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-bb5498f04e67628762395f015898ec02f62288d42af8092081cb58454b14301c | What is the number of ordered pairs of integers $(m,n)$ for which $mn>0$ and $m^3+n^3+99mn=33^3$? | [
"35"
] | [
"What is the number of ordered pairs of integers $(m,n)$ for which $mn>0$ and $m^3+n^3+99mn=33^3$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8adcb8fcdb9f45aa765118e3fae1d63fdf25a5f0c73e1b165095d679f1171961 | What is the number of pairs $(A, B)$ of sets such that $A \cup B = \{a_1, a_2, a_3\}$, where $A \neq B$? | [
"26"
] | [
"What is the number of pairs $(A, B)$ of sets such that $A \\cup B = \\{a_1, a_2, a_3\\}$, where $A \\neq B$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92f323e0305e0968b7c2e8bb05bf6a1249df8ead2a5e4e33bd9ba70560064ca4 | What is the number of partitions of a positive integer \( n \) into \( k \) integers such that no two integers in the partition are equal? Example: For \( n = 7 \) and \( k = 3 \), one valid partition is \( 1 + 2 + 4 \). | [
"p\\left(n - \\frac{k(k - 1)}{2}, k\\right)"
] | [
"What is the number of partitions of a positive integer \\( n \\) into \\( k \\) integers such that no two integers in the partition are equal?\n\nExample: For \\( n = 7 \\) and \\( k = 3 \\), one valid partition is \\( 1 + 2 + 4 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-aad23e14c4e13d3b7da62cc6f67f6169e78fed4cbd3f4f48ce7e8815544cb886 | What is the number of paths that go from $(0,0)$ in a random walk along the $X$ axis to the point $(2n,k)$ without crossing the $X$ axis? ($k$ is even and less than $2n$). Please provide a combinatorial solution using the reflection principle (not recursion). | [
" \\dbinom{2n}{n + \\frac{k}{2}} - \\dbinom{2n}{n + \\frac{k}{2} + 1} "
] | [
"What is the number of paths that go from $(0,0)$ in a random walk along the $X$ axis to the point $(2n,k)$ without crossing the $X$ axis? ($k$ is even and less than $2n$). Please provide a combinatorial solution using the reflection principle (not recursion)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-238f43132ee3f63b43c39d4d3dbfde1c63ccf5a7f051636566276bdb7b2e1186 | What is the number of perfect square and cube factors of \(15!\)? | [
"4"
] | [
"What is the number of perfect square and cube factors of \\(15!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1463f5b705535e49e42eed385c60c396336ba5d856364b591411d3950f1b9566 | What is the number of permutations of $n$ elements that have exactly $k$ fixed points? For example, there are 3 permutations of 3 elements with exactly 1 fixed point: 1. $\begin{bmatrix}1&2&3\\1&3&2\end{bmatrix}$ (1 is the fixed point) 2. $\begin{bmatrix}1&2&3\\3&2&1\end{bmatrix}$ (2 is the fixed point) 3. $\begin{bmat... | [
"\\(\\binom{n}{k} \\cdot !(n-k)\\)"
] | [
"What is the number of permutations of $n$ elements that have exactly $k$ fixed points? \n\nFor example, there are 3 permutations of 3 elements with exactly 1 fixed point:\n1. $\\begin{bmatrix}1&2&3\\\\1&3&2\\end{bmatrix}$ (1 is the fixed point)\n2. $\\begin{bmatrix}1&2&3\\\\3&2&1\\end{bmatrix}$ (2 is the fixed poi... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-80debdd1e37ec39d7a9d6c29b8a734ccd2c686e1ca0f88b35ba7a726712dacd8 | What is the number of permutations of \(1, \ldots, n\) with at least \(k\) fixed points? | [
"\\(\\sum_{j=k}^{n} \\binom{n}{j}(n - j)!\\sum_{i = 0}^{n - j}\\frac{(-1)^i}{i!}\\)"
] | [
"What is the number of permutations of \\(1, \\ldots, n\\) with at least \\(k\\) fixed points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-0e092618ebcef637228e193d07726a2da1e36a117b6cfb2c067405e9f00f8a98 | What is the number of permutations of \(1, \ldots, n\) with exactly \(k\) fixed points? The formula given is \(\binom{n}{k}(n - k)!\sum_{i = 0}^{n - k}\frac{(-1)^i}{i!}\). | [
"\\(\\binom{n}{k}(n - k)!\\sum_{i = 0}^{n - k}\\frac{(-1)^i}{i!}\\)"
] | [
"What is the number of permutations of \\(1, \\ldots, n\\) with exactly \\(k\\) fixed points? The formula given is \\(\\binom{n}{k}(n - k)!\\sum_{i = 0}^{n - k}\\frac{(-1)^i}{i!}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-2c93ba98ca6413e7187baaa876b41dee09b22c4454db8f7a9b7f329fcf645aa9 | What is the number of permutations of length $N$, consisting of all the numbers between $1$ and $N$, such that no number is in its original position? | [
"\\( D_n = n! \\sum_{r=0}^n \\frac{(-1)^r}{r!} \\)"
] | [
"What is the number of permutations of length $N$, consisting of all the numbers between $1$ and $N$, such that no number is in its original position?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-6e6c19e6ea98e8bbd49d31b3783c90881714af0904695f54bd5633656d9c7a65 | What is the number of permutations of n elements $a_1, a_2, ..., a_n$ such that no element remains in its original position? | [
"\\( n!\\left(\\sum_{i=0}^n \\frac{(-1)^i}{i!}\\right) \\)"
] | [
"What is the number of permutations of n elements $a_1, a_2, ..., a_n$ such that no element remains in its original position?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-295e46fea606feb8f39a0522dfe3b3258517aa658962658b6010ba02defb02c4 | What is the number of permutations of n elements such that no two consecutive numbers appear next to each other? | [
"a(n)"
] | [
"What is the number of permutations of n elements such that no two consecutive numbers appear next to each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-6de8468d69ee33c2b6fc7999a8685c21604c8836a026bf957c861ecffa578403 | What is the number of permutations of the set of elements of $S_n$ such that two consecutive permutations differ by an involution of the form $(i,j)$? | [
"n!"
] | [
"What is the number of permutations of the set of elements of $S_n$ such that two consecutive permutations differ by an involution of the form $(i,j)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-29ff6866d72e30c5f6b64761647265f471b71c2fd423f3a490c596e6e16b2ece | What is the number of points \( P \) strictly lying inside an equilateral triangle \( ABC \) such that the sum of the perpendicular distances from \( P \) to the three sides of the triangle is minimized? | [
"\\infty"
] | [
"What is the number of points \\( P \\) strictly lying inside an equilateral triangle \\( ABC \\) such that the sum of the perpendicular distances from \\( P \\) to the three sides of the triangle is minimized?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af3659ca6b906176baba57b30cc863128cabc43b2a689589c07740178ce4e112 | What is the number of points equidistant from a circle and two parallel tangents to the circle? | [
"3"
] | [
"What is the number of points equidistant from a circle and two parallel tangents to the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13a709eb181a13e179f501f0a6fe64f8e73914c3477cf3ec39c80613d724b86a | What is the number of positive divisors of $2008$? | [
"8"
] | [
"What is the number of positive divisors of $2008$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e0d0daeeec5b190d64819248281b34ea97301527ea3576375247361f94d9f0a | What is the number of positive integers $4 \leq a \leq 1000$ for which $a^2$ has its last two digits being both even? | [
"299"
] | [
"What is the number of positive integers $4 \\leq a \\leq 1000$ for which $a^2$ has its last two digits being both even?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 20
} | 40 | 0 | 0 |
omr-ws-1ea05cc10bd09ac93ca2b06c5e54842ca8b2dee240360ac56859c69eb61877b2 | What is the number of positive integers \( n \leq 154 \) such that the coefficient of \( x^n \) exists in the binomial expansion of \( (1 + x^7 + x^{14} + x^{21} + \cdots + x^{77})(1 + x^{11} + x^{22} + \cdots + x^{77}) \)? | [
"95"
] | [
"What is the number of positive integers \\( n \\leq 154 \\) such that the coefficient of \\( x^n \\) exists in the binomial expansion of \\( (1 + x^7 + x^{14} + x^{21} + \\cdots + x^{77})(1 + x^{11} + x^{22} + \\cdots + x^{77}) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c644ffc391b3c8ed738d28466ef28f1fb47dc79219faae569a41a4df3f703871 | What is the number of positive integral divisors of 104,060,401? | [
"5"
] | [
"What is the number of positive integral divisors of 104,060,401?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-28509b8f511cc5f689247241aae7f45fbec9e0249c93cae1ab470623d6034021 | What is the number of possibilities for placing 4 white towers and 4 black towers on an 8 * 8 chessboard so that in each row and column there are not two towers of the same color? | [
"2822400"
] | [
"What is the number of possibilities for placing 4 white towers and 4 black towers on an 8 * 8 chessboard so that in each row and column there are not two towers of the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-f51d386fa5bef56ce810dffc6d5d92c12552af0459f3a6bc04c344de30186ef7 | What is the number of real roots for the polynomial $f(x) = x^5 + 2x^4 + 5x^3 + 5x^2 + 4x - 2$? | [
"1"
] | [
"What is the number of real roots for the polynomial $f(x) = x^5 + 2x^4 + 5x^3 + 5x^2 + 4x - 2$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-918ff54e2956fd62989384af192c037cbaae1f49952142ee65b445ed9212e93f | What is the number of real roots of the equation $\sin x = x \cdot \cos x$? | [
"\\(\\infty\\)"
] | [
"What is the number of real roots of the equation $\\sin x = x \\cdot \\cos x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-d93a7d2326d96fbcb696dfc36f7265d56354a1a9917455ac37a47ef5482521ab | What is the number of real roots of the polynomial $p(x) = x(x-2)(x-4)(x-6) + 2$? | [
"4"
] | [
"What is the number of real roots of the polynomial $p(x) = x(x-2)(x-4)(x-6) + 2$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2c339041425453a7b91c04ce7a6c18b8e7c6f2661eb05eb071d7629d5a38a76 | What is the number of real solutions of $|2x - \lfloor x \rfloor| = 4$? | [
"4"
] | [
"What is the number of real solutions of $|2x - \\lfloor x \\rfloor| = 4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f247211540f6ded1c4092c22038baaf250109233c6ba4394811f31d42c3e5cc | What is the number of real solutions to the equation $\sqrt{x}=1+\sqrt{|2x-7|}$? | [
"2"
] | [
"What is the number of real solutions to the equation $\\sqrt{x}=1+\\sqrt{|2x-7|}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02382090170f9d79a7f01f95379f634238b169bb4f0e66e420919ee267c63cee | What is the number of rotationally distinct colorings of the faces of a cube using three colors? | [
"57"
] | [
"What is the number of rotationally distinct colorings of the faces of a cube using three colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60c10eb5aa04b1e779ee1170aaa71792f2072c3a5178f3763746ffcfc3042935 | What is the number of sequences of letters $a, b,$ and $c$ of length $7$ such that no two adjacent letters are the same and no two sub-sequences are equal? A sub-sequence is a consecutive set of letters from the sequence. | [
"0"
] | [
"What is the number of sequences of letters $a, b,$ and $c$ of length $7$ such that no two adjacent letters are the same and no two sub-sequences are equal? A sub-sequence is a consecutive set of letters from the sequence."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-2d161edb146f5796667f821bdfd005a9e000867637443cb2b86e4feac9c03c51 | What is the number of sides \( n \) of a regular polygon such that the ratio of the area of the polygon inscribed in a circle to the area of the polygon circumscribing the same circle is 3:4? | [
"6"
] | [
"What is the number of sides \\( n \\) of a regular polygon such that the ratio of the area of the polygon inscribed in a circle to the area of the polygon circumscribing the same circle is 3:4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e5a1d9176e464966572d6fcf03d1756fb436778c76c01cca2df92b7fe0a50f81 | What is the number of sides of a regular polygon for which the number of diagonals is 45? | [
"11"
] | [
"What is the number of sides of a regular polygon for which the number of diagonals is 45?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-b723e99f81e1c87c4cb51def57365c564a870a54f7969770f41236a0c19688f4 | What is the number of sides of the regular polygon with the largest number of sides whose interior angles measure an integer multiple of $7^\circ$? | [
"72"
] | [
"What is the number of sides of the regular polygon with the largest number of sides whose interior angles measure an integer multiple of $7^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff8b77e03f97a26dff78c14f137645aa5980ce7ffa236449ffb5bad223589d0e | What is the number of solutions in natural numbers to the equation $3x + 4y + 5z = 200$? | [
"314"
] | [
"What is the number of solutions in natural numbers to the equation $3x + 4y + 5z = 200$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-577b8300c249cdb09b52b13f8ad53c96f2da860546ec9f3c873b43bad2c15f29 | What is the number of solutions of the system $a^{2}+2b^{2}-c^{2}+2d^{2}+e^{2}=0$ and $5a^{2}+12b^{2}-3c^{2}+2d^{2}-e^{2}=0$ in $\mathbb{Z}/p\mathbb{Z}$? | [
"p^3"
] | [
"What is the number of solutions of the system $a^{2}+2b^{2}-c^{2}+2d^{2}+e^{2}=0$ and $5a^{2}+12b^{2}-3c^{2}+2d^{2}-e^{2}=0$ in $\\mathbb{Z}/p\\mathbb{Z}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7e26a6c030284f29fe28097aa373341064d79ac03f6013e5f480299316965bd1 | What is the number of square meters in the area of a circle with diameter $6$ meters? Express your answer in terms of $\pi$. | [
"9 \\pi"
] | [
"What is the number of square meters in the area of a circle with diameter $6$ meters? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98c64dd5339499aa910220a095a8802ccc26ff1889f726064373d1b0f463898a | What is the number of square meters in the area of a circle with diameter 4 meters? Express your answer in terms of $\pi$. | [
"4\\pi"
] | [
"What is the number of square meters in the area of a circle with diameter 4 meters? Express your answer in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0fe09a23c34721e887f4327184a96d144da1fd8cf17f4d841c4a5e55be9a7f06 | What is the number of square units in the area of trapezoid ABCD with vertices A(0,0), B(0,-2), C(4,0), and D(4,6)? | [
"16"
] | [
"What is the number of square units in the area of trapezoid ABCD with vertices A(0,0), B(0,-2), C(4,0), and D(4,6)?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f87b06c9a97e464c974db7a6d5b6e9da733443c9b31353367a9489d7d8f8a7fb | What is the number of square units in the region satisfying $|x| + |y| \leq 9$ and $-5 \leq x \leq 4$? | [
"121"
] | [
"What is the number of square units in the region satisfying $|x| + |y| \\leq 9$ and $-5 \\leq x \\leq 4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 4
} | 34 | 0 | 0 |
omr-ws-26a6a1d946e84cb8126b4b7058b0794ea96c502acab6fa42b1aa57c7fc9a6c99 | What is the number of strings consisting of $2a$s, $3b$s, and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$, and $c$ is not immediately followed by $a$? | [
"174"
] | [
"What is the number of strings consisting of $2a$s, $3b$s, and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$, and $c$ is not immediately followed by $a$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-89309ac4213b055bac78932be10a6a16070fdcbb6bea30a1a651b29b632fc0e3 | What is the number of subsets \(\{a_1, a_2, \ldots, a_{10}\}\) of \(\{2, 3, 4, \ldots, 1000\}\) such that \(\gcd(a_i, a_j) = 1\) for every \(1 \leq i \neq j \leq 10\)? | [
"\\dbinom{168}{10}"
] | [
"What is the number of subsets \\(\\{a_1, a_2, \\ldots, a_{10}\\}\\) of \\(\\{2, 3, 4, \\ldots, 1000\\}\\) such that \\(\\gcd(a_i, a_j) = 1\\) for every \\(1 \\leq i \\neq j \\leq 10\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-ae75b55da862d243ef0607e54271b46f20f6c37da18c194713add43a20921ed7 | What is the number of subsets of the set \( A = \{1, 2, 3, \ldots, 17\} \) such that no two elements in any subset have a difference of 4? | [
"6656"
] | [
"What is the number of subsets of the set \\( A = \\{1, 2, 3, \\ldots, 17\\} \\) such that no two elements in any subset have a difference of 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 3
} | 58 | 0 | 0 |
omr-ws-bb9167969ac73cd733069edd26bfa95577cb4b2af487dfdcad0d6b18833c73f0 | What is the number of subsets with 20 elements of a set with 52 elements? | [
"\\(\\binom{52}{20}\\)"
] | [
"What is the number of subsets with 20 elements of a set with 52 elements?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-7c81dea1596b2fe666f9bbc1f27ab2cbd695920f29cfb5ac8cc165c862338576 | What is the number of surjective maps from a set \( E \) to a set \( F \) with cardinalities \( n \) and \( m \) respectively? | [
"\\( m! \\cdot S(n, m) \\)"
] | [
"What is the number of surjective maps from a set \\( E \\) to a set \\( F \\) with cardinalities \\( n \\) and \\( m \\) respectively?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 24
} | 36 | 0 | 0 |
omr-ws-b3225cf3a978356c8f688051c9295ef555103136d9f26e55f3316df2b029849f | What is the number of terms with rational coefficients among the $1001$ terms in the expansion of $$\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?$$ | [
"167"
] | [
"What is the number of terms with rational coefficients among the $1001$ terms in the expansion of\n$$\\left(x\\sqrt[3]{2}+y\\sqrt{3}\\right)^{1000}?$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11677041e7b08d6d6e76d4dd8ae2c9c424e92cceb62c4a9e27f82ad6f6723f81 | What is the number of terms with rational coefficients among the 1001 terms in the expansion of \[ \left(x\sqrt[3]{2} + y\sqrt{3}\right)^{1000}? \] | [
"167"
] | [
"What is the number of terms with rational coefficients among the 1001 terms in the expansion of\n\\[\n\\left(x\\sqrt[3]{2} + y\\sqrt{3}\\right)^{1000}?\n\\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-e0fe191398749c7afb22e32f40d99c262dc511024f630459751d473da5955257 | What is the number of trees with \( n \) vertices where the degree of each vertex is no more than 4? | [
"A000602"
] | [
"What is the number of trees with \\( n \\) vertices where the degree of each vertex is no more than 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-983d99838c265784a7cf34b3a20148a852728640836ded774824d34b9022b834 | What is the number of units in the distance between $(2,5)$ and $(-6,-1)$? | [
"10"
] | [
"What is the number of units in the distance between $(2,5)$ and $(-6,-1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d028a50522505898a5fb286cd75a0bc62b9792d6f71442812ba81fd15382ecc | What is the number of units in the quotient ring $R=\mathbb{Z}/2020\mathbb{Z}$? | [
"800"
] | [
"What is the number of units in the quotient ring $R=\\mathbb{Z}/2020\\mathbb{Z}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-216d532a896b0b319b62d300c0c7be8d186e02401f4ada2b49fda4c14b3fc950 | What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? | [
"144"
] | [
"What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-f9a33db16bff686744f24792b4bd48f43361a9acda5572e83cb0ed7e2fad50eb | What is the number of ways to decompose the space $\mathbb{F}_2^n$ as an ordered direct sum of two subspaces? ($\mathbb{F}_2$ is the field of two elements) | [
"\\(2^{n^2}\\)"
] | [
"What is the number of ways to decompose the space $\\mathbb{F}_2^n$ as an ordered direct sum of two subspaces? ($\\mathbb{F}_2$ is the field of two elements)"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9390867141937481fc5963a63bea805185528378ac7097c06bee2a0676b59c84 | What is the number of ways to distribute $n$ balls into $m$ bins, where each bin can contain zero or more balls? | [
"\\((m+n-1)C(m-1)\\)"
] | [
"What is the number of ways to distribute $n$ balls into $m$ bins, where each bin can contain zero or more balls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-4a4c81a11cf3c231f6c745d027b69bb9cce17fea8cf4bda86d0dfceae8621424 | What is the number of ways to draft 15 recruits into 3 different regiments, with 5 recruits in each regiment? | [
"\\(\\frac{15!}{5!5!5!}\\)"
] | [
"What is the number of ways to draft 15 recruits into 3 different regiments, with 5 recruits in each regiment?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-c35981a291d1a02b38eb22b901ebca1c8e0df1d56250e1660b76e00403cdbc0e | What is the number of ways to fill an $m \times n$ array with -1 and +1 such that the product of the numbers in each row and each column is equal to -1? | [
"\\( 2^{(m-1)(n-1)} \\)"
] | [
"What is the number of ways to fill an $m \\times n$ array with -1 and +1 such that the product of the numbers in each row and each column is equal to -1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7cf977a3efd466ea93fa03fe98f053711ba835e51692c9b4e0a4894afc5c178f | What is the number of ways to go up n stairs if we can take one or two steps at a time? | [
"\\( F(n+1) \\)"
] | [
"What is the number of ways to go up n stairs if we can take one or two steps at a time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-ab67e9fcdd157f50d9b9f43887d985a4fcfbfae28a21dda239f96bbe54baa6ec | What is the number of ways to order the 26 letters of the alphabet so that no two vowels (a, e, i, o, and u) occur consecutively? | [
"\\(5! \\times 21! \\times \\binom{20}{5}\\)"
] | [
"What is the number of ways to order the 26 letters of the alphabet so that no two vowels (a, e, i, o, and u) occur consecutively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b85a10bd7de037a4d85d16ce24737060c0156aa314e8d9017f06e4c11a692339 | What is the number of ways to paint $n$ identical balls blue, green, or red such that the number of blue balls is even? | [
"\\dfrac{2n^2 + 8n + 7 + (-1)^n}{8}"
] | [
"What is the number of ways to paint $n$ identical balls blue, green, or red such that the number of blue balls is even?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-9cc2ee385d9136ad18b6544d686e23620d369dff22723b425e68cf2e4a88e09d | What is the number of ways to paint $n$ non-identical balls blue, green, or red such that the number of blue balls is even? | [
"\\dfrac{3^n + 1}{2}"
] | [
"What is the number of ways to paint $n$ non-identical balls blue, green, or red such that the number of blue balls is even?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62fc81b34b30fb3105cf94035432cef6258173cc775d8703251906524de75377 | What is the number of ways to place 6 rooks on an 8x8 chessboard such that no two rooks can attack each other? | [
"564480"
] | [
"What is the number of ways to place 6 rooks on an 8x8 chessboard such that no two rooks can attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-8c5f17c78317c871cc54e28f197813b4957a259ce902d7e24ca15657e8f7e592 | What is the number of ways to put $r$ distinguishable balls into $m$ distinguishable boxes such that each box contains at most 3 balls, given that $r \leq 3m$? | [
"r! \\cdot \\left[ x^r \\right] \\left(1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!}\\right)^m"
] | [
"What is the number of ways to put $r$ distinguishable balls into $m$ distinguishable boxes such that each box contains at most 3 balls, given that $r \\leq 3m$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-0dc2735937bde27828158147af1c8481dcceb17e5a087fdb13cf5b8310cd3d06 | What is the number of ways to write a sum of (at least two) positive integers that add up to \(n\), where different orders of the summands are not distinguished? | [
"p(n) - 1"
] | [
"What is the number of ways to write a sum of (at least two) positive integers that add up to \\(n\\), where different orders of the summands are not distinguished?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50d60fc72673032f77feb4155f45d4b0a9434b8d40e55f1658153d38aff472be | What is the number of zeroes in the product \(1^1 \times 2^2 \times 3^3 \times \cdots \times 100^{100}\)? | [
"1300"
] | [
"What is the number of zeroes in the product \\(1^1 \\times 2^2 \\times 3^3 \\times \\cdots \\times 100^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4634014b6232c5cc58b67e68ce0ee2c845be0741ab60cdf2124617a7ba255c97 | What is the number of zeros at the end of the integer \( 10^2! + 11^2! + \cdots + 99^2! \)? | [
"24"
] | [
"What is the number of zeros at the end of the integer \\( 10^2! + 11^2! + \\cdots + 99^2! \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-58cea8279b087bae5e5268099cb3a6247579cb5ffc6b0b4ee3d7af4cf108289f | What is the number whose sum of factors is 124? | [
"48 or 75"
] | [
"What is the number whose sum of factors is 124?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 10
} | 34 | 0 | 0 |
omr-ws-d3b8bc2a82f98a10e3b3da38581bab52588e11ebf01b0d6f8f69580c4c3e6eaa | What is the numerical coefficient of \( x^{27} \) in the complete expansion of \( (x^2 + 5x + 2)^5 \cdot (x^2 - 7x + 3)^9 \)? | [
"-38"
] | [
"What is the numerical coefficient of \\( x^{27} \\) in the complete expansion of \\( (x^2 + 5x + 2)^5 \\cdot (x^2 - 7x + 3)^9 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 18
} | 50 | 0 | 0 |
omr-ws-4a98b0eefcd6d13fc0ec4afa5a36e50dc58f8f4bb710697098b1685c23099f41 | What is the numerical measure of a parallelogram's fourth angle if its other three angles have degree measures $2x+30$, $3x+50$, and $4x-10$? | [
"110"
] | [
"What is the numerical measure of a parallelogram's fourth angle if its other three angles have degree measures $2x+30$, $3x+50$, and $4x-10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-0b5a5120ee7e7c8cd5e95f16847e93c506444c0ca4295ce1835bffe0e68d181c | What is the numerical value of $(\log_{s^2} m^5)(\log_{m^3} t^6)(\log_{t^5} s^8)$? | [
"8"
] | [
"What is the numerical value of $(\\log_{s^2} m^5)(\\log_{m^3} t^6)(\\log_{t^5} s^8)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eef8006bbe3593c62b58e935ea2b4f4e6a1257402fc0208110f4d111fc731aa6 | What is the numerical value of $\cot^4 \frac{\pi}{7} + \cot^4 \frac{2\pi}{7} + \cot^4 \frac{3\pi}{7}$? | [
"19"
] | [
"What is the numerical value of $\\cot^4 \\frac{\\pi}{7} + \\cot^4 \\frac{2\\pi}{7} + \\cot^4 \\frac{3\\pi}{7}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 9
} | 54 | 0 | 0 |
omr-ws-ba2b90f560a72cc69484d3237a9a1a362d6822d4f5417f33f2abd0e530288916 | What is the numerical value of \( k \) if the roots of the quadratic polynomial \( x^2 - 3(a-2)x + a^2 - 2a + 2k \) are rational? | [
"-0.4"
] | [
"What is the numerical value of \\( k \\) if the roots of the quadratic polynomial \\( x^2 - 3(a-2)x + a^2 - 2a + 2k \\) are rational?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-d19c6f3e7f3719e214821b361177ba7ec40a9e1cb0d6b7f67497dac4e5937f9c | What is the numerical value of the integral \( I = \int_{0}^{\infty} (\sqrt{x^2+a^2} - \sqrt{x^2-a^2}) \, dx \)? | [
"\\infty"
] | [
"What is the numerical value of the integral \\( I = \\int_{0}^{\\infty} (\\sqrt{x^2+a^2} - \\sqrt{x^2-a^2}) \\, dx \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-da4786a359f156ade7ebf5a86cf493f3de00d1f4cfcbb265180b6bf3d6000d7d | What is the one's digit of \( (2 + \sqrt{5})^{2008} \)? | [
"1"
] | [
"What is the one's digit of \\( (2 + \\sqrt{5})^{2008} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 47,
"genselect": 8
} | 55 | 0 | 0 |
omr-ws-de6e4b92c24fc8ca1f26ceb25559131c31e8966a958d3ac0bfbe80da21a31c06 | What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \cdots + 2009^{2009}?$ | [
"5"
] | [
"What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \\cdots + 2009^{2009}?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5560522b8c8d095118a89df56db66be83f044cafc1500b62267b3a49ab7badd2 | What is the ones digit of $7^{35}$ when written as an integer? | [
"3"
] | [
"What is the ones digit of $7^{35}$ when written as an integer?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f69fc17118725d4a6db7a80741ad50387ed00afe892b5c8d66eb3a5b2b93ea34 | What is the ones digit of \(222{,}222 - 22{,}222 - 2{,}222 - 222 - 22 - 2\)? | [
"2"
] | [
"What is the ones digit of \\(222{,}222 - 22{,}222 - 2{,}222 - 222 - 22 - 2\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45255b435f75f67e9e8d8e83097550418f4024e07abb14da98c90ac6437c9e0c | What is the ones digit of the number \(2015^2 + 2015^0 + 2015^1 + 2015^5\)? | [
"6"
] | [
"What is the ones digit of the number \\(2015^2 + 2015^0 + 2015^1 + 2015^5\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-083b2872012d1bac5c50bf882a416f396b2e0febff70407a27d5f21d7fa4763c | What is the only integer whose square is less than its double? | [
"1"
] | [
"What is the only integer whose square is less than its double?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ff74ce2ef99af92ad5ecabea46bfa3eea51b9fea041296aaeb3429596c72bd5 | What is the only natural number \( n \) such that \( 2^8 + 2^{11} + 2^n \) is a perfect square? | [
"12"
] | [
"What is the only natural number \\( n \\) such that \\( 2^8 + 2^{11} + 2^n \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-4a8e325e937a572318753c8d63c153ea0f41335ea8df71ed16863fa42c7d3eb5 | What is the only real number \( x > 1 \) which satisfies the equation \[ \log_2 x \log_4 x \log_6 x = \log_2 x \log_4 x + \log_2 x \log_6 x + \log_4 x \log_6 x? \] | [
"48"
] | [
"What is the only real number \\( x > 1 \\) which satisfies the equation\n\\[ \\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": 7
} | 23 | 0 | 0 |
omr-ws-203e268eacfb937a281d2300bb0e8cfed67b063c2cb2b7b5bd1c7804f7787faf | What is the order of $186$ modulo $407$? | [
"2"
] | [
"What is the order of $186$ modulo $407$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-cf7e44a14efdb334ae3e30582c5c19990d87c020d4666cfc85386b4c13cf037b | What is the order of 12 in the group U(29)? | [
"4"
] | [
"What is the order of 12 in the group U(29)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9061d7556a2b6adec9ea89097fa2fa48c86fab0de2543659ef30bbc3512586a0 | What is the order of magnitude of the sum $$\sum_{p\in\mathbb P, p\leq n}\frac{1}{(\log{p})^k},$$ where \( k \) is a given integer and \( \mathbb P \) denotes the set of prime numbers? | [
"\\(\\frac{n}{\\log(n)^{k+1}}\\)"
] | [
"What is the order of magnitude of the sum \n$$\\sum_{p\\in\\mathbb P, p\\leq n}\\frac{1}{(\\log{p})^k},$$\nwhere \\( k \\) is a given integer and \\( \\mathbb P \\) denotes the set of prime numbers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-f94fb07e90b6b13ecd7698a15af1e660c1b148882a1d4b045601e76f415c5946 | What is the order of the group $G$ of invertible $3 \times 3$ matrices over the field $\mathbb{Z}/2\mathbb{Z}$? | [
"168"
] | [
"What is the order of the group $G$ of invertible $3 \\times 3$ matrices over the field $\\mathbb{Z}/2\\mathbb{Z}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e740c65cf58c3215f55f94fea24c53bf0cf84dd04b0005f43ad2ff00330d335b | What is the order of the group $SL_2(\mathbb{Z} / n\mathbb{Z})$? | [
"\\(\\prod_{p^j \\| n} (p^2 - 1) p^{3j - 2}\\)"
] | [
"What is the order of the group $SL_2(\\mathbb{Z} / n\\mathbb{Z})$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-b81da099cefa33101a8a9e5ba53e5924c1b912fe76381447462e5ce92cc9a54b | What is the order of the group $\frac{\mathbb{Z}_{12} \times D_{4}}{\langle 3, \rho \rangle}$ where $\rho \in D_{4}$ is such that $\rho^{4} = e$? | [
"24"
] | [
"What is the order of the group $\\frac{\\mathbb{Z}_{12} \\times D_{4}}{\\langle 3, \\rho \\rangle}$ where $\\rho \\in D_{4}$ is such that $\\rho^{4} = e$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-2222fe9855a757fb79291a1bb64f858fac799a86a624b70e00cc180ea2e82f4b | What is the order of the group \( G \) with the presentation \( \left < a,b \mid a^{16} = b^6 = 1, bab^{-1} = a^3 \right > \)? | [
"48"
] | [
"What is the order of the group \\( G \\) with the presentation \\( \\left < a,b \\mid a^{16} = b^6 = 1, bab^{-1} = a^3 \\right > \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-471512b3904e9e22381ba8f605aea5dd60e3d6c51b076c6f89a8d067aedf4375 | What is the order of the normalizer of a Sylow p-subgroup of the simple group \( A_n \)? | [
"\\frac{1}{2} \\prod_{i=1}^k \\left( (p^i)!^{a_i} \\times a_i! \\right)"
] | [
"What is the order of the normalizer of a Sylow p-subgroup of the simple group \\( A_n \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcdd6b377be8c81996fc9bd69556e2747bf7bd70d953defe7c589fed0e496faf | What is the order of the splitting field of the polynomial $x^5 + x^4 + 1 = (x^2 + x + 1)(x^3 + x + 1)$ over $\mathbb{Z}_2$? | [
"8"
] | [
"What is the order of the splitting field of the polynomial $x^5 + x^4 + 1 = (x^2 + x + 1)(x^3 + x + 1)$ over $\\mathbb{Z}_2$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f51e00996d31e4ba4a1f6b85803f0a4ea5b585ed53884e8de05dd628c7b259c | What is the order of the symmetry group of the square? | [
"8"
] | [
"What is the order of the symmetry group of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f90a46816e251accf7eafc03c46a90d89316de7b07532876f478f5e93e31ea3a | What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$? | [
"(3,-1)"
] | [
"What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94cc3212a6b437d05beb0f13d9200cfa5f1c3a7ff12ea8ab988f3928584ea7f9 | What is the parity of \( N \) in \( N = \left\lfloor \frac{2002!}{2001 \times 2003} \right\rfloor \)? | [
"Even"
] | [
"What is the parity of \\( N \\) in \\( N = \\left\\lfloor \\frac{2002!}{2001 \\times 2003} \\right\\rfloor \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-495fd1d7e5fe78f0449a01973049116a5b03584dc83b62c7e9cc75e9f5a1c755 | What is the parity of \( N \) where \( N = \lfloor (2023!)^{\frac{1}{2023}} \rfloor \)? | [
"Even"
] | [
"What is the parity of \\( N \\) where \\( N = \\lfloor (2023!)^{\\frac{1}{2023}} \\rfloor \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-730babbedee2b6ff97ec342649c566b216eac9f7dd26ccbdc70da3b8f41d3bad | What is the partial fraction expansion of \( \frac{1}{e^z - 1} \)? Given that it has simple poles at \( 2k\pi i \) for \( k \in \mathbb{Z} \) with residues of 1, determine the partial fraction expansion. | [
"\\(-\\frac{1}{2} + \\frac{1}{z} + \\sum_{k=1}^{\\infty} \\left( \\frac{1}{z - 2\\pi i k} + \\frac{1}{z + 2\\pi i k} \\right)\\)"
] | [
"What is the partial fraction expansion of \\( \\frac{1}{e^z - 1} \\)? \n\nGiven that it has simple poles at \\( 2k\\pi i \\) for \\( k \\in \\mathbb{Z} \\) with residues of 1, determine the partial fraction expansion."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3cb95e99b58745b18dc3ea8405de9a801b034325eaa7e5c56800adb3ea3b34da | What is the perfect square trinomial for \(16x^4 - 56x^2 y + \_\_\_ = ( \_\_\_ - \_\_\_ )^2\)? | [
"\\(49y^2\\)"
] | [
"What is the perfect square trinomial for \\(16x^4 - 56x^2 y + \\_\\_\\_ = ( \\_\\_\\_ - \\_\\_\\_ )^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 18
} | 25 | 0 | 0 |
omr-ws-0b148ece9cf5a01ab8a210c2bc45ce27a3bd8717cfcd0e0366603299d44e6a6d | What is the perimeter of a regular dodecagon inscribed in a circle with radius 1? | [
"\\(6\\sqrt{6} - 6\\sqrt{2}\\)"
] | [
"What is the perimeter of a regular dodecagon inscribed in a circle with radius 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-95eb37f20279325921530eb98b35bd72e39e3589bf3c605b7f705852231c60c6 | What is the perimeter of a right-angled triangle if one of its sides is of length $47$ and the other two sides have integer lengths? | [
"2256"
] | [
"What is the perimeter of a right-angled triangle if one of its sides is of length $47$ and the other two sides have integer lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-96a802109582d9d9bd472a358223b7a53ee74120951a03fb54763fa501f2920a | What is the perimeter of a spiral constructed by connecting the midpoints of adjacent sides of a regular hexagon with side length 1, shading a triangle, and continuing the process as shown? | [
"\\(4 + 2\\sqrt{3}\\)"
] | [
"What is the perimeter of a spiral constructed by connecting the midpoints of adjacent sides of a regular hexagon with side length 1, shading a triangle, and continuing the process as shown?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-04522b61f0f06db6f466a4bf155067f818e23524d793f61ea03e65605a43b029 | What is the perimeter of a square with a side length of 4 cm? (Note: The original post seems to have a typo or unclear statement. Assuming the intent was to ask about the perimeter of a square with a given side length.) | [
"16"
] | [
"What is the perimeter of a square with a side length of 4 cm? \n\n(Note: The original post seems to have a typo or unclear statement. Assuming the intent was to ask about the perimeter of a square with a given side length.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3415df227e1552fd99a3473def81dd45c20cf2a8974d20c8d095fbfc7f047732 | What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$? | [
"16"
] | [
"What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0 \\le u \\le 1$, $0 \\le v \\le 1$, and $0 \\le w \\le 1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 6,
"tir": 14
} | 56 | 0 | 0 |
omr-ws-f7744e4198b945c2f7c6e4e1d7d1c2dc25bcf346698849219c28aa87b50625ee | What is the perimeter of the equilateral triangle ACE inscribed in a regular hexagon ABCDEF with side length 1? | [
"3"
] | [
"What is the perimeter of the equilateral triangle ACE inscribed in a regular hexagon ABCDEF with side length 1?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3988f05435eb7df93c55bb6a8ba116d95b41da8fc67184ac79507acd370cc22 | What is the perimeter of the next smallest Pythagorean right triangle with legs differing by 1? | [
"408"
] | [
"What is the perimeter of the next smallest Pythagorean right triangle with legs differing by 1?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2cdbfd1d0d89a581b965ca9bc215175cbc2705c8e04efadd12e4c202fa1e77b | What is the perimeter of the right triangle whose exradius of the hypotenuse is 30? | [
"60"
] | [
"What is the perimeter of the right triangle whose exradius of the hypotenuse is 30?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 1
} | 40 | 0 | 0 |
omr-ws-189201bdbce9a66c87cea8a84fb3a42f169c08a102affc300064c9fbf47802b4 | What is the perimeter, in cm, of quadrilateral $ABCD$ if $\overline{AB} \perp \overline{BC}$, $\overline{DC} \perp \overline{BC}$, $AB=9$ cm, $DC=4$ cm, and $BC=12$ cm? | [
"38"
] | [
"What is the perimeter, in cm, of quadrilateral $ABCD$ if $\\overline{AB} \\perp \\overline{BC}$, $\\overline{DC} \\perp \\overline{BC}$, $AB=9$ cm, $DC=4$ cm, and $BC=12$ cm?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 2
} | 34 | 0 | 0 |
omr-ws-d77a423405e8bf3db8f9ca014be498b8565ab981bb7b7b1819e836f1bf8df630 | What is the period of $\cos{(\cos{(x)})}$? | [
"$2\\pi$"
] | [
"What is the period of $\\cos{(\\cos{(x)})}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66504bd06d1a99e172fb1cf40d6af18bd08cbcf00b9cbe5cb01c7e4b6e250d73 | What is the period of $\sin^{-1}(\sin x)$? | [
"\\(2\\pi\\)"
] | [
"What is the period of $\\sin^{-1}(\\sin x)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
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