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-d7aec107dad2cfad2f4a18c47faf3b0f8d10356e9ee7e645a4b800f1b9b02b2e | A group of 30 students all took an examination. They were marked out of five, no one getting a 1. The sum of the marks they received was 93, the number of 3's being greater than the number of 5's and less than the number of 4's. In addition, the number of 4's was divisible by 10, and the number of 5's was even. How man... | [
"11 \"twos\", 7 \"threes\", 10 \"fours\" and 2 \"fives\""
] | [
"A group of 30 students all took an examination. They were marked out of five, no one getting a 1. The sum of the marks they received was 93, the number of 3's being greater than the number of 5's and less than the number of 4's. In addition, the number of 4's was divisible by 10, and the number of 5's was even. Ho... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a144569d9b483254afc3103744a58900a8d9d96bb1bdf3b5e78241a0d063890c | A group of 300 students has provisions for 45 days. After 15 days, 160 additional students are added. How many more days will the provisions last? | [
"19"
] | [
"A group of 300 students has provisions for 45 days. After 15 days, 160 additional students are added. How many more days will the provisions last?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 15
} | 21 | 0 | 0 |
omr-ws-ed4fd8ff0b3d56bab45864cd7d7a6fd36b8f1ccadafd61b6117ad7a39fa6d0ec | A group of 4 people orders from a restaurant menu that lists 3 main courses: fish, chicken, and steak. How many combinations of main-course orders are possible in this group? | [
"15"
] | [
"A group of 4 people orders from a restaurant menu that lists 3 main courses: fish, chicken, and steak. How many combinations of main-course orders are possible in this group?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-124845f2e3cfa399417e82967a192cce87c61f046d105da722aa181c3358defb | A group of 5 is to be chosen from 6 men and 4 women. In how many ways can the group be chosen if it contains a particular man X but excludes a particular woman Y? | [
"70"
] | [
"A group of 5 is to be chosen from 6 men and 4 women. In how many ways can the group be chosen if it contains a particular man X but excludes a particular woman Y?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-cc6ba8b664141bb3d40d8c3c8df864ac3206856ea5cadd534b1ec413bec14157 | A group of 50 people of different ages sit around a big round table. A pair of people \(A\) and \(B\) is called a newbie pair if \(A\) and \(B\) are not sitting next to each other, and on at least one of the arcs connecting \(A\) and \(B\) around the table edge, all people are older than \(A\) and \(B\). Determine the ... | [
"47"
] | [
"A group of 50 people of different ages sit around a big round table. A pair of people \\(A\\) and \\(B\\) is called a newbie pair if \\(A\\) and \\(B\\) are not sitting next to each other, and on at least one of the arcs connecting \\(A\\) and \\(B\\) around the table edge, all people are older than \\(A\\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-3617131fc2fef0488e428f7fd0d00e92b5fcd30c57c922f329d3db36ea906805 | A group of 6 friends went to the movies. Colin and Megan insist on sitting next to each other and Abe wants to sit next to them as well. Lee, Dani, and Libbey also want to sit together. How many ways can they be seated? | [
"48"
] | [
"A group of 6 friends went to the movies. Colin and Megan insist on sitting next to each other and Abe wants to sit next to them as well. Lee, Dani, and Libbey also want to sit together. How many ways can they be seated?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-880f7846ae62798b9c504144a279183689af224465afdb3d983e2063847ad4c2 | A group of 6 students forms study groups and service activity groups such that each group has exactly 3 members. For any pair of students, there are the same number of study groups and service activity groups that both students are members of. Given that there is at least one group and no three students belong to the s... | [
"8"
] | [
"A group of 6 students forms study groups and service activity groups such that each group has exactly 3 members. For any pair of students, there are the same number of study groups and service activity groups that both students are members of. Given that there is at least one group and no three students belong to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-9a18f5687f6cd9487fac4d01c294a728277fb71c9be65c06cb67fdca06069b42 | A group of 6 students forms study groups and service activity groups, each consisting of exactly 3 members. For any pair of students, the number of study groups they both belong to is the same as the number of service activity groups they both belong to. Given that there is at least one group and no three students belo... | [
"8"
] | [
"A group of 6 students forms study groups and service activity groups, each consisting of exactly 3 members. For any pair of students, the number of study groups they both belong to is the same as the number of service activity groups they both belong to. Given that there is at least one group and no three students... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 8,
"tir": 1
} | 42 | 0 | 0 |
omr-ws-fd48b46a542897e65401914bf53afca1735f0ae8e0f06d7f501a3714036aaf0e | A group of 61 students is divided into 3 subgroups. Two students play a game of chess (only once) if and only if they are in the same subgroup. Find a partition such that the total number of games is a multiple of 61. | [
"(61,0,0), (47,13,1), (33,26,2), (39,19,3), (52,5,4), (38,17,6), (30,24,7), (43,10,8), (29,21,11), (34,15,12), (25,20,16)"
] | [
"A group of 61 students is divided into 3 subgroups. Two students play a game of chess (only once) if and only if they are in the same subgroup. Find a partition such that the total number of games is a multiple of 61."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 22
} | 34 | 0 | 0 |
omr-ws-57d1a360eda41c56cd3475bbe4fa0e2010f562d7a4250cf10ce9f00d52346696 | A group of 7 pirates of different ages owned a total of 2009 coins. Initially, each pirate (except the youngest one) had one coin more than the next younger. Every day, a pirate was chosen and gave a coin to each of the other pirates. Find the largest possible number of coins a pirate can have after several days. | [
"1994"
] | [
"A group of 7 pirates of different ages owned a total of 2009 coins. Initially, each pirate (except the youngest one) had one coin more than the next younger. Every day, a pirate was chosen and gave a coin to each of the other pirates. Find the largest possible number of coins a pirate can have after several days."... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-41d96bda0f99fd12a2b36b1c0c618cec8670ec3f07e8e356f6e23bb1be756545 | A group of 8 participants, including Fred Flinstone, each have a number from 0 to 7 on their forehead, which they cannot see but can see on others. They must devise a strategy beforehand to ensure that at least one participant can correctly guess their own number when asked individually, without any communication durin... | [
"The strategy ensures at least one correct guess."
] | [
"A group of 8 participants, including Fred Flinstone, each have a number from 0 to 7 on their forehead, which they cannot see but can see on others. They must devise a strategy beforehand to ensure that at least one participant can correctly guess their own number when asked individually, without any communication ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-633f18ea294725e8aa2653cb54e2342cdb7dae13330dcb89d38c8ae7b33a7472 | A group of 9 students, \( s_1, s_2, \dots, s_9 \), is to be divided into three teams \( X, Y, \) and \( Z \) of sizes 2, 3, and 4, respectively. Suppose that \( s_1 \) cannot be selected for team \( X \), and \( s_2 \) cannot be selected for team \( Y \). Determine the number of ways to form such teams. | [
"665"
] | [
"A group of 9 students, \\( s_1, s_2, \\dots, s_9 \\), is to be divided into three teams \\( X, Y, \\) and \\( Z \\) of sizes 2, 3, and 4, respectively. Suppose that \\( s_1 \\) cannot be selected for team \\( X \\), and \\( s_2 \\) cannot be selected for team \\( Y \\). Determine the number of ways to form such te... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 16
} | 38 | 0 | 0 |
omr-ws-04702fa24d971e3bb5f90329fd9a2ea5c8fc13082fa49a62a132c86a25f7d43c | A group of N friends take R separate photos such that each photo contains at least one person and for each pair of photos, there is at least one person who appears on both. Find the largest possible value of R. | [
"\\( 2^{n-1} \\)"
] | [
"A group of N friends take R separate photos such that each photo contains at least one person and for each pair of photos, there is at least one person who appears on both. Find the largest possible value of R."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-e543df9b82b256c9f6f20baf11b9ffe930c2afc34978a10ae9cab1bbc7b27e0c | A group of \( m \times n \) people are to be placed into \( m \) distinguishable teams, each having \( n \) players. How many ways can this be accomplished? | [
"\\(\\frac{(mn)!}{(n!)^m}\\)"
] | [
"A group of \\( m \\times n \\) people are to be placed into \\( m \\) distinguishable teams, each having \\( n \\) players. How many ways can this be accomplished?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-a69b3416999aefe0a6ab60941d04837abb8f19831a03f4d3ddc594d1d38bfa80 | A group of \( n > 1 \) pirates of different ages owned a total of 2009 coins. Initially, each pirate (except the youngest one) had one coin more than the next younger. Find all possible values of \( n \). | [
"\\( n = 7, 41, 49 \\)"
] | [
"A group of \\( n > 1 \\) pirates of different ages owned a total of 2009 coins. Initially, each pirate (except the youngest one) had one coin more than the next younger. Find all possible values of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff03b673f1c9569be3fe7ae3abca831fe1b28ac132a37753d6cc0d1c13ece378 | A group of \( n \) friends wrote a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \), and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 of the 12 problems. An 8 x 4 table is created where the entry in the \( i... | [
"10"
] | [
"A group of \\( n \\) friends wrote a math contest consisting of eight short-answer problems \\( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \\), and four full-solution problems \\( F_1, F_2, F_3, F_4 \\). Each person in the group correctly solved exactly 11 of the 12 problems. An 8 x 4 table is created where the entry ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ec473ed7aa89006642f1a59c478d297a441a0826c6a7ab179599c8efdf0f1224 | A group of \( n \) married couples is seated around a circular table. The distance between a husband and wife is defined as the number of people sitting between them, measured either clockwise or counter-clockwise, whichever gives the smaller result. What is the average number of couples, per arrangement, where both me... | [
"\\(\\frac{2n}{2n-1}\\)"
] | [
"A group of \\( n \\) married couples is seated around a circular table. The distance between a husband and wife is defined as the number of people sitting between them, measured either clockwise or counter-clockwise, whichever gives the smaller result. What is the average number of couples, per arrangement, where ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 7
} | 38 | 0 | 0 |
omr-ws-d54c949f16b6cfc20ffb038209d09cb3420caed8fc0df8759cbd1a5bed9f6812 | A group of \( n \) people (\( n > 2 \)) play Rock-Paper-Scissors. Each player independently and randomly chooses Rock, Scissors, or Paper with equal probabilities. Let \( X \), \( Y \), and \( Z \) be the number of players who pick Rock, Scissors, and Paper, respectively, in one game. (a) Find the joint probability mas... | [
"0"
] | [
"A group of \\( n \\) people (\\( n > 2 \\)) play Rock-Paper-Scissors. Each player independently and randomly chooses Rock, Scissors, or Paper with equal probabilities. Let \\( X \\), \\( Y \\), and \\( Z \\) be the number of players who pick Rock, Scissors, and Paper, respectively, in one game.\n\n(a) Find the joi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b52ec1bea6952729dc048b518e1a9b2e4714a97fe388c9bf56e84a70342440a4 | A group of \( n \) people are standing in a circle, numbered consecutively clockwise from 1 to \( n \). Starting with person number 2, every other person is removed proceeding clockwise. Let \( j(n) \) denote the last person remaining. Find a simple way to compute \( j(n) \) for any positive integer \( n > 1 \). | [
"\\( j(n) = 2(n - 2^m) + 1 \\) where \\( 2^m \\) is the largest power of 2 less than or equal to \\( n \\)."
] | [
"A group of \\( n \\) people are standing in a circle, numbered consecutively clockwise from 1 to \\( n \\). Starting with person number 2, every other person is removed proceeding clockwise. Let \\( j(n) \\) denote the last person remaining. Find a simple way to compute \\( j(n) \\) for any positive integer \\( n ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-8275d2e485d4e5b8bd2cdbcaf4cacfa946e9c8ecbd0885c62c23f887cb271a78 | A group of \( n \) people, including Caroline and her best friend Catherine, stand in random order around a circle, with each order being equally likely. If the probability that Catherine is adjacent to Caroline is \( \frac{1}{9} \), then what is the value of \( n \)? | [
"19"
] | [
"A group of \\( n \\) people, including Caroline and her best friend Catherine, stand in random order around a circle, with each order being equally likely. If the probability that Catherine is adjacent to Caroline is \\( \\frac{1}{9} \\), then what is the value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f97d925087aeedcebb3b09ff2324f47fc3b85b614e36ed54c999c2212af005e | A group of \(T + 60\) students compete at an ARML site. At that site, a total of 50 students got at least one of individual questions 9 and 10 correct, 48 students got question 9 correct, and an astonishing 22 students got question 10 correct! Compute the probability that a randomly selected student from this group got... | [
"\\dfrac{1}{3}"
] | [
"A group of \\(T + 60\\) students compete at an ARML site. At that site, a total of 50 students got at least one of individual questions 9 and 10 correct, 48 students got question 9 correct, and an astonishing 22 students got question 10 correct! Compute the probability that a randomly selected student from this gr... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-e10f81c3e8ffa4b80af99355069d8185ddc581a74a05fce05976b41025141112 | A group of \(n \geq 3\) people sit at a round table, where \(n\) is an odd number. Some people shake hands with each other in such a way that no two pairs of people shake hands with each other more than once. What is the maximum possible number of handshakes? | [
"\\dfrac{n(n-1)}{2}"
] | [
"A group of \\(n \\geq 3\\) people sit at a round table, where \\(n\\) is an odd number. Some people shake hands with each other in such a way that no two pairs of people shake hands with each other more than once. What is the maximum possible number of handshakes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-4168c1e47e6f6e2c7338c2beb8478f7e54667c77959f8171b47f7558e613250e | A group of children forms a circle and each child starts with an even number of pieces of candy. Each child then gives half of what he has to his right-hand neighbour. After the transaction, if a child has an odd number of pieces, he or she will receive an extra piece from an external source. Let $n$ be the number of c... | [
"8"
] | [
"A group of children forms a circle and each child starts with an even number of pieces of candy. Each child then gives half of what he has to his right-hand neighbour. After the transaction, if a child has an odd number of pieces, he or she will receive an extra piece from an external source. Let $n$ be the number... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15ea06448a2077913019bb8a9cdc45e7f0479a11e3bdbc7507db6497b0598480 | A group of children is sitting around a round table, each starting with an even number of candies. Each turn, each child gives half of their candies to the child on their right. If a child has an odd number of candies after a turn, the teacher gives them an extra candy. Let $N$ be the number of children and let $C_i$ b... | [
"42"
] | [
"A group of children is sitting around a round table, each starting with an even number of candies. Each turn, each child gives half of their candies to the child on their right. If a child has an odd number of candies after a turn, the teacher gives them an extra candy. Let $N$ be the number of children and let $C... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4748bba995b75793ad018b6e768e28688d31bc8e874953429d1d7cde5c5130ac | A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a s... | [
"945"
] | [
"A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, an... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-183c146695686e486c1255cf6ed7ebe668e4040a502d0deb9892a9289aec43e5 | A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a s... | [
"945"
] | [
"A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, an... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-79789eb733bfdbe63e0658510e66f14e5f65ef4f16b001af594235d4e9f09c4c | A group of clerks is assigned the task of sorting 1775 files. Each clerk sorts at a constant rate of 30 files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a simil... | [
"945"
] | [
"A group of clerks is assigned the task of sorting 1775 files. Each clerk sorts at a constant rate of 30 files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-fff2c63e25a8cc8ec5051de57241e0b3b8edb7a09d63f004ba53fd9a03f62411 | A group of fifteen students is to be divided into four groups of sizes $4$, $4$, $4$, and $3$. Find the number of ways to do this. | [
"\\(\\frac{\\binom{15}{4}\\binom{11}{4}\\binom{7}{4}}{3!}\\)"
] | [
"A group of fifteen students is to be divided into four groups of sizes $4$, $4$, $4$, and $3$. Find the number of ways to do this."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-854476a6fcc7010db67157a52a3acf71ff800b4298e2128d7b6fa2c39324a5a8 | A group of friends bought a crystal bowl. When 4 of them were not present, each of the others had to pay an extra $36 to cover the cost. Later, 3 of the missing friends arrived, and after dividing the cost evenly again, each of those who initially paid got $30 back. How many people are in the group, and how much did th... | [
"\\( x = 10 \\), \\( d = 540 \\)"
] | [
"A group of friends bought a crystal bowl. When 4 of them were not present, each of the others had to pay an extra $36 to cover the cost. Later, 3 of the missing friends arrived, and after dividing the cost evenly again, each of those who initially paid got $30 back. How many people are in the group, and how much d... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-5e791c01184cae3828c19df1ede285e558693f6dd1650a1e9cc93517aaf818dc | A group of friends pools their money to buy a car. If there had been one fewer friend, each friend would have spent 72 dollars more than they did. If there had been two more friends, each friend would have spent 72 dollars less than they did. How many dollars did the car cost? | [
"864"
] | [
"A group of friends pools their money to buy a car. If there had been one fewer friend, each friend would have spent 72 dollars more than they did. If there had been two more friends, each friend would have spent 72 dollars less than they did. How many dollars did the car cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-181a92b2b0c70a5b1aadca377940d5db92bcca51f966ab4954872191b9d0f4d5 | A group of hikers traveled a 76 km road in 10 days, never traveling more than 16 km in two consecutive days, but traveling at least 23 km in three consecutive days. Find the maximum number of kilometers the hikers may have traveled in one day. | [
"9"
] | [
"A group of hikers traveled a 76 km road in 10 days, never traveling more than 16 km in two consecutive days, but traveling at least 23 km in three consecutive days. Find the maximum number of kilometers the hikers may have traveled in one day."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-1bf03b2340af8eb8c4fc2974fc6f33e9a8dd9b4556109d40bc85e8c193c71d74 | A group of mathematicians is attending a conference. A mathematician is $k$-content if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in the same room, they are all at least $3k + 1$-content. Let $f(n, k)$ be t... | [
"1"
] | [
"A group of mathematicians is attending a conference. A mathematician is $k$-content if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in the same room, they are all at least $3k + 1$-content. Let $f(n, k)$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca65e464f6079d71ac92ca60961c7330f69d3560390d3c864caf1cd5601ee62c | A group of mechanics have inspected a lot of old vehicles. They determined that 45 vehicles have engine problems, 30 have electrical faults, and 35 have oil leaks. There are exactly 10 vehicles that have engine problems and electrical faults, 10 with engine problems and oil leakage, and 10 with electrical faults and oi... | [
"100"
] | [
"A group of mechanics have inspected a lot of old vehicles. They determined that 45 vehicles have engine problems, 30 have electrical faults, and 35 have oil leaks. There are exactly 10 vehicles that have engine problems and electrical faults, 10 with engine problems and oil leakage, and 10 with electrical faults a... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-132971cb208dc03d3772eb90c74764901cc1c7ced4f11dcb58306270afcfa220 | A group of men and women went to a pizza restaurant where each pizza is made up of 13 small slices. Each man can eat up to 8 or 9 slices while each woman can eat up to 3 or 4 slices. If 6 pizzas weren't enough and 7 pizzas were too much, compute the largest amount of people (sum of men and women) who came to the restau... | [
"30"
] | [
"A group of men and women went to a pizza restaurant where each pizza is made up of 13 small slices. Each man can eat up to 8 or 9 slices while each woman can eat up to 3 or 4 slices. If 6 pizzas weren't enough and 7 pizzas were too much, compute the largest amount of people (sum of men and women) who came to the r... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-a64a6ad87b18f5483b8c39e7998c95dcc0e0ee62c2409b0f72830bef7ee542dd | A group of observers each entered the shop at an arbitrary moment and remained for 7 hours, ensuring that at least one observer was present at any given time during the 24-hour period. Each observer noted that exactly 70 tons of steel were smelted during their 7-hour shift. What is the maximal amount of steel that coul... | [
"240 tons"
] | [
"A group of observers each entered the shop at an arbitrary moment and remained for 7 hours, ensuring that at least one observer was present at any given time during the 24-hour period. Each observer noted that exactly 70 tons of steel were smelted during their 7-hour shift. What is the maximal amount of steel that... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-d0ab4450e85e2443dd60f3be83e8c3678adc89df1db2ce12ea15b4b5c307ea16 | A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present? | [
"6"
] | [
"A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a25c91282451535abd925821be68ca0639cb89bb7c84c402debe5a98b74a8ae | A group of people is divided over two buses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$. Let $n$ be the number of seats on one bus and $T_n = \frac{n(n+1)}{2}$ be the $n$-th triangular number. Find the sum of all possible values ... | [
"3"
] | [
"A group of people is divided over two buses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\\frac{1}{2}$. Let $n$ be the number of seats on one bus and $T_n = \\frac{n(n+1)}{2}$ be the $n$-th triangular number. Find the sum of all possible ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-08de355bfb5d7ba33212b1985388aecbf0d74de406a0925c135249905b69989d | A group of people was marching in a parade. Initially, they formed a perfect square with the same number of rows and columns. Then, they changed the formation to a rectangle with five more rows than before. How many people were marching in the parade? | [
"400"
] | [
"A group of people was marching in a parade. Initially, they formed a perfect square with the same number of rows and columns. Then, they changed the formation to a rectangle with five more rows than before. How many people were marching in the parade?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-ab22c092165003483ca2030489289b734d88c714cdb777b574864d25e35b7e56 | A group of people was to share equally the cost of a bus trip. Due to unforeseen circumstances, five of them were unable to make it, and the remaining people had to pay an extra \$ each. How many people were supposed to share the cost originally? | [
"45"
] | [
"A group of people was to share equally the cost of a bus trip. Due to unforeseen circumstances, five of them were unable to make it, and the remaining people had to pay an extra \\$ each. How many people were supposed to share the cost originally?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d861eefa851a4c0ad80a2dcb36734b025a537e10ae6a6774a4a1896854b47b1c | A group of pupils is arranged in a circle around their teacher, each with a positive number of sweets. The game begins when the teacher nominates a pupil. If the pupil has an odd number of sweets, he takes one more sweet from the teacher and then gives half of his sweets to the pupil on his right. If he has an even num... | [
"(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)"
] | [
"A group of pupils is arranged in a circle around their teacher, each with a positive number of sweets. The game begins when the teacher nominates a pupil. If the pupil has an odd number of sweets, he takes one more sweet from the teacher and then gives half of his sweets to the pupil on his right. If he has an eve... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-491cb92210c39652257fe85c1c0c4659d24a3df06e43d76115c50269ed5697e9 | A group of students count from $00$ to $99$ by saying one digit at a time. When they reach $99$, the counting restarts from $00$. Csongor enters the room and listens to the counting. How many digits must Csongor hear at least to determine the current number in the counting sequence? | [
"2"
] | [
"A group of students count from $00$ to $99$ by saying one digit at a time. When they reach $99$, the counting restarts from $00$. Csongor enters the room and listens to the counting. How many digits must Csongor hear at least to determine the current number in the counting sequence?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d74f01fa6db6d113f749430fea6df88fe8b7ce7827254d82ee80a702f9ebbd92 | A group of students decided to buy a radio priced between Rs170 and Rs195. At the last moment, 2 students backed out, causing the remaining students to pay Re 1 more than they initially planned. Find the price of the radio and the number of students. | [
"20"
] | [
"A group of students decided to buy a radio priced between Rs170 and Rs195. At the last moment, 2 students backed out, causing the remaining students to pay Re 1 more than they initially planned. Find the price of the radio and the number of students."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53783e32877def324f44c11c62abe83f41915d5785c5cbbe08ddc176e639f7d0 | A group of students goes out to lunch. If two have burritos and five have tacos, the bill will be $19.50. If five have burritos and two have tacos, the bill will be $22.50. Find the price of each taco and each burrito. | [
"\\( t = 2.50 \\), \\( b = 3.50 \\)"
] | [
"A group of students goes out to lunch. If two have burritos and five have tacos, the bill will be $19.50. If five have burritos and two have tacos, the bill will be $22.50. Find the price of each taco and each burrito."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 32
} | 38 | 0 | 0 |
omr-ws-fa7a94fe9d5cbdddd1d5b1035f9badbb13bce132e27524795ca7e7153a1bd24c | A group of students with a total weight of 28000 kilograms needs to be transported to the second floor using an elevator that can hold a maximum of 1000 kilograms. The heaviest student weighs less than 133 kilograms. Determine the minimum number of trips the elevator must make to transport all the students. | [
"28"
] | [
"A group of students with a total weight of 28000 kilograms needs to be transported to the second floor using an elevator that can hold a maximum of 1000 kilograms. The heaviest student weighs less than 133 kilograms. Determine the minimum number of trips the elevator must make to transport all the students."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0cd0755a4677802934117c39b70f4e7c6214a01ccb75d05129fcc0531313a475 | A group of tourists includes both boys and girls. If at least one boy and one girl must be chosen as volunteers, there are exactly 49 ways to pick a group of 3 volunteers. Compute the total number of tourists. | [
"9"
] | [
"A group of tourists includes both boys and girls. If at least one boy and one girl must be chosen as volunteers, there are exactly 49 ways to pick a group of 3 volunteers. Compute the total number of tourists."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac336dd5c0634b3b18c37d4724d06258c6d8cce1dd97901f187d5ea6a4fd1ad9 | A group of tourists takes 10 buses on an outgoing trip and 8 buses on a return trip, with each bus carrying at least 1 tourist. Let $x$ be the minimum number of tourists such that each of them has taken a bus on the return trip that has more people than the bus they took on the outgoing trip. Find the value of $x$. | [
"3"
] | [
"A group of tourists takes 10 buses on an outgoing trip and 8 buses on a return trip, with each bus carrying at least 1 tourist. Let $x$ be the minimum number of tourists such that each of them has taken a bus on the return trip that has more people than the bus they took on the outgoing trip. Find the value of $x$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8356e35d6f131ef95dece5b98ef4a2039a835f27024b622236066a85790f3cfa | A group of tourists takes 10 buses on an outgoing trip and 8 buses on a return trip, with each bus carrying at least 1 tourist. Let $x_i$ be the number of tourists on the $i$-th outgoing bus and $y_i$ be the number of tourists on the $i$-th return bus. Determine the minimum number of tourists $k$ such that for at least... | [
"4"
] | [
"A group of tourists takes 10 buses on an outgoing trip and 8 buses on a return trip, with each bus carrying at least 1 tourist. Let $x_i$ be the number of tourists on the $i$-th outgoing bus and $y_i$ be the number of tourists on the $i$-th return bus. Determine the minimum number of tourists $k$ such that for at ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-2f6d661c3bfb185b4443cd148a288104594dd3e558687b07cd66bbd13b0116ed | A group of women working together at the same rate can build a wall in 45 hours. They joined the work over a period of time one by one at equal intervals, and each stayed until the work was completed. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work? | [
"75"
] | [
"A group of women working together at the same rate can build a wall in 45 hours. They joined the work over a period of time one by one at equal intervals, and each stayed until the work was completed. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woman work?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e19d834bc29bc7f50845e66f8e0f7370a453aed2534e5c277494348344c477c2 | A group of women working together at the same rate can build a wall in 45 hours. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed until the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first woma... | [
"75"
] | [
"A group of women working together at the same rate can build a wall in 45 hours. They joined the work over a period of time, one by one, at equal intervals. Once at work, each one stayed until the work was complete. If the first woman worked 5 times as many hours as the last woman, for how many hours did the first... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-fb83f8931d0456803fecfbb9d492b7d36c70eb2ba8edfa396c84e425f7643823 | A guidance counselor is planning schedules for 30 students. Sixteen students want to take French, 16 want to take Spanish, and 11 want to take Latin. Five students want to take both French and Latin, and of these, 3 also want to take Spanish. Five students want only Latin, and 8 want only Spanish. How many students wan... | [
"7"
] | [
"A guidance counselor is planning schedules for 30 students. Sixteen students want to take French, 16 want to take Spanish, and 11 want to take Latin. Five students want to take both French and Latin, and of these, 3 also want to take Spanish. Five students want only Latin, and 8 want only Spanish. How many student... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-d326b2e79177e6a17fbab467cc35e5b2f51d61ab8adf2c2de70d378f5e1dc05e | A gym class of size $N$ is split into three "even" groups. The probability that two specific friends, $A$ and $B$, are in the same group is $\frac{13}{121}$, even if another student is out sick. Find $N$. | [
"39"
] | [
"A gym class of size $N$ is split into three \"even\" groups. The probability that two specific friends, $A$ and $B$, are in the same group is $\\frac{13}{121}$, even if another student is out sick. Find $N$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b663fae07700383c8ac47fd9a723ea99b5a6522da27b77a1a3e775bdcdf1ab90 | A gym will admit a maximum of 5000 members. The cost for 100 members to join is \$200 per year. For every 20 additional members who join, the cost decreases by \$5 per year. How many members should join to maximize the gym's income? | [
"450"
] | [
"A gym will admit a maximum of 5000 members. The cost for 100 members to join is \\$200 per year. For every 20 additional members who join, the cost decreases by \\$5 per year. How many members should join to maximize the gym's income?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a6b3bff02c76e05e250457fe0521df757889c4d2c3c1b464a3917e358ac2d152 | A hacker intercepts two encrypted messages \( m^{d_1} \) and \( m^{d_2} \) where \( d_1 \) and \( d_2 \) are coprime. Given \( d_1 = 17 \) and \( d_2 = 23 \), and knowing that \( m^{d_1} = 123456789 \) and \( m^{d_2} = 987654321 \), find the value of \( m \) modulo \( 1000000007 \). | [
"262933486"
] | [
"A hacker intercepts two encrypted messages \\( m^{d_1} \\) and \\( m^{d_2} \\) where \\( d_1 \\) and \\( d_2 \\) are coprime. Given \\( d_1 = 17 \\) and \\( d_2 = 23 \\), and knowing that \\( m^{d_1} = 123456789 \\) and \\( m^{d_2} = 987654321 \\), find the value of \\( m \\) modulo \\( 1000000007 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c99f1cc9b9a498ffd79054b9afab3ca867f768ad355c47851d3afa7c81a5a52 | A hadron travels along the number line beginning at the number $1$, making one move every Planck time unit (in either direction). What is the expected number of Planck time units for it to reach the number $8$ given that the hadron cannot arrive at a number less than or equal to $-2$? | [
"77"
] | [
"A hadron travels along the number line beginning at the number $1$, making one move every Planck time unit (in either direction). What is the expected number of Planck time units for it to reach the number $8$ given that the hadron cannot arrive at a number less than or equal to $-2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-5b3cba540903c8f3b0ec4d90f934960976452403ccfc1c292ccb27789b71d6ec | A hailstone number \( n = d_1d_2 \ldots d_k \) is a positive integer with distinct nonzero digits such that \( d_m < d_{m-1} \) if \( m \) is even and \( d_m > d_{m-1} \) if \( m \) is odd for \( m = 1, 2, \ldots, k \) (and \( d_0 = 0 \)). Let \( a \) be the number of four-digit hailstone numbers and \( b \) be the num... | [
"3528"
] | [
"A hailstone number \\( n = d_1d_2 \\ldots d_k \\) is a positive integer with distinct nonzero digits such that \\( d_m < d_{m-1} \\) if \\( m \\) is even and \\( d_m > d_{m-1} \\) if \\( m \\) is odd for \\( m = 1, 2, \\ldots, k \\) (and \\( d_0 = 0 \\)). Let \\( a \\) be the number of four-digit hailstone numbers... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82ed6c583947614d223a7efebc45143948d24cf27d0a776c49f8fc326a9fb01d | A half-circle is inscribed in a right triangle such that its center lies on the hypotenuse. The center of the half-circle divides the hypotenuse into segments of lengths 30 and 40. Find the length of the arc of the half-circle between the points where it touches the legs of the triangle. | [
"\\( 12\\pi \\)"
] | [
"A half-circle is inscribed in a right triangle such that its center lies on the hypotenuse. The center of the half-circle divides the hypotenuse into segments of lengths 30 and 40. Find the length of the arc of the half-circle between the points where it touches the legs of the triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 24
} | 47 | 0 | 0 |
omr-ws-1481194507067b3dda065a056ec81c4a3dd85975cb9954170e06ceb67c573811 | A hallway has 8 offices on one side and 5 offices on the other side. A worker randomly starts in one office and then randomly goes to a second and then a third office (all three different). Find the probability that the worker crosses the hallway at least once. | [
"\\(\\frac{10}{13}\\)"
] | [
"A hallway has 8 offices on one side and 5 offices on the other side. A worker randomly starts in one office and then randomly goes to a second and then a third office (all three different). Find the probability that the worker crosses the hallway at least once."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-cf142b56ab17bff4e7f5f7506813db65498fe00a044a21d69a4ebb5a5caf0a69 | A hallway of width a turns into another hallway of width b at a right angle. What is the length of the longest stick I can carry through this hallway? | [
"\\(\\frac{3}{4}\\)"
] | [
"A hallway of width a turns into another hallway of width b at a right angle. What is the length of the longest stick I can carry through this hallway?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ecbd09e582f3b766afc56b3a422e00d5e46670e48cfd1391b02f9d35d39929e | A hand of four cards of the form \( (c, c, c + 1, c + 1) \) is called a tractor. Vinjai has a deck consisting of four of each of the numbers 7, 8, 9, and 10. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor. | [
"\\(\\frac{27}{455}\\)"
] | [
"A hand of four cards of the form \\( (c, c, c + 1, c + 1) \\) is called a tractor. Vinjai has a deck consisting of four of each of the numbers 7, 8, 9, and 10. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-ccb5d1c54d9980d1d15b89461c85c41ae7b7df5b7ce4278d44b12011b5e77537 | A happiness number is a positive integer whose cube has 13 times more positive divisors than the number itself. How many positive divisors does a happiness number have? | [
"40"
] | [
"A happiness number is a positive integer whose cube has 13 times more positive divisors than the number itself. How many positive divisors does a happiness number have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-d01b5149787564d8b1ec474d1122bd914ba7706334a29fc13f3485c9cb5653fa | A hardware supplier manufactures 3 kinds of clamps: types A, B, and C. The supplier must make 20 more type C clamps than the total of types A and B, and twice as many type B clamps as type A clamps. The shop must produce 260 clamps per day. How many of each type are made per day? | [
"40"
] | [
"A hardware supplier manufactures 3 kinds of clamps: types A, B, and C. The supplier must make 20 more type C clamps than the total of types A and B, and twice as many type B clamps as type A clamps. The shop must produce 260 clamps per day. How many of each type are made per day?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-096bc51d78d14dac8aa784ecb27f0e9e33aa26c7077b4d38a0363dd378087906 | A hardware supplier manufactures 3 kinds of clamps; types A, B, and C. Production restrictions force it to make 20 more type C clamps than the total of the other types and twice as many type B clamps as type A clamps. The shop must produce 260 clamps per day. How many of each type are made per day? How many type A clam... | [
"40"
] | [
"A hardware supplier manufactures 3 kinds of clamps; types A, B, and C. Production restrictions force it to make 20 more type C clamps than the total of the other types and twice as many type B clamps as type A clamps. The shop must produce 260 clamps per day. How many of each type are made per day?\n\nHow many typ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff2df29e5ae58d321ddf685bfd336d60555cf47000a4bb604dd876f0e537a405 | A hare is 50 of her own leaps ahead of a grey-hound. The hare takes 4 leaps while the grey-hound takes 3 leaps. In one leap, the hare goes $1 \frac{3}{4}$ meters and the grey-hound goes $2 \frac{3}{4}$ meters. In how many leaps will the grey-hound overtake the hare? | [
"210"
] | [
"A hare is 50 of her own leaps ahead of a grey-hound. The hare takes 4 leaps while the grey-hound takes 3 leaps. In one leap, the hare goes $1 \\frac{3}{4}$ meters and the grey-hound goes $2 \\frac{3}{4}$ meters. In how many leaps will the grey-hound overtake the hare?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-96d773c0943c082de9f4174285f4e315f419487159435d0643fce84c896e3fe0 | A hare is 50 of her own leaps ahead of a grey-hound. While the hare takes 4 leaps, the grey-hound takes 3 leaps. In one leap, the hare goes 7/4 meters and the grey-hound goes 11/4 meters. In how many leaps will the grey-hound overtake the hare? | [
"210"
] | [
"A hare is 50 of her own leaps ahead of a grey-hound. While the hare takes 4 leaps, the grey-hound takes 3 leaps. In one leap, the hare goes 7/4 meters and the grey-hound goes 11/4 meters. In how many leaps will the grey-hound overtake the hare?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-0631cd5e3b58ad0bcffa6e3ae22e7e48610e938b640ea1c001a286fe2c5846c2 | A has 2 pennies and B has 1 penny. They match their pennies until one has all three. Let \( X \) be a random variable denoting the number of steps taken to win the game. What is the distribution of \( X \)? | [
"\\( X_n = \\frac{1}{4} \\left( \\frac{3}{4} \\right)^n \\)"
] | [
"A has 2 pennies and B has 1 penny. They match their pennies until one has all three. Let \\( X \\) be a random variable denoting the number of steps taken to win the game. What is the distribution of \\( X \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-408dc8a6fd677b9ab4828cbcd048db9dc0e865832f74ea1a64ab4acb8deccdf5 | A has 7 blue socks and 7 red socks mixed in a drawer. Blindfolded, A takes two socks from the drawer. B looks at the socks, and if they have the same color, B gives A 1 point. A keeps drawing socks until the drawer is empty. Find the probability that A’s score is at most 2. | [
"\\dfrac{184}{429}"
] | [
"A has 7 blue socks and 7 red socks mixed in a drawer. Blindfolded, A takes two socks from the drawer. B looks at the socks, and if they have the same color, B gives A 1 point. A keeps drawing socks until the drawer is empty. Find the probability that A’s score is at most 2."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d0e19f812b305712a1692a38310b664c90bb032a48f597f1e9c5d29b6c15a6f | A haystack contains 999 pieces of hay and 1 needle. Person A picks out one item every 1 second, and Person B picks out two items every 2 seconds. They start at the same time. Who is more likely to get the needle? | [
"B"
] | [
"A haystack contains 999 pieces of hay and 1 needle. Person A picks out one item every 1 second, and Person B picks out two items every 2 seconds. They start at the same time. Who is more likely to get the needle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-d27cef5d6dbeb75f75622d7f3a58059f742785b843cd27b94fc56d6c29bfdfd4 | A haystack contains 999 pieces of hay and 1 needle. Person A picks out one item every 1 second, while Person B picks out two items every 2 seconds. Both start at the same time. Who is more likely to get the needle? | [
"Equal probability"
] | [
"A haystack contains 999 pieces of hay and 1 needle. Person A picks out one item every 1 second, while Person B picks out two items every 2 seconds. Both start at the same time. Who is more likely to get the needle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 22
} | 31 | 0 | 0 |
omr-ws-048940391c9101fe4be503f58c3e28e9dabb5e3467dd0e5fa924052aecb45705 | A heap of balls consists of one thousand 10-gram balls and one thousand 9.9-gram balls. We wish to pick out two heaps of balls with equal numbers of balls in them but different total weights. What is the minimum number of weighings required to do this? | [
"2"
] | [
"A heap of balls consists of one thousand 10-gram balls and one thousand 9.9-gram balls. We wish to pick out two heaps of balls with equal numbers of balls in them but different total weights. What is the minimum number of weighings required to do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-9547b2bf6ead6d85c2a4d81f6682dd30f6409d3a5ed8385f1628650fceb763bf | A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the top of a building 120 ft high. How much work is done in pulling half the rope to the top of the building? | [
"468.75 ft-lb"
] | [
"A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the top of a building 120 ft high. How much work is done in pulling half the rope to the top of the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 4
} | 25 | 0 | 0 |
omr-ws-255d606b673a2c34e4f9aed52517d14ced0e028ef6c992067d67e13e5f46eb91 | A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the top of a building 120 ft high. How much work is done in pulling the rope to the top of the building? | [
"625 ft-lb"
] | [
"A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the top of a building 120 ft high. How much work is done in pulling the rope to the top of the building?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-1dd84fb766433c57906db8ebd4c485e98e6fb8d0b6971e8e1e71ac499f2bf870 | A helicopter flew 20km on a bearing of 040 degrees from point A to point B. From point B, it flew 70km due east (bearing of 090 degrees) to point C. From point C, it flew 20km on a bearing of 220 degrees to reach point D. What is the size of angle ADC? | [
"130 degrees",
"130^\\circ"
] | [
"A helicopter flew 20km on a bearing of 040 degrees from point A to point B. From point B, it flew 70km due east (bearing of 090 degrees) to point C. From point C, it flew 20km on a bearing of 220 degrees to reach point D. What is the size of angle ADC?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 19
} | 58 | 0 | 0 |
omr-ws-b2836d89c4d8f1b3071333b42526ba0b0fdf6df183937fe9440e8938df441cf6 | A helicopter is directly above an oil tanker at a height of 5000 km. The pilot spots an oil platform with an angle of depression of 26 degrees. What is the distance between the oil tanker and the oil platform? | [
"10252"
] | [
"A helicopter is directly above an oil tanker at a height of 5000 km. The pilot spots an oil platform with an angle of depression of 26 degrees. What is the distance between the oil tanker and the oil platform?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-ad7241794b1a5410005abaf839921cd6791fd1fceb0d16ea0955ad12600cfc14 | A helicopter went 7 km from point A to point B. Then it flew 6 km from B to C on a bearing of 290 degrees. If Angle ABC = 130 degrees, solve for the following: a. The bearing of B from A. b. The bearing of B from C. c. The bearing of C from A. d. The bearing of A from C. | [
"a. 340°, b. 110°, c. 317°, d. 137°"
] | [
"A helicopter went 7 km from point A to point B. Then it flew 6 km from B to C on a bearing of 290 degrees. If Angle ABC = 130 degrees, solve for the following:\na. The bearing of B from A.\nb. The bearing of B from C.\nc. The bearing of C from A.\nd. The bearing of A from C."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-195f7d7bc0305bbbe3eb14d1b97746e698e3a914afc3921a61449cbf045f53d5 | A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}{5}$. Compute the angle at the vertex of the cone. | [
"\\(\\theta = \\tan^{-1} \\left( \\frac{5\\sqrt{26}}{156} \\right)\\)"
] | [
"A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\\frac{18}{5}$. Compute the angle at the vertex of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9c6a893035c555bfca360ddeac2a7c94f3c2f4dba0c64a820029c61dcf97a0d | A hemisphere is inscribed in a cone so that its great circle lies in the base of the cone. Determine the vertex angle of the axial section of the cone if the ratio of the total surface area of the cone to the surface area of the hemisphere is $18: 5$. | [
"2\\arcsin\\frac{5}{6} \\text{ or } 2\\arcsin\\frac{1}{6}"
] | [
"A hemisphere is inscribed in a cone so that its great circle lies in the base of the cone. Determine the vertex angle of the axial section of the cone if the ratio of the total surface area of the cone to the surface area of the hemisphere is $18: 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b929522e368c87441dc40a65a3f0286e5ace92a5faceac907f8b3b1d9f3c8e00 | A hemisphere is inscribed in a regular quadrangular pyramid so that its base is parallel to the base of the pyramid and the spherical surface is in contact with it. Determine the total surface area of the pyramid, if its lateral faces are inclined to the base at an angle $\alpha $ and the radius of the sphere is equal ... | [
"$S_{total}=\\frac{8r^2(1+\\cot\\alpha )^2\\cos^2 \\frac{\\alpha }{2}}{\\cos\\alpha } =\\frac{4r^2\\sin^2(45^\\circ+\\alpha )}{\\cos\\alpha \\sin^2 \\frac{\\alpha }{2}}$"
] | [
"A hemisphere is inscribed in a regular quadrangular pyramid so that its base is parallel to the base of the pyramid and the spherical surface is in contact with it. Determine the total surface area of the pyramid, if its lateral faces are inclined to the base at an angle $\\alpha $ and the radius of the sphere is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e224e5694791a9d7aa66b99c61dc8d6ad136c2f8279395f67496dbe496021f3 | A hemisphere is inscribed in a regular quadrangular pyramid so that its base lies on the base of the pyramid and the spherical surface touches the lateral faces of the pyramid. Find the ratio of the surface of the hemisphere to the total surface area of the pyramid and the volume of the hemisphere, if the lateral faces... | [
"$q=\\frac{3\\pi }{8}\\sin 2\\alpha \\tan\\frac{\\alpha }{2}$; $V=\\frac{\\pi m^3 \\sin^3 \\alpha }{96\\sin^6 \\left(45^\\circ-\\frac{\\alpha }{2}\\right)}$"
] | [
"A hemisphere is inscribed in a regular quadrangular pyramid so that its base lies on the base of the pyramid and the spherical surface touches the lateral faces of the pyramid. Find the ratio of the surface of the hemisphere to the total surface area of the pyramid and the volume of the hemisphere, if the lateral ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-307f3ef79d6bac06d2ae0c9b1a6a67fd15af4684ec7687ecf5b9c9c7eab94274 | A hemisphere is inscribed in a regular tetrahedron, the edge of which is equal to unity, so that three faces of the tetrahedron are tangent to its spherical surface, and the fourth serves as the plane of the diameter. Determine the total surface area of the hemisphere. | [
"$4\\pi /27$"
] | [
"A hemisphere is inscribed in a regular tetrahedron, the edge of which is equal to unity, so that three faces of the tetrahedron are tangent to its spherical surface, and the fourth serves as the plane of the diameter. Determine the total surface area of the hemisphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6d1b52cb7fa9e934490d7fb470917b06bf8e1bf07618ee47fc349450624414d6 | A hemisphere is mounted on the base of a right circular cone such that the circular part of the hemisphere coincides with the circular base of the cone. What is the possible maximum total surface area in terms of its total volume? | [
" \\left( \\frac{\\sqrt{145} + 16}{8} \\right) \\left( \\frac{24}{25} \\right)^{2/3} \\pi^{1/3} V^{2/3} "
] | [
"A hemisphere is mounted on the base of a right circular cone such that the circular part of the hemisphere coincides with the circular base of the cone. What is the possible maximum total surface area in terms of its total volume?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f23d8da8446aa972cac79936b1cd1c326386ce1056aa5bacbfbaeb673b9085d5 | A hemisphere of radius $2$ is lying on the ground with its flat side down. An equilateral triangle is inscribed into the circle that is the base of this hemisphere. Compute the surface area of that portion of the hemisphere which lies directly above the triangle. | [
"\\(2\\pi\\)"
] | [
"A hemisphere of radius $2$ is lying on the ground with its flat side down. An equilateral triangle is inscribed into the circle that is the base of this hemisphere. Compute the surface area of that portion of the hemisphere which lies directly above the triangle."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 23
} | 27 | 0 | 0 |
omr-ws-65e532d9267942584b448f67efce1c22c3e30d82746c98df1a6fe38c8bc12854 | A hemisphere with radius 12 is divided into equal sectors by planes going through the center and perpendicular to the figure's flat face. If the angle between any two adjacent planes is $60^\circ$, find the surface area of one such sector of the hemisphere (including all faces). | [
"\\(144\\pi\\)"
] | [
"A hemisphere with radius 12 is divided into equal sectors by planes going through the center and perpendicular to the figure's flat face. If the angle between any two adjacent planes is $60^\\circ$, find the surface area of one such sector of the hemisphere (including all faces)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-ca14c16fd5b4ee53730cef44d450f56fe266d882ca4e103b46351d15fb70b6e8 | A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 27 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet? | [
"\\frac{1}{3}"
] | [
"A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 27 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21e1ff9b1194173a64c5842facbd0cd2c9e4486f66a55ab098113c3cab7ca30a | A hemispherical tank with a radius of 5 feet is full of water. Given that water weighs 62.5 lb per cubic foot, find the work required to pump the water out of the tank. | [
"\\dfrac{390625}{24} \\pi"
] | [
"A hemispherical tank with a radius of 5 feet is full of water. Given that water weighs 62.5 lb per cubic foot, find the work required to pump the water out of the tank."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-3af42f6e30be7793aec12afa0557fd85a453a2d85d05bad448720b8bbba35ad7 | A heptagon $ABCDEFG$ contains a segment $XY$ of length $6-2\sqrt{3}$ in its interior such that the pentagon $ABCXY$ can be rotated around $A$ to form the pentagon $AYXFG$, and this can be further rotated around $F$ to form the pentagon $CDEFX$. What is the area of the heptagon $ABCDEFG$? | [
"36 - 12\\sqrt{3}"
] | [
"A heptagon $ABCDEFG$ contains a segment $XY$ of length $6-2\\sqrt{3}$ in its interior such that the pentagon $ABCXY$ can be rotated around $A$ to form the pentagon $AYXFG$, and this can be further rotated around $F$ to form the pentagon $CDEFX$. What is the area of the heptagon $ABCDEFG$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-193ebedfe66c6bc3742a301c5cb079514596708c2df56ec533b79356af4dde45 | A heptagon is dissected into convex pentagons and hexagons such that each of its vertices belongs to at least two smaller polygons. Let $N$ be the minimum number of polygons in such a dissection. Find the value of $N$. | [
"13"
] | [
"A heptagon is dissected into convex pentagons and hexagons such that each of its vertices belongs to at least two smaller polygons. Let $N$ be the minimum number of polygons in such a dissection. Find the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-88430e4fc836a52f9778ac19d14c0e0d1d97f99791d815475391071c27cdc81a | A heptagon is dissected into convex pentagons and hexagons such that each of its vertices belongs to at least two smaller polygons. Let $N$ be the number of polygons in such a tessellation. Find the minimum possible value of $N$. | [
"13"
] | [
"A heptagon is dissected into convex pentagons and hexagons such that each of its vertices belongs to at least two smaller polygons. Let $N$ be the number of polygons in such a tessellation. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-acbd0a01650b5fa5022e3d993da01270e3ba98bf13f89b0fa410afe43c32b5b7 | A heptagon is divided into convex pentagons and hexagons such that each vertex of the heptagon is a vertex of at least two of these polygons (excluding the heptagon itself). If $P$ is the number of pentagons and $H$ is the number of hexagons, find the minimum possible value of $P+H$. | [
"14"
] | [
"A heptagon is divided into convex pentagons and hexagons such that each vertex of the heptagon is a vertex of at least two of these polygons (excluding the heptagon itself). If $P$ is the number of pentagons and $H$ is the number of hexagons, find the minimum possible value of $P+H$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c400a1f36e5f7d6e57d5a87cee7ba43b2ac29c4d1ef35a11279fccc88d8fe721 | A herd stampeded d feet at r feet per minute and arrived 7 minutes late. How fast should the herd have stampeded to arrive on time? | [
"\\(\\frac{60dr}{60d-7r}\\)"
] | [
"A herd stampeded d feet at r feet per minute and arrived 7 minutes late. How fast should the herd have stampeded to arrive on time?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-bac7b5e7ef81cb5e914605e0a3f2348893c7401411dd23e87c2e900e51bf8577 | A hexagon $ABCDEF$ is inscribed in a circle and satisfies $AB \cdot CD \cdot EF = BC \cdot DE \cdot FA$. The points $B$ and $B_1$ are symmetrical over the line $AC$; $D$ and $D_1$ are symmetrical over $CE$; and $F$ and $F_1$ are symmetrical over $EA$. Calculate the ratio of the areas of $\triangle BDF$ and $\triangle B... | [
"1"
] | [
"A hexagon $ABCDEF$ is inscribed in a circle and satisfies $AB \\cdot CD \\cdot EF = BC \\cdot DE \\cdot FA$. The points $B$ and $B_1$ are symmetrical over the line $AC$; $D$ and $D_1$ are symmetrical over $CE$; and $F$ and $F_1$ are symmetrical over $EA$. Calculate the ratio of the areas of $\\triangle BDF$ and $\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-d47d23c90a8d9777d5e6fa1db163c683000ad0d78e9136703e996f0891d4bf8f | A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be the intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, respectively. Given that $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, and $\angle ESF=35^{\circ}$. Let $T$ be the intersection of $BE$ and $CF$.... | [
"95°"
] | [
"A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be the intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, respectively. Given that $\\angle BPC=50^{\\circ}$, $\\angle CQD=45^{\\circ}$, $\\angle DRE=40^{\\circ}$, and $\\angle ESF=35^{\\circ}$. Let $T$ be the intersection of $... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-7b7e2524fa06817ed37510b964e5efdaf55cde58323d99661b9e34c5c5b9369e | A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$, and $A_3A_6$ concur. Let $l_i$ be the radical axis of the circles $A_iA_{i+1}A_{i-2}$ and $A_iA_{i-1}A_{i+2}$ (with $A_i$ and $A_{i+6}$ coinciding). If the point of concurrency of the lines $l_i$ for $i=1,\cdots,6$ is $... | [
"0"
] | [
"A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$, and $A_3A_6$ concur. Let $l_i$ be the radical axis of the circles $A_iA_{i+1}A_{i-2}$ and $A_iA_{i-1}A_{i+2}$ (with $A_i$ and $A_{i+6}$ coinciding). If the point of concurrency of the lines $l_i$ for $i=1,\\cdots,6... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-a563b7e9f69145639751465445323853608d6b22736ec5ddfb96ba5f2dad2678 | A hexagon ABCDEF is inscribed in a circle with side lengths AB = 2, BC = 3, CD = 4, DE = 4, EF = 3, and FA = 2. What is the radius of the circumscribed circle? | [
"\\sqrt{13}"
] | [
"A hexagon ABCDEF is inscribed in a circle with side lengths AB = 2, BC = 3, CD = 4, DE = 4, EF = 3, and FA = 2. What is the radius of the circumscribed circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-4730970041130225c8d851c3dc5c41ac90ba67675f4160c4d745e62ba96140c1 | A hexagon \(ABCDEF\) is inscribed in a circle of radius \(R = \frac{7\sqrt{3}}{3}\). It is known that \(\angle A = \angle C = \angle E\) and \(AB = CD = EF = 3\). Find the area of the hexagon \(ABCDEF\). | [
"\\dfrac{47\\sqrt{3}}{2}"
] | [
"A hexagon \\(ABCDEF\\) is inscribed in a circle of radius \\(R = \\frac{7\\sqrt{3}}{3}\\). It is known that \\(\\angle A = \\angle C = \\angle E\\) and \\(AB = CD = EF = 3\\). Find the area of the hexagon \\(ABCDEF\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c69eb65a1da309fb71b77091ff617a5a08f78ece3f404d7e617bd6900258f946 | A hexagon \(ABCDEF\) is inscribed in a circle of radius \(R\) with center \(O\) such that \(AB = CD = EF = R\). Let \(P\), \(Q\), and \(R\) be the second points of intersection of the circumcircles of \(\triangle OBC\), \(\triangle ODE\), and \(\triangle OFA\) respectively. Calculate the circumradius of \(\triangle PQR... | [
"\\dfrac{R}{2}"
] | [
"A hexagon \\(ABCDEF\\) is inscribed in a circle of radius \\(R\\) with center \\(O\\) such that \\(AB = CD = EF = R\\). Let \\(P\\), \\(Q\\), and \\(R\\) be the second points of intersection of the circumcircles of \\(\\triangle OBC\\), \\(\\triangle ODE\\), and \\(\\triangle OFA\\) respectively. Calculate the cir... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-563a5c96aa07c72d016c912b979ac1668f143231e015238efde42fff5a395f6f | A hexagon \(A_1A_2 \ldots A_6\) is inscribed in a regular hexagon, with one vertex of the former hexagon lying on each side of the latter. Three nonadjacent sides \(A_1A_2\), \(A_3A_4\), and \(A_5A_6\) of the first hexagon are extended to form a triangle. If the vertices of this triangle lie on the lines containing the... | [
"\\dfrac{27\\sqrt{3}}{16} s^2"
] | [
"A hexagon \\(A_1A_2 \\ldots A_6\\) is inscribed in a regular hexagon, with one vertex of the former hexagon lying on each side of the latter. Three nonadjacent sides \\(A_1A_2\\), \\(A_3A_4\\), and \\(A_5A_6\\) of the first hexagon are extended to form a triangle. If the vertices of this triangle lie on the lines ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-301aa66bcbac1ae5b1facc2893c0621395a9c8152043e2378a3d9dccfc6dfbbf | A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$, and $7$, respectively. Find the lengths of the remaining two sides. | [
"1 and 8"
] | [
"A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$, and $7$, respectively. Find the lengths of the remaining two sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 15
} | 18 | 0 | 0 |
omr-ws-5e45abc076d8ab67151fe1bc91640eef92ef47f4d948f8f6b51f03e87a05c2e2 | A hexagon has interior angles of 120 degrees each and sides of various lengths. Moving counter-clockwise, the consecutive sides are 8 units, 7 units, 12 units, and 6 units. Determine the lengths of the next two sides. | [
"9 and 10"
] | [
"A hexagon has interior angles of 120 degrees each and sides of various lengths. Moving counter-clockwise, the consecutive sides are 8 units, 7 units, 12 units, and 6 units. Determine the lengths of the next two sides."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-95dc950c04d803e58a078b3a64ea58834092adf51547341f7a34f7f87517109e | A hexagon inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$. | [
"272"
] | [
"A hexagon inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 9
} | 57 | 0 | 0 |
omr-ws-9626381abaf6f4c5e15cad1e51acd1ce4f969cb50dfe49b1f0a504478d6a9bb8 | A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac{m}{n}... | [
"409"
] | [
"A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\\frac... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
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