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-904cad754f6794b2cc14bf15603c97b952cf402c21e1db770d1ce932ab56dd62 | What are the conditions for \( n \) so that \( a, b, c, n, \) and \( d \) are all positive integers in the expressions: \[ a^2 + b^2 = c^2 \] \[ (a+n)^2 + (b+n)^2 = d^2 \] | [
"\\( n = p^2 + 2s^2 \\) where \\( p \\) and \\( s \\) are integers chosen by us.\n\nThis solution is derived from the system of equations provided in Post 7, which gives a specific form for \\( n \\) in terms of integers \\( p \\) and \\( s \\)."
] | [
"What are the conditions for \\( n \\) so that \\( a, b, c, n, \\) and \\( d \\) are all positive integers in the expressions:\n\\[ a^2 + b^2 = c^2 \\]\n\\[ (a+n)^2 + (b+n)^2 = d^2 \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1f89b4947b38b4b3142aa8a1806b27fe3adc2d954820152507a252aca7e1b4f | What are the conditions for an ordered (clockwise or counterclockwise) degree sequence to be valid for a maximal outerplanar graph (a planar triangulated convex polygon) with \( v \) vertices? Given conditions: 1. The sum of the degrees must be equal to \( 4v - 6 \). 2. Each degree \( d \) must satisfy \( 2 \leq d \leq... | [
"\\text{The degree sequence sums to }4v - 6\\text{, each degree is between 2 and }v - 1\\text{, has exactly two 2s (three if }v = 3\\text{), and no two 2s are adjacent.}"
] | [
"What are the conditions for an ordered (clockwise or counterclockwise) degree sequence to be valid for a maximal outerplanar graph (a planar triangulated convex polygon) with \\( v \\) vertices? \n\nGiven conditions:\n1. The sum of the degrees must be equal to \\( 4v - 6 \\).\n2. Each degree \\( d \\) must satisfy... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-8cada0182a7e6a4691ecb8253fe4827a5a466da8e47c8b41bd8908b415ee5bcf | What are the conditions for an ordered degree sequence (clockwise or counterclockwise) to be valid for a maximal outerplanar graph (a planar triangulated convex polygon) with \( v \) vertices? Given conditions: 1. The sum of the degrees must be equal to \( 4v - 6 \). 2. Each degree \( d \) must satisfy \( 2 \leq d \leq... | [
"\\text{Each vertex of degree 2 is adjacent (in the cyclic order) to two vertices that are adjacent to each other.}"
] | [
"What are the conditions for an ordered degree sequence (clockwise or counterclockwise) to be valid for a maximal outerplanar graph (a planar triangulated convex polygon) with \\( v \\) vertices? \n\nGiven conditions:\n1. The sum of the degrees must be equal to \\( 4v - 6 \\).\n2. Each degree \\( d \\) must satisfy... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8827a17433b890f638b935ad14423375e5fbfa76aee79383b29cea5943588b68 | What are the conditions for the prime number $ p $ so that the polynomial $ 2x^2 - p $ is irreducible over $ \mathbb{Q} $? | [
"\\( p > 2 \\)"
] | [
"What are the conditions for the prime number $ p $ so that the polynomial $ 2x^2 - p $ is irreducible over $ \\mathbb{Q} $?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c73234fb18a4ebce751b99e78108fc755027a33e83ba50b8d8dd006470138b6b | What are the conditions for the quadratic equation $ax^2 + bx + c = 0$ to have all roots as natural numbers? | [
"\\( b \\) and \\( c \\) are integers, \\( b^2 - 4c = B^2 \\) for some integer \\( B \\), and \\( b \\) and \\( B \\) have the same parity."
] | [
"What are the conditions for the quadratic equation $ax^2 + bx + c = 0$ to have all roots as natural numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-d8afcd39d53f9d57981213f3a029569a326ced9d629ec0d7985efb86bdaa422d | What are the conditions for the series $\sum_{n=0}^{\infty}\frac{(-1)^n}{an+b}$ to converge, where $a$ and $b$ are real numbers and $an+b \ne 0$ for any $n\in\mathbb{Z}_{\ge 0}$? | [
"The series converges if $a \\neq 0$ and $an+b \\neq 0$ for any $n \\in \\mathbb{Z}_{\\ge 0}$."
] | [
"What are the conditions for the series $\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{an+b}$ to converge, where $a$ and $b$ are real numbers and $an+b \\ne 0$ for any $n\\in\\mathbb{Z}_{\\ge 0}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-b493f2d8a7fdaa789780973abc3012e05fc24da1b03a629077cfdf247a3517c4 | What are the conditions for the series $\sum_{n=0}^{\infty}\frac{1}{an+b}$ to converge, where $a$ and $b$ are real numbers and $an+b \ne 0$ for any $n\in\mathbb{Z}_{\ge 0}$? | [
"The series $\\sum_{n=0}^{\\infty}\\frac{1}{an+b}$ does not converge."
] | [
"What are the conditions for the series $\\sum_{n=0}^{\\infty}\\frac{1}{an+b}$ to converge, where $a$ and $b$ are real numbers and $an+b \\ne 0$ for any $n\\in\\mathbb{Z}_{\\ge 0}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-51ae0698be03692955ffd43431120669f0cc327c24bffe04fc0191f51fff7594 | What are the conditions on a group $G$ such that there exists an integer $n$ for which $G$ is isomorphic to a subgroup of $A_n$ (the alternating group of $n$ elements)? | [
"\\( G \\) must be finite."
] | [
"What are the conditions on a group $G$ such that there exists an integer $n$ for which $G$ is isomorphic to a subgroup of $A_n$ (the alternating group of $n$ elements)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-c36254b22704d12ccfb5ebbb4dcae82bbbcdc54087f479eab454b872083ac175 | What are the conditions on a positive integer \( n \) such that for every positive integer \( k \), there exists a positive integer \( m \) satisfying \( n \mid m^4 + m^3 + m^2 + k \)? | [
"\\( n = 2 \\)"
] | [
"What are the conditions on a positive integer \\( n \\) such that for every positive integer \\( k \\), there exists a positive integer \\( m \\) satisfying \\( n \\mid m^4 + m^3 + m^2 + k \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 26
} | 26 | 0 | 0 |
omr-ws-02b32a6b9b174b72fda742a0353c5b808c69ee01bb491fde6c7adcf70bd1028c | What are the conditions on a vector function $F: \mathbb{R}^3 \to \mathbb{R}^3$ such that the integral $\int_{-\infty}^{\infty} F(r) e^{||x-r||} d^3r = 0$ for all vectors $x \in \mathbb{R}^3$? | [
"F(r) = 0 \\text{ for all } r \\in \\mathbb{R}^3"
] | [
"What are the conditions on a vector function $F: \\mathbb{R}^3 \\to \\mathbb{R}^3$ such that the integral $\\int_{-\\infty}^{\\infty} F(r) e^{||x-r||} d^3r = 0$ for all vectors $x \\in \\mathbb{R}^3$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7caa6de95cf0d601f429d93d1cb854ed49ce38244c7de10d12e479b29c59cec6 | What are the conditions on an odd integer \( n \) such that \( n^4 \) can be expressed as the sum of \( n \) odd perfect squares? | [
"n \\equiv 1 \\pmod{8}"
] | [
"What are the conditions on an odd integer \\( n \\) such that \\( n^4 \\) can be expressed as the sum of \\( n \\) odd perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-da2856e9d420954facc534ea76851b26c65f32cc9b25db57ed0085b9105e91da | What are the conditions on positive integers \(a\) and \(b\) such that there exists a \(k\) for which \[ a^k \equiv -1 \pmod{b} \] is true? | [
"\\((a, b) = 1\\) and \\(\\phi(b)\\) is even."
] | [
"What are the conditions on positive integers \\(a\\) and \\(b\\) such that there exists a \\(k\\) for which\n\\[ a^k \\equiv -1 \\pmod{b} \\]\nis true?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-740beb8724b2b710165e5722b3dc04c26f1c35bbd8f227d0a45f4c88e4353e03 | What are the conditions on the sides and angles of a trapezoid so that a circle can be inscribed inside it? | [
"\\(AB + CD = BC + DA\\)"
] | [
"What are the conditions on the sides and angles of a trapezoid so that a circle can be inscribed inside it?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-03ffe5f2e0d526f9d33284c92d9f9e0c664b0f861584f89fcba02711d8f78531 | What are the conditions that triangle $ABC$ has to satisfy in order for triangle $A'B'C'$ to be equilateral, where points $A', B',$ and $C'$ are symmetric to points $A, B,$ and $C$ with respect to the lines $BC, CA,$ and $AB$ respectively? | [
"\\(ABC\\) is equilateral or isosceles with the apex angle being \\(30\\) or \\(150\\) degrees."
] | [
"What are the conditions that triangle $ABC$ has to satisfy in order for triangle $A'B'C'$ to be equilateral, where points $A', B',$ and $C'$ are symmetric to points $A, B,$ and $C$ with respect to the lines $BC, CA,$ and $AB$ respectively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-079ab6811bd0610d4ce37b67f94c00b369e9e0d2a9553e86e2ede4ac2910281f | What are the consecutive positive integers that sum to 171966464? | [
"4194324"
] | [
"What are the consecutive positive integers that sum to 171966464?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f37e5d42d0042003543816ff6960fd75f14746e983c4f3c93de67252229151d1 | What are the constraint equations for the problem where the total amount of coffee beans used for hot coffee, cold brew, and espresso cannot exceed 100 pounds? | [
"\\( 5x + 7y + 12z \\leq 100 \\)"
] | [
"What are the constraint equations for the problem where the total amount of coffee beans used for hot coffee, cold brew, and espresso cannot exceed 100 pounds?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5d1701f238f582e94474e8e7d8bf0e96a0167fb1f0c0deb39dde82b9201af4e | What are the coordinates of N', the image of N(5,-3) under a reflection in the y-axis? | [
"(-5, -3)"
] | [
"What are the coordinates of N', the image of N(5,-3) under a reflection in the y-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-389b97fb6dacd5d58b491dab13fa42a396b0f19609e535439c4293f32d227ada | What are the coordinates of first base, second base, and third base on a Little League baseball diamond, where the origin is at home plate, the positive x-axis is in the direction from home plate to first base, and the positive y-axis is in the direction from home plate to third base? The diamond is a square with sides... | [
"First base: \\((60, 0)\\), Second base: \\((60, 60)\\), Third base: \\((0, 60)\\)"
] | [
"What are the coordinates of first base, second base, and third base on a Little League baseball diamond, where the origin is at home plate, the positive x-axis is in the direction from home plate to first base, and the positive y-axis is in the direction from home plate to third base? The diamond is a square with ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-1b31a0cbf10dc7f91c28afbd1eb1e60c1087154213757960c131c18480ecb21f | What are the coordinates of the feet of the perpendiculars from a point \( P \) to the sides of a triangle in barycentric coordinates? | [
"(1 - v, v, 0)"
] | [
"What are the coordinates of the feet of the perpendiculars from a point \\( P \\) to the sides of a triangle in barycentric coordinates?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-77bab460b125b7debf7c9db23aa5de79eb9c7b8001ad46cc70d842af758780da | What are the coordinates of the foci of the ellipse given by the equation $3x^2 + 4y^2 - 6x + 16y + 7 = 0$? | [
"\\((2, -2)\\) and \\((0, -2)\\)"
] | [
"What are the coordinates of the foci of the ellipse given by the equation $3x^2 + 4y^2 - 6x + 16y + 7 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-fbfbfe6539e1d172cef8c2710c7b2aa7bd67597222c4ff4e88692acf832f0eaf | What are the coordinates of the maximum point on the graph of the equation \(x^2 + y^3 - 8x - 12y - 4 = 0\)? | [
"\\( (4, 2\\sqrt[3]{2} + \\sqrt[3]{4}) \\)"
] | [
"What are the coordinates of the maximum point on the graph of the equation \\(x^2 + y^3 - 8x - 12y - 4 = 0\\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-a805d4f1186aa987f072d08eac528760dddc3d5f1d40858e410a273fe8a38efd | What are the critical points of the parametric equations \( x = 2^t - t^2 \) and \( y = 3^t - t^3 \)? | [
"\\text{There are no critical points.}"
] | [
"What are the critical points of the parametric equations \\( x = 2^t - t^2 \\) and \\( y = 3^t - t^3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-aaa50dc2c45d2f0687717168ddd15bb73d6226cfce0092bc719ed483becf8ac6 | What are the cube roots of $27i$? | [
"\\(\\frac{3\\sqrt{3}}{2}+\\frac{3i}{2}, \\frac{-3\\sqrt{3}}{2}+\\frac{3i}{2}, -3i\\)"
] | [
"What are the cube roots of $27i$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 29
} | 35 | 0 | 0 |
omr-ws-261e5b36e07734d5acb1bef5052454bba14accae9b2e9a83fe7b27fc52511004 | What are the dimensions of the box of largest volume that can be made from a 10 by 18 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides? | [
"\\left( \\dfrac{26 + 2\\sqrt{61}}{3}\\, \\text{cm}, \\dfrac{2 + 2\\sqrt{61}}{3}\\, \\text{cm}, \\dfrac{14 - \\sqrt{61}}{3}\\, \\text{cm} \\right)"
] | [
"What are the dimensions of the box of largest volume that can be made from a 10 by 18 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bf813b000b73d483a1295389f00d6e879700f11848683effb6c36e5647fd902 | What are the dimensions of the parallelepiped with the maximum volume that can be inscribed in a sphere of radius 1? | [
"\\(\\frac{2\\sqrt{3}}{3}\\)"
] | [
"What are the dimensions of the parallelepiped with the maximum volume that can be inscribed in a sphere of radius 1?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-b38a0dd9366001172d180d14e5f1fb5d293e4037b9ba60e55f1eebe7c43b3429 | What are the dimensions of the subspaces \( X^k \), where \( X^k = \text{Span} \{ (AB - BA)^k : A, B \in \mathbb{R}^{n \times n} \} \)? | [
"n^2"
] | [
"What are the dimensions of the subspaces \\( X^k \\), where \\( X^k = \\text{Span} \\{ (AB - BA)^k : A, B \\in \\mathbb{R}^{n \\times n} \\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-a228438b0a2524a9896969c8ae5c70d508646a98792e88a792a6238e70fa2089 | What are the dimensions of the woodlot (in mm) on each of the two maps, given that the woodlot is 2x3km in size and Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m? | [
"Map A: 250mm x 375mm, Map B: 90.9mm x 136.3mm"
] | [
"What are the dimensions of the woodlot (in mm) on each of the two maps, given that the woodlot is 2x3km in size and Map A has a scale of 1cm: 80m and Map B has a scale of 5mm: 110m?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-6094884e3a7cb4c0a3dcf6dc3aaa73714172c9180094ad2007e01efbbe0a08f9 | What are the divisors of a negative integer, like $-6$? | [
"$\\{ \\pm 1, \\pm 2, \\pm 3, \\pm 6 \\}$"
] | [
"What are the divisors of a negative integer, like $-6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-e192e345f43d2ad8cd463b2f88bfdefacdb6bdf187ddb50ff788cbbd7592cbce | What are the domain and range of the Chebyshev polynomials $T_{n}(x) = \cos ( n \arccos x)$? | [
"Domain: \\([-1, 1]\\), Range: \\([-1, 1]\\)"
] | [
"What are the domain and range of the Chebyshev polynomials $T_{n}(x) = \\cos ( n \\arccos x)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 13
} | 26 | 0 | 0 |
omr-ws-f119283575cacfada2b6cffd07cb41b34d4010f81dcecc8f438edd909daa59d1 | What are the domain and range of the function \( y = x^2 + 3 \)? | [
"Domain: All real numbers, Range: \\( y \\geq 3 \\)"
] | [
"What are the domain and range of the function \\( y = x^2 + 3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-cb44dd5488f0b847352d90f31c3d67a9c508264c29e137723e90bd6bdbb90b8b | What are the double cosets of \( S_{n-1} \setminus S_{n-1} / S_{n-1} \)? | [
"1"
] | [
"What are the double cosets of \\( S_{n-1} \\setminus S_{n-1} / S_{n-1} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 3
} | 13 | 0 | 0 |
omr-ws-77b38e6ef5e50a6734db1a6ee5ae07a6f42906236f478611af0d704cf8fa3c6d | What are the dual space $c^{\ast}$ and the bidual space $c^{\ast\ast}$ of the space $c$ of all convergent complex sequences? | [
"\\ell^\\infty \\oplus \\mathbb{C}"
] | [
"What are the dual space $c^{\\ast}$ and the bidual space $c^{\\ast\\ast}$ of the space $c$ of all convergent complex sequences?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-2621491b55eb493b6b94a7f0cd8889f950aef98321efa0b33070669be2efaa29 | What are the eigenvalues of a matrix with -1 on the principal diagonal and 1 elsewhere? | [
"\\( n-2 \\) and \\( -2 \\) (with multiplicity \\( n-1 \\))"
] | [
"What are the eigenvalues of a matrix with -1 on the principal diagonal and 1 elsewhere?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 31
} | 37 | 0 | 0 |
omr-ws-47706bfe0b5b28a634b32fbe7edf5c8028616917f83062a6d6431e50167e6b9c | What are the elements of the ring \( R = \mathbb{Z} + \mathbb{Z}\left[\frac{1+\sqrt{-11}}{2}\right] \)? | [
"\\( R = \\left\\{ a + b \\left( \\frac{1+\\sqrt{-11}}{2} \\right) \\mid a, b \\in \\mathbb{Z} \\right\\} \\)"
] | [
"What are the elements of the ring \\( R = \\mathbb{Z} + \\mathbb{Z}\\left[\\frac{1+\\sqrt{-11}}{2}\\right] \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 20
} | 32 | 0 | 0 |
omr-ws-03001bf706807d05de62fc4836a48c96c6b0bc17c13f1e6e59db03f447c3d9af | What are the endomorphisms \( f: M_n(K) \to M_n(K) \) such that for every matrix \( M \in M_n(K) \), \( M \) is similar to \( f(M) \)? | [
"\\( f(X) = AXB \\) or \\( f(X) = A^t XB \\) for some invertible matrices \\( A \\) and \\( B \\)."
] | [
"What are the endomorphisms \\( f: M_n(K) \\to M_n(K) \\) such that for every matrix \\( M \\in M_n(K) \\), \\( M \\) is similar to \\( f(M) \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-91e476e4fec02196d7a5819af1db53ebeb8a74163b7f3a13c88e892c5a0ac96b | What are the equations of the asymptotes of \( y = \tan(x) \) between \( 0 \) and \( 2\pi \)? | [
"\\( x = \\frac{\\pi}{2} \\) and \\( x = \\frac{3\\pi}{2} \\)"
] | [
"What are the equations of the asymptotes of \\( y = \\tan(x) \\) between \\( 0 \\) and \\( 2\\pi \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-07c23044e21692d7c20615398bf2f17c68676f0c294288558fc4096d30f57611 | What are the equations of the tangent lines to the circle \((x-2)^2+(y-7)^2=25\) at the point \((6,12)\)? | [
"\\(y = 12\\) and \\(y = -\\frac{40x}{9} + \\frac{116}{3}\\)"
] | [
"What are the equations of the tangent lines to the circle \\((x-2)^2+(y-7)^2=25\\) at the point \\((6,12)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-a97dfa7745498841b641eb35f16413a7ed7cdd1a341e3d6123bc9909af8a3603 | What are the extreme points of the unit ball of $ C(X)$, where $ X$ is a compact Hausdorff space? | [
"\\( |f(x)| = 1 \\) identically."
] | [
"What are the extreme points of the unit ball of $ C(X)$, where $ X$ is a compact Hausdorff space?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-8af4dc368d35b4e03b8218d28b248933208efd9fc63348bf6b191bbca3f37a72 | What are the extreme points of the unit ball of $ l^p$? | [
"\n\\begin{cases}\n\\{ \\pm e_n \\}_{n=1}^\\infty & \\text{if } p = 1, \\\\\n\\{ x \\in \\ell^p : \\|x\\|_p = 1 \\} & \\text{if } 1 < p < \\infty, \\\\\n\\{ x \\in \\ell^\\infty : |x_n| = 1 \\text{ for all } n \\} & \\text{if } p = \\infty.\n\\end{cases}\n"
] | [
"What are the extreme points of the unit ball of $ l^p$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-335e2bd318c35a76fcf6b0d5c87d1d195e2ce12342da4d27f48e420a35e1af57 | What are the factors of $4$? | [
"\\(1, 2, 4\\)"
] | [
"What are the factors of $4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-0504b5de31814487a3a05924c0a92024d5827f97bbcd23389afab3bb35934a2c | What are the factors of 899? | [
"899"
] | [
"What are the factors of 899?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-a13683dc3075a379d93fd0cf1068efff7ff2162c18ba6a7b1fbadc6647438344 | What are the factors of the expression \(16x^2 - 72xy + 81y^2 - 12x + 27y\)? | [
"\\((4x - 9y - 3)(4x - 9y)\\)"
] | [
"What are the factors of the expression \\(16x^2 - 72xy + 81y^2 - 12x + 27y\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-ec960b296e1b17870a49ec3090ba94e13961686d353dc60ebfd3a1bb4ab7571e | What are the final $n$ digits of $a^b$ given that $a$, $b$, and $n$ are all positive integers? | [
"a^b \\mod 10^n"
] | [
"What are the final $n$ digits of $a^b$ given that $a$, $b$, and $n$ are all positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e90dcf44750f32a2f64fa9583417acca1f644fe5850027d38c5631b3d6f860f | What are the first 100 decimal places of \((7 + \sqrt{50})^{100}\)? | [
"9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999"
] | [
"What are the first 100 decimal places of \\((7 + \\sqrt{50})^{100}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 4,
"tir": 20
} | 57 | 0 | 0 |
omr-ws-c43db83c173cf46356397e030e4e491c16ea9397c1a4ff227c524084f0fade01 | What are the first 100 digits after the decimal point of $(\sqrt{10}+3)^{2010}$? (Calculator allowed) | [
"9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999"
] | [
"What are the first 100 digits after the decimal point of $(\\sqrt{10}+3)^{2010}$? (Calculator allowed)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-6e93a9ee347d6e4546fb5683172a8a35c7970485307dd5435d02d61d14cb6d95 | What are the first 3 digits of $2^{(1234!)!}$? | [
"100"
] | [
"What are the first 3 digits of $2^{(1234!)!}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-220d35b6f097ae667c5d75e39d6486cb6903527a05fbd573a872a336021e21d5 | What are the first 3 terms of the binomial expansion of $ \sqrt{9 - x} $? | [
"\\( 3 - \\frac{x}{6} + \\frac{x^2}{216} \\)"
] | [
"What are the first 3 terms of the binomial expansion of $ \\sqrt{9 - x} $?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-2bdab15db5e0a4176cf3ce4a76c311eb1a7579710b91babb73c7384e87e7e67e | What are the first differences and second differences of the function $y = -2x^2 + 1$? | [
"First differences: \\(-2, -6, -10, -14\\); Second differences: \\(-4\\)"
] | [
"What are the first differences and second differences of the function $y = -2x^2 + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 25
} | 34 | 0 | 0 |
omr-ws-ff13b0583507af01fc049c0a14833faaa51d5ceed86ad869202734ec019f0eb1 | What are the first digits before and after the decimal point of $(\sqrt{2} + \sqrt{3})^{1980}$? | [
"9"
] | [
"What are the first digits before and after the decimal point of $(\\sqrt{2} + \\sqrt{3})^{1980}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-6e4234c11e0513d4b616ddf61050cd0f535b98d02361fe529f19cdc71d72a8d8 | What are the first five digits of the number $2^{3141592653}$? | [
"51355"
] | [
"What are the first five digits of the number $2^{3141592653}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 18
} | 21 | 0 | 0 |
omr-ws-066281b6ad32675fe7d771cda6496700e6f9dee70a407322b83885509641b4bb | What are the first three digits after the decimal point in \((10^{2002} + 1)^{10/7}\)? You may use the extended binomial theorem: \((x + y)^r = x^r \left(1 + r \frac{y}{x} + \frac{r(r-1)}{2!} \left(\frac{y}{x}\right)^2 + \frac{r(r-1)(r-2)}{3!} \left(\frac{y}{x}\right)^3 + \cdots \right)\) for \(r\) real and \(\left|\fr... | [
"428"
] | [
"What are the first three digits after the decimal point in \\((10^{2002} + 1)^{10/7}\\)?\n\nYou may use the extended binomial theorem: \\((x + y)^r = x^r \\left(1 + r \\frac{y}{x} + \\frac{r(r-1)}{2!} \\left(\\frac{y}{x}\\right)^2 + \\frac{r(r-1)(r-2)}{3!} \\left(\\frac{y}{x}\\right)^3 + \\cdots \\right)\\) for \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 10
} | 42 | 0 | 0 |
omr-ws-25f7fafad83f4a1e3abc0c4989f7d00ff487fc5a1ae4c10b0d8d41c497327e65 | What are the first three digits of \(1999^{2000}\)? | [
"422"
] | [
"What are the first three digits of \\(1999^{2000}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 23
} | 43 | 0 | 0 |
omr-ws-12502352c0216d849d9bac517524791e1c234664ee30fa8bfe6e532a6552a639 | What are the first three digits of the number \(1^1 + 2^2 + 3^3 + 4^4 + \dots + 999^{999} + 1000^{1000}\)? | [
"100"
] | [
"What are the first three digits of the number \\(1^1 + 2^2 + 3^3 + 4^4 + \\dots + 999^{999} + 1000^{1000}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ace170ea026582619ddafb366f126c88fcfd11c195063ea00b97856bedcd737 | What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2002} + 1\right)^{10/7}?$ | [
"428"
] | [
"What are the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{2002} + 1\\right)^{10/7}?$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-78bf488c8ba952d2f276c51eac5b7f86e4a74040eae2f9adeb6ca30450a872a8 | What are the first three digits to the right of the decimal point in the decimal representation of the number $(10^{2002}+1)^{\frac{10}{7}}$? | [
"428"
] | [
"What are the first three digits to the right of the decimal point in the decimal representation of the number $(10^{2002}+1)^{\\frac{10}{7}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-1009271b181177f039d6151bd3a7d850922190042d36179820d328859280b1d9 | What are the formulas for finding the number of edges, faces, and vertices of a polyhedron given any two of these quantities? | [
"\\( V - E + F = 2 \\) and derived formulas \\( E = V + F - 2 \\), \\( F = E - V + 2 \\), \\( V = E - F + 2 \\)"
] | [
"What are the formulas for finding the number of edges, faces, and vertices of a polyhedron given any two of these quantities?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-c3a10b649bdda5da49c3d6db7c7bca12f1ff6cea47e55d3bf3c262778a788bdb | What are the four distinct positive integers with the smallest sum that can be placed on the sides of two identical tetrahedral dice so that the sum of the two bottom numbers has ten distinct possibilities? | [
"1, 2, 5, 7"
] | [
"What are the four distinct positive integers with the smallest sum that can be placed on the sides of two identical tetrahedral dice so that the sum of the two bottom numbers has ten distinct possibilities?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-c3311e44b1d41769380916cf80f743c2bb45cfd4792a607c6cd253a376381fb6 | What are the fourth roots of the complex number $1+\sqrt{15}i$? | [
"\\(\\frac{\\sqrt{4+\\sqrt{10}}+i\\sqrt{4-\\sqrt{10}}}{2}\\), \\(\\frac{-\\sqrt{4+\\sqrt{10}}-i\\sqrt{4-\\sqrt{10}}}{2}\\), \\(\\frac{\\sqrt{4-\\sqrt{10}}-i\\sqrt{4+\\sqrt{10}}}{2}\\), \\(\\frac{-\\sqrt{4-\\sqrt{10}}+i\\sqrt{4+\\sqrt{10}}}{2}\\)"
] | [
"What are the fourth roots of the complex number $1+\\sqrt{15}i$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 23
} | 35 | 0 | 0 |
omr-ws-84f3aab2ce9d12bceade93ae971f3dc9ef5510da71a603b0c9c7bb8d7d3c9282 | What are the functions $f$ satisfying $f(f(x))=e^x$ for all $x>1$? | [
"\\text{No such functions exist in elementary form, but non-elementary solutions exist.}"
] | [
"What are the functions $f$ satisfying $f(f(x))=e^x$ for all $x>1$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7f44e2c438d09e1656d8b03057d6625c4d8609d9ba5fb29cc7a335a8265565d | What are the generators and relations of the group $GL(n, \mathbb{Z}_2)$? | [
"GL(n, \\mathbb{Z}_2) \\text{ is generated by all transvections } t_{ij} \\text{ with relations } t_{ij}^2 = 1, \\, t_{ij}t_{jk}t_{ij} = t_{jk}t_{ik}, \\text{ and } t_{ij}t_{kl} = t_{kl}t_{ij} \\text{ for disjoint pairs}"
] | [
"What are the generators and relations of the group $GL(n, \\mathbb{Z}_2)$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-51cfab9a47d7ab794887b4340cb621e32df3aade2eb131d3ff14ba8e364d87a6 | What are the generators of \(\mathbb{Z}\)? | [
"\\(1\\) and \\(-1\\)"
] | [
"What are the generators of \\(\\mathbb{Z}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-b4ddf3fa624e7d0161c33d41ef9e4f988bcb78d2aca5999cc28f5fbb35e1dfd3 | What are the global sections of $ O(l)$ on $ \mathbb{P}^{n}$ for $ l < 0$? | [
"0"
] | [
"What are the global sections of $ O(l)$ on $ \\mathbb{P}^{n}$ for $ l < 0$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6ffbd59cc43535a25f4d930b8ea43a63fecd1dadac9e2e462ae2a7a09d74c77 | What are the global sections of $ O(l)$ on $ \mathbb{P}^{n}$ for $ l = 0$? | [
"Constant functions."
] | [
"What are the global sections of $ O(l)$ on $ \\mathbb{P}^{n}$ for $ l = 0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-82e7a68a4aa733a64484ae8baaf15269dba56977f637c22ecb34ca9c634377d8 | What are the global sections of $ O(l)$ on $ \mathbb{P}^{n}$ for $ l > 0$? | [
"Homogeneous polynomials in \\( x_{0},\\ldots,x_{n} \\) of degree \\( l \\)."
] | [
"What are the global sections of $ O(l)$ on $ \\mathbb{P}^{n}$ for $ l > 0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7ee34087ab5ad3e00a58a95f6e3b723149cc41c9ca281b1ee9b152db43ef184b | What are the highest degree polynomials \( P(x) \) and \( Q(x) \) such that \( P(x) - Q(x) = 1 \) and both polynomials factorize fully over the integers? Provide an example for cubic polynomials: \[ P(x) = (x+1)(4x+1)(4x+1) \] \[ Q(x) = x(4x+3)(4x+3) \] where \( P(x) - Q(x) = 1 \). | [
"Q(x) = x(4x+3)^2"
] | [
"What are the highest degree polynomials \\( P(x) \\) and \\( Q(x) \\) such that \\( P(x) - Q(x) = 1 \\) and both polynomials factorize fully over the integers? Provide an example for cubic polynomials:\n\\[ P(x) = (x+1)(4x+1)(4x+1) \\]\n\\[ Q(x) = x(4x+3)(4x+3) \\]\nwhere \\( P(x) - Q(x) = 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-da5aa5377a6af459b3a79e0fc602710a5e67c397e929079b33b1a0f266fc6d2c | What are the holomorphic functions $f$ on an open connected subset $\omega \subset \mathbb{C}$ such that the function $g:\omega \to \mathbb{C}$ defined by $g(z) = \text{Re}(z)f(z)$ is also holomorphic? | [
"\\( f \\equiv 0 \\)"
] | [
"What are the holomorphic functions $f$ on an open connected subset $\\omega \\subset \\mathbb{C}$ such that the function $g:\\omega \\to \\mathbb{C}$ defined by $g(z) = \\text{Re}(z)f(z)$ is also holomorphic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-00036ee498bc7d2811ec5ac6f0873b25ec2eb33966583a27dc02928b2e8916d8 | What are the ideals in $\mathbb{Z}[\sqrt{-29}]$ that contain the element 30? | [
"48"
] | [
"What are the ideals in $\\mathbb{Z}[\\sqrt{-29}]$ that contain the element 30?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-43024f28ca0c35960f764155538a04358a8c1ea40c901cc715c0d5fb6f990b6e | What are the imaginary cube roots of 1? | [
"\\(-\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i\\) and \\(-\\frac{1}{2} - \\frac{\\sqrt{3}}{2}i\\)"
] | [
"What are the imaginary cube roots of 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 27
} | 29 | 0 | 0 |
omr-ws-8d37d0853899a20ad1b3aac6bce7d9cf920d2e90a8178ab8f450c1e599b7f3da | What are the infinite abelian groups that are isomorphic to each of their non-trivial homomorphic images? | [
"$\\mathbb{Z}(p^\\infty)$ for some prime $p$"
] | [
"What are the infinite abelian groups that are isomorphic to each of their non-trivial homomorphic images?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-01d50239d23bc846cd199ce24ddec478d7e4b4125cc8974b4d68774e170a1c4f | What are the initial solutions for the recurrence relation for the number of n-digit sequences over the alphabet {0, 1, 2} containing two consecutive 1’s? | [
"0"
] | [
"What are the initial solutions for the recurrence relation for the number of n-digit sequences over the alphabet {0, 1, 2} containing two consecutive 1’s?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-236abcd1cbc051c0aa6382f6b2addc99c03893f3b587788deb16680544b878ef | What are the integer solutions to $(2x-y)(x-2y)=-x-y$? | [
"(2, 2)"
] | [
"What are the integer solutions to $(2x-y)(x-2y)=-x-y$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee09317c20789aa54107a7f1d63056feef02e0a280a6a05519be4aff54bbd6f3 | What are the integer solutions to the equation \(x^2 + 2 = y^3\)? | [
"(\\pm 5, 3)"
] | [
"What are the integer solutions to the equation \\(x^2 + 2 = y^3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-43bfb680a501fc7d7102c98f9badc1ee8a9003daff1eb772a2e53a3fc38d1efc | What are the integers $n$ such that $1 \leq n < 200$ and $37 < P(n) < 45$? | [
"\\( 58, 67, 76, 85, 158, 167, 176, 185 \\)"
] | [
"What are the integers $n$ such that $1 \\leq n < 200$ and $37 < P(n) < 45$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 27
} | 35 | 0 | 0 |
omr-ws-ee898d154da58fbea3239dc2a347401aa7f974fbac8a1a42709bb4bb439d25d9 | What are the interesting properties of a group of order 2022? (Note: 2022 = 2 × 3 × 337). For example, such a group cannot be simple, it must have subgroups of order 2, 3, and 337. Additionally, there is a unique (hence normal) Sylow 337-subgroup. What other interesting facts can be derived about such a group? | [
"\\text{The group is solvable, has a normal Sylow 337-subgroup, and is a semidirect product of } C_{337} \\text{ with a group of order 6 (cyclic or } S_3\\text{).}"
] | [
"What are the interesting properties of a group of order 2022? (Note: 2022 = 2 × 3 × 337). For example, such a group cannot be simple, it must have subgroups of order 2, 3, and 337. Additionally, there is a unique (hence normal) Sylow 337-subgroup. What other interesting facts can be derived about such a group?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-5ef0dd09f7ad48fcaad8a96b4321fe339d2e1c66b7575d197ab3e86d431e69bc | What are the irreducible elements of $\mathbb{Z}((X))$? | [
"\\pm p X^k \\text{ where } p \\text{ is a prime in } \\mathbb{Z} \\text{ and } k \\in \\mathbb{Z}"
] | [
"What are the irreducible elements of $\\mathbb{Z}((X))$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-8c8d07f1a60e7c0532733e06dca11f9c111ee5a70cbd7af37838b851044264c1 | What are the last 2 digits of $\sum_{n=1}^{100} n!$ when expressed in the octal number system? | [
"30_8"
] | [
"What are the last 2 digits of $\\sum_{n=1}^{100} n!$ when expressed in the octal number system?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99a430059a8d969cc2ec493d83458243d41c20bc39b1a210e68f91b84d81d38f | What are the last 2 digits of \(2^{{{{3}^4}^\cdots}^{2012}}\)? | [
"52"
] | [
"What are the last 2 digits of \\(2^{{{{3}^4}^\\cdots}^{2012}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-623390da3987ae7080df195fb90265ca2c763b0dbceb1d783ce47d487f47018a | What are the last 2 nonzero digits of \(90!\)? | [
"12"
] | [
"What are the last 2 nonzero digits of \\(90!\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-4c0a1a7750910d4fb9fea340648857ac486daad5d04d0deda9bf5e543206243f | What are the last 3 digits of $111^{89}$? | [
"391"
] | [
"What are the last 3 digits of $111^{89}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-c70604b98966f695ecae31d9db4aec1e55a16b010b85a76905b49153a079009c | What are the last 3 digits of $2^{(1234!)!}$? | [
"001"
] | [
"What are the last 3 digits of $2^{(1234!)!}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f7f306d5477be5480a3127c37cbb264ec710d8ba0497039ab616f3eeec74058 | What are the last 3 digits of $3^{2016}$? | [
"721"
] | [
"What are the last 3 digits of $3^{2016}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-9d6ca16c938749d8649e70a90578cc068edef8833848adedb8f7700b78c74739 | What are the last 3 digits of $3^{3^{3^3}}$? | [
"387"
] | [
"What are the last 3 digits of $3^{3^{3^3}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-323def485498d8a740c1e2844ee1d84501be5d25c290c770576cbd6cc18b3bf8 | What are the last 3 digits of $62^{62^{62}}$? | [
"936"
] | [
"What are the last 3 digits of $62^{62^{62}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-b2cca823fbce67ef8286381f9a58b0257a5be973aa8bc7e8f43e0baace87de50 | What are the last 3 digits of the 2020th term in the increasing sequence of positive multiples of 3 that are one less than a perfect square? | [
"440"
] | [
"What are the last 3 digits of the 2020th term in the increasing sequence of positive multiples of 3 that are one less than a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86fd943ac0ab17d698818d8578e381c2cfb0240f434353599f9dc10bd1c69f5b | What are the last 3 digits of the number \((555^{555})^{(555^{555})}\)? | [
"875"
] | [
"What are the last 3 digits of the number \\((555^{555})^{(555^{555})}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67a3666a1485c941aba328fbca24b11e5eff9b9b31eb6aa5fbb68e0c0e0ec7f0 | What are the last 4 digits of $7^{2012}$? | [
"7201"
] | [
"What are the last 4 digits of $7^{2012}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-2e51f5aa8a826bbaf8b710a7e3bb7aab645c7baf5fe6f08bba0ad5854a26d6b7 | What are the last 4 digits of the 81st Fermat number? | [
"8497"
] | [
"What are the last 4 digits of the 81st Fermat number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 20
} | 23 | 0 | 0 |
omr-ws-b3698c0dc130d83e6b6fcf1ba99bd76063779d937c8d8dbc88f4f8041cbf82da | What are the last 5 digits of the product \(1 \times 3 \times 5 \times \cdots \times 2019\)? | [
"71875"
] | [
"What are the last 5 digits of the product \\(1 \\times 3 \\times 5 \\times \\cdots \\times 2019\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-04454a24aef5b85987f29fcd937f87dc357c89c8c36e69dec887cf227f9a4e3f | What are the last 8 digits of the product $11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111$? | [
"19754321"
] | [
"What are the last 8 digits of the product $11 \\times 101 \\times 1001 \\times 10001 \\times 100001 \\times 1000001 \\times 111$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-ecee3c8b7e821d6356c343e2b07d211bd09d18d5ebea04aae72b1f15c5bf94d0 | What are the last five digits of $54321^{43210}$? | [
"95201"
] | [
"What are the last five digits of $54321^{43210}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-93cb3daa9121ffd9557ff890f70ac77ee78828a6e3abec941c72cc5ff5c25581 | What are the last four digits of $1(1!) + 2(2!) + 3(3!) + \cdots + 2013(2013!)$? | [
"9999"
] | [
"What are the last four digits of $1(1!) + 2(2!) + 3(3!) + \\cdots + 2013(2013!)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a29ac37f511bd2dc173be3b96f3307d1b4941bcb2963007521649b193fdb3239 | What are the last four digits of $2001^{2001}$? | [
"2001"
] | [
"What are the last four digits of $2001^{2001}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f58fd911cc2dbb565929fc031a05c75316af4e3f38474e28c62d6a5bf2d02668 | What are the last four digits of $2^{27653} - 1$? | [
"2991"
] | [
"What are the last four digits of $2^{27653} - 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-3b4b95bfea6bac30adef9e9894dc8a18b6c2f0d8dcb4f180dc7ed3f951a6c794 | What are the last four digits of $9^{9^9}$? | [
"7289"
] | [
"What are the last four digits of $9^{9^9}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-5722547c88be254bb03e9f8a3c131d9b9eda7f5e6f141b965530ce2effdfcc95 | What are the last four digits of the number \( N = 9^1 + 9^2 + \ldots + 9^{400} \)? | [
"0000"
] | [
"What are the last four digits of the number \\( N = 9^1 + 9^2 + \\ldots + 9^{400} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-86a15dae83860650d0e6133bc417ea23671f22233ede1c7a3724d109c98d873e | What are the last three digits of $1201^{1202}$? | [
"401"
] | [
"What are the last three digits of $1201^{1202}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-2a1e910589c7051077fa43d9cf5a8c6d5a876a313eaa5b64d6a8aef3a334b310 | What are the last three digits of $1337^{1337}$? | [
"617"
] | [
"What are the last three digits of $1337^{1337}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-a6ff2fd90d489c7c10155099c95171d2d5ce8bcca167f1ab90b63e957467bd86 | What are the last three digits of $2003^{2002^{2001}}$? | [
"241"
] | [
"What are the last three digits of $2003^{2002^{2001}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 42
} | 106 | 0 | 0 |
omr-ws-24a5eb274835e65772a963c9cb7100859f34a02f43d2ce4769944bb28bd33f72 | What are the last three digits of $2013^{2060}$ when written in base 7? | [
"622"
] | [
"What are the last three digits of $2013^{2060}$ when written in base 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-720d46739f07a92dde6fd0cec3c77c177eab65db4b209a6f9e8f20691424c470 | What are the last three digits of $3^{210}$? | [
"049"
] | [
"What are the last three digits of $3^{210}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.