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-cc338bd1e912066609f5a75047cda5f932f26859ad00accfeb2c499c6ece8efe | Among the integers from $0$ to $1995$, two numbers have been crossed out. Determine which two numbers must be crossed out so that the length of the longest arithmetic progression among the remaining $1994$ numbers is minimized. | [
"0 and 1995"
] | [
"Among the integers from $0$ to $1995$, two numbers have been crossed out. Determine which two numbers must be crossed out so that the length of the longest arithmetic progression among the remaining $1994$ numbers is minimized."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4a8e3a2213af430c7007bba46676a6f4adfec0c52cc10c91a708ab5f0bc2e3d | Among the lines that pass through point \( O \), how many divide the area of the figure made up of 3 semicircles into two equal parts? | [
"1"
] | [
"Among the lines that pass through point \\( O \\), how many divide the area of the figure made up of 3 semicircles into two equal parts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-41938b4abd289fd6f09d141048f7a749d4dfc0868dd577f239e384ace2f6665b | Among the lines that pass through point \( O \), how many divide the perimeter of the figure made up of 3 semicircles into two equal parts? | [
"2"
] | [
"Among the lines that pass through point \\( O \\), how many divide the perimeter of the figure made up of 3 semicircles into two equal parts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-bba64ec152e44eb9ac74cb9da9d61e03dda8054b4b4a2889dc2b3d2048b9119b | Among the natural numbers between 1 and 1,000,000, are there more numbers that can be represented as a sum of a perfect square and a positive perfect cube, or numbers that cannot be represented in this way? | [
"\\text{More numbers cannot be represented}"
] | [
"Among the natural numbers between 1 and 1,000,000, are there more numbers that can be represented as a sum of a perfect square and a positive perfect cube, or numbers that cannot be represented in this way?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-179c5a58c81afe4aabded8897892d8beab095d01774422117b6842843dcac123 | Among the natural numbers from $1$ to $1000000$, which are greater: those that can be represented as $2x^2 - 3y^2$ or those that can be represented as $10xy - x^2 - y^2$ with integer numbers $x$ and $y$? | [
"\\(2x^2 - 3y^2\\) appears more often."
] | [
"Among the natural numbers from $1$ to $1000000$, which are greater: those that can be represented as $2x^2 - 3y^2$ or those that can be represented as $10xy - x^2 - y^2$ with integer numbers $x$ and $y$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-6b7b667074c580ff32143d6d43dd1a27f2ec7ac2ef83a47f0ede0db50007d69c | Among the natural numbers from 1 to 1,000,000, inclusive, which are there more of: numbers that can be represented as the sum of a perfect square and a positive perfect cube, or numbers that cannot be? | [
"\\text{Numbers that cannot be represented}"
] | [
"Among the natural numbers from 1 to 1,000,000, inclusive, which are there more of: numbers that can be represented as the sum of a perfect square and a positive perfect cube, or numbers that cannot be?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15806f48b4194bf32614ae7ced17bdae8cc90b86616e1dc10f07fec004e8d56f | Among the numbers $1, 2, 3, \ldots, 1000000$, let $X$ be the number of numbers that can be expressed as the sum of a perfect square and a perfect cube. Compare $X$ and $1000000 - X$. | [
"\\( X < 1000000 - X \\)"
] | [
"Among the numbers $1, 2, 3, \\ldots, 1000000$, let $X$ be the number of numbers that can be expressed as the sum of a perfect square and a perfect cube. Compare $X$ and $1000000 - X$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 28
} | 46 | 0 | 0 |
omr-ws-8de7acd4fb9cd57c54ff43874ee9e87e990d987122836f2fe26017351be7f602 | Among the numbers $10^1+1, 10^2+1, \ldots, 10^{2010}+1$, calculate the maximum number of primes that can exist in this sequence. | [
"2"
] | [
"Among the numbers $10^1+1, 10^2+1, \\ldots, 10^{2010}+1$, calculate the maximum number of primes that can exist in this sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b376f66bae20991349b7f4d30908d06522c8f838f796778303fa889f4280351c | Among the numbers $11n + 10^{10}$, where $1 \le n \le 10^{10}$ and $n$ is an integer, how many are perfect squares? | [
"44801"
] | [
"Among the numbers $11n + 10^{10}$, where $1 \\le n \\le 10^{10}$ and $n$ is an integer, how many are perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-7ad5e3e195b7c9f2a99e1699df47e14694736ce1aeface511bd3b61902a01497 | Among the numbers 1, 2, 3, ..., n, find the largest integer \( k \) such that there exists a number divisible by \( 2^k \) but not by \( 2^{k+1} \). Calculate the sum of all such numbers divisible by \( 2^k \) but not by \( 2^{k+1} \) for \( n = 1024 \). | [
"1024"
] | [
"Among the numbers 1, 2, 3, ..., n, find the largest integer \\( k \\) such that there exists a number divisible by \\( 2^k \\) but not by \\( 2^{k+1} \\). Calculate the sum of all such numbers divisible by \\( 2^k \\) but not by \\( 2^{k+1} \\) for \\( n = 1024 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d763f1fbf0877ca05340c8f4fe173c2ac41d3c60c2ad689fd8896b9a3427f89c | Among the numbers from $1$ to $999$, which integers $n$ satisfy $1 \leq n < 100$ and $P(n) = 12$? | [
"26, 34, 43, 62"
] | [
"Among the numbers from $1$ to $999$, which integers $n$ satisfy $1 \\leq n < 100$ and $P(n) = 12$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 26
} | 29 | 0 | 0 |
omr-ws-2d4106d83dcaba116806c343df4f0e71b37cb6789a5ae7a8ec53ff71b90489c2 | Among the numbers from 1 to 2018, how many numbers must be picked to ensure that there are two numbers such that the larger number is not more than twice the smaller number? | [
"12"
] | [
"Among the numbers from 1 to 2018, how many numbers must be picked to ensure that there are two numbers such that the larger number is not more than twice the smaller number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9d512d662b694f70722568b0a3ade0bfe2a275692cea3ebf23208ff13cbac6d1 | Among the pairs $(m, n)$ that satisfy $m^3 - 4mn^2 = 8n^3 - 2m^2n$, find those for which $m + n^2 = 3$. | [
"\\((-6, -3), (2, 1), (-6, 3), (2, -1)\\)"
] | [
"Among the pairs $(m, n)$ that satisfy $m^3 - 4mn^2 = 8n^3 - 2m^2n$, find those for which $m + n^2 = 3$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-ecfdf27ca2eeae0a6422be2d1c1cd494f6271705568b3ab5965b0f31a246a208 | Among the palindromes from $10,000$ to $999,999$, determine how many have an odd sum of digits and how many have an even sum of digits. Also, find how many times the palindromes with an odd sum of digits outnumber those with an even sum of digits. | [
"\\dfrac{1}{3}"
] | [
"Among the palindromes from $10,000$ to $999,999$, determine how many have an odd sum of digits and how many have an even sum of digits. Also, find how many times the palindromes with an odd sum of digits outnumber those with an even sum of digits."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-beea17c1fd1a062e3f53396c477372d4c8d7d02bb0bac29b5d832f9c004d31f3 | Among the plane points with both coordinates in the set \(\{1, 2, \ldots, 106\}\), some points have been marked such that for every two marked points \((x, y)\) and \((x', y')\), at least one of the following conditions is met: 1. \(x > x' - 10\) and \(y > y' - 10\); 2. \(x' > x - 10\) and \(y' > y - 10\). Determine th... | [
"121"
] | [
"Among the plane points with both coordinates in the set \\(\\{1, 2, \\ldots, 106\\}\\), some points have been marked such that for every two marked points \\((x, y)\\) and \\((x', y')\\), at least one of the following conditions is met:\n1. \\(x > x' - 10\\) and \\(y > y' - 10\\);\n2. \\(x' > x - 10\\) and \\(y' >... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-bfa10e2f5274c0d9b4140b6b70eaf18310952e1a896c56e699acbb45377233f4 | Among the points \(A, B, C, D\) no three are collinear. The lines \(AB\) and \(CD\) intersect at \(E\), and \(BC\) and \(DA\) intersect at \(F\). Let \(P\) be the intersection point of the circles with diameters \(AC\) and \(BD\), if they intersect. Find the ratio \(\frac{EP}{FP}\) in terms of the coordinates of \(A, B... | [
"\\dfrac{EA \\cdot EC}{EB \\cdot ED}"
] | [
"Among the points \\(A, B, C, D\\) no three are collinear. The lines \\(AB\\) and \\(CD\\) intersect at \\(E\\), and \\(BC\\) and \\(DA\\) intersect at \\(F\\). Let \\(P\\) be the intersection point of the circles with diameters \\(AC\\) and \\(BD\\), if they intersect. Find the ratio \\(\\frac{EP}{FP}\\) in terms ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e485fa05dd0e7099770c6e95656733ef20f5f9466a8311f1bd0a7ded685da37 | Among the points \(A, B, C, D\) no three are collinear. The lines \(AB\) and \(CD\) intersect at \(E\), and \(BC\) and \(DA\) intersect at \(F\). Let \(P\) be the intersection point of the circles with diameters \(AC\), \(BD\), and \(EF\) if they intersect at a common point. If such a point \(P\) exists, calculate the ... | [
"360"
] | [
"Among the points \\(A, B, C, D\\) no three are collinear. The lines \\(AB\\) and \\(CD\\) intersect at \\(E\\), and \\(BC\\) and \\(DA\\) intersect at \\(F\\). Let \\(P\\) be the intersection point of the circles with diameters \\(AC\\), \\(BD\\), and \\(EF\\) if they intersect at a common point. If such a point \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-62dc4f9095e17bba30aaf4649da605ee7cb127325b84b8cb15a034a80bd61f16 | Among the points corresponding to numbers $1, 2, \ldots, 2n$ on a real line, $n$ are colored blue and $n$ are colored red. Let $a_1 < a_2 < \ldots < a_n$ be the blue points and $b_1 > b_2 > \ldots > b_n$ be the red points. Calculate the value of the sum \[ |a_1 - b_1| + |a_2 - b_2| + \ldots + |a_n - b_n| \] for $n = 5$... | [
"25"
] | [
"Among the points corresponding to numbers $1, 2, \\ldots, 2n$ on a real line, $n$ are colored blue and $n$ are colored red. Let $a_1 < a_2 < \\ldots < a_n$ be the blue points and $b_1 > b_2 > \\ldots > b_n$ be the red points. Calculate the value of the sum\n\n\\[ |a_1 - b_1| + |a_2 - b_2| + \\ldots + |a_n - b_n| \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e754a666c2d899a3b5bd8544a5aae4c865c20021a932f3895297be00f2ca084 | Among the points corresponding to numbers $1, 2, \ldots, 2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1 < a_2 < \ldots < a_n$ be the blue points and $b_1 > b_2 > \ldots > b_n$ be the red points. Calculate the value of the sum $$|a_1 - b_1| + \ldots + |a_n - b_n|$$ for $n = 10$. | [
"100"
] | [
"Among the points corresponding to numbers $1, 2, \\ldots, 2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1 < a_2 < \\ldots < a_n$ be the blue points and $b_1 > b_2 > \\ldots > b_n$ be the red points. Calculate the value of the sum\n$$|a_1 - b_1| + \\ldots + |a_n - b_n|$$\nfor $n = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48e1ecb2fe8191bff6677570617e783a23393c24be49bfe5803109a574fba295 | Among the points corresponding to numbers $1, 2, \ldots, 2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1, a_2, \ldots, a_n$ be the blue points and $b_1, b_2, \ldots, b_n$ be the red points. Calculate the value of the sum $\mid a_1 - b_1 \mid + \mid a_2 - b_2 \mid + \cdots + \mid a_n - b_n \mid$ f... | [
"100"
] | [
"Among the points corresponding to numbers $1, 2, \\ldots, 2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1, a_2, \\ldots, a_n$ be the blue points and $b_1, b_2, \\ldots, b_n$ be the red points. Calculate the value of the sum $\\mid a_1 - b_1 \\mid + \\mid a_2 - b_2 \\mid + \\cdots + \\mid a_n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bcd0484b016ae178cf8b1752d5479bf3370b544b3cca2bd759d906693f41f3a | Among the points of the plane with both coordinates in the set \(\{1, 2, \ldots, 106\}\), some points were marked. For every two marked points \((x, y)\) and \((x', y')\), at least one of the following conditions is met: 1. \(x > x' - 10\) and \(y > y' - 10\); 2. \(x' > x - 10\) and \(y' > y - 10\). What is the highest... | [
"1060"
] | [
"Among the points of the plane with both coordinates in the set \\(\\{1, 2, \\ldots, 106\\}\\), some points were marked. For every two marked points \\((x, y)\\) and \\((x', y')\\), at least one of the following conditions is met:\n1. \\(x > x' - 10\\) and \\(y > y' - 10\\);\n2. \\(x' > x - 10\\) and \\(y' > y - 10... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-5359f8982773e43638e412b35030c9625220919df4b88e21e73340e861dff5f1 | Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime? | [
"\\(\\frac{2}{5}\\)"
] | [
"Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-121d84efa5a91a32a6908f836d70c20f5970777772716ac95d45f01bd362eaa8 | Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order? | [
"\\(2005 \\cdot 1002 \\cdot 4009\\)"
] | [
"Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-12b6c01d9ec6cb12e4e9e8319ba76b16e345eaa50f8a6c3897402b22a9c35bd6 | Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order? | [
"\\(2005^2\\)"
] | [
"Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2fb30dad36fefab82c1f66bab57bdf22ebd6cdd40518630a973fd911ca5d10f7 | Among the quadratic trinomials $x^2 + px + q$ where $p$ and $q$ are integers and $1 \leq p, q \leq 1997$, determine whether there are more trinomials with integer roots or more trinomials without real roots. | [
"More trinomials without real roots."
] | [
"Among the quadratic trinomials $x^2 + px + q$ where $p$ and $q$ are integers and $1 \\leq p, q \\leq 1997$, determine whether there are more trinomials with integer roots or more trinomials without real roots."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-3245165fdd00c78f5641a22111a00cf55cbf1c7563ee127dd175029807045141 | Among the spatial points \( A, B, C, D \), at most two of them are aparted at a distance greater than \( 1 \). Find the maximum value of the expression: \[ g(A,B,C,D) = AB + BC + AD + CA + DB + DC. \] | [
"6"
] | [
"Among the spatial points \\( A, B, C, D \\), at most two of them are aparted at a distance greater than \\( 1 \\). Find the maximum value of the expression:\n\\[ g(A,B,C,D) = AB + BC + AD + CA + DB + DC. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f07955b13d9049daafa1eea8b4792428315bbab75f3ccadfb9ed5c9ab326c46 | Among the square roots of the first $2015$ natural numbers, determine the maximum possible number of elements that can form an arithmetic sequence. | [
"44"
] | [
"Among the square roots of the first $2015$ natural numbers, determine the maximum possible number of elements that can form an arithmetic sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6ab57ef63f9427c60b6d8a85ddfa17da50f902d457ad9acba7081512e826d68 | Among the straight lines determined by each pair of vertices of a cube, how many pairs of non-coplanar lines which are not perpendicular are there? | [
"96"
] | [
"Among the straight lines determined by each pair of vertices of a cube, how many pairs of non-coplanar lines which are not perpendicular are there?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-e71b2b58eb9c224a14441516e64e435b0bac64ebaa6ab24208da19a5d1f96354 | Among the tetrahedrons formed by choosing 4 vertices of a rectangular parallelepiped, find those which have two faces that are isosceles triangles. Express the lengths of their edges as a function of the length \(a\) of the smallest edge. | [
"\\(a\\), \\(a\\sqrt{2}\\), \\(a\\sqrt{3}\\)"
] | [
"Among the tetrahedrons formed by choosing 4 vertices of a rectangular parallelepiped, find those which have two faces that are isosceles triangles. Express the lengths of their edges as a function of the length \\(a\\) of the smallest edge."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-88444bfaf7841e71e0975ebe0be365ce4b8c1c47825967311bf8ecee52e382ad | Among the triangles that have a side of length $5$ m and an opposite angle of $30^\circ$, determine the triangle with the maximum area. Calculate the values of the other two angles and the area of the triangle. | [
"\\(\\frac{25\\sqrt{3}}{4}\\)"
] | [
"Among the triangles that have a side of length $5$ m and an opposite angle of $30^\\circ$, determine the triangle with the maximum area. Calculate the values of the other two angles and the area of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-11ac270faa2e5ac6bc23dfb4c4d74bedeffdfaa291fd49604724deb91bee29fb | Among the triangles with integer sides and a perimeter of 12 cm, which triangle has the maximum area? | [
"\\( (4, 4, 4) \\)"
] | [
"Among the triangles with integer sides and a perimeter of 12 cm, which triangle has the maximum area?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9f0269a0a8941bec9af6a633a50cfc76cee60d30a1360404f6192287d61bc963 | Among the triplets $(a, b, c)$ of natural numbers that satisfy $a + b + c = 2009$, which triplet has the highest product? | [
"(669, 670, 670)"
] | [
"Among the triplets $(a, b, c)$ of natural numbers that satisfy $a + b + c = 2009$, which triplet has the highest product?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-d4a9aeb3a8d7b50dc6aa9dc0695918a683a1f402d43dd3c5cbbe3b80e127d211 | Among the unbiased estimators identified in Problem 1, which one has the smallest variance? | [
"\\(\\hat{\\theta_5} = \\overline{Y}\\)"
] | [
"Among the unbiased estimators identified in Problem 1, which one has the smallest variance?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-9ad11835508a45d6230ade243d994b270401f80420eebbfaa3c7c378b00ccc44 | Among the vertices of a regular 11-gon, how many ways can a triple of vertices be chosen so that the triangle formed by these vertices is acute? | [
"165"
] | [
"Among the vertices of a regular 11-gon, how many ways can a triple of vertices be chosen so that the triangle formed by these vertices is acute?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 4
} | 20 | 0 | 0 |
omr-ws-c0590901eb49a38403c0d0423e3871cd7d13ece26efdfb865964f3a2bd01d43b | Among the vertices of a regular 17-gon, how many ways can a triple of vertices be chosen so that the triangle formed by these vertices is acute? | [
"204"
] | [
"Among the vertices of a regular 17-gon, how many ways can a triple of vertices be chosen so that the triangle formed by these vertices is acute?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-52040b379ae871808a309dc20c389a694793bed08d83b3df23807794d1bb86b9 | Among three similar coins, one is counterfeit and it is not known whether it is lighter or heavier than a genuine coin. Using two weightings on a pan balance, identify the counterfeit coin and determine whether it is lighter or heavier than a genuine coin. | [
"\n\\text{First weigh coin A vs B.} \\\\\n\\text{If equal, weigh C vs A; if C ≠ A, C is counterfeit (heavier or lighter).} \\\\\n\\text{If A > B, weigh A vs C. If A > C, A is heavier; else, B is lighter.} \\\\\n\\text{If B > A, weigh B vs C. If B > C, B is heavier; else, A is lighter.}"
] | [
"Among three similar coins, one is counterfeit and it is not known whether it is lighter or heavier than a genuine coin. Using two weightings on a pan balance, identify the counterfeit coin and determine whether it is lighter or heavier than a genuine coin."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-ccf38ed226c6ac9e803a8cbfbde95f72f9c684ef3c70f472fd9bfcf05ab95d24 | Among triangles with one side of a given length \( L \) and a given area \( S \), determine all of those for which the product of the lengths of the three altitudes is maximum. | [
"The triangles that maximize the product of the altitudes are isosceles triangles with the vertex on the perpendicular bisector of the side of length \\( L \\), or right triangles if \\( 2S > L^2 \\)."
] | [
"Among triangles with one side of a given length \\( L \\) and a given area \\( S \\), determine all of those for which the product of the lengths of the three altitudes is maximum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-b39aedabb56897059d94dbd75d0c3f6efde11356c0dd04b53c0a9884e8eb2783 | Among twelve similar coins, one is counterfeit and it is not known whether it is lighter or heavier than a genuine coin. Using three weightings on a pan balance, identify the counterfeit coin and determine whether it is lighter or heavier than a genuine coin. | [
"The counterfeit coin can be identified and determined as lighter or heavier in three weighings by systematically comparing groups and analyzing the balance outcomes."
] | [
"Among twelve similar coins, one is counterfeit and it is not known whether it is lighter or heavier than a genuine coin. Using three weightings on a pan balance, identify the counterfeit coin and determine whether it is lighter or heavier than a genuine coin."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47290c714465d803c5222c49034c9e54e7b6de778e9e2eb29ff202e22e5546ce | Amongst $2n$ unique people, each person may watch anime or not. It is known that: 1. The number of anime-watchers does not exceed $n$. 2. A number of anime-watchers also read manga, such that those that read manga must also watch anime. What is the probability that the number of people that read manga is equal to the n... | [
"\\(\\frac{\\binom{2n}{n}}{2^{2n}}\\)"
] | [
"Amongst $2n$ unique people, each person may watch anime or not. It is known that:\n1. The number of anime-watchers does not exceed $n$.\n2. A number of anime-watchers also read manga, such that those that read manga must also watch anime.\nWhat is the probability that the number of people that read manga is equal ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e69c246f6a41133c941723034ec1cabb66a7e2e8dfa8073514b9b94cf2f2ab5 | Amy Gutmann and Liz Magill are organizing a party for 1000 people and want to know the social network of the 1000 people. A social network is a set of ordered pairs representing friendships. If $N$ is the total number of social networks possible among the 1000 people, what is $\log_2 N$? | [
"500500"
] | [
"Amy Gutmann and Liz Magill are organizing a party for 1000 people and want to know the social network of the 1000 people. A social network is a set of ordered pairs representing friendships. If $N$ is the total number of social networks possible among the 1000 people, what is $\\log_2 N$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53e2a70e0795cbe6ea2fb585618506acea2dfb55aba17994f35e875b78be7d86 | Amy adds some positive integers together and gets 17. Bella multiplies those positive integers to get N. Compute the least possible value that cannot be N. | [
"518"
] | [
"Amy adds some positive integers together and gets 17. Bella multiplies those positive integers to get N. Compute the least possible value that cannot be N."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-96e1520cc3da87b22d63321e0f0563dc8eca9f112a1b80bb050d070dbb95d465 | Amy adds some positive numbers together and gets 17. Bella multiplies those same numbers together and gets N. Compute the least positive integer that cannot be N. | [
"518"
] | [
"Amy adds some positive numbers together and gets 17. Bella multiplies those same numbers together and gets N. Compute the least positive integer that cannot be N."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-a856cf39d232d4b5fa153d230fe4c379e1e1df6eccf1ed6ebf8ea925a3e14763 | Amy and Bec play a game with three piles, each containing 2020 stones. Players take turns, with Amy going first. Each move consists of choosing one of the piles, removing the unchosen pile(s) from the game, and then dividing the chosen pile into 2 or 3 non-empty piles. A player loses if they are unable to make a move. ... | [
"7"
] | [
"Amy and Bec play a game with three piles, each containing 2020 stones. Players take turns, with Amy going first. Each move consists of choosing one of the piles, removing the unchosen pile(s) from the game, and then dividing the chosen pile into 2 or 3 non-empty piles. A player loses if they are unable to make a m... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-bce155814e5bb4463c0a57257d6c99aaa1b6da685a33b01947fb91c463098bf2 | Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form a cylindrical tube. Amy tapes the two 8-inch sides together without overlap. Belinda tapes the two 6-inch sides together without overlap. What is $\pi$ times the positive difference of the volumes of the two tubes? | [
"24"
] | [
"Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form a cylindrical tube. Amy tapes the two 8-inch sides together without overlap. Belinda tapes the two 6-inch sides together without overlap. What is $\\pi$ times the positive difference of the volumes of the two tubes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-469f3a92f55e8a26f03ad5ce4033013a5c2ae89dcfc852afcdc109c73461d1ca | Amy and Bob play a game by filling numbers from 1 to 99 into 99 equally spaced points on a circle. Amy starts, and they take turns filling in the numbers. The player who makes three adjacent points form an arithmetic sequence loses. Who will win the game? | [
"Bob"
] | [
"Amy and Bob play a game by filling numbers from 1 to 99 into 99 equally spaced points on a circle. Amy starts, and they take turns filling in the numbers. The player who makes three adjacent points form an arithmetic sequence loses. Who will win the game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-ae22be82d4715cd8e39090323bd3f1b7bf9f99594abbd26d587ff7c8e6911c60 | Amy and Bob play a game where they alternate turns, with Amy going first. At the start, there are 20 cookies on a red plate and 14 on a blue plate. A legal move consists of either eating two cookies from one plate or moving one cookie from the red plate to the blue plate (but not from the blue plate to the red plate). ... | [
"1"
] | [
"Amy and Bob play a game where they alternate turns, with Amy going first. At the start, there are 20 cookies on a red plate and 14 on a blue plate. A legal move consists of either eating two cookies from one plate or moving one cookie from the red plate to the blue plate (but not from the blue plate to the red pla... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51074850dafd95c992e6f3866be11d718c6f906179fdd9c582f9eeed5c243ea2 | Amy and Bob take turns choosing numbers from the set $\{0, 1, 2, \ldots, 81\}$, with Amy choosing first. Each player chooses one number from the remaining numbers each turn. After all 82 numbers are chosen, let $A$ be the sum of the numbers Amy chooses and $B$ be the sum of the numbers Bob chooses. Amy aims to maximize... | [
"41"
] | [
"Amy and Bob take turns choosing numbers from the set $\\{0, 1, 2, \\ldots, 81\\}$, with Amy choosing first. Each player chooses one number from the remaining numbers each turn. After all 82 numbers are chosen, let $A$ be the sum of the numbers Amy chooses and $B$ be the sum of the numbers Bob chooses. Amy aims to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-282feb558a994cd8409eb553f5fe7489fcde9c9a266c2e6e3747c607dbf5e3dc | Amy and Eric are playing a game where some numbers are written on the board. Amy goes first. On each player's turn, they may take any two numbers whose GCD is not already on the board and write the GCD of those two numbers on the board. The first player who is unable to write a number smaller than any number currently ... | [
"3"
] | [
"Amy and Eric are playing a game where some numbers are written on the board. Amy goes first. On each player's turn, they may take any two numbers whose GCD is not already on the board and write the GCD of those two numbers on the board. The first player who is unable to write a number smaller than any number curre... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-ff8d24d77cdaf49270c0d999e4aa281e6f32106056382c698a85a11028d1619b | Amy distributes candies on Halloween according to the following rule: the first child receives a 23rd of the candies plus one candy, the second child receives a 23rd of the remaining candies plus two candies, and so on. All candies are given away and each child receives the same amount of candies. How many children vis... | [
"23"
] | [
"Amy distributes candies on Halloween according to the following rule: the first child receives a 23rd of the candies plus one candy, the second child receives a 23rd of the remaining candies plus two candies, and so on. All candies are given away and each child receives the same amount of candies. How many childre... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fadf4c386747398a156a48b232a2225715ec242b7d351acce10edf4bd8bd1ea3 | Amy has $5 in quarters, dimes, and nickels, where the total value in quarters is more than the total value of the dimes, and the total value of the dimes is more than the total value of the nickels. What is the maximum number of coins Amy could have? | [
"55"
] | [
"Amy has $5 in quarters, dimes, and nickels, where the total value in quarters is more than the total value of the dimes, and the total value of the dimes is more than the total value of the nickels. What is the maximum number of coins Amy could have?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-00ef507c7ff97081166e2b6f1c7fd52fa2cff5b899f26505b13d23ca75a62431 | Amy has 21 square tiles, some red and some black. She arranges them in a rectangle such that no two black tiles share a border. What is the largest possible number of black tiles Amy can have? | [
"11"
] | [
"Amy has 21 square tiles, some red and some black. She arranges them in a rectangle such that no two black tiles share a border. What is the largest possible number of black tiles Amy can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc4c40fa2e4bfd7ef550db1d2a9dc7b352a78cfe7d896420f56ce7debe5b2fd6 | Amy has a number of rocks such that the mass of each rock, in kilograms, is a positive integer. The sum of the masses of the rocks is $2018$ kilograms. Amy realizes that it is impossible to divide the rocks into two piles of $1009$ kilograms each. What is the maximum possible number of rocks that Amy could have? | [
"1009"
] | [
"Amy has a number of rocks such that the mass of each rock, in kilograms, is a positive integer. The sum of the masses of the rocks is $2018$ kilograms. Amy realizes that it is impossible to divide the rocks into two piles of $1009$ kilograms each. What is the maximum possible number of rocks that Amy could have?"
... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 45
} | 45 | 0 | 0 |
omr-ws-382e797abef12fc030f5bf2cb36aa1895c6ea96efb8208d6f680682e33626e44 | Amy has decided to make a $5$ by $5$ matrix with entries as positive integers. The product of the $5$ numbers in any row, column, or diagonal is equal to a constant $k$. If $8000 < k < 32000$, what is the value of $k$? | [
"10000"
] | [
"Amy has decided to make a $5$ by $5$ matrix with entries as positive integers. The product of the $5$ numbers in any row, column, or diagonal is equal to a constant $k$. If $8000 < k < 32000$, what is the value of $k$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bec616d5c33e6e051c02f43d6692be89671f42b4f92d75808d433ec1891e1ac | Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \( x \), the sum of... | [
"2.5"
] | [
"Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \\( x \\), the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 7,
"tir": 1
} | 29 | 0 | 0 |
omr-ws-11d71dda2ac9b49f1197a9e2a84c1b53a3d68ccc4405892d1bd8c5dfd9bd0b27 | Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \(x\), the sum of t... | [
"\\dfrac{5}{2}"
] | [
"Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates \\(x\\), the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-b59534ddec25a590fbced26c4a801c496d62b4638e80a71ebfe4c6ffea730042 | Amy is traveling on the $xy$-plane where her motion is described by the equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant 3 mph, the magnitude of her velocity as she approaches (1,−1) can be expressed as $\frac{\sqrt{a + be^4}}{c + de^2}$. Find $\frac{ac}{bd}$. | [
"-1"
] | [
"Amy is traveling on the $xy$-plane where her motion is described by the equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant 3 mph, the magnitude of her velocity as she approaches (1,−1) can be expressed as $\\frac{\\sqrt{a + be^4}}{c + de^2}$. Find $\\frac{ac}{bd}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d58f44a2667798d05dd8c3d184cf04a89d0c07426849c57683a49ff0ea730f05 | Amy painted a dart board over a square clock face using the "hour positions" as boundaries. If \( t \) is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and \( q \) is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then wha... | [
"\\(2 \\sqrt{3} - 2\\)"
] | [
"Amy painted a dart board over a square clock face using the \"hour positions\" as boundaries. If \\( t \\) is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and \\( q \\) is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'cloc... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 6
} | 24 | 0 | 0 |
omr-ws-27f3c9798589b3d507d23acfb1e5075518795730aaa5c537c65cfe95481976a6 | Amy skis down a hill at a constant speed of 60 km/h to cover 1500 meters from the top to the bottom. After skiing down, she immediately gets on the lift and skis down again, meeting David who has completed a full circuit on the lift. If a seat passes through the bottom of the lift every 15 seconds, calculate the number... | [
"24"
] | [
"Amy skis down a hill at a constant speed of 60 km/h to cover 1500 meters from the top to the bottom. After skiing down, she immediately gets on the lift and skis down again, meeting David who has completed a full circuit on the lift. If a seat passes through the bottom of the lift every 15 seconds, calculate the n... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 3
} | 21 | 0 | 0 |
omr-ws-b21ded8f01dc777ab94b176bdc55b030da4c6348784cc9c71fd24ae74f778237 | Amy starts at a point and runs 50 meters south-west to her first clue. Mei runs to her first clue, which is 120 meters north-west of Amy’s first clue. Mei then makes a quarter turn clockwise and runs the same distance to her second clue as she ran to her first clue. How far is Mei’s second clue from the starting point,... | [
"184"
] | [
"Amy starts at a point and runs 50 meters south-west to her first clue. Mei runs to her first clue, which is 120 meters north-west of Amy’s first clue. Mei then makes a quarter turn clockwise and runs the same distance to her second clue as she ran to her first clue. How far is Mei’s second clue from the starting p... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-776f136361b7a9d43d271ac93126da9f43bb7663794abdf96c55a80726bc2bbb | Amy starts with a piece of paper with the number $0$ on it. On each of $\binom{6+4}{6}$ days, she starts at the lower left corner of a $4 \times 6$ grid and travels to the upper right corner, only going right or up at each intersection. When she gets there, she counts the number of blocks, $b$, that lie above her path,... | [
"\\dbinom{10}{4}_q"
] | [
"Amy starts with a piece of paper with the number $0$ on it. On each of $\\binom{6+4}{6}$ days, she starts at the lower left corner of a $4 \\times 6$ grid and travels to the upper right corner, only going right or up at each intersection. When she gets there, she counts the number of blocks, $b$, that lie above he... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1799036036202896cbd4ad4d9c72d6f2218c2c4a9aa57d5418af1ae15e6b5899 | Amy the Ant is crawling in 3D space, starting at the origin $(0, 0, 0)$. Every move she can move $1$ step in any of the $6$ directions. What is the probability that she lands on the point $(8, 7, 9)$ after $30$ moves? | [
"0"
] | [
"Amy the Ant is crawling in 3D space, starting at the origin $(0, 0, 0)$. Every move she can move $1$ step in any of the $6$ directions. What is the probability that she lands on the point $(8, 7, 9)$ after $30$ moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-6213b4618062f287f2783f1bc43ea7398b532cea656672be5a9900f385e8f46a | Amy works at a toy manufacturing company and her job is to make toy cars. After 5 hours, she made 4 toy cars. After 24 hours, she made 18 toy cars. Starting with the 5th hour, she makes the toy cars at a constant rate. During which hour will she make the 40th toy car? | [
"54"
] | [
"Amy works at a toy manufacturing company and her job is to make toy cars. After 5 hours, she made 4 toy cars. After 24 hours, she made 18 toy cars. Starting with the 5th hour, she makes the toy cars at a constant rate. During which hour will she make the 40th toy car?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a4689087d590a1079d6513e5c6cf4b70f66a21567cac2274f3317c9f2f61339a | Amy works for 36 hours per week for 10 weeks during the summer, making $\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\$3000$, how many hours per week must she work? | [
"12"
] | [
"Amy works for 36 hours per week for 10 weeks during the summer, making $\\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\\$3000$, how many hours per week must she work?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cb071983d24217aabd2c797034b59301e65928268a2709bf3a25bb05e975dd8 | Amy writes a \(1\) or a \(-1\) in each cell of an \(n \times 2\) grid, where \(n\) is not a multiple of \(4\). Below each column, she writes the product of the \(2\) entries in that column in blue, and to the right of each row, she writes the product of the \(n\) entries in that row in green. Given that the sum of the ... | [
"0"
] | [
"Amy writes a \\(1\\) or a \\(-1\\) in each cell of an \\(n \\times 2\\) grid, where \\(n\\) is not a multiple of \\(4\\). Below each column, she writes the product of the \\(2\\) entries in that column in blue, and to the right of each row, she writes the product of the \\(n\\) entries in that row in green. Given ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-20bdec3840b6e80055e954fcf653eb5cbd6b0440725b66cef5bcad124bd15b67 | Amy, Ben, and Carol each sit at distinct vertices of an equilateral triangle. Diana, Emily, and Frank each sit at distinct midpoints of the sides of the same triangle. Each person wears either a red baseball cap or a blue baseball cap. In how many ways can they choose the colors of their baseball caps so that no three ... | [
"18"
] | [
"Amy, Ben, and Carol each sit at distinct vertices of an equilateral triangle. Diana, Emily, and Frank each sit at distinct midpoints of the sides of the same triangle. Each person wears either a red baseball cap or a blue baseball cap. In how many ways can they choose the colors of their baseball caps so that no t... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-971c3fd964e7bd4b9ccc8bdab93cce6f2d84f7c40929e13ad84234f46bd9be6b | Amy, Ben, and Carol each sit at distinct vertices of an equilateral triangle. Diana, Emily, and Frank each sit at distinct midpoints of the sides of the same triangle. Each person wears either a red baseball cap or a blue baseball cap. In how many ways can they choose the colors of their baseball caps so that no three ... | [
"12"
] | [
"Amy, Ben, and Carol each sit at distinct vertices of an equilateral triangle. Diana, Emily, and Frank each sit at distinct midpoints of the sides of the same triangle. Each person wears either a red baseball cap or a blue baseball cap. In how many ways can they choose the colors of their baseball caps so that no t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-d117ddd877074ca826e3e5aa969c5a77516c9088b1a25e7623778d3f6ed28f8a | Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the first person to roll one of his favorite numbers wins. Amy's favorite number is 1, Ben's favorite numbers are 2 and 3, and Chime's are 4, 5, and 6. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won after Chi... | [
"\\(\\frac{5}{13}\\)"
] | [
"Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the first person to roll one of his favorite numbers wins. Amy's favorite number is 1, Ben's favorite numbers are 2 and 3, and Chime's are 4, 5, and 6. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won afte... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a28871b77783141ae22cffb2ff481d2305aba21bd186216fc2cba97e4cd46402 | Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible? | [
"132"
] | [
"Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those girls but disliked by the third. In how many different ways is this possible?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-c7eed770c02c300222306b06d912bb3475d8fe3d6836a6523bf5c831f72485c2 | Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? | [
"132"
] | [
"Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 63,
"genselect": 8,
"tir": 14
} | 85 | 0 | 0 |
omr-ws-a6e0a00ccb9037eb0cf7a8d9a43a1f969e609426dca4f213f1f2a54202811f32 | Amy, Beth, and Jo listen to four different songs. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? | [
"132"
] | [
"Amy, Beth, and Jo listen to four different songs. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 10
} | 42 | 0 | 0 |
omr-ws-0cee1215e190c90c2854f7e638093d16376377740a2834ff92b01e0cee173068 | Amy, Bob, Cordelia, Desmond, and Eve are visiting two parks. Independently, they each choose a random time between 3:00 and 4:00, they will randomly choose one of the two parks to go to, and stay there until 4:00. What is the probability that at 3:30, at least one of the parks contains at least 2 people? | [
"\\dfrac{21}{32}"
] | [
"Amy, Bob, Cordelia, Desmond, and Eve are visiting two parks. Independently, they each choose a random time between 3:00 and 4:00, they will randomly choose one of the two parks to go to, and stay there until 4:00. What is the probability that at 3:30, at least one of the parks contains at least 2 people?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a3b4b26c200debcfc3056a19b7001c1d79d6b91c6a719fb5d28bd0ee3a5b8f5 | Amy, Bomani, and Charlie started working at $1{:}00 \text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group and was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \text{ PM}$... | [
"1:25 PM"
] | [
"Amy, Bomani, and Charlie started working at $1{:}00 \\text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group and was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \\tex... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-f21d3b8298871c9b9351327bf5a1f24c5cdb41d0b3b66d3c4e587a077dacc298 | Amélia and Beatriz play battleship on a $2n \times 2n$ board. Amélia places $n^2$ submarines on the intersections of $n$ chosen lines and $n$ chosen columns. What is the minimum number of cells Beatriz must choose to ensure at least one submarine will explode? | [
"\\(3n+1\\)"
] | [
"Amélia and Beatriz play battleship on a $2n \\times 2n$ board. Amélia places $n^2$ submarines on the intersections of $n$ chosen lines and $n$ chosen columns. What is the minimum number of cells Beatriz must choose to ensure at least one submarine will explode?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c439f223ec862dbcec6ad44d4462891578cf0a85ad6484c7f2b19446f35c18c9 | An "Iberian" set of positive integers is defined as a subset of \(\{2, 3, 4, \ldots, 2018\}\) such that whenever \(m\) and \(n\) belong to the set, then \(\gcd(m, n)\) also belongs to the set. An "Iberian" set is "Olympic" if it is not contained in any other Iberian set. Find all the Olympic Iberian sets that contain t... | [
"11"
] | [
"An \"Iberian\" set of positive integers is defined as a subset of \\(\\{2, 3, 4, \\ldots, 2018\\}\\) such that whenever \\(m\\) and \\(n\\) belong to the set, then \\(\\gcd(m, n)\\) also belongs to the set. An \"Iberian\" set is \"Olympic\" if it is not contained in any other Iberian set. Find all the Olympic Iber... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-cc0d0d1932c61f47500fc41795e1b598262a9d419c8fe4aafd37600b20c152d1 | An "almost square" is defined as a number that can be written as a product of two consecutive natural numbers. Let $N$ be an almost square. Find the smallest positive integer $k$ such that $N = \frac{a(a+1)}{b(b+1)}$ for some almost squares $a(a+1)$ and $b(b+1)$, where $b = a - k$. Calculate $k$ for $N = 56$. | [
"42"
] | [
"An \"almost square\" is defined as a number that can be written as a product of two consecutive natural numbers. Let $N$ be an almost square. Find the smallest positive integer $k$ such that $N = \\frac{a(a+1)}{b(b+1)}$ for some almost squares $a(a+1)$ and $b(b+1)$, where $b = a - k$. Calculate $k$ for $N = 56$."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-4b4512346e433069a8865097afc47a257819945a0e794e265a2d350b2e692e88 | An "eye" is formed by points \( A, B, C, A', B', C' \) where \( AB, AC, A'B', A'C' \) are tangent to the circle of quadrilateral \( BB'C'C \) with center \( O \). Points \( A, O, \) and \( A' \) are collinear. Given that \( AB = AC = A'B' = A'C' = 1 \) cm and the area of the circle is \( \pi / 4 \) cm\(^2\), determine ... | [
"\\(\\sqrt{5}\\)"
] | [
"An \"eye\" is formed by points \\( A, B, C, A', B', C' \\) where \\( AB, AC, A'B', A'C' \\) are tangent to the circle of quadrilateral \\( BB'C'C \\) with center \\( O \\). Points \\( A, O, \\) and \\( A' \\) are collinear. Given that \\( AB = AC = A'B' = A'C' = 1 \\) cm and the area of the circle is \\( \\pi / 4 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-06546c0ab6cca927ee9c51bcf52203484f0734d3eda940fec65c79005b8183fe | An "unfair" coin has a $2/3$ probability of turning up heads. If the coin is tossed $7$ times, what is the probability that the total number of heads is even? Express your answer as a rational number in lowest terms. | [
"\\(\\frac{1093}{2187}\\)"
] | [
"An \"unfair\" coin has a $2/3$ probability of turning up heads. If the coin is tossed $7$ times, what is the probability that the total number of heads is even? Express your answer as a rational number in lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-afe2738d4cf1caebfe56327379193363bcf7ff41decff4393384fb28386f7fc7 | An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even? | [
"\\(\\frac{1}{2} \\left( 1 + \\left( \\frac{1}{3} \\right)^{50} \\right)\\)",
"\\(\\frac{1}{2}\\left(1 + \\frac{1}{3^{50}}\\right)\\)",
"\\(\\frac{3^{50} + 1}{2 \\cdot 3^{50}}\\)"
] | [
"An \"unfair\" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 112,
"tir": 88
} | 200 | 0 | 0 |
omr-ws-a928671537d050f7f4e6d3d4250e241312a7e6321dcdbea73ee638ca9db1df7b | An "unfair" coin has a $\frac{2}{3}$ probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even? | [
"\\(\\frac{1}{2} \\left(1 + \\left(\\frac{1}{3}\\right)^{50}\\right)\\)"
] | [
"An \"unfair\" coin has a $\\frac{2}{3}$ probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ce5be5a46241525ab94889cf7461913dbcdcbbca0c7e6d72bb9b48132eecd702 | An $ m \times n \times p$ rectangular box has half the volume of an $(m + 2) \times (n + 2) \times (p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m \leq n \leq p$. What is the largest possible value of $p$? | [
"130"
] | [
"An $ m \\times n \\times p$ rectangular box has half the volume of an $(m + 2) \\times (n + 2) \\times (p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m \\leq n \\leq p$. What is the largest possible value of $p$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 10
} | 31 | 0 | 0 |
omr-ws-132cf6ac127f0f709ebe123d6cd9ce21b111da2e40a8b3b3b01df092c2989377 | An $11 \times 11$ chessboard is covered with one $\boxed{ }$ shaped tile and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the possible positions for the $\boxed{ }$ shaped tile. | [
"\\text{All squares where the sum of the row and column numbers is a multiple of 3}"
] | [
"An $11 \\times 11$ chessboard is covered with one $\\boxed{ }$ shaped tile and forty $\\boxed{ }\\boxed{ }\\boxed{ }$ shaped tiles. Determine the possible positions for the $\\boxed{ }$ shaped tile."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-6400b41967632b6746a6f9abe66ab3f26722d19976de8cf255da602972caf1b4 | An $8 \times 8$ array consists of the numbers $1, 2, \ldots, 64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal? | [
"88"
] | [
"An $8 \\times 8$ array consists of the numbers $1, 2, \\ldots, 64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0734ede95c98966fe60ca2a953fc1dd6183b255255699ebebbbb691da6fc59ee | An $8 \times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Let $d$ be the minimum distance between any two marked points or between any marked point and the edge of the board. Calculate the maximum possible value of $d^2$. | [
"5"
] | [
"An $8 \\times 8$ board is divided into unit squares. Ten of these squares have their centers marked. Let $d$ be the minimum distance between any two marked points or between any marked point and the edge of the board. Calculate the maximum possible value of $d^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-a2f7ff3275f9e70c073bb0cc43dd9ee706bd672d83b9f2e4a8e658321e69252c | An $8 \times 8$ board is tiled with 13 rods (1 by $n$ rectangles) so that all of it is covered without overlap. Find the maximum possible value of the product of the lengths of the 13 rods. | [
"5^{12} \\times 4"
] | [
"An $8 \\times 8$ board is tiled with 13 rods (1 by $n$ rectangles) so that all of it is covered without overlap. Find the maximum possible value of the product of the lengths of the 13 rods."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-51fee22f8d8e6e70a0107922ffc802ea94aafc59e73f2f8fad73d6f818033a1a | An $8 \times 8$ checkerboard has its corners trimmed to form a circular dartboard with a diameter equal to the original length of the side of the board. A dart is thrown at the circle randomly. What is the probability that the dart lands in a square that is entirely contained in the circle? | [
"64%"
] | [
"An $8 \\times 8$ checkerboard has its corners trimmed to form a circular dartboard with a diameter equal to the original length of the side of the board. A dart is thrown at the circle randomly. What is the probability that the dart lands in a square that is entirely contained in the circle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-51ce2aaade23dde9fade98ee87ce8abedd960d763fbb3fd8b9c008e122dce7a0 | An $8 \times 8$ chessboard has all four of its corner squares removed. How many ways can 8 rooks be placed on the board so that no two rooks attack each other from their respective positions? (A rook can attack any square that is horizontal or vertical to its current location.) | [
"518400"
] | [
"An $8 \\times 8$ chessboard has all four of its corner squares removed. How many ways can 8 rooks be placed on the board so that no two rooks attack each other from their respective positions? (A rook can attack any square that is horizontal or vertical to its current location.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-397b1585f457966f2ce4304bad1dd256f1d27c044f04a0be84e73125eb7fa6c1 | An $8 \times 8$ chessboard is completely tiled by $2 \times 1$ dominoes. A king's tour is defined as a sequence of moves where the king visits each cell exactly once. Let $N$ be the number of such king's tours where every move transitions the king from one cell of a domino to the other cell of the same domino. Calculat... | [
"0"
] | [
"An $8 \\times 8$ chessboard is completely tiled by $2 \\times 1$ dominoes. A king's tour is defined as a sequence of moves where the king visits each cell exactly once. Let $N$ be the number of such king's tours where every move transitions the king from one cell of a domino to the other cell of the same domino. C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a2152dda851165636df5b44e7f602bea973aa3f2259e3c23920651dfc23630e7 | An $8 \times 8$ chessboard is completely tiled by $2 \times 1$ dominoes. Let $T$ be the number of distinct king's tours that visit each cell exactly once, moving from one cell of a domino to the other cell of the same domino in consecutive moves. Find the remainder when $T$ is divided by $1000$. | [
"0"
] | [
"An $8 \\times 8$ chessboard is completely tiled by $2 \\times 1$ dominoes. Let $T$ be the number of distinct king's tours that visit each cell exactly once, moving from one cell of a domino to the other cell of the same domino in consecutive moves. Find the remainder when $T$ is divided by $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-d020712b10bb8152469edce392478b43a79e67dfcbb8b065e54fb0a6ff825d66 | An $8 \times 8$ chessboard is completely tiled by $2 \times 1$ dominoes. Positive integers are placed in all cells of the board such that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. Let $S$ be the sum of the number... | [
"7",
"9"
] | [
"An $8 \\times 8$ chessboard is completely tiled by $2 \\times 1$ dominoes. Positive integers are placed in all cells of the board such that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. Let $S$ be the sum of the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-a711187d2f3b9083958e98c966446ba000585fa7a8ce6224ad072e5a0db0efbc | An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$. Let $N$ be the number of pairs of dominoes that form a $2 \times 2$ square. Calculate the minimum possible value of $N$. | [
"8"
] | [
"An $8 \\times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \\times 2$. Let $N$ be the number of pairs of dominoes that form a $2 \\times 2$ square. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6e39895542573f452fec7453df83d9f0206bf03d174328b861172a40b7b2c9b4 | An $8 \times 8$ grid is tiled with $2 \times 1$ dominoes, each either entirely black or entirely white. The number of black and white unit squares in each column are given by $b_{(c_1)}, b_{(c_2)},\dots, b_{(c_8)}$ and $w_{(c_1)}, w_{(c_2)}, \dots, w_{(c_8)}$, respectively. Similarly, the number of black and white squa... | [
"88"
] | [
"An $8 \\times 8$ grid is tiled with $2 \\times 1$ dominoes, each either entirely black or entirely white. The number of black and white unit squares in each column are given by $b_{(c_1)}, b_{(c_2)},\\dots, b_{(c_8)}$ and $w_{(c_1)}, w_{(c_2)}, \\dots, w_{(c_8)}$, respectively. Similarly, the number of black and w... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-313807e099f66a30d2fe7b786c46c17736b2ff09e1260b309c48005bcbcabace | An $8 \times 8$ square grid is divided into $1 \times 1$ squares by corridors. Each vertex has a switch that toggles the illumination of all adjacent corridors. The watchman starts in a corner of a completely unlit grid. Determine the minimum number of switches the watchman must toggle to ensure he can pass from any sw... | [
"40"
] | [
"An $8 \\times 8$ square grid is divided into $1 \\times 1$ squares by corridors. Each vertex has a switch that toggles the illumination of all adjacent corridors. The watchman starts in a corner of a completely unlit grid. Determine the minimum number of switches the watchman must toggle to ensure he can pass from... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-c6803d42ee93881f80eec2a2ca18fcfe493e037f8d9976bc99da2bb68b6428e5 | An $8 \times 8$ table is divided into $64$ unit squares. In some of these unit squares, a diagonal is drawn. If none of these diagonals have any common point, what is the maximum number of diagonals that can be drawn? | [
"36"
] | [
"An $8 \\times 8$ table is divided into $64$ unit squares. In some of these unit squares, a diagonal is drawn. If none of these diagonals have any common point, what is the maximum number of diagonals that can be drawn?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-08bd97f51f3938a282f157306ab93a0ff5bc2e72465f1c4621d63f3ed3412e48 | An $8 \times 8$ table is initially empty. Two cells are considered neighbors if they share an edge or a vertex. One fills all $64$ cells with numbers one at a time by writing in an empty cell the number equal to the quantity of its empty neighbor cells. Calculate the sum of the $64$ written numbers. | [
"210"
] | [
"An $8 \\times 8$ table is initially empty. Two cells are considered neighbors if they share an edge or a vertex. One fills all $64$ cells with numbers one at a time by writing in an empty cell the number equal to the quantity of its empty neighbor cells. Calculate the sum of the $64$ written numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa0fe8a43346754a4c3a3bfd9bb792401493c8e86c4df4d20793109b45054f32 | An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. What is the area of the region common to both the rectangle and the circle? | [
"\\(2\\pi + 4\\)"
] | [
"An $8$ by $2\\sqrt{2}$ rectangle has the same center as a circle of radius $2$. What is the area of the region common to both the rectangle and the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 30
} | 78 | 0 | 0 |
omr-ws-3a1e04ad04e66e1b1171a0d2284031566ae45930f5899b95b6bb73e8714cb556 | An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? | [
"\\( 80 - 20\\pi \\)"
] | [
"An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f17c444b179da53f1133202acdb8b39b0d9e2526b9481d7b2094b0e20869ea8f | An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\frac{1}{2}$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? | [
"\\(80 - 20\\pi\\)"
] | [
"An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\\frac{1}{2}$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-5e84e38913c522251347fc597fa61cf6c9ab38be700812b23a3ee587eb47b745 | An $8' \times 10'$ table sits in the corner of a square room. The table needs to be moved to a new position without tilting it or taking it apart. What is the smallest integer value of $S$ (the side length of the square room in feet) for which the table can be moved as desired? Note: The table must fit within the room ... | [
"13"
] | [
"An $8' \\times 10'$ table sits in the corner of a square room. The table needs to be moved to a new position without tilting it or taking it apart. What is the smallest integer value of $S$ (the side length of the square room in feet) for which the table can be moved as desired?\n\nNote: The table must fit within ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-42977905ba6251d6a2972937e61927f03b5b450e6ed02df30db852ca9b66df89 | An $L$-region of size $n$ is defined as an $L$-shaped region with two sides of length $2n$ and four sides of length $n$, and an $L$-tile is defined as a tile with the same shape as an $L$-region of size $1$ (i.e., a $2 \times 2$ square with one $1 \times 1$ square missing). Determine the number of ways to tile an $L$-r... | [
"16"
] | [
"An $L$-region of size $n$ is defined as an $L$-shaped region with two sides of length $2n$ and four sides of length $n$, and an $L$-tile is defined as a tile with the same shape as an $L$-region of size $1$ (i.e., a $2 \\times 2$ square with one $1 \\times 1$ square missing). Determine the number of ways to tile a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-79f67ad40369864ad785bad4bf0f3d4b0bab482d047a28427ffaaadf4e67cd3b | An $\alpha-$triangle is defined with base $c = x$ and sides $a = x\sqrt[2000]{110^{123}}$ and $b = x \sqrt[1000]{110}$. A move consists of choosing a side and making it the base for a new $\alpha-$triangle. Starting with a base length $c = 1$, how many distinct ways are there to make an $\alpha-$triangle with base leng... | [
"0"
] | [
"An $\\alpha-$triangle is defined with base $c = x$ and sides $a = x\\sqrt[2000]{110^{123}}$ and $b = x \\sqrt[1000]{110}$. A move consists of choosing a side and making it the base for a new $\\alpha-$triangle. Starting with a base length $c = 1$, how many distinct ways are there to make an $\\alpha-$triangle with... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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