18f28a08 | What is the holomorphism group of $\mathbb{Z}_n$, denoted by $Hol(\mathbb{Z}_n)$, where $Hol(G) = G \rtimes Aut(G)$, the semi-direct product via the identity map of $Aut(G) \rightarrow Aut(G)$? | [
"$\\mathbb{Z}_n \\rtimes (\\mathbb{Z}_n)^*$"
] | [
"What is the holomorphism group of $\\mathbb{Z}_n$, denoted by $Hol(\\mathbb{Z}_n)$, where $Hol(G) = G \\rtimes Aut(G)$, the semi-direct product via the identity map of $Aut(G) \\rightarrow Aut(G)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b30ca87b7dd41ec0b8063e1a3f987e23caa468b8d0ba4a15afc39db21af3a25d | What is the hundreds digit of $(20! - 15!)$? | [
"0"
] | [
"What is the hundreds digit of $(20! - 15!)$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e5627c95ec2b5eca041662183beb079114e4be62e883f37d1336df731c8b758 | What is the hundreds digit of $2011^{2011}$? | [
"6"
] | [
"What is the hundreds digit of $2011^{2011}$?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 24
} | 55 | 0 | 0 |
omr-ws-f846f97c95fe9acabdc8a3d072c7cb853ae438a68f9524b28c91d43e7fef3852 | What is the hundreds digit of $2015^{1337}$? | [
"3"
] | [
"What is the hundreds digit of $2015^{1337}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed47220475eed0a5d6f8b1c14856f1150d604c2d7ed100cfb071434e83d9aa6f | What is the image (reflection) of the curve \( y^2 = 4x \) about the line \( x + y + 2 = 0 \)? | [
"(x + 2)^2 = -4(y + 2)"
] | [
"What is the image (reflection) of the curve \\( y^2 = 4x \\) about the line \\( x + y + 2 = 0 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ee42fdbb9116431e256fda3c7f9f03a8bf13ae14867436a20507fa4d1d2ffbc1 | What is the image of $\sqrt{2}$ under an involutive automorphism of $\mathbb{C}$? Are there only two options: the same or the opposite? | [
"\\(\\sqrt{2}\\)"
] | [
"What is the image of $\\sqrt{2}$ under an involutive automorphism of $\\mathbb{C}$? Are there only two options: the same or the opposite?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8dc9a863d0994f2cd18bca625072fa046e09678d56b32733b00d42a723e7bf7b | What is the image of (-2,5) after a reflection in the origin? | [
"(2, -5)"
] | [
"What is the image of (-2,5) after a reflection in the origin?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-f475bbb53697e595c4c3067e2eb3f3b4951f670668f79919726c6ecd84179909 | What is the image of the curve \( y^2 = 4x \) about the line \( x + y + 2 = 0 \)? | [
"\\( (x+2)^2 = -4(y+2) \\)"
] | [
"What is the image of the curve \\( y^2 = 4x \\) about the line \\( x + y + 2 = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-74baee8f222b9b1bf78e3a97a8271cde2c787a720b6ef18cfd6867b636c6d406 | What is the image of the first quadrant under the map $w = z^2$ where $z = x + iy$? Specifically, determine the transformation of the coordinates $(u, v)$ where $u = x^2 - y^2$ and $v = 2xy$. | [
"v \\geq 0"
] | [
"What is the image of the first quadrant under the map $w = z^2$ where $z = x + iy$? Specifically, determine the transformation of the coordinates $(u, v)$ where $u = x^2 - y^2$ and $v = 2xy$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-c32f9d4f20580d423cfa3f7f8fb12f62375e77520b26c5e4470fa962b49fe1d6 | What is the image of the open unit disc \( D \) under the sine function, i.e., \( \sin(D) \)? | [
" \\left\\{ w \\in \\mathbb{C} \\,\\big|\\, | \\arcsin(w) | < 1 \\right\\} "
] | [
"What is the image of the open unit disc \\( D \\) under the sine function, i.e., \\( \\sin(D) \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-0e103b098fcc3deedbd34dddace387bfb2eaf067a48e2fae855633e467ca4622 | What is the image of the open unit disc \( D \) under the tangent function, i.e., \( \tan(D) \)? | [
"\\mathbb{C} \\setminus \\{ iy \\mid |y| \\geq \\tanh(1) \\}"
] | [
"What is the image of the open unit disc \\( D \\) under the tangent function, i.e., \\( \\tan(D) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-5e6ce851bb5def84967c59a74a6d3f284ae588a390f1f8fbc51ea31474f7dd75 | What is the image of the open unit disc under a map of the form \( f(z) = \frac{1}{z} (z-a_1)^{b_1} (z-a_2)^{b_2} \cdots (z-a_n)^{b_n} \), where \( |a_i| = 1 \), \( 0 < b_i < 1 \), and the sum of the \( b_i \)'s is 2? | [
"\\mathbb{C} \\setminus \\{0\\}"
] | [
"What is the image of the open unit disc under a map of the form \\( f(z) = \\frac{1}{z} (z-a_1)^{b_1} (z-a_2)^{b_2} \\cdots (z-a_n)^{b_n} \\), where \\( |a_i| = 1 \\), \\( 0 < b_i < 1 \\), and the sum of the \\( b_i \\)'s is 2?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-f83c4303d7daa82ae14f60e60ba4f6f6c5d0e69c21edf5f0d268a737599a3a34 | What is the image of the point (1, 0) after a clockwise rotation of 30 degrees? | [
"\\(\\left( \\frac{\\sqrt{3}}{2}, \\frac{1}{2} \\right)\\)"
] | [
"What is the image of the point (1, 0) after a clockwise rotation of 30 degrees?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-008f6c147424326a9176abc3e50474360c8293386aee123910aa1f18129dafb7 | What is the image of the point \((m, n)\) after reflection across the line \(y = 2x\)? | [
"\\(\\left(\\frac{-3m+4n}{5}, \\frac{4m+3n}{5}\\right)\\)"
] | [
"What is the image of the point \\((m, n)\\) after reflection across the line \\(y = 2x\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-c346ad8a015e4ca96727cb2eab09298806468b8efd69c9427ffd70e8ac35e95f | What is the image of the unit circle under the transformation \( z \mapsto (z-1)^2 \) in the complex plane? | [
"(x^2 + y^2 - 2x)^2 = 4(x^2 + y^2)"
] | [
"What is the image of the unit circle under the transformation \\( z \\mapsto (z-1)^2 \\) in the complex plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70f9169c9714d5dbc8466d97deebf205ea8f664dd07959417a9dfc53ffe4f428 | What is the imaginary part of $\tan^{-1}(5i/3)$? | [
"\\ln 2"
] | [
"What is the imaginary part of $\\tan^{-1}(5i/3)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-337dd640b44684590c5302eff63dfc653e50dcb78b9520d06bb24c1029a45de9 | What is the imaginary part of the product of the 10th roots of \(7 - 8i\)? | [
"8"
] | [
"What is the imaginary part of the product of the 10th roots of \\(7 - 8i\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9f67398f381d353a0a804eae0fb956bf1c743b1e87d2648f55fbce41a68e32c | What is the imaginary part of the square root of \(4 - 3i\) which has a positive real part? | [
"\\(-\\frac{\\sqrt{2}}{2}\\)"
] | [
"What is the imaginary part of the square root of \\(4 - 3i\\) which has a positive real part?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-4db7e03c8570a9688282e238b2b6f4c0492400a450434a07a7466f6f25ed7caf | What is the indefinite integral of $r e^{r}$ with respect to $r$? | [
"e^{r}(r - 1) + C"
] | [
"What is the indefinite integral of $r e^{r}$ with respect to $r$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-333e95ebd1c26dee9a945df3cc770f1ad03f91d2e87f023b02a912125232c80f | What is the indefinite integral of \( x^2 e^{-ax^2 + bx + c} \)? | [
" -\\frac{2 a x + b}{4 a^2} e^{-a x^2 + b x + c} + \\frac{\\sqrt{\\pi} (2a + b^2) e^{c + \\frac{b^2}{4a}}}{8 a^{5/2}} \\text{erf}\\left( \\sqrt{a} \\left( x - \\frac{b}{2a} \\right) \\right) + C "
] | [
"What is the indefinite integral of \\( x^2 e^{-ax^2 + bx + c} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-c3b7fc2b6d510bae8a391163ad4f961ffc91fa20c8324c3298edeecc5b4745dd | What is the indefinite integral of \(\frac{\ln(x)}{1+x}\)? | [
"\\ln(x) \\ln(1+x) + \\text{Li}_2(-x) + C"
] | [
"What is the indefinite integral of \\(\\frac{\\ln(x)}{1+x}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e7777b30b86d1812555d5f5b93ae4b8d608cfb8b44cfdffee03891dca36e1f0e | What is the indefinite integral of the floor function $\int [x] \, dx$? | [
"nx + C for x in [n, n+1)"
] | [
"What is the indefinite integral of the floor function $\\int [x] \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8911c0826c2b0b3965cb3f76b48714c7127bd70dc8e6afe873cc8923ba33e66d | What is the induction of the alternating representation of \( A_3 \) to \( S_3 \)? | [
"The induction of the alternating representation of \\( A_3 \\) to \\( S_3 \\) is the direct sum of the trivial representation and the 2-dimensional irreducible representation of \\( S_3 \\)."
] | [
"What is the induction of the alternating representation of \\( A_3 \\) to \\( S_3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-331c49932b70ad4097e64e45ca547bb30ed9723bed75d9b4d1408f78bb383e33 | What is the infimum (greatest lower bound) of the circumradius of a triangle that contains a point \( P \) in its interior, where the distances from \( P \) to the three vertices are 3, 4, and 5? | [
"\\(\\frac{37}{10}\\)"
] | [
"What is the infimum (greatest lower bound) of the circumradius of a triangle that contains a point \\( P \\) in its interior, where the distances from \\( P \\) to the three vertices are 3, 4, and 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 3
} | 6 | 0 | 0 |
omr-ws-c93b919d2b400873e474ab7f677ebdd091402ff3ea47726134a0380c0de1e5a5 | What is the infimum of \[g(a,b,c,d)=\dfrac{1}{a^2+4} + \dfrac{1}{b^2+4} + \dfrac{1}{c^2+4} + \dfrac{1}{d^2+4}\] in the set \( S = \{(a,b,c,d) : a,b,c,d \geq 0, ab + ac + ad + bc + bd + cd = 6\} \)? | [
"\\(\\frac{7}{10}\\)"
] | [
"What is the infimum of \n\\[g(a,b,c,d)=\\dfrac{1}{a^2+4} + \\dfrac{1}{b^2+4} + \\dfrac{1}{c^2+4} + \\dfrac{1}{d^2+4}\\]\nin the set \\( S = \\{(a,b,c,d) : a,b,c,d \\geq 0, ab + ac + ad + bc + bd + cd = 6\\} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 9
} | 56 | 0 | 0 |
omr-ws-2717ecfb94952b22c24a0b9b7690e29f2a2be30e71b11f46e1e23f9b6f3d6888 | What is the instantaneous rate of change at \( x = 2 \) of the function \( f(x) = \frac{x^2 - 4}{x - 2} \)? | [
"2"
] | [
"What is the instantaneous rate of change at \\( x = 2 \\) of the function \\( f(x) = \\frac{x^2 - 4}{x - 2} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4eb1669ae56558065cfb9cd9996ff28026af20fa22e4b9d340ff6345a79025ef | What is the integer part of \(\left(2 + \sqrt{3}\right)^{2019}\)? | [
"\\((2+\\sqrt{3})^{2019} + (2-\\sqrt{3})^{2019} - 1\\)"
] | [
"What is the integer part of \\(\\left(2 + \\sqrt{3}\\right)^{2019}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-a5de2ada504f3064cac23dd581756081c3bda824e6072db57b77001c7103247c | What is the integer part of \[ \frac{1}{\frac{1}{1990} + \frac{1}{1991} + \cdots + \frac{1}{2008}}? \] | [
"105"
] | [
"What is the integer part of \n\n\\[\n\\frac{1}{\\frac{1}{1990} + \\frac{1}{1991} + \\cdots + \\frac{1}{2008}}?\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-775990f037b19ca3f0437ec8fef8df5e2faf0c45e4ccb247d8fd6c65e95647e8 | What is the integer value of $y$ in the arithmetic sequence $2^2, y, 2^4$? | [
"10"
] | [
"What is the integer value of $y$ in the arithmetic sequence $2^2, y, 2^4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8bf9c09b070ca5a9f32a96952ddd68105b2a0af530f9ef08b141b6cb18a0e38 | What is the integral from $1$ to $x^2$ of $\sqrt{1+t^3}$? | [
"\\(2x \\sqrt{1+x^6}\\)"
] | [
"What is the integral from $1$ to $x^2$ of $\\sqrt{1+t^3}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb17b4df6d02b9a11c19764b6c9ef0995562d6b0b8c170bbdda8fc40fe2ead0a | What is the integral of $\cos(x) \ln(x)$? | [
"\\ln(x) \\sin(x) - \\text{Si}(x) + C"
] | [
"What is the integral of $\\cos(x) \\ln(x)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1718e95bead94bf99a2f18f28b3a1ec0d7345784c2b1a10d5eb9a606e50c2295 | What is the integral of $\frac{1}{\sin(x)}$? | [
"\\(\\log \\left| \\tan \\frac{x}{2} \\right| + C\\)"
] | [
"What is the integral of $\\frac{1}{\\sin(x)}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-af4c99aede29b304e37183e7af603d48a2640c7093d1e3ff5f7bbcf0a7246e97 | What is the integral of $\frac{1}{x \ln(x)} \, dx$? | [
"\\(\\ln|\\ln(x)| + C\\)"
] | [
"What is the integral of $\\frac{1}{x \\ln(x)} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-2410fe6582f8d2878989ecb4e266eec5f09fc9bd4099593abb232ad2613bd18e | What is the integral of $\ln(x)\ln(1-x)$ from 0 to 1 with respect to $x$? | [
"\\(2 - \\frac{\\pi^2}{6}\\)"
] | [
"What is the integral of $\\ln(x)\\ln(1-x)$ from 0 to 1 with respect to $x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a08d2f8121d1948d5fb7e984dae2ac3030320e331a041a3d8203f6332ac874b2 | What is the integral of $cos(x)cos(x^2)$ from $x=0$ to infinity? | [
"\\(\\frac{1}{2} \\sqrt{\\frac{\\pi}{2}} \\left(\\sin \\frac{1}{4} + \\cos \\frac{1}{4} \\right)\\)"
] | [
"What is the integral of $cos(x)cos(x^2)$ from $x=0$ to infinity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-959741799ad4e48db907333ab981499de0bce9b23fa8cd605de6c60045f10749 | What is the integral of $|\cos x|$ with respect to $x$? | [
"\\( F(x) = (-1)^k \\sin x + 2k + C, \\quad \\text{for } -\\frac{\\pi}{2} + \\pi k \\le x \\le \\frac{\\pi}{2} + \\pi k, \\ k \\in \\mathbb{Z} \\)"
] | [
"What is the integral of $|\\cos x|$ with respect to $x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 16
} | 39 | 0 | 0 |
omr-ws-fb98f1349e620c80733e311119c0eb569995c1c71ca70ccaf9d27da2129415da | What is the integral of \( \frac{1}{x \ln(x)} \)? | [
"\\ln|\\ln(x)| + C"
] | [
"What is the integral of \\( \\frac{1}{x \\ln(x)} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-377ecc9ad7cc10b9a99234c65dacd9fda840761f6895690fb214b44f3aa72dd9 | What is the integral of \( x^x \, dx \)? | [
"\\sum_{n=0}^{\\infty} \\frac{x^{n+1} (\\ln x)^n}{n! (n+1)} + C"
] | [
"What is the integral of \\( x^x \\, dx \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-5517d66d290c2b28e2fa52f957dc2bc55064b1e7ff98cbf69582833cec0a10ee | What is the integral of \((3 + x^3)^{-1.66}\)? | [
"\\frac{x \\, _2F_1\\left(\\frac{1}{3}, \\frac{83}{50}; \\frac{4}{3}; -\\frac{x^3}{3}\\right)}{3^{83/50}} + C"
] | [
"What is the integral of \\((3 + x^3)^{-1.66}\\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-2f3dc742b594d2bc503463946ad13377129e43ac9938c54ae83e4797985d6a1c | What is the integral of \(1 - x \cot x\)? | [
"x - x \\ln |\\sin x| + \\int \\ln |\\sin x| \\, dx + C"
] | [
"What is the integral of \\(1 - x \\cot x\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-5807c88a24cb9e19583c31af2a3087d2d5be06d64b8c96f91e301bfdb85dbbbe | What is the integral of \(\frac{\cos x}{x}\)? Please show the workings. | [
"\\(\\pi\\)"
] | [
"What is the integral of \\(\\frac{\\cos x}{x}\\)? Please show the workings."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f72a23adf9af890875735d7a6b997cede004301bfdb77dcd0b3aa24550eb748 | What is the integral of the function \(\frac{a^x}{x}\), where \(a\) is a constant? | [
"\\(Ei(x \\ln a)\\)"
] | [
"What is the integral of the function \\(\\frac{a^x}{x}\\), where \\(a\\) is a constant?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-32ee21b3abea3400e6a11e679b95e49536ae0302b0b071d3ec9adca444109a8d | What is the integral part of \( M \) if \[ M = \sqrt{2012 \times \sqrt{2013 \times \sqrt{2014 \times \sqrt{\cdots \sqrt{(2012^2-3) \times \sqrt{(2012^2-2) \times \sqrt{(2012^2-1)}}}}}}} \]? | [
"2012"
] | [
"What is the integral part of \\( M \\) if \n\\[ M = \\sqrt{2012 \\times \\sqrt{2013 \\times \\sqrt{2014 \\times \\sqrt{\\cdots \\sqrt{(2012^2-3) \\times \\sqrt{(2012^2-2) \\times \\sqrt{(2012^2-1)}}}}}}} \\]?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-c92cc97a51097ced479b699d925c1f3fedfc713dfb24520a58bdd8624d4eecba | What is the interior of the set of diagonalizable matrices in \( M_n(\mathbb{C}) \)? | [
"The interior consists of diagonalizable matrices with distinct eigenvalues."
] | [
"What is the interior of the set of diagonalizable matrices in \\( M_n(\\mathbb{C}) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a2fbd5ce2a15443208092ca5e6cc443dc4252b13a0b846d6f4db32376ea84f00 | What is the interior of the set of diagonalizable matrices in \( M_n(\mathbb{R}) \)? | [
"The interior consists of matrices with distinct eigenvalues."
] | [
"What is the interior of the set of diagonalizable matrices in \\( M_n(\\mathbb{R}) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-b4e2201e4ee28c04da960aa8e6c39c8dd94e1fc2aff52a72f812e46efa7ed3f1 | What is the intersection of the lines given by $2y=-x+3$ and $-y=5x+1$? Enter the answer as an ordered pair. | [
"\\left(-\\frac{5}{9}, \\frac{16}{9}\\right)"
] | [
"What is the intersection of the lines given by $2y=-x+3$ and $-y=5x+1$? Enter the answer as an ordered pair."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ec2ed77a919f155c162be3e0a2d40f6425cea853cb948cb3c6177c2a0f9f13eb | What is the intersection of the lines given by $y=-4x$ and $y-2=12x$? Express your answer as an ordered pair, with both coordinates expressed as common fractions. | [
"\\left(-\\frac{1}{8}, \\frac{1}{2}\\right)"
] | [
"What is the intersection of the lines given by $y=-4x$ and $y-2=12x$? Express your answer as an ordered pair, with both coordinates expressed as common fractions."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4c30473eb5d8a4923c282874a39f5b5faab88ccbb860cffb365bbf5d1d13c03 | What is the intersection of the open intervals \((-1/n, 1 + 1/n]\) as \(n\) ranges over the positive integers? | [
"[0,1]"
] | [
"What is the intersection of the open intervals \\((-1/n, 1 + 1/n]\\) as \\(n\\) ranges over the positive integers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-0ebf3a59ecfc7ee101a4d3c2497082174116c68c55967591e68cb6f3c7f8c935 | What is the intersection of the open intervals \((-1/n, 1 + 1/n]\) for all positive integers \(n\)? | [
"[0,1]"
] | [
"What is the intersection of the open intervals \\((-1/n, 1 + 1/n]\\) for all positive integers \\(n\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-e5eb24acdd0d4dd4c912725d2cefb0dcacc1e7bc5d9e1eb04c1b742fb9e63411 | What is the intersection point of the line $y = 2x + 5$ and the line perpendicular to it that passes through the point $(5, 5)$? | [
"(1, 7)"
] | [
"What is the intersection point of the line $y = 2x + 5$ and the line perpendicular to it that passes through the point $(5, 5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73fa1ae15f9bbccb2d8509468ba130bc830935817110708ed58510b8c6c3fd1b | What is the interval for the diameter of a round hole that can be completely covered by two square boards, each with a side length of 1 metre? | [
"\\((0, 4 - 2\\sqrt{2}]\\)"
] | [
"What is the interval for the diameter of a round hole that can be completely covered by two square boards, each with a side length of 1 metre?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1832e51d0d7654081ae954ff916626f44c67c358936eff259df2b83be8c9e72 | What is the interval of positive $ k $ such that the quartic $ x^4 + 10x^3 + kx^2 + 10x + 1 $ has all real roots? | [
"[18, 27]"
] | [
"What is the interval of positive $ k $ such that the quartic $ x^4 + 10x^3 + kx^2 + 10x + 1 $ has all real roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a68efc7c056567d89fd9b32e7fa8f4d89458e57737a910c49a1021b31da2ce82 | What is the interval of validity for the initial value problem $(x-100)y' + 2y = \ln |x|$, with the initial condition $y(1) = 5$? | [
"\\(]0, 100[\\)"
] | [
"What is the interval of validity for the initial value problem $(x-100)y' + 2y = \\ln |x|$, with the initial condition $y(1) = 5$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-8a933dd25aa758ef23028a1e9ed06fb30150e2a7632d947fec28fcca60f2845e | What is the inverse Laplace transform of \( F(s) = s \)? | [
"\\( \\delta^{\\prime}(t) \\)"
] | [
"What is the inverse Laplace transform of \\( F(s) = s \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-17be9f92ed8ea091c9cdb01269b12a6aa362fa9914f092c11d434301ece8043c | What is the inverse function $f^{-1}(x)$ given $f(x) = x + \lfloor x \rfloor$? | [
"\\( f^{-1}(x) = x - \\left\\lfloor \\frac{x}{2} \\right\\rfloor \\)"
] | [
"What is the inverse function $f^{-1}(x)$ given $f(x) = x + \\lfloor x \\rfloor$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-c734a8e623349b12f5fb9f3322eab415bff8e5910b4faf2fec8713da6fad89c9 | What is the inverse function of $f(x) = x^3 + x$? | [
"\\( f^{-1}(x) = \\sqrt[3]{\\sqrt{\\left(\\frac{x}{2}\\right)^2 + \\frac{1}{27}} + \\frac{x}{2}} - \\sqrt[3]{\\sqrt{\\left(\\frac{x}{2}\\right)^2 + \\frac{1}{27}} - \\frac{x}{2}} \\)"
] | [
"What is the inverse function of $f(x) = x^3 + x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 20
} | 68 | 0 | 0 |
omr-ws-7f190e40e468a7ec4902ed69e7ac229f0b7ca90c20351aa85eebbad5288a6940 | What is the inverse function of \( Y = 3x + \cos(x) \)? | [
"x = 3y + \\cos(y)"
] | [
"What is the inverse function of \\( Y = 3x + \\cos(x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-fc759e49d5cb4eb244bf2a9ada35c6f6e332d6b0d30bedcc90bfb8980db2e8ff | What is the inverse function of \( f(x) = x^3 + x \)? | [
"\\( y=\\sqrt[3]{\\frac{x}{2}\\pm\\sqrt{\\frac{x^2}{4}+\\frac{1}{27}}}+\\sqrt[3]{\\frac{x}{2}\\mp\\sqrt{\\frac{x^2}{4}+\\frac{1}{27}}} \\)"
] | [
"What is the inverse function of \\( f(x) = x^3 + x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-89ba831c9081d2d871da84f17fc6bcba971b99e12723562ff6d6d53bd26dac2e | What is the inverse matrix of the matrix $A[k]_{m,n}=e^{(2\pi imn)/k}$ as a function of $k$? | [
"\\( A^{-1} = \\frac{1}{k} \\overline{A} \\)"
] | [
"What is the inverse matrix of the matrix $A[k]_{m,n}=e^{(2\\pi imn)/k}$ as a function of $k$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-611846b2d869c908d23278fa55ab05cee068cc1b485ed52997b7a5612b4ebbe3 | What is the inverse of $f(x)=4-5x$? | [
"\\frac{4-x}{5}"
] | [
"What is the inverse of $f(x)=4-5x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bfe31218da00fa9ad087610715076bf8efb97d91ddc48bfcf0c1bb68a7648255 | What is the inverse of the function \( f(x) = 2^{\frac{1}{x}} + 3^{\frac{1}{x}} + 5^{\frac{1}{x}} \)? | [
"f^{-1}(x) \\text{ cannot be expressed with elementary functions}"
] | [
"What is the inverse of the function \\( f(x) = 2^{\\frac{1}{x}} + 3^{\\frac{1}{x}} + 5^{\\frac{1}{x}} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-81c64f235a5c96c2a1573f3030397c374fce3d704a109ad6947dd8e93f99caa9 | What is the inverse of the function \( f(x) = 3 + x + e^x \)? | [
"\\( f^{-1}(y) = y - 3 - W(e^{y-3}) \\)"
] | [
"What is the inverse of the function \\( f(x) = 3 + x + e^x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-866344be9628474e9d6ad16d76570d4da31b365ec9fee9ef86cdd7b4c7ef6b2c | What is the inverse of the function \( f: (2, +\infty) \to \mathbb{R} \), defined by \( f(x) = \frac{2x^2 - 7x - 9}{x - 2} \)? | [
"f^{-1}(x) = \\dfrac{x + 7 + \\sqrt{x^2 - 2x + 121}}{4}"
] | [
"What is the inverse of the function \\( f: (2, +\\infty) \\to \\mathbb{R} \\), defined by \\( f(x) = \\frac{2x^2 - 7x - 9}{x - 2} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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