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-3e1fe4714e27d4432e19f0dd22f54300f1675e0511068ecde7d5b692e7f5dca4 | Among 2013 runners on a straight road 4 meters wide, where no two runners are closer than 2 meters from each other at any given moment, let $d$ be the maximum possible distance between any two runners. Calculate the minimum value of $d$. | [
"1340"
] | [
"Among 2013 runners on a straight road 4 meters wide, where no two runners are closer than 2 meters from each other at any given moment, let $d$ be the maximum possible distance between any two runners. Calculate the minimum value of $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-327985284676dfcbf6af157bd6fd6cb4a94cb1e61fd9adeead2c86479292de98 | Among 2015 circles with radius $1$ lying on a plane, find the minimum number $k$ such that one can always select a collection $C$ of $k$ circles where either every pair of two circles in $C$ intersects or every pair of two circles in $C$ does not intersect. | [
"43"
] | [
"Among 2015 circles with radius $1$ lying on a plane, find the minimum number $k$ such that one can always select a collection $C$ of $k$ circles where either every pair of two circles in $C$ intersects or every pair of two circles in $C$ does not intersect."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-6c501069a3c0e2db2e29d0a3899a3966bf36787be3eee83215bb132edb8d5b8a | Among 2016 circles with radius 1 lying on the plane, let $C$ be the largest collection of circles such that either every pair of two circles in $C$ intersect or every pair of two circles in $C$ does not intersect. Find the minimum possible value of $|C|$. | [
"45"
] | [
"Among 2016 circles with radius 1 lying on the plane, let $C$ be the largest collection of circles such that either every pair of two circles in $C$ intersect or every pair of two circles in $C$ does not intersect. Find the minimum possible value of $|C|$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7054ef5800a54620687defc452b7ddf4cd1e82c27b0d6263b2b88b4764c1c105 | Among 24 coins, identical in appearance, two are counterfeit: one being heavier and one being lighter than a real one. Determine the minimal number of weightings needed to ascertain whether the combined weight of the two counterfeit coins is equal to the combined weight of two genuine coins. If the minimal number of we... | [
"22"
] | [
"Among 24 coins, identical in appearance, two are counterfeit: one being heavier and one being lighter than a real one. Determine the minimal number of weightings needed to ascertain whether the combined weight of the two counterfeit coins is equal to the combined weight of two genuine coins. If the minimal number ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3fdcc41dc1b75594024866807e40a9cfb71403c0d181d17c86551e978a58954 | Among 28 integers chosen from the interval $[104, 208]$, find the maximum number of distinct prime divisors these integers can have. | [
"no_valid_answer_found"
] | [
"Among 28 integers chosen from the interval $[104, 208]$, find the maximum number of distinct prime divisors these integers can have."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b05d6caef8f6fdf9077e1cf1caf99e14386e11b5518cd1c6e399995cfa30369 | Among 2n-1 positive integers, find the smallest possible value of n for which it is guaranteed that there exists a subset of n integers whose sum is divisible by n. | [
"1"
] | [
"Among 2n-1 positive integers, find the smallest possible value of n for which it is guaranteed that there exists a subset of n integers whose sum is divisible by n."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50aa3f44e2be99c8a6f29c607b81059f28f90fc723022f37140bd02dee72c6d2 | Among 33 balls, there are 2 radioactive ones. You can put several balls in a detector that will indicate if both radioactive balls are among the balls tested. What is the smallest number of times you need to use the detector to certainly find at least one of the radioactive balls? | [
"3"
] | [
"Among 33 balls, there are 2 radioactive ones. You can put several balls in a detector that will indicate if both radioactive balls are among the balls tested. What is the smallest number of times you need to use the detector to certainly find at least one of the radioactive balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b04614fbddf5368577cc00596f8f45ba245281d681813889a49b93c52c4f8ac | Among 400 athletes, every 3 months, 100 of them are randomly selected for testing for illegal steroids. In one year, what is the probability that a specific athlete, Breanna, is tested at least once? | [
"\\dfrac{175}{256}"
] | [
"Among 400 athletes, every 3 months, 100 of them are randomly selected for testing for illegal steroids. In one year, what is the probability that a specific athlete, Breanna, is tested at least once?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d29a2d95f5d2248d010d3066d24307a22b074ccc892873b5f6aac465af6c2750 | Among 400 different points chosen inside a unit cube, determine the minimum radius $r$ such that there exists a sphere of radius $r$ containing at least four of these points. | [
"\\dfrac{\\sqrt{3}}{10}"
] | [
"Among 400 different points chosen inside a unit cube, determine the minimum radius $r$ such that there exists a sphere of radius $r$ containing at least four of these points."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-5ba8f3bd221db280f23dcfc9511787ef477720d6da54a9900a22aeb14aa92ad8 | Among 49 students who each solve 3 problems with scores ranging from 0 to 7, determine the minimum number of pairs of students (A, B) such that for every problem, the score of student A is greater than or equal to the score of student B. | [
"1"
] | [
"Among 49 students who each solve 3 problems with scores ranging from 0 to 7, determine the minimum number of pairs of students (A, B) such that for every problem, the score of student A is greater than or equal to the score of student B."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-4cc990723d81bcfc1a060bf0376d0d46b5f4110f8da39114b21716a23b29994e | Among 49 students who have competed in solving 3 problems, each scored with an integer from 0 to 7, let $N$ be the number of pairs of students $(A, B)$ such that for every problem, student $A$ has won at least as many points as student $B$. Calculate the minimum possible value of $N$. | [
"2"
] | [
"Among 49 students who have competed in solving 3 problems, each scored with an integer from 0 to 7, let $N$ be the number of pairs of students $(A, B)$ such that for every problem, student $A$ has won at least as many points as student $B$. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-7871a2443dcd057a1cb74db1179e8bd043f23c4cb9ed885dd6a3d611de8fcfcd | Among 49 students who took a math test consisting of three problems, each scored between 0 and 7. Let $N$ be the number of pairs of students $(A, B)$ such that for each problem, the score of student $A$ is greater than or equal to that of student $B$. Calculate the minimum possible value of $N$. | [
"2"
] | [
"Among 49 students who took a math test consisting of three problems, each scored between 0 and 7. Let $N$ be the number of pairs of students $(A, B)$ such that for each problem, the score of student $A$ is greater than or equal to that of student $B$. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcc9413db71d6a7380812bd07b1d8ca2bc021b67411e435257fcbc9fce21aca1 | Among 49 students who took a test with 3 questions, each scoring an integer from 0 to 7, let $N$ be the number of pairs of students $(A, B)$ such that student $A$ scored more than student $B$ in each question. Calculate the minimum possible value of $N$. | [
"0"
] | [
"Among 49 students who took a test with 3 questions, each scoring an integer from 0 to 7, let $N$ be the number of pairs of students $(A, B)$ such that student $A$ scored more than student $B$ in each question. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-ed45c54ef08002e1712b8b18f09ddd0bdfc65543dbe63219c45d36a0b08c5498 | Among 5 different lines that intersect at a point, how many angles can be formed? | [
"20"
] | [
"Among 5 different lines that intersect at a point, how many angles can be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-dbc6b786ed4d105608401fc35c2cd61befb5587ded7896012fe4c35412c63ea3 | Among 500 points inside a unit square, let \(A_1, A_2, \ldots, A_{12}\) be 12 distinct points such that the sum of the distances \(A_1A_2 + A_2A_3 + \cdots + A_{11}A_{12}\) is minimized. Calculate the maximum possible value of this minimized sum. | [
"\\dfrac{1}{2}"
] | [
"Among 500 points inside a unit square, let \\(A_1, A_2, \\ldots, A_{12}\\) be 12 distinct points such that the sum of the distances \\(A_1A_2 + A_2A_3 + \\cdots + A_{11}A_{12}\\) is minimized. Calculate the maximum possible value of this minimized sum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-dd884b3a70f33b180420a40513d0271e771d48bb4d03b2d00e350e718c196b51 | Among 5000 movie fans, determine the maximum number of groups of the second kind that can be formed such that the remaining groups are of the first kind, with each group of the first kind having members who have all watched the same movie, and each group of the second kind having members who have each watched a unique ... | [
"5000"
] | [
"Among 5000 movie fans, determine the maximum number of groups of the second kind that can be formed such that the remaining groups are of the first kind, with each group of the first kind having members who have all watched the same movie, and each group of the second kind having members who have each watched a un... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-5731fe2334fa167ece1c337f4a1e40997aac203eaa8dc7cba960c1fe8c08877c | Among 51 points inside a square of side length 1, find the minimum radius $r$ such that there exist three points that can be covered by a circle of radius $r$. Express your answer as a reduced fraction. | [
"\\dfrac{\\sqrt{2}}{10}"
] | [
"Among 51 points inside a square of side length 1, find the minimum radius $r$ such that there exist three points that can be covered by a circle of radius $r$. Express your answer as a reduced fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec3f6704e653679d3d8733a3ab4f5acd5ea253843d62d76550b4a06f6a341a1a | Among 64 people where each person knows at least 6 others, determine the maximum number of people that can be arranged around a round table such that each person sits between two people they know. | [
"64"
] | [
"Among 64 people where each person knows at least 6 others, determine the maximum number of people that can be arranged around a round table such that each person sits between two people they know."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-2882556b9d2447c2bcb70542b9a48b52b245cc379592bc9b4718657d989d8a0b | Among 7 boys seated randomly in a row, what is the probability that exactly two boys are seated between Bill and George? | [
"\\(\\frac{4}{21}\\)"
] | [
"Among 7 boys seated randomly in a row, what is the probability that exactly two boys are seated between Bill and George?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5f3277512127e790c0b25657dc57213495122d154823b18c83617fe3568aa957 | Among 70 distinct numbers chosen from the integers between 1 and 200, calculate the maximum number of pairs \((a, b)\) such that \(a - b = 4\), \(5\), or \(9\). | [
"192"
] | [
"Among 70 distinct numbers chosen from the integers between 1 and 200, calculate the maximum number of pairs \\((a, b)\\) such that \\(a - b = 4\\), \\(5\\), or \\(9\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-228b537df0ffe827950db693a271226eea59f7b5b9a61522839929b491b514c0 | Among 89 points lying inside a circle of radius 5, calculate the minimum number of pairs of points whose distance from each other is less than $\sqrt{2}$. | [
"39"
] | [
"Among 89 points lying inside a circle of radius 5, calculate the minimum number of pairs of points whose distance from each other is less than $\\sqrt{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-b39a7c90fa81a63adf4fe0cd18a7b833e7bec0e18a48e74b7ed7722e40a9ca61 | Among 9 apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making 2 weighings on a two-pan balance? | [
"2 weighings"
] | [
"Among 9 apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making 2 weighings on a two-pan balance?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 12
} | 21 | 0 | 0 |
omr-ws-b93bb59220f27e5af77505153dbaff49fa786972958a0993d4ba4fa70f5a2e22 | Among 9 mathematicians, in any set of 3 mathematicians, at least 2 of them can speak the same language. Each mathematician can speak at most 3 languages. Determine the minimum number of mathematicians who must speak the same language. | [
"4"
] | [
"Among 9 mathematicians, in any set of 3 mathematicians, at least 2 of them can speak the same language. Each mathematician can speak at most 3 languages. Determine the minimum number of mathematicians who must speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-4982e45d557795541686f67cb08d000ec7eb0b06e7993c81e4bb0b4c0b347a9c | Among 90 distinct positive integers, let $a$ and $b$ be two integers whose least common multiple is the smallest possible value greater than 2024. Find the minimum possible value of $\text{lcm}(a, b)$. | [
"2025"
] | [
"Among 90 distinct positive integers, let $a$ and $b$ be two integers whose least common multiple is the smallest possible value greater than 2024. Find the minimum possible value of $\\text{lcm}(a, b)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7db606a3b9cae01871978206ffdffa83cc843ff8ee793e216be1c1a1c37fce39 | Among 999 points chosen in the plane, let \( D \) be the set of all distinct distances between these points. Find the minimum possible value of \( |D| \). | [
"499"
] | [
"Among 999 points chosen in the plane, let \\( D \\) be the set of all distinct distances between these points. Find the minimum possible value of \\( |D| \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-9261d8112934eef188263046cbd85a1128474eef6f484bbb9006beacf093b83c | Among N people, every pair of them has an odd number of friends in common. Let $f(x)$ denote the number of friends person $x$ has. Calculate the sum of all possible values of $f(1) + f(2) + \cdots + f(N)$. | [
"n(n-1)"
] | [
"Among N people, every pair of them has an odd number of friends in common. Let $f(x)$ denote the number of friends person $x$ has. Calculate the sum of all possible values of $f(1) + f(2) + \\cdots + f(N)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-decfd6ec2aa3bc8aa189c8d3090de3bc7087115708b826271fa982fc7d727ed4 | Among N people, every pair of them has an odd number of friends in common. Let $f(x)$ denote the number of friends person $x$ has. Calculate the sum of all possible values of $f(x)$ for any person $x$. | [
"N-1"
] | [
"Among N people, every pair of them has an odd number of friends in common. Let $f(x)$ denote the number of friends person $x$ has. Calculate the sum of all possible values of $f(x)$ for any person $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-43b920cab1f97ceed54efb48382a754f8c55092064d98f624fdd2b6ad6e123ed | Among N positive integers chosen arbitrarily, what is the smallest possible value of N to ensure that there exist two integers whose sum or difference is divisible by 111? | [
"57"
] | [
"Among N positive integers chosen arbitrarily, what is the smallest possible value of N to ensure that there exist two integers whose sum or difference is divisible by 111?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f173fab21a42fae6161b02e4a66e3fc7590fefb3a7fdbcf1dbaccb5bc533485f | Among Vlad's 25 colleagues, each has a unique number of friends, and every colleague has at least one friend. Determine how many friends Vlad has. | [
"13"
] | [
"Among Vlad's 25 colleagues, each has a unique number of friends, and every colleague has at least one friend. Determine how many friends Vlad has."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ad29af73cc37251b5380eb50961c73b6b10b4eda12ad43c810e4385f47c4b6d | Among \( N \) coins, \( N - 1 \) are authentic with the same weight and one is fake, with a different weight than the others. Using a two-plate scale, determine all \( N \) for which it is possible to find the fake coin and determine if it is heavier or lighter than the authentic ones, with certainty. Each time it can ... | [
"N \\geq 3"
] | [
"Among \\( N \\) coins, \\( N - 1 \\) are authentic with the same weight and one is fake, with a different weight than the others. Using a two-plate scale, determine all \\( N \\) for which it is possible to find the fake coin and determine if it is heavier or lighter than the authentic ones, with certainty. Each t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-aca068db95ba54b349528bbd407b8172deffd3bf3c7054d78a33f871aadca419 | Among \( n > 2 \) points on the plane (not all on a line), every point is assigned a number from the set \(\{-1, 0, 1\}\) so that the numbers assigned to the points lying on a straight line sum to \( 0 \). If there are \( k \) points on a line and \( k > 2 \), find the sum of all possible values of \( k \) for which it... | [
"0"
] | [
"Among \\( n > 2 \\) points on the plane (not all on a line), every point is assigned a number from the set \\(\\{-1, 0, 1\\}\\) so that the numbers assigned to the points lying on a straight line sum to \\( 0 \\). If there are \\( k \\) points on a line and \\( k > 2 \\), find the sum of all possible values of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-cc846a7e561bbe5732f38ad5c596c046b80902c8bae09aaa1122c6e2d4cc8927 | Among \( n \) apparently equal coins, less than half of which are fake and differ in weight from the regular coins, determine the minimum number of weighings required to guarantee finding at least one regular coin using a balance without scale. Let \( f(n) \) represent this minimum number of weighings. Calculate \( f(1... | [
"3"
] | [
"Among \\( n \\) apparently equal coins, less than half of which are fake and differ in weight from the regular coins, determine the minimum number of weighings required to guarantee finding at least one regular coin using a balance without scale. Let \\( f(n) \\) represent this minimum number of weighings. Calcula... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-959a624e5b745fd3e63bb84c63db7f2ce1312c7c711583fc1d897e2550d849fc | Among \( n \) apparently equal coins, less than half of which are fakes differing from the regular coins by weight, determine the minimum number of weighings required to guarantee finding at least one regular coin using a balance without a scale. Express your answer in terms of \( n \). | [
"\\lceil \\log_3 n \\rceil"
] | [
"Among \\( n \\) apparently equal coins, less than half of which are fakes differing from the regular coins by weight, determine the minimum number of weighings required to guarantee finding at least one regular coin using a balance without a scale. Express your answer in terms of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60af7702583e2635effdf6a84da31e5f6bc9455f9e3bf317e990054595ca6b82 | Among \( n \) consecutive natural numbers, there are more multiples of 2022 than multiples of 2021. What is the largest \( n \) for which this is possible? | [
"4041"
] | [
"Among \\( n \\) consecutive natural numbers, there are more multiples of 2022 than multiples of 2021. What is the largest \\( n \\) for which this is possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-a8f6a285e58004dc15c1414b913738a5808719b65c04d3361cb5666fdb939326 | Among \( n+1 \) positive integers less than or equal to \( 2n \), let \( S \) be the set of all such integers. Define \( f(n) \) as the minimum number of pairs \((p, q)\) in \( S \) such that \( p \mid q \). Calculate \( f(10) \). | [
"1"
] | [
"Among \\( n+1 \\) positive integers less than or equal to \\( 2n \\), let \\( S \\) be the set of all such integers. Define \\( f(n) \\) as the minimum number of pairs \\((p, q)\\) in \\( S \\) such that \\( p \\mid q \\). Calculate \\( f(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95a6f024de949fc85dbba1333fa07f783f5edc789b35bbca2ecbf4ef94a06dba | Among \( n+1 \) positive integers less than or equal to \( 2n \), let \( S \) be the set of all such integers. Define \( f(n) \) as the minimum number of pairs of co-prime numbers that must exist in any subset of \( S \) of size \( n+1 \). Calculate \( f(10) \). | [
"5"
] | [
"Among \\( n+1 \\) positive integers less than or equal to \\( 2n \\), let \\( S \\) be the set of all such integers. Define \\( f(n) \\) as the minimum number of pairs of co-prime numbers that must exist in any subset of \\( S \\) of size \\( n+1 \\). Calculate \\( f(10) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52f232aeaaddbe031e3df737e7575cbeabac43333c243c9ece214dad78e02550 | Among \( n+2 \) integers chosen from the set \( \{1, \ldots, 2n\} \), calculate the minimum number of pairs of consecutive integers that must exist. | [
"3"
] | [
"Among \\( n+2 \\) integers chosen from the set \\( \\{1, \\ldots, 2n\\} \\), calculate the minimum number of pairs of consecutive integers that must exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a85dd93f07c784d4042462a2386a07b149ade0a35898a5371d6977ee87bec7a | Among all $2^{15}$ possible initial sequences of 15 chocolates (where each chocolate is either milk or dark), how many sequences allow Barbara to have a winning strategy in the game where players alternately eat a positive number of consecutive chocolates starting from the leftmost, such that the number of chocolates e... | [
"8320"
] | [
"Among all $2^{15}$ possible initial sequences of 15 chocolates (where each chocolate is either milk or dark), how many sequences allow Barbara to have a winning strategy in the game where players alternately eat a positive number of consecutive chocolates starting from the leftmost, such that the number of chocola... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac73b88d8d8e3ad370fbd238c90ee17da30007e17dc08e90de9c4db310db20a2 | Among all 14-digit numbers in base 10 that contain exactly 7 ones and 7 zeros and start with a 1, how many of them are divisible by 7? | [
"246"
] | [
"Among all 14-digit numbers in base 10 that contain exactly 7 ones and 7 zeros and start with a 1, how many of them are divisible by 7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-320e27b86265e61b52a684275f8ef3efcd414a521aded982fe579669ed42656a | Among all 7-digit numbers formed using the digits 1, 2, 3, 4, 5, 6, 7 exactly once, what is the position of the number 3654721 when these numbers are listed in ascending order? | [
"2010"
] | [
"Among all 7-digit numbers formed using the digits 1, 2, 3, 4, 5, 6, 7 exactly once, what is the position of the number 3654721 when these numbers are listed in ascending order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-e038212e3c82f7fd2aba94c7d6af4b9be9b482229c9f9352c7d6b8d08773b068 | Among all curves joining a given point (0,b) on the y-axis to a point on the x-axis and enclosing a given area S together with the x-axis, find the curve that generates the least area when rotated about the x-axis. The integrand is given by $y(1+y'^2)^\frac{1}{2}-\lambda y$, where the Lagrange multiplier is $2\pi\lambd... | [
"\\(\\frac{x}{a} + \\frac{y}{b} = 1\\) where \\(2S = ab\\)"
] | [
"Among all curves joining a given point (0,b) on the y-axis to a point on the x-axis and enclosing a given area S together with the x-axis, find the curve that generates the least area when rotated about the x-axis. The integrand is given by $y(1+y'^2)^\\frac{1}{2}-\\lambda y$, where the Lagrange multiplier is $2\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 3
} | 16 | 0 | 0 |
omr-ws-73ef7bab413eb987312735630cddc6b1bbd6e2a0c968ede76379e080c69d807e | Among all cyclic quadrilaterals inscribed in a circle of radius $R$ with one of its angles equal to $120^\circ$, consider the one with the maximum possible area. | [
"\\(2\\sqrt{3}R^2\\)"
] | [
"Among all cyclic quadrilaterals inscribed in a circle of radius $R$ with one of its angles equal to $120^\\circ$, consider the one with the maximum possible area."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c7c59b54cd9f7d8a66d615a24cdf87bfb677ad5d274fceca6b2e72f4bcefb8a | Among all cyclic quadrilaterals inscribed in a circle with radius $R$, find the maximum possible area $A_{\text{max}}$ and express it in terms of $R$. | [
"2R^2"
] | [
"Among all cyclic quadrilaterals inscribed in a circle with radius $R$, find the maximum possible area $A_{\\text{max}}$ and express it in terms of $R$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b753f2bcc8b483258883a53439271affd92f7f671932257508ea41c8b8c5845 | Among all eight-digit multiples of four, are there more numbers that contain the digit $1$ or more numbers that do not contain the digit $1$ in their decimal representation? | [
"more numbers with the digit $1$"
] | [
"Among all eight-digit multiples of four, are there more numbers that contain the digit $1$ or more numbers that do not contain the digit $1$ in their decimal representation?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-95ad675d214975ed32a85d971c08d025a0bb36a5662d222675d725d8a144a50b | Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation? | [
"More with the digit 1"
] | [
"Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1db3c09b7dfb41cf254945f2d377c469cecd36da48113789b739d7ce28e53704 | Among all equilateral triangles $XYZ$ such that $X$ is on side $AB$, $Y$ is on the extension of $BC$ (closer to $B$), and $Z$ is on the extension of $AC$ (closer to $A$) of an acute triangle $ABC$ where $AB$ is the smallest side, which equilateral triangle has the smallest perimeter? | [
"Pedal triangle of \\( S' \\)"
] | [
"Among all equilateral triangles $XYZ$ such that $X$ is on side $AB$, $Y$ is on the extension of $BC$ (closer to $B$), and $Z$ is on the extension of $AC$ (closer to $A$) of an acute triangle $ABC$ where $AB$ is the smallest side, which equilateral triangle has the smallest perimeter?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04a2d1b4d60c931378c10d28ef7640228c511ed626cd4404e41fb5e6d8d443cc | Among all equilateral triangles with vertices on the sides \(AB\), \(AC\), and \(BC\) of triangle \(ABC\), let \(P_{\text{min}}\) be the perimeter of the equilateral triangle with the least perimeter, where the vertices \(F_1\), \(F_2\), and \(F_3\) are defined as the projections of the isogonal conjugate of the Fermat... | [
"\\dfrac{3}{2}"
] | [
"Among all equilateral triangles with vertices on the sides \\(AB\\), \\(AC\\), and \\(BC\\) of triangle \\(ABC\\), let \\(P_{\\text{min}}\\) be the perimeter of the equilateral triangle with the least perimeter, where the vertices \\(F_1\\), \\(F_2\\), and \\(F_3\\) are defined as the projections of the isogonal c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1413e209e071705ccd26ad3141df5bcbed32c02c12a5accb6b17e34e6007bd81 | Among all figures with a constant circumference of $20\pi$, calculate the area of the figure that maximizes the enclosed area. | [
"100\\pi"
] | [
"Among all figures with a constant circumference of $20\\pi$, calculate the area of the figure that maximizes the enclosed area."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d706a88a9a2b2b56dba77ad5d5a0c52b9692a7ec8045ca0d4e2420d31a63a24 | Among all four-digit numbers with 3 as their thousands digit, how many have exactly two identical digits? | [
"270"
] | [
"Among all four-digit numbers with 3 as their thousands digit, how many have exactly two identical digits?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec6367783d3b4307f41f982a6be34906d816b13a39037d818d61400c98862aa5 | Among all hyperplane sections through the center of an ellipsoid in $\mathbb{R}^n$, which section has the largest surface area? | [
"\\text{The hyperplane orthogonal to the shortest principal axis}"
] | [
"Among all hyperplane sections through the center of an ellipsoid in $\\mathbb{R}^n$, which section has the largest surface area?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bc36342c5622baef439ebaccde74099d3695137492eb5be32864a4bdcfbb5f5 | Among all inscribed n-gons in a fixed circle of radius $R$, let $A_{\text{max}}$ be the maximum possible area. For a regular n-gon inscribed in the same circle, calculate the ratio $\frac{A_{\text{max}}}{nR^2}$. Express your answer in terms of $n$. | [
"\\dfrac{\\sin\\left(\\dfrac{2\\pi}{n}\\right)}{2}"
] | [
"Among all inscribed n-gons in a fixed circle of radius $R$, let $A_{\\text{max}}$ be the maximum possible area. For a regular n-gon inscribed in the same circle, calculate the ratio $\\frac{A_{\\text{max}}}{nR^2}$. Express your answer in terms of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-258948a1adbebfc35c58436097e946d9b9743efe785d309e6f3a22fff45a39b2 | Among all monic polynomials of degree $n$, let $\widehat{p}(x)$ be the unique polynomial that minimizes $d_n = \int_{-1}^{1} |p(x)|^2 dx$. Find the value of $d_n$ for $n=3$. | [
"\\dfrac{8}{175}"
] | [
"Among all monic polynomials of degree $n$, let $\\widehat{p}(x)$ be the unique polynomial that minimizes $d_n = \\int_{-1}^{1} |p(x)|^2 dx$. Find the value of $d_n$ for $n=3$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4111cd7ad5eb175d7fdaf28f998b71228629245ba8d7af1306e4bd0d3bf6330b | Among all n-gons with a perimeter of 100 units, find the maximum possible area of a regular n-gon as a function of n. Express your answer in simplest radical form. | [
"\\dfrac{2500}{n \\tan\\left(\\dfrac{\\pi}{n}\\right)}"
] | [
"Among all n-gons with a perimeter of 100 units, find the maximum possible area of a regular n-gon as a function of n. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d09a9abe0ae0539efb132f06bd2a729e357c5f69f218508de1289f7ba30f1edc | Among all natural number solutions \((x, y)\) of the equation \(\frac{x + y}{2} + \sqrt{xy} = 54\), find the solution with the largest \(x\). | [
"(75, 3)"
] | [
"Among all natural number solutions \\((x, y)\\) of the equation \\(\\frac{x + y}{2} + \\sqrt{xy} = 54\\), find the solution with the largest \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-59d108565dc3a1b34a7f2bab4d611979c26ccb0275dab97a6ccd91dea8fb3200 | Among all nonnegative integer polynomials \( P(x) \) of degree \( n \) such that \( P(x) \geq 0 \) for \( x \in [0,1] \), find the polynomial(s) that minimize \( \int_0^1 P(x) \, dx \). | [
"x^n"
] | [
"Among all nonnegative integer polynomials \\( P(x) \\) of degree \\( n \\) such that \\( P(x) \\geq 0 \\) for \\( x \\in [0,1] \\), find the polynomial(s) that minimize \\( \\int_0^1 P(x) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-938f0e02d022e9683063b3f0b69d667b88b994d6f3279bd8daa244243e9652aa | Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$. | [
"\\(-6\\)"
] | [
"Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-054c56b0e861fc778760b665129d1d0dfd06c3665c2367aa42ec98812665c7c5 | Among all ordered triples of integers $(x, y, z)$ that satisfy $x + y + z = 8$ and $x^3 + y^3 + z^3 = 134$, what is the maximum possible value of $|x| + |y| + |z|$? | [
"10"
] | [
"Among all ordered triples of integers $(x, y, z)$ that satisfy $x + y + z = 8$ and $x^3 + y^3 + z^3 = 134$, what is the maximum possible value of $|x| + |y| + |z|$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58891bf7cce60f6ab93aa8501421b21629d774972fd4bab6195a4809dcf745d2 | Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | [
"\\frac{1}{20}"
] | [
"Among all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, y \\le 10 \\pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$."
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-fa545988924d0cfb43af120582370cc7df7e75f61dda76aa15c7435eda797c0b | Among all plane sections through the center of an ellipsoid in $\mathbb{R}^3$, which section has the largest perimeter? | [
"\\text{The section containing the two longest principal axes of the ellipsoid}"
] | [
"Among all plane sections through the center of an ellipsoid in $\\mathbb{R}^3$, which section has the largest perimeter?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-fda2f03b1a442bfce3426ffd8764a359ac1f564d5ed6be75a708c3b43666929a | Among all planes passing through a fixed point \((a, b, c)\) with \(a, b, c > 0\) and meeting the positive parts of the three coordinate axes, find the one such that the tetrahedron bounded by it and the coordinate planes has minimal volume. | [
"\\dfrac{x}{3a} + \\dfrac{y}{3b} + \\dfrac{z}{3c} = 1"
] | [
"Among all planes passing through a fixed point \\((a, b, c)\\) with \\(a, b, c > 0\\) and meeting the positive parts of the three coordinate axes, find the one such that the tetrahedron bounded by it and the coordinate planes has minimal volume."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-92b4730a267211c051ad6f966c6c8f2235229e3b42694ee6e525e9ce45bf5b71 | Among all point lattices on the plane that intersect every closed convex region of unit width, which lattice has the largest area for its fundamental parallelogram? | [
"\\dfrac{3\\sqrt{3}}{8}"
] | [
"Among all point lattices on the plane that intersect every closed convex region of unit width, which lattice has the largest area for its fundamental parallelogram?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-8bdc163f2212da34928a93d0fa989c359aeef9b80df68b8f3129a8e70f7d8832 | Among all polynomials \( P(x) \) with integer coefficients for which \( P(-10) = 145 \) and \( P(9) = 164 \), compute the smallest possible value of \( |P(0)| \). | [
"25"
] | [
"Among all polynomials \\( P(x) \\) with integer coefficients for which \\( P(-10) = 145 \\) and \\( P(9) = 164 \\), compute the smallest possible value of \\( |P(0)| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-bebc5fa15ad8efca193fc2fec4b9b368251caef737befd46636439ed4d5f7ecf | Among all possible representations of the positive integer \( n \) as \( n = \sum_{i=1}^k a_i \) with positive integers \( k \) and \( a_1 < a_2 < \cdots < a_k \), for which representation will the product \( \prod_{i=1}^k a_i \) be maximum? | [
"2 + 3 + 4 + \\cdots + m \\text{ (adjusted if necessary)}"
] | [
"Among all possible representations of the positive integer \\( n \\) as \\( n = \\sum_{i=1}^k a_i \\) with positive integers \\( k \\) and \\( a_1 < a_2 < \\cdots < a_k \\), for which representation will the product \\( \\prod_{i=1}^k a_i \\) be maximum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-3e90214ecb26d623b4f22292e018803b819ca70284b6f62be0fa00a20d9167e7 | Among all prisms with a volume of $1000\pi$ cubic units, determine the minimum possible surface area of a right circular cylinder. Express your answer as $a\pi + b\pi\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find $a+b+c$. | [
"302"
] | [
"Among all prisms with a volume of $1000\\pi$ cubic units, determine the minimum possible surface area of a right circular cylinder. Express your answer as $a\\pi + b\\pi\\sqrt{c}$, where $a$, $b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find $a+b+c$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7435cbadd572bbf482b604d9a40c48dc9dd4794a7b40876db56ed2ba6127e37f | Among all quadrilaterals circumscibed about a given circle with radius $r$, find the minimum possible area of such a quadrilateral and express it in terms of $r$. | [
"4r^2"
] | [
"Among all quadrilaterals circumscibed about a given circle with radius $r$, find the minimum possible area of such a quadrilateral and express it in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-465c0e28de81142b22c219192e0840ae5fa32cb1e4f358268b44a912cfdd8567 | Among all quadrilaterals circumscribed about a given circle of radius $r$, let $P_{\text{min}}$ denote the perimeter of the quadrilateral with the smallest perimeter. If $r = 3$, find $P_{\text{min}}$. | [
"24"
] | [
"Among all quadrilaterals circumscribed about a given circle of radius $r$, let $P_{\\text{min}}$ denote the perimeter of the quadrilateral with the smallest perimeter. If $r = 3$, find $P_{\\text{min}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c38c9d66d8ed859d1643d23eda72f010756ec9b1b4ac88d31097cbcd3c9e4b3b | Among all quadrilaterals inscribed in a unit square, determine all of those, if any, whose area is exactly \( \frac{1}{2} \). (Note: There is a vertex of the quadrilateral on each of the 4 sides of the square; these cannot be vertices of the square). | [
"\\text{All such quadrilaterals with either } a = c \\text{ or } b = d"
] | [
"Among all quadrilaterals inscribed in a unit square, determine all of those, if any, whose area is exactly \\( \\frac{1}{2} \\). (Note: There is a vertex of the quadrilateral on each of the 4 sides of the square; these cannot be vertices of the square)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dff308cc6d8268b98e731312d9a9c031891b2a0650dad2afa0ae0bb53a4ffe3d | Among all quadrilaterals whose interiors lie inside a semi-circle of radius \( r \), determine if there exists one (or more) with the maximum area. If so, identify the shape and calculate the area. | [
"\\(\\frac{3\\sqrt{3}}{4}r^2\\)"
] | [
"Among all quadrilaterals whose interiors lie inside a semi-circle of radius \\( r \\), determine if there exists one (or more) with the maximum area. If so, identify the shape and calculate the area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 22
} | 75 | 0 | 0 |
omr-ws-5fb4469225925f0cbe3f009b8608ba7f7a45954ac07041209118964eb920f5c5 | Among all quadrilaterals with a perimeter of 20 units, find the maximum possible area. Express your answer in simplest radical form if necessary. | [
"25"
] | [
"Among all quadrilaterals with a perimeter of 20 units, find the maximum possible area. Express your answer in simplest radical form if necessary."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6c90497cffcc72b109e7c59ead87e52378fc00eeba2502a5dbe56851d3e19fd4 | Among all quadrilaterals with a perimeter of 20, find the maximum possible area and express it as a fraction in simplest form. What is the sum of the numerator and the denominator of this fraction? | [
"26"
] | [
"Among all quadrilaterals with a perimeter of 20, find the maximum possible area and express it as a fraction in simplest form. What is the sum of the numerator and the denominator of this fraction?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fe20fb6aa31a4075aa05c45db90fa0dc8ef56f52750bee5ad795dcf9a31b0f7 | Among all quadrilaterals with a perimeter of 20, find the maximum possible area and express it in the form $A = k^2$. What is the value of $k$? | [
"5"
] | [
"Among all quadrilaterals with a perimeter of 20, find the maximum possible area and express it in the form $A = k^2$. What is the value of $k$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afa9e545dbb6b3e28ff2ef1fa5ffc5306c215363b36483986219ea94e1541a54 | Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a decimal rounded to the nearest tenth. | [
"100.0"
] | [
"Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a decimal rounded to the nearest tenth."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9a3d92d71cacc63b060b1f7769bd1c5387b0ac0028a66d4026dbf88269a4048 | Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a fraction in simplest form. What is the sum of the numerator and the denominator of this fraction? | [
"101"
] | [
"Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a fraction in simplest form. What is the sum of the numerator and the denominator of this fraction?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a065c8d9dbd0360f23700f112f97e86a651001badd157c870fda6e7129776ce | Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a simplified fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? | [
"101"
] | [
"Among all quadrilaterals with a perimeter of 40, find the maximum possible area and express it as a simplified fraction $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efd9086ce4d4abfba99a7b1ae7efd93773b7f904bfc345a2917f1980c7319e18 | Among all quadrilaterals with a perimeter of 40, find the maximum possible area. | [
"100"
] | [
"Among all quadrilaterals with a perimeter of 40, find the maximum possible area."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ec420885b469f30768b1d6237d26e2fc164c4d5b245097bf3b98acd0ceeb118 | Among all quadrilaterals with an area of 1, find the minimum possible value for the sum of the lengths of the sides and diagonals. | [
"\\( 4 + 2\\sqrt{2} \\)"
] | [
"Among all quadrilaterals with an area of 1, find the minimum possible value for the sum of the lengths of the sides and diagonals."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f4792c017d0b0ed29f89b472c0f77be8c6c7963ff36470d0718931b2bae69c9a | Among all quadrilaterals with diagonals of lengths 10 and 14, and the angle between them being 60 degrees, find the minimum possible perimeter and express it in the form $\sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Calculate $a + b$. | [
"592"
] | [
"Among all quadrilaterals with diagonals of lengths 10 and 14, and the angle between them being 60 degrees, find the minimum possible perimeter and express it in the form $\\sqrt{a} + \\sqrt{b}$, where $a$ and $b$ are positive integers. Calculate $a + b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-689669c9801abc1ce055c147d42f54ca4e86846568bfe08c9451feb5186367e0 | Among all quadrilaterals with given side lengths \(AB = a\), \(BC = b\), \(CD = c\), and \(DA = d\), find the one with the greatest area. | [
"\\(\\sqrt{(s-a)(s-b)(s-c)(s-d)}\\) where \\(s = \\frac{a + b + c + d}{2}\\)"
] | [
"Among all quadrilaterals with given side lengths \\(AB = a\\), \\(BC = b\\), \\(CD = c\\), and \\(DA = d\\), find the one with the greatest area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d11c91ce5d9c257ef39f7b246f3cd76d29a5f248f7f9d52f97ed634859589fd1 | Among all quadrilaterals with side lengths $a$, $b$, $c$, and $d$, let $Q(a, b, c, d)$ denote the maximum possible area. Given $a=5$, $b=12$, $c=13$, and $d=10$, find the value of $Q(5, 12, 13, 10)$. | [
"20\\sqrt{21}"
] | [
"Among all quadrilaterals with side lengths $a$, $b$, $c$, and $d$, let $Q(a, b, c, d)$ denote the maximum possible area. Given $a=5$, $b=12$, $c=13$, and $d=10$, find the value of $Q(5, 12, 13, 10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bce407d1b4db8396966d167e4e04666e294c223d968698b148e2d49e0595112e | Among all quadrilaterals with three equal sides of length 5, let $A$ be the area of the quadrilateral with the largest possible area. Find the value of $A$. | [
"\\dfrac{75\\sqrt{3}}{4}"
] | [
"Among all quadrilaterals with three equal sides of length 5, let $A$ be the area of the quadrilateral with the largest possible area. Find the value of $A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-80a862bb6f3e5fb0453af3d5918a706491a95ee628d898a9a5ebd1683dc60898 | Among all rectangles with an area of 2008 square units, what is the smallest possible value of the sum of the lengths of the diagonals? | [
"\\( 4\\sqrt{1004} \\)"
] | [
"Among all rectangles with an area of 2008 square units, what is the smallest possible value of the sum of the lengths of the diagonals?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-23088e208390e519b4f18f95fc2d1d2b657f2b7af4d045ed70ccd25446e31165 | Among all rectangles with the same diagonal length $d$, find the rectangle with the maximum area. | [
"\\(\\frac{d^2}{2}\\)"
] | [
"Among all rectangles with the same diagonal length $d$, find the rectangle with the maximum area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-2a1a6b106117df82f29c4d80202176fb8c4c3dbcf61ca27e1eca2f431b53049a | Among all regular polygons with \(n \geq 3\) sides and a fixed perimeter \(P\), let \(A_n\) denote the area of the inscribed polygon. Find the value of \(\frac{A_6}{A_3}\). | [
"\\dfrac{3}{2}"
] | [
"Among all regular polygons with \\(n \\geq 3\\) sides and a fixed perimeter \\(P\\), let \\(A_n\\) denote the area of the inscribed polygon. Find the value of \\(\\frac{A_6}{A_3}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbcbad593bdb93a8fbb102f25510567fa293639a38d346d736e719d51ce4da01 | Among all tetrahedra with a given volume \( V = 1 \), find the minimum possible value of the surface area \( S \). Express your answer in terms of \( V \). | [
"2 \\cdot 3^{7/6} V^{2/3}"
] | [
"Among all tetrahedra with a given volume \\( V = 1 \\), find the minimum possible value of the surface area \\( S \\). Express your answer in terms of \\( V \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4b79e60b9f094d83ae5dba82c7b4d75f21306c2945e1ab29a80bddc0227a9e7 | Among all tetrahedra with a volume of 1, let $A$ be the surface area of the regular tetrahedron and $B$ be the surface area of any other tetrahedron. Determine the value of $A^3 - B^3$. | [
"0"
] | [
"Among all tetrahedra with a volume of 1, let $A$ be the surface area of the regular tetrahedron and $B$ be the surface area of any other tetrahedron. Determine the value of $A^3 - B^3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-cf67e573985e6a1c36dd58d2e798df309e2ae72ed7f103c1d72da830555c222c | Among all tetrahedrons that have a unit ball as an inner tangent ball, which one has the minimum volume? | [
"8\\sqrt{3}"
] | [
"Among all tetrahedrons that have a unit ball as an inner tangent ball, which one has the minimum volume?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f1c3b3e718fa780fa1e21385c5ac8b77290c4fa21bbd02b32bbdc2008229b29 | Among all the 6-letter words that can be formed using the letters of the word $\text{GOOGLE}$, what is the probability that the letter 'O' remains between the two 'G's in a randomly selected word? | [
"\\(\\frac{1}{6}\\)"
] | [
"Among all the 6-letter words that can be formed using the letters of the word $\\text{GOOGLE}$, what is the probability that the letter 'O' remains between the two 'G's in a randomly selected word?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-8d698b9607c617ba16a7ecf6afdb4fc80ce9758a76cfe7e6b03bae2efacd492c | Among all the differences of the form $36^m-5^n$ where $m$ and $n$ are natural numbers find the one having the smallest absolute value. | [
"$\\pm 22$"
] | [
"Among all the differences of the form $36^m-5^n$ where $m$ and $n$ are natural numbers find the one having the smallest absolute value."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-844a3d8dd49caa0ec5eb4ee7630bfe305bbcb59af212a615c56c1ab0bdd40bca | Among all the natural numbers with four digits that can be formed using only the digits from the set \( C = \{1, 2, 3, 4, 5, 6\} \), find how many are multiples of 4. | [
"324"
] | [
"Among all the natural numbers with four digits that can be formed using only the digits from the set \\( C = \\{1, 2, 3, 4, 5, 6\\} \\), find how many are multiples of 4."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-192eb4159e7d967d30e33769f0e7ea3713776ea79a1f1e13fac08321dbf669db | Among all the natural numbers with three different digits that can be formed using only the digits from the set \( C = \{1, 2, 3, 4, 5, 6\} \), find how many are multiples of 3. | [
"48"
] | [
"Among all the natural numbers with three different digits that can be formed using only the digits from the set \\( C = \\{1, 2, 3, 4, 5, 6\\} \\), find how many are multiples of 3."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-1913acf6d6ffb9a270feb86ca3417d20b028d5a905ab2e3cd4cba4aa1ddd0fbd | Among all the planes that are tangent to the surface $x(yz)^2 = 1$, find the one that is farthest from the origin. | [
"2^{\\frac{2}{5}} x + 2^{\\frac{9}{10}} y + 2^{\\frac{9}{10}} z = 5"
] | [
"Among all the planes that are tangent to the surface $x(yz)^2 = 1$, find the one that is farthest from the origin."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-f6c3360ab78726231e697e3c58bf8654afb24a5a785b9daa2644743e1732ade7 | Among all the right triangles of area $S$ find that for which the area of the circumscribed circle is the least. | [
"The area of the circle circumscribed about the isosceles right triangle with legs $\\sqrt {2S}$ is the least"
] | [
"Among all the right triangles of area $S$ find that for which the area of the circumscribed circle is the least."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17c8558595b9d814dc7f28874a8fe7d0ae5ebb509855c8914f30642ba8a7029e | Among all the roots of \[z^8 - z^6 + z^4 - z^2 + 1 = 0,\]the maximum imaginary part of a root can be expressed as $\sin \theta,$ where $-90^\circ \le \theta \le 90^\circ.$ Find $\theta.$ | [
"54^\\circ"
] | [
"Among all the roots of\n\\[z^8 - z^6 + z^4 - z^2 + 1 = 0,\\]the maximum imaginary part of a root can be expressed as $\\sin \\theta,$ where $-90^\\circ \\le \\theta \\le 90^\\circ.$ Find $\\theta.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ab3dd4777acc61cfcc01c550ee263bb1ff51bad4b7e3d6420db908952a1b4e5 | Among all the solutions, real and complex, of the system of equations \[ x^2 + y^2 + x + y = 8 \] \[ xy + x + y = 5 \] what is the least possible real part of \( x \)? | [
"-3"
] | [
"Among all the solutions, real and complex, of the system of equations\n\n\\[ x^2 + y^2 + x + y = 8 \\]\n\\[ xy + x + y = 5 \\]\n\nwhat is the least possible real part of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-6a4f54fd83ea5c964224054e475b49e4bd3454c19075395004254b7c0f9a7b8b | Among all the words of length $n$ in $\{H,T\}^n$, which one takes the least amount of time to appear on average? | [
"H^{n-1}T"
] | [
"Among all the words of length $n$ in $\\{H,T\\}^n$, which one takes the least amount of time to appear on average?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-4afe61eabced45fe21f205466afc860351c5ec39b48ac1ca6812bc3fa04a3ac3 | Among all third-degree polynomials $P(x)$ with nonnegative integer coefficients such that $P(1) = 20$, determine the polynomial for which the value of $P(3)/P(2)$ is maximized. | [
"\\( \\frac{27}{8} \\)"
] | [
"Among all third-degree polynomials $P(x)$ with nonnegative integer coefficients such that $P(1) = 20$, determine the polynomial for which the value of $P(3)/P(2)$ is maximized."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c7746b9b3cc7208172fe09772e780ad434067d1a95d5f96ab60376c436563ff | Among all third-degree polynomials $P(x)$ with nonnegative integer coefficients such that $P(1) = 20$, determine the polynomial for which the value of $P(4)$ is minimized. | [
"1280"
] | [
"Among all third-degree polynomials $P(x)$ with nonnegative integer coefficients such that $P(1) = 20$, determine the polynomial for which the value of $P(4)$ is minimized."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cab9086747b1ffa9641fabc8bbf481a990295d1352dbbca65284a4810b9d6cc | Among all triangles $ABC,$ find the maximum value of $\sin A + \sin B \sin C.$ | [
"\\frac{1 + \\sqrt{5}}{2}"
] | [
"Among all triangles $ABC,$ find the maximum value of $\\sin A + \\sin B \\sin C.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52c1e7853e52ce5c0d9f22d1ac46d28f79c0c57887f30f4b4913d40e0e355f06 | Among all triangles ABC that share side AB and have equal areas, determine the position of point C that results in the triangle with the smallest perimeter. | [
"C \\text{ lies on the perpendicular bisector of } AB",
"Point C should be such that AC = BC."
] | [
"Among all triangles ABC that share side AB and have equal areas, determine the position of point C that results in the triangle with the smallest perimeter."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 2,
"tir": 10
} | 40 | 0 | 0 |
omr-ws-58736d17f04fac2f7f5dc709c7dcbfb6ab0c76ef98a29c448eed14ea987cae05 | Among all triangles \(ABC\) such that \(A \in C(O_a, r_a)\), \(B \in C(O_b, r_b)\), and \(C \in C(O_c, r_c)\), calculate the maximum possible area of triangle \(ABC\) when the circles \(C(O_a, r_a)\), \(C(O_b, r_b)\), and \(C(O_c, r_c)\) have radii \(r_a = 3\), \(r_b = 4\), and \(r_c = 5\) respectively, and their cente... | [
"84"
] | [
"Among all triangles \\(ABC\\) such that \\(A \\in C(O_a, r_a)\\), \\(B \\in C(O_b, r_b)\\), and \\(C \\in C(O_c, r_c)\\), calculate the maximum possible area of triangle \\(ABC\\) when the circles \\(C(O_a, r_a)\\), \\(C(O_b, r_b)\\), and \\(C(O_c, r_c)\\) have radii \\(r_a = 3\\), \\(r_b = 4\\), and \\(r_c = 5\\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
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