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-ceba77045d0576c787d130835997f2f519feb2bbdc71adb2480723079c38eade | Write the number 40320 as a product of consecutive natural numbers. | [
"\\(8 \\cdot 7 \\cdot 6 \\cdot 5 \\cdot 4 \\cdot 3 \\cdot 2 \\cdot 1\\) or \\(2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 \\cdot 8\\)"
] | [
"Write the number 40320 as a product of consecutive natural numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d15cd6ac37a23f73cfb581f2fe8aca9248d3651eba8573b0f03f442c0dd529d5 | Write the number of square feet that one gallon of glue will cover as a function of \( d \) (Use 1 ft³ = 7.5 gal). | [
"\\dfrac{2}{15d}"
] | [
"Write the number of square feet that one gallon of glue will cover as a function of \\( d \\) (Use 1 ft³ = 7.5 gal)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-44f3e14c1dc34b122d58f71e54fd6a7176507c781be11e38bc93651473af6d56 | Write the numbers $1$ through $9$ in a $14 \times 14$ grid such that the numbers in consecutive cells (horizontally, vertically, or diagonally) are relatively prime. Let $N$ be the maximum number of times any single number appears in the grid. Find the minimum possible value of $N$. | [
"28"
] | [
"Write the numbers $1$ through $9$ in a $14 \\times 14$ grid such that the numbers in consecutive cells (horizontally, vertically, or diagonally) are relatively prime. Let $N$ be the maximum number of times any single number appears in the grid. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-84449d2f687b12bee70816b25e589eccb593a5f9d1c4d641bf4550d67d6afb9c | Write the numbers $1$ to $15$ in a row such that the sum of every two adjacent numbers is a perfect square. How many ways can this be done? | [
"2"
] | [
"Write the numbers $1$ to $15$ in a row such that the sum of every two adjacent numbers is a perfect square. How many ways can this be done?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6f13b0cc78ccbded4d1792d9f15f8a2214b522789b34f8b20491ea92f0527db | Write the numbers $1, 2, 3, \ldots, 2012$ on a circle such that each number is a multiple of the difference between its two adjacent numbers. | [
"1, 2, 3, \\ldots, 2012"
] | [
"Write the numbers $1, 2, 3, \\ldots, 2012$ on a circle such that each number is a multiple of the difference between its two adjacent numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d737c2893f8dbd55095a3cf108a3ca847b4d5a585ade802fa6d5ee3d9edc2fbf | Write the numbers $1, 2, 3, \ldots, 2013$ on a circle such that each number is a multiple of the difference between its two adjacent numbers. | [
"1, 2013, 2, 2012, 3, 2011, \\ldots, 1007, 1006, 1008"
] | [
"Write the numbers $1, 2, 3, \\ldots, 2013$ on a circle such that each number is a multiple of the difference between its two adjacent numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-4819979ceecf9b339f85ad6eba915a259e3094660aa0cbba1521fda76efb579e | Write the numbers $1, 2, 3, \ldots, 8$ on a circle such that each number is a multiple of the difference between its two adjacent numbers. | [
"\\(1, 7, 8, 5, 3, 2, 4, 6\\)"
] | [
"Write the numbers $1, 2, 3, \\ldots, 8$ on a circle such that each number is a multiple of the difference between its two adjacent numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-4929528e97a00b9d2a8dbc6f79b01c6d060a3c5371b5a88676c5b2770f765ded | Write the numbers 1 through 2012 in a line to form the number \( A = 12345 \ldots 20112012 \). Starting with the first digit of \( A \), multiply it by 2 and add the second digit. Continue this process with the result by multiplying by 2 and adding the next digit, and so on until the last digit of \( A \). This process... | [
"4"
] | [
"Write the numbers 1 through 2012 in a line to form the number \\( A = 12345 \\ldots 20112012 \\). Starting with the first digit of \\( A \\), multiply it by 2 and add the second digit. Continue this process with the result by multiplying by 2 and adding the next digit, and so on until the last digit of \\( A \\). ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-9ff78c9ed2d9cd50e0a493640620876128e050491f0ad57836046e2418e227c2 | Write the numbers 1, 1. Write their sum between them to get 1, 2, 1. Repeat this procedure 100 times. What is the sum of the numbers obtained after 100 iterations? | [
"\\( 2^{101} - 1 \\)"
] | [
"Write the numbers 1, 1. Write their sum between them to get 1, 2, 1. Repeat this procedure 100 times. What is the sum of the numbers obtained after 100 iterations?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8283b1a95b7fa47bfb2c130ab2e097a6a1a8ba36712ea33354ac79ffc6ee812d | Write the numbers 1, 1. Write their sum between them to get 1, 2, 1. Repeat this procedure 100 times. What is the sum of the numbers obtained after 100 repetitions? | [
"\\( 3^{100} + 1 \\)"
] | [
"Write the numbers 1, 1. Write their sum between them to get 1, 2, 1. Repeat this procedure 100 times. What is the sum of the numbers obtained after 100 repetitions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-2c5aa38ac2701d18974c5bb72845bd65abe82da914b97497d2e92b1e73e0d788 | Write the numbers 1, 2, …, n in a row, in increasing order. In the next row, write the same numbers in decreasing order. Consider the difference of any two numbers, one of which is lying below the other. Determine the sum of the absolute values of all such differences. | [
"\\(\\lfloor \\frac{n^2}{2} \\rfloor\\)"
] | [
"Write the numbers 1, 2, …, n in a row, in increasing order. In the next row, write the same numbers in decreasing order. Consider the difference of any two numbers, one of which is lying below the other. Determine the sum of the absolute values of all such differences."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3bbe479e009d735fae47c09f6649b8ac30d9f7d0d8d9f4e820b63f6173527bbb | Write the numbers from $1$ to $16$ in a circle such that for every triple $a, b, c$ which goes in succession, the condition $b^2 - ac \equiv 0 \pmod{17}$ is satisfied. | [
"1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6"
] | [
"Write the numbers from $1$ to $16$ in a circle such that for every triple $a, b, c$ which goes in succession, the condition $b^2 - ac \\equiv 0 \\pmod{17}$ is satisfied."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71b2ec24ab24c1ee59b9075c7f243df7f7dab2c0c8e07eacff02c2f9fdc231a8 | Write the numbers from $1$ to $16$ in the cells of a $4 \times 4$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of the sides of the original $4 \times 4$ square, the sum of th... | [
"\n\\begin{array}{cccc}\n2 & 8 & 5 & 15 \\\\\n6 & 10 & 1 & 11 \\\\\n7 & 13 & 16 & 12 \\\\\n3 & 9 & 14 & 4 \\\\\n\\end{array}\n"
] | [
"Write the numbers from $1$ to $16$ in the cells of a $4 \\times 4$ square so that:\n1. Each cell contains exactly one number;\n2. Each number is written exactly once;\n3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of the sides of the original $4 \\times 4$ square, the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc54ea84b93e01480ec573173b718ea58fbbdc5203fccd94b89611435d812e53 | Write the parametric equation of a line that is perpendicular to the line given by $(-4+6t, 3+2t, -1-7t)$. The direction vector of the given line is $(6, 2, -7)$. Determine the values of $p$, $q$, and $r$ such that the new line intersects the given line and is perpendicular to it. | [
"\\( p = -4, q = 3, r = -1 \\)"
] | [
"Write the parametric equation of a line that is perpendicular to the line given by $(-4+6t, 3+2t, -1-7t)$. The direction vector of the given line is $(6, 2, -7)$. Determine the values of $p$, $q$, and $r$ such that the new line intersects the given line and is perpendicular to it."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-c2f19e57bdeacd62cd5bc34c6cfc5de655be0e486835354fe922479892922a6e | Write the parametric equations of the line that goes through the point (6, -2, 1) and is perpendicular to both lines given by: \[ [x, y, z] = [1, 4, -2] + t[3, -1, 1] \] and \[ [x, y, z] = [9, 5, -3] + s[1, -3, 7]. \] | [
"\\( x = 6 + t, y = -2 + 5t, z = 1 + 2t \\)"
] | [
"Write the parametric equations of the line that goes through the point (6, -2, 1) and is perpendicular to both lines given by:\n\\[ [x, y, z] = [1, 4, -2] + t[3, -1, 1] \\]\nand\n\\[ [x, y, z] = [9, 5, -3] + s[1, -3, 7]. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-30cad7190be90b2ea3f1a1c23a9739f72e31da3fde3dfc224b49e322d12af903 | Write the partial fraction decomposition for the rational expression $\frac{1}{a^{2}-x^{2}}$. Check your result algebraically. Then assign a value to the constant $a$ and check the result graphically. | [
"\\(\\frac{1}{2a} \\left( \\frac{1}{a + x} + \\frac{1}{a - x} \\right)\\)"
] | [
"Write the partial fraction decomposition for the rational expression $\\frac{1}{a^{2}-x^{2}}$. Check your result algebraically. Then assign a value to the constant $a$ and check the result graphically."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-d1a7f468dae5b3cdba6cec0bf5520143bc51fe5bcf757afab1f4be4176d53acb | Write the partial fraction decomposition of \(\frac{x^2 + 4}{x^2 - 3x + 2}\). | [
"\\(1 + \\frac{-8}{x-2} + \\frac{5}{x-1}\\)"
] | [
"Write the partial fraction decomposition of \\(\\frac{x^2 + 4}{x^2 - 3x + 2}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-b97c4aadc627098fd4066d6d0c0363e86cf2a9e59f3358c6123ed4f5e7db4b2a | Write the permutation $(1 3 4 2)$ as the product of transpositions and evaluate the corresponding matrix product using the definition $P = \sum e_{\sigma(i), i}$ for permutation matrices. | [
"\\(\\begin{pmatrix} 0 & 0 & 1 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0 \\end{pmatrix}\\)"
] | [
"Write the permutation $(1 3 4 2)$ as the product of transpositions and evaluate the corresponding matrix product using the definition $P = \\sum e_{\\sigma(i), i}$ for permutation matrices."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-898ff9814554000679740a41ba9bd7ff4796d5c25fa80a7b97d091f86c5d8906 | Write the permutation $(1235)(413)$ as a product of disjoint cycles. | [
"\\((15)(234)\\)"
] | [
"Write the permutation $(1235)(413)$ as a product of disjoint cycles."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-b0c9a8f8b76cf829f612f9b22d177daa0ce0b0e99a065e14fd3446f9a30b3a51 | Write the permutation $(13256)(23)(46512)$ as a product of disjoint cycles. | [
"\\((15)(234)(6)\\)"
] | [
"Write the permutation $(13256)(23)(46512)$ as a product of disjoint cycles."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d343d02acde23b2a99299ab644b0c81e89b51772694b8d2f6be67aa5a5b63a1c | Write the polar equation for a hyperbola with eccentricity \( \frac{3}{2} \) and directrix \( x = -2 \). | [
"r = \\dfrac{6}{2 - 3 \\cos \\theta}"
] | [
"Write the polar equation for a hyperbola with eccentricity \\( \\frac{3}{2} \\) and directrix \\( x = -2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1122dda1d506a4a4a1f2b21cdef86b04f82f54cccfc64885f3eea0f060be5416 | Write the polynomial $2(a^2 + b^2 + c^2) + abc + 8 - 5(a + b + c)$ with $a, b, c > 0$ as a sum of squares. | [
"(a - 2)^2 + (b - 2)^2 + (c - 2)^2 + (a - 1)(b - 1) + (b - 1)(c - 1) + (c - 1)(a - 1)"
] | [
"Write the polynomial $2(a^2 + b^2 + c^2) + abc + 8 - 5(a + b + c)$ with $a, b, c > 0$ as a sum of squares."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-6a611e38e55252b284f74dce029f97f131540b1743db00290bd63051d0f6c2ef | Write the polynomial $\prod_{j=1}^{1996}(1+jx^{3^j})=\sum_{k=0}^m a_kx^{p_k}$ where $p_k$ is increasing and $a_k$ is non-zero. Find the coefficient $a_{1994}$. | [
"0"
] | [
"Write the polynomial $\\prod_{j=1}^{1996}(1+jx^{3^j})=\\sum_{k=0}^m a_kx^{p_k}$ where $p_k$ is increasing and $a_k$ is non-zero. Find the coefficient $a_{1994}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-a9d8a222a4a8e571e880b2e965c4bb028b58fcd6719c503ccb0ffe5bce5950c7 | Write the polynomial $a^2 + b^2 + c^2 + 2abc + 1 - 2(ab + bc + ca)$ with $a, b, c > 0$ as a sum of squares. | [
"(ab - 1)^2 + (bc - 1)^2 + (ca - 1)^2"
] | [
"Write the polynomial $a^2 + b^2 + c^2 + 2abc + 1 - 2(ab + bc + ca)$ with $a, b, c > 0$ as a sum of squares."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-df19fdadd25d2a451e965ba2aecd5f2103ca80b2489911863daf7a9fbb0b2612 | Write the polynomial \( A(x) = x^5 - x^4 + x^3 - x^2 + x - 1 \) as a product of irreducible factors over \( \mathbb{Q} \). | [
"\\( (x-1)(x^2 - x + 1)(x^2 + x + 1) \\)"
] | [
"Write the polynomial \\( A(x) = x^5 - x^4 + x^3 - x^2 + x - 1 \\) as a product of irreducible factors over \\( \\mathbb{Q} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-56626f40def947983a8bcfa387b05ff094abb8e22cdc59203b1ae85ab5208a02 | Write the polynomial \( H = a^5c + 2a^3c - ac^2 + 4c + ac - 4 \) as a product of another polynomial. | [
"(a^3c + ac - 4)(a^2 + 2)"
] | [
"Write the polynomial \\( H = a^5c + 2a^3c - ac^2 + 4c + ac - 4 \\) as a product of another polynomial."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-23c28400a29592e914a1926dac35e014a9a9751667ad3a77b29669672c6f50ba | Write the polynomial \( P(x) = x^6 - 2x^5 + x^2 - x - 2 \) as a product of three non-constant integer polynomials. | [
"(x - 2)(x^2 + x + 1)(x^3 - x^2 + 1)"
] | [
"Write the polynomial \\( P(x) = x^6 - 2x^5 + x^2 - x - 2 \\) as a product of three non-constant integer polynomials."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c709d2bbd764a1398984f3ada9396ffd87d65baa6d272bf48cbd8a9bb6d418d6 | Write the polynomial \( f(X) = X^{2005} - 2005X + 2004 \) as a product of irreducible polynomials in \( \mathbb{Z}[X] \). | [
"(X - 1)^2 \\left( X^{2003} + 2X^{2002} + 3X^{2001} + \\cdots + 2004 \\right)"
] | [
"Write the polynomial \\( f(X) = X^{2005} - 2005X + 2004 \\) as a product of irreducible polynomials in \\( \\mathbb{Z}[X] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-1c4de43d4d198f4c29f6d738a00ef5f448c1cd3c772167979732caa2fa7674b6 | Write the polynomial \(P(x) = x^{2003} + 2x^{2002} + \ldots + 2003x + 2004\) as a product of irreducible polynomials in \(\mathbb{Z}[x]\). | [
"P(x)"
] | [
"Write the polynomial \\(P(x) = x^{2003} + 2x^{2002} + \\ldots + 2003x + 2004\\) as a product of irreducible polynomials in \\(\\mathbb{Z}[x]\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-5f5bdf3d5e5e5cd84db2acb4a5858afc338ec0321148138159fc3570aee666e7 | Write the polynomial \(\prod_{n=1}^{1996}(1+nx^{3^n}) = \sum_{n=0}^m a_nx^{k_n}\), where the \(k_n\) are in increasing order, and the \(a_n\) are nonzero. Find the coefficient \(a_{1996}\). | [
"665280"
] | [
"Write the polynomial \\(\\prod_{n=1}^{1996}(1+nx^{3^n}) = \\sum_{n=0}^m a_nx^{k_n}\\), where the \\(k_n\\) are in increasing order, and the \\(a_n\\) are nonzero. Find the coefficient \\(a_{1996}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-d95caed3311fe9a8f9bf2870c15f0857b8d0269f12ba205fbe2959f865f057e5 | Write the positive integers $\{1, 2, 3, \ldots, n\}$ on a blackboard. Every turn, replace an integer $k$ with $k+1$ and $k+2$, but ensure $k$ does not appear more than $k$ times. Determine the smallest value of $n$ such that it is impossible to make every element on the blackboard greater than $n$. | [
"5"
] | [
"Write the positive integers $\\{1, 2, 3, \\ldots, n\\}$ on a blackboard. Every turn, replace an integer $k$ with $k+1$ and $k+2$, but ensure $k$ does not appear more than $k$ times. Determine the smallest value of $n$ such that it is impossible to make every element on the blackboard greater than $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-3f3cbc266367b8f171d7b36d7f3ecf722d8c66de296d010bb370649095168b9b | Write the positive integers \(\{1, 2, 3, \ldots, n\}\) on a blackboard. Every turn, replace an integer \(k\) with \(k+1\) and \(k+2\), but \(k\) cannot appear on the blackboard more than \(k\) times. Determine the smallest value of \(n\) such that it is impossible to perform a finite series of replacements to make ever... | [
"5"
] | [
"Write the positive integers \\(\\{1, 2, 3, \\ldots, n\\}\\) on a blackboard. Every turn, replace an integer \\(k\\) with \\(k+1\\) and \\(k+2\\), but \\(k\\) cannot appear on the blackboard more than \\(k\\) times. Determine the smallest value of \\(n\\) such that it is impossible to perform a finite series of rep... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-691e67135529348f15e09c74b883f5a15c33dd7c034bc26c7dbc5a0e96e97e09 | Write the positive integers in a spiral as follows: \[ \begin{matrix} 7 & 8 & 9 & 10\\ 6 & 1 & 2 & \cdots\\ 5 & 4 & 3 & \cdots \end{matrix}\] Place a white pawn on every square whose number is divisible by 3 or 5. Can a white queen, starting at 1, make a single move of a million squares? (A queen may move along any row... | [
"Yes"
] | [
"Write the positive integers in a spiral as follows:\n\\[ \\begin{matrix}\n7 & 8 & 9 & 10\\\\\n6 & 1 & 2 & \\cdots\\\\\n5 & 4 & 3 & \\cdots\n\\end{matrix}\\]\nPlace a white pawn on every square whose number is divisible by 3 or 5. Can a white queen, starting at 1, make a single move of a million squares? (A queen m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-2ec41d779e6951cf1bafc6b37cbddc3ac7c40d9b67223c3d4ede46f7ebaedd04 | Write the quadratic equation \( f(t) = -2t^2 + 4t + 1 \) in standard form. Identify the vertex, axis of symmetry, and x-intercepts. | [
"Vertex \\((1, 3)\\), Axis of Symmetry \\( x = 1 \\), X-intercepts \\( 1 \\pm \\sqrt{\\frac{3}{2}} \\)"
] | [
"Write the quadratic equation \\( f(t) = -2t^2 + 4t + 1 \\) in standard form. Identify the vertex, axis of symmetry, and x-intercepts."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-0230f535f4f44be0e06f91a957b5b482a99935a92286770ab0b92933956de9e4 | Write the result in scientific notation: $\frac{420000000 \times 0.00075}{0.00000036 \times 250}$. | [
"3.5 \\times 10^9"
] | [
"Write the result in scientific notation: $\\frac{420000000 \\times 0.00075}{0.00000036 \\times 250}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1ce0a1b76e29bbcf42753511546fc9b9780fef2219117645d45197dc9f9b23e | Write the roots of the equation $x^2 + x + 2 = 0$ in the form of a polynomial of the 7th root of unity. | [
"\\(\\zeta + \\zeta^2 + \\zeta^4\\) and \\(\\zeta^3 + \\zeta^5 + \\zeta^6\\)"
] | [
"Write the roots of the equation $x^2 + x + 2 = 0$ in the form of a polynomial of the 7th root of unity."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-82de973130d41c291c0697b9d7f0e87a51f12a42fd3407ed1d653165517e71c2 | Write the second-degree Taylor polynomial for \( f' \) about 4 and use it to approximate \( f'(4.3) \). | [
"\\(-0.54\\)"
] | [
"Write the second-degree Taylor polynomial for \\( f' \\) about 4 and use it to approximate \\( f'(4.3) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4866104e5305d0ee94db13fe0f3383944f3e99439d4586323f49b11b29099bdf | Write the solutions of the equation $z^2 + z + 2 = 0$ in terms of the lowest degree polynomial of $\alpha$, where $\alpha$ is a 7th root of unity. | [
"\\( \\alpha^4 + \\alpha^2 + \\alpha \\)"
] | [
"Write the solutions of the equation $z^2 + z + 2 = 0$ in terms of the lowest degree polynomial of $\\alpha$, where $\\alpha$ is a 7th root of unity."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-6edf43dfefee7428b566d9a203352f3d37f8530288484d4d160e6ef84613c6ee | Write the statement below in a symmetric form: If $a \ge 1 \ge b \ge c \ge 0$ and $a^2 + b^2 + c^2 = 3$, then $$abc + \frac{1}{\sqrt{2}}(a - c) \ge 1.$$ | [
"Let \\(a, b, c\\) be real numbers, and let \\( x = \\max\\{a, b, c\\}, \\ y = \\min\\{a, b, c\\} \\). If \\( x \\ge 1 \\ge a + b + c - x - y \\ge y \\ge 0 \\) and \\( x^2 + y^2 + z^2 = 3 \\), then \\( abc + \\frac{1}{\\sqrt{2}}(x - y) \\ge 1 \\)."
] | [
"Write the statement below in a symmetric form:\nIf $a \\ge 1 \\ge b \\ge c \\ge 0$ and $a^2 + b^2 + c^2 = 3$, then\n$$abc + \\frac{1}{\\sqrt{2}}(a - c) \\ge 1.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7f9e6ef4d308264818d019932839bb3d5e16790735203fb2f4fe79dd0614152b | Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients. | [
"\\(\\frac{1}{2n^2 + 14n + 24}\\)"
] | [
"Write the sum $\\sum_{i=0}^{n}{\\frac{(-1)^i\\cdot\\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 11
} | 58 | 0 | 0 |
omr-ws-f699dedb2d7933d14b5da43eeb5efa771cd49dbfb6e983e4deb224755ab39c21 | Write the sum $\sum_{k=2}^{63}\frac{k^3-k}{k^6-1}$ in the form $\frac{l}{m}$, where $l$ and $m$ are relatively prime integers. | [
"\\(\\frac{2015}{12099}\\)"
] | [
"Write the sum $\\sum_{k=2}^{63}\\frac{k^3-k}{k^6-1}$ in the form $\\frac{l}{m}$, where $l$ and $m$ are relatively prime integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 30
} | 77 | 0 | 0 |
omr-ws-a2843655867cce0ecd3057b0241d6fc14673a8b6192054586eacc320fbe45115 | Write the sum \( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p-1} \) where \( p \) is a prime number as a fraction in lowest terms in the form \( \frac{A_p}{B_p} \). Make a conjecture for the value of \( A_p \pmod{p^2} \). | [
"\\( A_p \\equiv 0 \\pmod{p^2} \\)"
] | [
"Write the sum \\( 1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{p-1} \\) where \\( p \\) is a prime number as a fraction in lowest terms in the form \\( \\frac{A_p}{B_p} \\). Make a conjecture for the value of \\( A_p \\pmod{p^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-03d2ea166779a9fe98fb372d605c0a223e101953f748b3b238848d7166899b47 | Write the sum \( \sum_{k=0}^{n} \frac{(-1)^k \binom{n}{k}}{k^3 + 9k^2 + 26k + 24} \) in the form \( \frac{p(n)}{q(n)} \), where \( p \) and \( q \) are polynomials with integer coefficients. | [
"\\(\\frac{1}{2(n+3)(n+4)}\\)"
] | [
"Write the sum \\( \\sum_{k=0}^{n} \\frac{(-1)^k \\binom{n}{k}}{k^3 + 9k^2 + 26k + 24} \\) in the form \\( \\frac{p(n)}{q(n)} \\), where \\( p \\) and \\( q \\) are polynomials with integer coefficients."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 19
} | 64 | 0 | 0 |
omr-ws-75a456d8b5a98e6ded9f0f030fe1155f47c63879f4af251549c5b7d61ab8759f | Write the sum \(\sum_{i=0}^{n} \frac{(-1)^i \cdot \binom{n}{i}}{i^3 + 9i^2 + 26i + 24}\) as a ratio of two explicitly defined polynomials with integer coefficients. | [
"\\dfrac{1}{2n^2 + 14n + 24}"
] | [
"Write the sum \\(\\sum_{i=0}^{n} \\frac{(-1)^i \\cdot \\binom{n}{i}}{i^3 + 9i^2 + 26i + 24}\\) as a ratio of two explicitly defined polynomials with integer coefficients."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4eae58d186bb3d638b33786af341c1d60203e22cf7fb891bd5d40c34e36c66ec | Write the sum \[ \sum_{k=0}^{n} \frac{(-1)^{k} \binom{n}{k}}{k^{3} + 9k^{2} + 26k + 24} \] in the form \(\frac{p(n)}{q(n)}\) where \(p\) and \(q\) are polynomials with integer coefficients. | [
"\\(\\frac{1}{2(n+3)(n+4)}\\)"
] | [
"Write the sum\n\n\\[\n\\sum_{k=0}^{n} \\frac{(-1)^{k} \\binom{n}{k}}{k^{3} + 9k^{2} + 26k + 24}\n\\]\n\nin the form \\(\\frac{p(n)}{q(n)}\\) where \\(p\\) and \\(q\\) are polynomials with integer coefficients."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 8
} | 62 | 0 | 0 |
omr-ws-534e9b55445a5235906a19bac032a3c6b983ecda68f2fae083db553edca831e9 | Write the transformation of coordinates $(X, Y)$ to $(x, y)$ given by $x = X \cos \theta - Y \sin \theta$ and $y = X \sin \theta - Y \cos \theta$ in matrix form. | [
"\\(\\begin{pmatrix}\n\\cos\\theta & -\\sin\\theta \\\\ \n\\sin\\theta & \\cos\\theta\n\\end{pmatrix}\\)"
] | [
"Write the transformation of coordinates $(X, Y)$ to $(x, y)$ given by $x = X \\cos \\theta - Y \\sin \\theta$ and $y = X \\sin \\theta - Y \\cos \\theta$ in matrix form."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30d52d03dae5a03bba442be06a27c24d9286adefb74758334dd9987245f36641 | Write the variational formulation of the following problem and indicate the test space. \[ \frac{\partial C}{\partial t}(x,t) = \frac{\partial^2 C}{\partial x^2}(x,t) + e^{-x}, \quad 0 < x < \pi, \, t > 0 \] with boundary conditions: \[ C(0,t) = C(\pi,t) = 0, \quad \forall t > 0 \] and initial condition: \[ C(x,0) = \s... | [
"\\text{Test space: } H_0^1(0, \\pi)"
] | [
"Write the variational formulation of the following problem and indicate the test space.\n\\[\n\\frac{\\partial C}{\\partial t}(x,t) = \\frac{\\partial^2 C}{\\partial x^2}(x,t) + e^{-x}, \\quad 0 < x < \\pi, \\, t > 0\n\\]\nwith boundary conditions:\n\\[\nC(0,t) = C(\\pi,t) = 0, \\quad \\forall t > 0\n\\]\nand init... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-dfe1407af8b487b61e47cf9a887a63936585b0885c3004bffec45a6b39111902 | Write the vector equation of the line $L$, which is the intersection of the planes $x + y = 0$ and $y + z = 0$. Find $(a, b, c)$ and $(p, q, r)$ such that $$L = \{(a, b, c) + \lambda(p, q, r) \mid \lambda \in \mathbb{R}\}.$$ | [
"\\((a, b, c) = (0, 0, 0)\\) and \\((p, q, r) = (1, -1, 1)\\)"
] | [
"Write the vector equation of the line $L$, which is the intersection of the planes $x + y = 0$ and $y + z = 0$. Find $(a, b, c)$ and $(p, q, r)$ such that $$L = \\{(a, b, c) + \\lambda(p, q, r) \\mid \\lambda \\in \\mathbb{R}\\}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 26
} | 36 | 0 | 0 |
omr-ws-59ce86ecca635557a973b85e209f67041044d056f152b127baf10eb8e38b5dda | Write vector \( \mathbf{a} = \mathbf{i} - 3\mathbf{j} + \mathbf{k} \) as the sum of a vector in the direction of \( \mathbf{b} = 5\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} \) and a vector in a direction perpendicular to \( \mathbf{b} \). | [
"\\mathbf{a} = \\left( -\\mathbf{i} - \\frac{4}{5}\\mathbf{j} + \\frac{3}{5}\\mathbf{k} \\right) + \\left( 2\\mathbf{i} - \\frac{11}{5}\\mathbf{j} + \\frac{2}{5}\\mathbf{k} \\right)"
] | [
"Write vector \\( \\mathbf{a} = \\mathbf{i} - 3\\mathbf{j} + \\mathbf{k} \\) as the sum of a vector in the direction of \\( \\mathbf{b} = 5\\mathbf{i} + 4\\mathbf{j} - 3\\mathbf{k} \\) and a vector in a direction perpendicular to \\( \\mathbf{b} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2bb1be23be3411051e399f5cfeb64fc6c33d7b6386d5b5314a20dde10a030c25 | Write, but do not evaluate, an integral expression that represents the maximum value of $g(x)$ on the interval $0 \leq x \leq \pi$, where $f(x) = \int_0^{3x} \sqrt{4+t^2} \, dt$ and $g(x) = f(\sin x)$. | [
"\\(\\int_{0}^{3} \\sqrt{4 + t^2}\\ dt\\)"
] | [
"Write, but do not evaluate, an integral expression that represents the maximum value of $g(x)$ on the interval $0 \\leq x \\leq \\pi$, where $f(x) = \\int_0^{3x} \\sqrt{4+t^2} \\, dt$ and $g(x) = f(\\sin x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-8f22feabe6629c4c1e2b809a11df45f9d9f1f8d08fe8a87a5d37ec6ff9b6b05a | Writing down all the integers from 19 to 92 consecutively forms a large integer \( \mathrm{N} = 192021 \ldots 909192 \). If \( \mathrm{N} \) is divisible by \( 3^k \), what is the maximum value of \( k \)? | [
"1"
] | [
"Writing down all the integers from 19 to 92 consecutively forms a large integer \\( \\mathrm{N} = 192021 \\ldots 909192 \\). If \\( \\mathrm{N} \\) is divisible by \\( 3^k \\), what is the maximum value of \\( k \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-1853b1d5a1f293710ac5098d6c646afdc6c5a4f4df8bdc42a9f0e983cf6e691d | Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called singular if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there? | [
"4"
] | [
"Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called singular if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-4b7cad25e8a66acb891a924836d17e72fe8f4bd61f4504a85aea0e519245b5ba | Wurmple's favorite number is the sum of all eight two-digit multiples of a certain number, but the last digit is smudged. What is Wurmple's favorite number? | [
"432"
] | [
"Wurmple's favorite number is the sum of all eight two-digit multiples of a certain number, but the last digit is smudged. What is Wurmple's favorite number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-20d15f69972c49a14c850df6ea414385235e1553c7d35266b772dff26c375c08 | X and Y are two different integers, greater than 1, with a sum less than or equal to 100, and Y is greater than X. S knows the sum X+Y, and P knows the product X*Y. Both S and P are perfect logicians and know the information in these two sentences. The following conversation occurs: - S says, "P does not know X and Y."... | [
"\\( X = 4, Y = 13 \\)"
] | [
"X and Y are two different integers, greater than 1, with a sum less than or equal to 100, and Y is greater than X. S knows the sum X+Y, and P knows the product X*Y. Both S and P are perfect logicians and know the information in these two sentences. The following conversation occurs:\n- S says, \"P does not know X ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-1a97527b848b305b48971771b7a4d635db6357f84add2c7a2a4b157a4fb3ff81 | X works alone for \( \frac{5}{12} \) of the time it would take Y alone to finish the work, and then Y finishes the remaining work alone. It took 52 minutes more than if X and Y had worked together from the beginning. How much time did it take to complete the work? (Note: The work done by Y is twice the work done by X w... | [
"116"
] | [
"X works alone for \\( \\frac{5}{12} \\) of the time it would take Y alone to finish the work, and then Y finishes the remaining work alone. It took 52 minutes more than if X and Y had worked together from the beginning. How much time did it take to complete the work? (Note: The work done by Y is twice the work don... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf019e5202485e04004603b34ca1dcc269757a6d496226253049cfa983341ea0 | X, Y, and Z are playing a game where each has an equal probability of winning each set. The first to win 3 sets is the winner. Given that X won the first set, find the probability that Y will be the winner. | [
"$\\frac{153}{729}$"
] | [
"X, Y, and Z are playing a game where each has an equal probability of winning each set. The first to win 3 sets is the winner. Given that X won the first set, find the probability that Y will be the winner."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 4
} | 11 | 0 | 0 |
omr-ws-a719dffccfd60347c2826052816f4b28dbeb586070bc5782b705ce1fccb2eeee | XYZ is an equilateral triangle with side length 2. ZV and YW are medians, and PQRS is a square. Find the ratio of the area of PQRS to the area of triangle XYZ. | [
"\\(\\frac{52-16\\sqrt{3}}{121\\sqrt{3}}\\)"
] | [
"XYZ is an equilateral triangle with side length 2. ZV and YW are medians, and PQRS is a square. Find the ratio of the area of PQRS to the area of triangle XYZ."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-e524790050582022b1dba9f71f637f4deaab55368e37c799d3f101c5da03f832 | Xander and Marcus play an ultimate game of Starcraft on Thanksgiving. The two bases are located 60 meters away along a straight course, so that any units running between the bases will fight with units coming the other way. Xander begins to churn out zerglings at a constant rate of 10 per minute. Each one, after spawni... | [
"Marcus wins after 17 seconds."
] | [
"Xander and Marcus play an ultimate game of Starcraft on Thanksgiving. The two bases are located 60 meters away along a straight course, so that any units running between the bases will fight with units coming the other way. Xander begins to churn out zerglings at a constant rate of 10 per minute. Each one, after s... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-1fb5399daf0f48cc2de0f65ed63533fe1bbf565103d1512be7569c92276f57e8 | Xanthia buys hot dogs that come in packages of six, and she buys hot dog buns that come in packages of eight. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns? | [
"4"
] | [
"Xanthia buys hot dogs that come in packages of six, and she buys hot dog buns that come in packages of eight. What is the smallest number of hot dog packages she can buy in order to be able to buy an equal number of hot dogs and hot dog buns?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-143dad1930fb7b6894e538e53c6cf00c1cd1576c38edd84ba4497d4adda9b31f | Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? | [
"135"
] | [
"Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b74f850550029cbeb1c756b7f27f4ad579020ddc803c2c5e10d9537019a09ee2 | Xavier is playing a game with buckets and coins. At the beginning, he has a single empty bucket. Before the game starts, Xavier rolls a fair six-sided die repeatedly. Every time the number rolled on the die is a multiple of $3$, Bobby puts a coin in Xavier's bucket. As soon as Xavier rolls a number that is not a multip... | [
"$\\sqrt{e}$"
] | [
"Xavier is playing a game with buckets and coins. At the beginning, he has a single empty bucket. Before the game starts, Xavier rolls a fair six-sided die repeatedly. Every time the number rolled on the die is a multiple of $3$, Bobby puts a coin in Xavier's bucket. As soon as Xavier rolls a number that is not a m... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-2e626dfe1e74cafe4e5c8a4fa51a8e379ae41024ed0bd4039467a9c30cb55bb1 | Xavier multiplied four digits (not necessarily distinct) and obtained a number that ends in 7. Determine the possible values of the sum of these four digits. List all possibilities. | [
"10, 14, 18, 22, 26, 30"
] | [
"Xavier multiplied four digits (not necessarily distinct) and obtained a number that ends in 7. Determine the possible values of the sum of these four digits. List all possibilities."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-e0e63b9fe978e5db50928ea6c0bb7e07e9f26fbd2e4c65f93c94923c0dd7e67e | Xavier, Yvonne, and Zach play badminton. Two players play at a time, and the third person plays the winner of the previous game (ties are not possible). Eventually, Xavier played 10 games while Yvonne played 21 games. How many games did Zach play? | [
"11"
] | [
"Xavier, Yvonne, and Zach play badminton. Two players play at a time, and the third person plays the winner of the previous game (ties are not possible). Eventually, Xavier played 10 games while Yvonne played 21 games. How many games did Zach play?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-1a634ace25fdda2c5e907d57930f47ef128e962625f8291b69678b41d929b13d | Xavier, Yvonne, and Zelda are playing a game where they take turns rolling a 4-sided die. Xavier goes first, Yvonne second, and Zelda third. A player who rolls a 4 wins and the game stops. A player who rolls a 1 drops out of the game. If there are two players left after this, they continue to play by the same rules. If... | [
"\\(\\frac{11}{28}\\)"
] | [
"Xavier, Yvonne, and Zelda are playing a game where they take turns rolling a 4-sided die. Xavier goes first, Yvonne second, and Zelda third. A player who rolls a 4 wins and the game stops. A player who rolls a 1 drops out of the game. If there are two players left after this, they continue to play by the same rule... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-3ddaba40cde7dccae7eda0bb9395fd02f0a52c20b9317ccdb242d5dcb2cd346c | Xenia and Sergey play a game where Xenia thinks of a positive integer \( N \) not exceeding 5000. She then fixes 20 distinct positive integers \( a_1, a_2, \cdots, a_{20} \) such that for each \( k = 1, 2, \cdots, 20 \), the numbers \( N \) and \( a_k \) are congruent modulo \( k \). By a move, Sergey tells Xenia a set... | [
"2"
] | [
"Xenia and Sergey play a game where Xenia thinks of a positive integer \\( N \\) not exceeding 5000. She then fixes 20 distinct positive integers \\( a_1, a_2, \\cdots, a_{20} \\) such that for each \\( k = 1, 2, \\cdots, 20 \\), the numbers \\( N \\) and \\( a_k \\) are congruent modulo \\( k \\). By a move, Serge... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5af3429d1aea63c463dbf428f69db03c916d1fa3011aee1c0bca2766597bf2e9 | Xiaotou is a car salesman who makes 100% profit on his cars. He sold 69 cars this week with an average price of $16200. No car sold for less than $5000 and no car sold for more than $60000. Each car's price was distinct and ended in two zeroes. What is the greatest possible median of the prices of the cars sold? | [
"$16200"
] | [
"Xiaotou is a car salesman who makes 100% profit on his cars. He sold 69 cars this week with an average price of $16200. No car sold for less than $5000 and no car sold for more than $60000. Each car's price was distinct and ended in two zeroes. What is the greatest possible median of the prices of the cars sold?"
... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d382d73f4522aae7bbc00f2e52e3fe7aec67f865f9dd96e661fc3cc418bd0743 | Xiaoyu is collecting a pyramid structure of red and green apples. On the top level, she puts 1 red apple. On the second level, she puts 4 green apples, and on the third level she puts 9 red apples. If she follows this pattern and makes the pyramid 20 rows high, how many green apples would be in her pyramid? | [
"1540"
] | [
"Xiaoyu is collecting a pyramid structure of red and green apples. On the top level, she puts 1 red apple. On the second level, she puts 4 green apples, and on the third level she puts 9 red apples. If she follows this pattern and makes the pyramid 20 rows high, how many green apples would be in her pyramid?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cba34712f1e5a51c7a31ad638740e22834af10781be454a64c9ce2bab208c06b | Ximena lists the whole numbers from 1 to 30 and adds them. Emilio copies Ximena's numbers but replaces each occurrence of the digit 2 with the digit 1, then adds his numbers. How much larger is Ximena's sum than Emilio's? | [
"103"
] | [
"Ximena lists the whole numbers from 1 to 30 and adds them. Emilio copies Ximena's numbers but replaces each occurrence of the digit 2 with the digit 1, then adds his numbers. How much larger is Ximena's sum than Emilio's?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6825c497b5c71ea981fd2805e5d4f6b176100daf06520b1b4a102f101d0dc066 | Xnopyt is rolling a 6-sided die, trying to maximize the value of $\frac{\text{value of last roll}}{\text{number of rolls}}.$ Once he deems a value satisfactory, he stops. With optimal strategy, let $A$ be the expected value of the number of rolls he does. Find $\lfloor 2^{16}A\rfloor.$ | [
"83740"
] | [
"Xnopyt is rolling a 6-sided die, trying to maximize the value of $\\frac{\\text{value of last roll}}{\\text{number of rolls}}.$ Once he deems a value satisfactory, he stops. With optimal strategy, let $A$ be the expected value of the number of rolls he does. Find $\\lfloor 2^{16}A\\rfloor.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-fdc4ea53c5078af52a213a442603292c0d8b0219100c90d3bc19532490d411b9 | Xonathan Jue goes to the casino with exactly \$1000. Each week, he has a $1/3$ chance of breaking even and a $2/3$ chance of losing \$500. Evaluate the expected number of weeks before he loses all his money. | [
"3"
] | [
"Xonathan Jue goes to the casino with exactly \\$1000. Each week, he has a $1/3$ chance of breaking even and a $2/3$ chance of losing \\$500. Evaluate the expected number of weeks before he loses all his money."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-431bd635e259ce842e8f9d88d0b75078b1517f8dc8ffe5cbc42c86626bf6f5e6 | Yamo and Sky are sharing \( n \) coins. Each time Yamo picks up some coins and tells Sky how many, while Sky decides who will get these coins. If someone gets \( p \) times of money or all the coins are taken, the sharing is over. How many coins can Yamo get at least? (In the specific case, \( n = 100 \) and \( p = 9 \... | [
"46"
] | [
"Yamo and Sky are sharing \\( n \\) coins. Each time Yamo picks up some coins and tells Sky how many, while Sky decides who will get these coins. If someone gets \\( p \\) times of money or all the coins are taken, the sharing is over. How many coins can Yamo get at least? (In the specific case, \\( n = 100 \\) and... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a6ec20ae6def8d3df15c3e8369eb88f9245095bf2476ac478e611bcb0b28d8c | Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to hi... | [
"\\frac{3}{4}"
] | [
"Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his hom... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c10d38ca4f5ede17706eb9fa174a4e1d7e47eae37413d04df40a510e334f403 | Yan takes 90 seconds to walk up an up escalator and 150 seconds to walk down the same escalator. Walking at the same pace, how long would it take Yan to climb the escalator if the escalator was turned off? How long would it take Yan to ride up the escalator if she wasn't walking but the escalator was on? | [
"225/2 seconds and 450 seconds"
] | [
"Yan takes 90 seconds to walk up an up escalator and 150 seconds to walk down the same escalator. Walking at the same pace, how long would it take Yan to climb the escalator if the escalator was turned off? How long would it take Yan to ride up the escalator if she wasn't walking but the escalator was on?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-20369cd35be334323d91a56a2970f66ae73d377a7567f5d8baabac38ea30e8dd | Yana has a four-function calculator with $1$ displayed on the screen. At each step, she multiplies the displayed number by a nonzero single-digit integer chosen at random. The expected number of steps until the displayed number leaves a remainder of $2$ when divided by $11$ can be expressed as $\frac{m}{n}$ for relativ... | [
"89"
] | [
"Yana has a four-function calculator with $1$ displayed on the screen. At each step, she multiplies the displayed number by a nonzero single-digit integer chosen at random. The expected number of steps until the displayed number leaves a remainder of $2$ when divided by $11$ can be expressed as $\\frac{m}{n}$ for r... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-b73629ea1e96ddd969a8334e6f2f92f630ddd91200d2703f1b353bc42504f000 | Yann and Camille go to a restaurant. If there are 10 items on the menu, and each orders one dish, how many different combinations of meals can Yann and Camille order? (Note that they are allowed to order the same dish, and that it does matter who orders what.) | [
"100"
] | [
"Yann and Camille go to a restaurant. If there are 10 items on the menu, and each orders one dish, how many different combinations of meals can Yann and Camille order? (Note that they are allowed to order the same dish, and that it does matter who orders what.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d999103d7547ea4f2455555d11c6eb58b487613d6930479f81e96b64c72016fc | Yasha writes all positive integers from 1 to $99^2$ in the cells of a $99 \times 99$ table (each number once). Grisha selects several cells such that no two selected cells share a common side and sums the numbers in these cells. Find the largest sum Grisha can guarantee to achieve. | [
"24014900"
] | [
"Yasha writes all positive integers from 1 to $99^2$ in the cells of a $99 \\times 99$ table (each number once). Grisha selects several cells such that no two selected cells share a common side and sums the numbers in these cells. Find the largest sum Grisha can guarantee to achieve."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b4822870a0bd5fd6efc02c4f1cb541fee6d0583921d1ca28d19ad2f8c656dd7c | Yasha's robo-dog moves distances that follow a geometric sequence. After one second, it jumps forward one foot. A second later, it jumps back 3 feet. At the end of 6 seconds, how far is the dog from its starting point? | [
"182"
] | [
"Yasha's robo-dog moves distances that follow a geometric sequence. After one second, it jumps forward one foot. A second later, it jumps back 3 feet. At the end of 6 seconds, how far is the dog from its starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd91f46d00d50f87f3d19cfcb057fd2a2b13a8072b69810e7d2414008aa42c64 | Yatta and Yogi play a game with a pile of $n$ stones. On each turn, a player can remove $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones, but not $4$ stones. The player who removes the last stone loses. Yogi goes first and has a winning position. For how many positive integers $n$ from $100$ to $2008$ inclusive does Yogi ha... | [
"1613"
] | [
"Yatta and Yogi play a game with a pile of $n$ stones. On each turn, a player can remove $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones, but not $4$ stones. The player who removes the last stone loses. Yogi goes first and has a winning position. For how many positive integers $n$ from $100$ to $2008$ inclusive does Yo... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7cfb17a61e609c6638d341bc2756ed149d701bb22686a8dd2c1202bf476c14d | Yeechi has a deck of cards consisting of the 2 through 5 of hearts and the 2 through 5 of spades. She deals two cards (at random) to each of four players. What is the probability that no player receives a pair? | [
"\\(\\frac{4}{7}\\)"
] | [
"Yeechi has a deck of cards consisting of the 2 through 5 of hearts and the 2 through 5 of spades. She deals two cards (at random) to each of four players. What is the probability that no player receives a pair?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 22
} | 32 | 0 | 0 |
omr-ws-f533a85525c6f014e6a328310ed9ff5def331d802a2209c1d56e36bdd5f8a900 | Yeechi has a deck of cards consisting of the 2-5 of hearts and the 2-5 of spades. She deals 2 cards at random to four players. What is the probability that no player receives a pair? | [
"\\(\\frac{1}{840}\\)"
] | [
"Yeechi has a deck of cards consisting of the 2-5 of hearts and the 2-5 of spades. She deals 2 cards at random to four players. What is the probability that no player receives a pair?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0b0118ee92485838f8dd9c746a6651d311e780ab12927ec707c0c85bf98eb5e | Yeechi has a deck of cards containing the 2 through 5 of hearts and the 2 through 5 of spades. She deals two cards (at random) to each of four players. How many total hands are there? | [
"2520"
] | [
"Yeechi has a deck of cards containing the 2 through 5 of hearts and the 2 through 5 of spades. She deals two cards (at random) to each of four players. How many total hands are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-6c956a2a0e0696f55804c8b35c34d215feb767e59769cc60a624227658399822 | Yesterday, Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian? | [
"150"
] | [
"Yesterday, Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f59611b1f46ad2e66d11e9b94a7c919e9160a785bf2181e9672908fecd00402 | Yikai wants to know Liang’s secret code, which is a 6-digit integer \( x \). Furthermore, let \( d(n) \) denote the digital sum of a positive integer \( n \). For instance, \( d(14) = 5 \) and \( d(3) = 3 \). It is given that \[ x + d(x) + d(d(x)) + d(d(d(x))) = 999868. \] Find \( x \). | [
"999820"
] | [
"Yikai wants to know Liang’s secret code, which is a 6-digit integer \\( x \\). Furthermore, let \\( d(n) \\) denote the digital sum of a positive integer \\( n \\). For instance, \\( d(14) = 5 \\) and \\( d(3) = 3 \\). It is given that \n\\[ x + d(x) + d(d(x)) + d(d(d(x))) = 999868. \\]\nFind \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-655447c176abee607f6084b547b3f9e50c9c8f2f2dd770b87dab2e26ebb1d155 | Yizheng and Jennifer are playing a game of ping-pong where the winner of a match is given one point. The game is won by reaching 11 points before the opponent reaches 10 points, or by being the first to have a score two points higher than the opponent's score if the score is tied at 10-to-10. The probability that Jenni... | [
"0.1552",
"\\(\\frac{9}{58}\\)"
] | [
"Yizheng and Jennifer are playing a game of ping-pong where the winner of a match is given one point. The game is won by reaching 11 points before the opponent reaches 10 points, or by being the first to have a score two points higher than the opponent's score if the score is tied at 10-to-10. The probability that ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 7,
"tir": 66
} | 119 | 0 | 0 |
omr-ws-94b0635f831648dbfb291561e8083218d3bd59022cc41a588e3045a9ec35dee2 | Yizheng and Jennifer are playing a game of ping-pong where the winner of a match is given one point. The game is won by reaching 11 points before the opponent reaches 10 points, or by reaching a score two points higher than the opponent if the score is tied at 10-to-10. The probability that Jennifer wins any one match ... | [
"\\(\\frac{9}{58}\\)"
] | [
"Yizheng and Jennifer are playing a game of ping-pong where the winner of a match is given one point. The game is won by reaching 11 points before the opponent reaches 10 points, or by reaching a score two points higher than the opponent if the score is tied at 10-to-10. The probability that Jennifer wins any one m... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-24c124a7a7e33cc4d82d7ab6c3425a7d41e8211c527d5ce7011282d57dd8cdd1 | Yizheng and Jennifer are playing a game of ping-pong. The winner of a match is given one point. If one person reaches 11 points before the other can reach 10 points, that person wins. If the score is tied 10-to-10, the game continues until one person’s score is two more than the other’s, at which point the person with ... | [
"\\dfrac{9}{58}"
] | [
"Yizheng and Jennifer are playing a game of ping-pong. The winner of a match is given one point. If one person reaches 11 points before the other can reach 10 points, that person wins. If the score is tied 10-to-10, the game continues until one person’s score is two more than the other’s, at which point the person ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc696ceab48b216d4de5312c88c0c216039af65a61bbc7a71824d68fdaa3a872 | Yofro has a list of 5 positive integers. He selects a number \( A \) from the list, then selects a number \( B \) from the list, with replacement. The expected value of \( A \cdot B \) is 4. If permutations of the same list are considered distinct, how many lists are possible? | [
"126"
] | [
"Yofro has a list of 5 positive integers. He selects a number \\( A \\) from the list, then selects a number \\( B \\) from the list, with replacement. The expected value of \\( A \\cdot B \\) is 4. If permutations of the same list are considered distinct, how many lists are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21dd864fbee12089e2dfbb31e650a599134001812179b1cb5999c5de8a15d79a | Yolanda puts some money aside for an MP3 Player and saves an additional \$5.50 each month thereafter. If she saves \$64.50 after 1 year, how much did she originally put aside? Show a formula using the arithmetic sequence method. | [
"4"
] | [
"Yolanda puts some money aside for an MP3 Player and saves an additional \\$5.50 each month thereafter. If she saves \\$64.50 after 1 year, how much did she originally put aside? Show a formula using the arithmetic sequence method."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 14
} | 32 | 0 | 0 |
omr-ws-ec96cb647c895b6d8fcc00f8b57c80fd0b841e90b780cd630a13cd7509f8b872 | Yor and Fiona are playing a match of tennis. The first player to win 6 games wins the match. Yor has currently won 2 games, while Fiona has won 0 games. Yor has a probability of $\frac{2}{3}$ to win each game, while Fiona has a probability of $\frac{1}{3}$ to win each game. Compute the probability that Fiona wins the m... | [
"835"
] | [
"Yor and Fiona are playing a match of tennis. The first player to win 6 games wins the match. Yor has currently won 2 games, while Fiona has won 0 games. Yor has a probability of $\\frac{2}{3}$ to win each game, while Fiona has a probability of $\\frac{1}{3}$ to win each game. Compute the probability that Fiona win... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-bf0ac6b0aafa68c11cc39fb876537ceb72d5f360e84468b7ac3bfc349a7e6932 | Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table after th... | [
"S \\left(1 - \\frac{4}{9} \\sin^2\\left(\\frac{\\pi}{M}\\right)\\right)^n"
] | [
"Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table aft... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-54b6410d3eb55905609d52f5d5f15486691a1789d9bb3e3c1d48ac1b9e2b047e | Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table after th... | [
"S \\left( \\dfrac{4}{7} + \\dfrac{3}{7} \\left( \\dfrac{2}{9} \\right)^n \\right)"
] | [
"Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table aft... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-a6afcb98ba55808d6897fdd68d0712f45b0abba4ee649c2651504af332fb7b50 | Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table after th... | [
"S \\left( \\dfrac{7}{9} \\right)^n"
] | [
"Yoshi's son marks the points that divide the sides into $3$ equal parts and cuts out figures from the table that are equal to these corners and relies on the marked points. If the table falls apart at some point, then Yoshi's son will continue to cut corners from each part. Find the area of the resulting table aft... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-c3e006c2e978581f42d0a6a8acb87ecfeb9de6a73d32a2ea3751cca6e2623b98 | You alternate tossing two weighted coins. The first coin you toss has a \(\frac{2}{3}\) chance of landing heads and a \(\frac{1}{3}\) chance of landing tails; the second coin has a \(\frac{1}{4}\) chance of landing heads and a \(\frac{3}{4}\) chance of landing tails. What is the probability that you will toss two heads... | [
"\\dfrac{13}{33}"
] | [
"You alternate tossing two weighted coins. The first coin you toss has a \\(\\frac{2}{3}\\) chance of landing heads and a \\(\\frac{1}{3}\\) chance of landing tails; the second coin has a \\(\\frac{1}{4}\\) chance of landing heads and a \\(\\frac{3}{4}\\) chance of landing tails. What is the probability that you wi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-33f5ae9cccd2101784a085cf9d36e575b7adfa53c6b7e678ec5374161803e19b | You and I are playing a game starting with 2008 bricks. On each turn, a player can take out between 1 and 100 bricks (inclusive). The player who takes the last brick wins. Assuming both players play perfectly and you move first, what move should you make to ensure a win? | [
"89"
] | [
"You and I are playing a game starting with 2008 bricks. On each turn, a player can take out between 1 and 100 bricks (inclusive). The player who takes the last brick wins. Assuming both players play perfectly and you move first, what move should you make to ensure a win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-bf66fc24b48e03df514a753d018c578b6c50a2be950f28fddc050b6343a9387a | You and I each have $14$ dollars. I flip a fair coin repeatedly; if it comes up heads, I pay you a dollar, but if it lands tails, you pay me a dollar. On average, how many times will I flip the coin before one of us runs out of money? | [
"196"
] | [
"You and I each have $14$ dollars. I flip a fair coin repeatedly; if it comes up heads, I pay you a dollar, but if it lands tails, you pay me a dollar. On average, how many times will I flip the coin before one of us runs out of money?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-247bdf82150ef14c4b2cca96bc14a4f2f3fb4623f187511cb62aa6bca909db2b | You and I each have 14 dollars. I flip a fair coin repeatedly; if it comes up heads I pay you a dollar, but if it lands tails you pay me a dollar. On average, how many times will I flip the coin before one of us runs out of money? | [
"196"
] | [
"You and I each have 14 dollars. I flip a fair coin repeatedly; if it comes up heads I pay you a dollar, but if it lands tails you pay me a dollar. On average, how many times will I flip the coin before one of us runs out of money?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-396b2b77a46bdba5b2ad060a4a763fb86533344aa9cdbee91927b980effcd74b | You and five of your friends line up in a random order to get tickets, one at a time. There are sixteen tickets available, labeled \(1\) through \(16\). Your friends do not care which ticket they get, and will take one at random; but your favorite number is \(13\), so if ticket \(\# 13\) is available, you will take it.... | [
"\\dfrac{27}{32}"
] | [
"You and five of your friends line up in a random order to get tickets, one at a time. There are sixteen tickets available, labeled \\(1\\) through \\(16\\). Your friends do not care which ticket they get, and will take one at random; but your favorite number is \\(13\\), so if ticket \\(\\# 13\\) is available, you... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-4dd9df79770d7fd3243c288f043b9f78ad05dd8a1a5b9b9b9c691345f4a42603 | You and nine friends initially spend $4000$ dollars on tickets to attend a concert. Six friends cancel last minute due to the flu. You and the remaining friends split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay? | [
"25%"
] | [
"You and nine friends initially spend $4000$ dollars on tickets to attend a concert. Six friends cancel last minute due to the flu. You and the remaining friends split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have to pay?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-b5e1fac6f0f511ae45dd5d6137f5fdff40c9de5ab058320a560f537299ceb729 | You and three friends each randomly pick a number from 1 to 5. If any of your friends pick the same number as you, you lose. What are the odds that you win? | [
"\\(64:61\\)"
] | [
"You and three friends each randomly pick a number from 1 to 5. If any of your friends pick the same number as you, you lose. What are the odds that you win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-43ac804c267994d45e91b39c975b6293ead16e760e710b85d9390278e5c68985 | You and your $k$ friends are sharing a box of $n$ chocolates. You split them equally among everyone else, and then give yourself the remainder. The particular combination of $n$ and $k$ is called generous if you have equal to or less chocolates than each of your friends has. Identify any pattern to determine which comb... | [
"n \\leq (k + 1) \\left\\lfloor \\frac{n}{k} \\right\\rfloor"
] | [
"You and your $k$ friends are sharing a box of $n$ chocolates. You split them equally among everyone else, and then give yourself the remainder. The particular combination of $n$ and $k$ is called generous if you have equal to or less chocolates than each of your friends has. Identify any pattern to determine which... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
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