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-cbe0d636fc06acc829f7f5214cc57e4257a3262af30a2526b15b8b2ac4c81481 | At a party with six married couples, the 12 guests sit down randomly around a large round table. What is the probability that at least one person is not sitting next to his/her spouse? (Assume each seating configuration is equally likely.) | [
"\\(\\frac{5204}{5205}\\)"
] | [
"At a party with six married couples, the 12 guests sit down randomly around a large round table. What is the probability that at least one person is not sitting next to his/her spouse? (Assume each seating configuration is equally likely.)"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-1ef355d8aa1d6094933de861ffdc91028a902be5470a30f9634876fc8be85383 | At a party with ten attendees, each writes a function from the real numbers to the real numbers on a piece of paper and puts it into a hat. It is known that applying these functions successively to the number $0$ in any order always results in $1$. If Daniel’s favorite function is of the form $f(x) = mx + b$, where $m$... | [
"10"
] | [
"At a party with ten attendees, each writes a function from the real numbers to the real numbers on a piece of paper and puts it into a hat. It is known that applying these functions successively to the number $0$ in any order always results in $1$. If Daniel’s favorite function is of the form $f(x) = mx + b$, wher... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-575addf974983213fc6a1a778432e34ebdc820e5351ec4d98d4ebe27cc1c0589 | At a party, 100 men each hang up their coat on a rack. At the end of the party, each man retrieves his coat in sequence. The first man takes a random coat, and each subsequent man takes his own coat if it is available or a random coat if it is not. Determine the probability that the last man takes his own coat. | [
"\\dfrac{1}{2}"
] | [
"At a party, 100 men each hang up their coat on a rack. At the end of the party, each man retrieves his coat in sequence. The first man takes a random coat, and each subsequent man takes his own coat if it is available or a random coat if it is not. Determine the probability that the last man takes his own coat."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-586f0d1ea9c90daa62e00075c2fd329484a466281898d33d4e37d4fe63951079 | At a party, 25 elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Calculate the minimum number of groups of three elves that can be chosen such that each group includes at least two elves who give a presen... | [
"25"
] | [
"At a party, 25 elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Calculate the minimum number of groups of three elves that can be chosen such that each group includes at least two elves who give a p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-798fc8f160dcdeb9c611e87c2e566a079c62c7f95660367d5d4c52151746f7bd | At a party, Mr. and Mrs. Brown and three other couples are present. Each person shook hands with some of the others, but no one shook hands with their own spouse. Mr. Brown found that everyone gave him a different number of handshakes. How many people did Mrs. Brown shake hands with? | [
"3"
] | [
"At a party, Mr. and Mrs. Brown and three other couples are present. Each person shook hands with some of the others, but no one shook hands with their own spouse. Mr. Brown found that everyone gave him a different number of handshakes. How many people did Mrs. Brown shake hands with?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cac4566803800e1c3b6cad76b138bdbac23719d34d7bf5cafff84326be3bc781 | At a party, Ted's age is 15 years less than twice Sally's age. The sum of their ages is 54. How old is Ted? | [
"31"
] | [
"At a party, Ted's age is 15 years less than twice Sally's age. The sum of their ages is 54. How old is Ted?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d13f863e763d0e5c15147d7bcc3934fe105343a21efa04382f01d7eb626bf31 | At a party, \( n \) people arrive wearing hats, which they check at the door. As they are leaving, each person takes a hat at random without looking. What is the expected number of cycles? (A cycle being the minimal group of people who collectively have each other's hats) | [
"\\(1 + \\frac{1}{2} + \\frac{1}{3} + \\ldots + \\frac{1}{n}\\)"
] | [
"At a party, \\( n \\) people arrive wearing hats, which they check at the door. As they are leaving, each person takes a hat at random without looking. What is the expected number of cycles? (A cycle being the minimal group of people who collectively have each other's hats)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-698b39df0ca159d184f23b878c2357f7d0cafa548767213a47de0fdb005e6906 | At a party, a group of 20 people needs to be seated at 4 tables with exactly 5 people at each table. A seating arrangement is called successful if any two people at the same table are friends. Given that successful seating arrangements exist, what is the greatest possible number of pairs of friends in this group? | [
"40"
] | [
"At a party, a group of 20 people needs to be seated at 4 tables with exactly 5 people at each table. A seating arrangement is called successful if any two people at the same table are friends. Given that successful seating arrangements exist, what is the greatest possible number of pairs of friends in this group?"... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f57b8c61e5c5d85af5f63d7e3839581339a2a6504290b235c80929f5b1b64825 | At a party, a married couple invited 9 other married couples. Some guests know each other and do not shake hands, while others shake hands if they have never met before. After the party, the landlord asked all 19 attendees (including his wife and the 18 invited guests) how many times they shook hands, and their answers... | [
"9"
] | [
"At a party, a married couple invited 9 other married couples. Some guests know each other and do not shake hands, while others shake hands if they have never met before. After the party, the landlord asked all 19 attendees (including his wife and the 18 invited guests) how many times they shook hands, and their an... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04a4be25130a16fb12c455a625a39d45b6e760b2054ffa88edf49d2711b9bd29 | At a party, among any 4 people, there are either 3 who are all friends among each other or 3 who are not friends with each other. Suppose there are 10 people at the party. What is the minimum number of groups needed to separate these people such that everyone in each group is either friends with everyone else in the sa... | [
"4"
] | [
"At a party, among any 4 people, there are either 3 who are all friends among each other or 3 who are not friends with each other. Suppose there are 10 people at the party. What is the minimum number of groups needed to separate these people such that everyone in each group is either friends with everyone else in t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-7d70f511c7088d2275653f3f5f46bf651eb0de1a60695054c4e7176511e57831 | At a party, among any 4 people, there are either 3 who are all friends with each other or 3 who are not friends with each other. Suppose the party has 10 people. Let $A$ be the number of ways to separate all the people into two groups such that everyone in the first group is friends with each other and everyone in the ... | [
"11"
] | [
"At a party, among any 4 people, there are either 3 who are all friends with each other or 3 who are not friends with each other. Suppose the party has 10 people. Let $A$ be the number of ways to separate all the people into two groups such that everyone in the first group is friends with each other and everyone in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9dc3e97d7a5f45ce4ead9c4ac4fea33b5c1e129ac19f1a2c3df6fac1c32fede | At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Let $G$ be the set of girls and $B$ be the set of boys. Define a function $f: G \times B \to \{0, 1\}$ such that $f(g, b) = 1$ if girl $g$ dances with boy $b$ and $f(g, b) = 0$ otherwise. Calculate the minimum possible val... | [
"0"
] | [
"At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Let $G$ be the set of girls and $B$ be the set of boys. Define a function $f: G \\times B \\to \\{0, 1\\}$ such that $f(g, b) = 1$ if girl $g$ dances with boy $b$ and $f(g, b) = 0$ otherwise. Calculate the minimum pos... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-dc6d215dd2f0e4652ce8b1cb279d0574f010789de0590a1b28d4d3ab40d87a70 | At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party? | [
"18"
] | [
"At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 6
} | 54 | 0 | 0 |
omr-ws-0858ae8b6af3f4403b823e5e38de3019d72a580a43dc9466e41303286604ebb8 | At a party, each man shakes hands with another man only once, and each woman shakes hands with another woman only once. No men and women shake hands with each other. If the number of handshakes among men is 7 and the number of handshakes among women is 7, and there are more men than women, find the number of men presen... | [
"4"
] | [
"At a party, each man shakes hands with another man only once, and each woman shakes hands with another woman only once. No men and women shake hands with each other. If the number of handshakes among men is 7 and the number of handshakes among women is 7, and there are more men than women, find the number of men p... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-93f990789ad59492ef79a3c015522c92707f4d9c7869c9c4107b6c0639dffd20 | At a party, each person knew exactly 22 others. For any pair of people X and Y who knew one another, there was no other person at the party that they both knew. For any pair of people X and Y who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party? | [
"100"
] | [
"At a party, each person knew exactly 22 others. For any pair of people X and Y who knew one another, there was no other person at the party that they both knew. For any pair of people X and Y who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 17
} | 34 | 0 | 0 |
omr-ws-8de76bd1f58f4360a29742ad778e22d39cf74bbf94d552b86783442b9e9e65c7 | At a party, each person knows exactly 22 others. For any pair of people who know each other, there is no other person at the party that they both know. For any pair of people who do not know each other, there are exactly 6 other people that they both know. How many people were at the party? | [
"100"
] | [
"At a party, each person knows exactly 22 others. For any pair of people who know each other, there is no other person at the party that they both know. For any pair of people who do not know each other, there are exactly 6 other people that they both know. How many people were at the party?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-06aa409287eca0a0dac502926502f8a5ef1c69a476b2b7e49a17511dd6ba3217 | At a party, each person shakes hands with exactly 50 other people. For each pair of people who shake hands with each other, there is exactly 10 other person who shakes hands with both of them. For each pair of people who do not shake hands with each other, there are exactly 16 other people who shake hands with both of ... | [
"41"
] | [
"At a party, each person shakes hands with exactly 50 other people. For each pair of people who shake hands with each other, there is exactly 10 other person who shakes hands with both of them. For each pair of people who do not shake hands with each other, there are exactly 16 other people who shake hands with bot... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a74aa7f806601b9f839948849d4d4357b2559b72fad17de8b321988376a3833a | At a party, each person shakes hands with exactly three other people and no two people shake hands with each other more than once. If fewer than fifteen handshakes take place, what is the maximum number of people who can be at the party? | [
"8"
] | [
"At a party, each person shakes hands with exactly three other people and no two people shake hands with each other more than once. If fewer than fifteen handshakes take place, what is the maximum number of people who can be at the party?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f11f61dbeea620f8b0b61b3d1f2e6cfe36e048d800009d2339d67b80c940078 | At a party, each person shakes hands with some of the others. If there are 20 people at the party and the total number of handshakes is 95, find the sum of all possible values for the number of people who shook hands an odd number of times. | [
"110"
] | [
"At a party, each person shakes hands with some of the others. If there are 20 people at the party and the total number of handshakes is 95, find the sum of all possible values for the number of people who shook hands an odd number of times."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-c817308907bb5b8ee466db050ff541d4445c7521212f0f877b9f2110d4a06a32 | At a party, each person shakes hands with some of the others. Let $N$ be the total number of people at the party, and let $O$ be the number of people who shake hands an odd number of times. If $N = 20$, find the sum of all possible values of $O$. | [
"110"
] | [
"At a party, each person shakes hands with some of the others. Let $N$ be the total number of people at the party, and let $O$ be the number of people who shake hands an odd number of times. If $N = 20$, find the sum of all possible values of $O$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7348c40d6bf7e35d1d17fe7f8d7f5ecc630c1d031009a73e6847ba34337e9bc | At a party, each person shakes hands with some of the others. Let $n$ be the total number of people at the party. If $S$ is the sum of the number of handshakes made by each person with an odd number of handshakes, find the remainder when $S$ is divided by $2$. | [
"0"
] | [
"At a party, each person shakes hands with some of the others. Let $n$ be the total number of people at the party. If $S$ is the sum of the number of handshakes made by each person with an odd number of handshakes, find the remainder when $S$ is divided by $2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-634e54d5db23dbfaac267cde9bccfd535b4cfbc4e952bc937f6e8aea1609f195 | At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? | [
"64"
] | [
"At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39ffa89c6a77f55102fad98018898e60db2d21d2a4b80e4ec4f20dcc73b15d30 | At a party, every woman dances with at least one man, and no man dances with every woman. If there are 10 women and 10 men at the party, find the minimum number of distinct pairs (M, W) where M is a man and W is a woman such that M dances with W, given that there exist men M and M' and women W and W' such that M dances... | [
"10"
] | [
"At a party, every woman dances with at least one man, and no man dances with every woman. If there are 10 women and 10 men at the party, find the minimum number of distinct pairs (M, W) where M is a man and W is a woman such that M dances with W, given that there exist men M and M' and women W and W' such that M d... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-caa97986ebc38b18489f6f6120dddfe111c82b588707bf187e7cbb98bb2262fb | At a party, five friends each hang their coats on a coat rack. When they leave, Ryan hands out the coats in a random order. What is the probability that at least three of them receive the correct coat? | [
"\\(\\frac{11}{120}\\)"
] | [
"At a party, five friends each hang their coats on a coat rack. When they leave, Ryan hands out the coats in a random order. What is the probability that at least three of them receive the correct coat?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-4c4582321659e012bad7b43dc10f137d391d7f2d0d8aef8455b7724c4ab9f5f3 | At a party, if one-half of the Martians became Earthlings, there would be 295 eyes at the party. If one-half of the Earthlings became Martians, there would be 410 heads at the party. How many Martians and how many Earthlings were at the party, given that Martians have 11 eyes and 7 heads, and Earthlings have 2 eyes and... | [
"30 Martians and 50 Earthlings."
] | [
"At a party, if one-half of the Martians became Earthlings, there would be 295 eyes at the party. If one-half of the Earthlings became Martians, there would be 410 heads at the party. How many Martians and how many Earthlings were at the party, given that Martians have 11 eyes and 7 heads, and Earthlings have 2 eye... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 28
} | 36 | 0 | 0 |
omr-ws-ed0df6464c17b6a6b83ac5667fa415410d26a83bfcec9871a1911c24ee908c3e | At a party, nine women each know at most three languages. Any two women can converse in at least one language. Determine the minimum number of women who must know a specific language so that there are at least five women who can converse in that language. | [
"5"
] | [
"At a party, nine women each know at most three languages. Any two women can converse in at least one language. Determine the minimum number of women who must know a specific language so that there are at least five women who can converse in that language."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-195ca782edf6ef425566d07a7f0ae8d9a1d3926fbe08d12c0d23675f290fa39f | At a party, no boy danced with all the girls and every girl danced with at least one boy. Let \( N \) be the number of distinct pairs of couples \((b, f)\) and \((b', f')\) such that boy \( b \) danced with girl \( f \) and boy \( b' \) danced with girl \( f' \), but boy \( b \) did not dance with girl \( f' \) and boy... | [
"1"
] | [
"At a party, no boy danced with all the girls and every girl danced with at least one boy. Let \\( N \\) be the number of distinct pairs of couples \\((b, f)\\) and \\((b', f')\\) such that boy \\( b \\) danced with girl \\( f \\) and boy \\( b' \\) danced with girl \\( f' \\), but boy \\( b \\) did not dance with ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8326606c7dd8369f44fde96d259018f8599c25cda5bece75d58d43cfbf8eff55 | At a party, persons \( A, B, C, D \) each bring a present and randomly receive one of the presents brought by the others. What is the probability that person \( A \) or person \( B \) receives the present they brought? | [
"\\(\\frac{5}{12}\\)"
] | [
"At a party, persons \\( A, B, C, D \\) each bring a present and randomly receive one of the presents brought by the others. What is the probability that person \\( A \\) or person \\( B \\) receives the present they brought?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-ceb4a6f77ecd08d1d8fa4d4231943d5a1496671c0004e2c323e4d7b5dd230f4d | At a party, there are $2011$ people sitting around a circular table, each with a glass of fruit juice. They clink glasses every second, following these rules: - They do not clink glasses crosswise. - At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at least until ev... | [
"2011"
] | [
"At a party, there are $2011$ people sitting around a circular table, each with a glass of fruit juice. They clink glasses every second, following these rules:\n- They do not clink glasses crosswise.\n- At each point of time, everyone can clink glasses with at most one other person.\nHow many seconds pass at least ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-439d6a8f3720fc05fdba0695c336ba2dfd637a473166b549497180663ed27dcc | At a party, there are $mn$ people arranged in an $m \times n$ table. Some $k$ of these people are policemen, and the rest are guests, including potential gangsters. Every policeman knows all the gangsters. Two people see each other if they are in the same row or column. Each policeman reports to the sheriff whether the... | [
"m + n - 1"
] | [
"At a party, there are $mn$ people arranged in an $m \\times n$ table. Some $k$ of these people are policemen, and the rest are guests, including potential gangsters. Every policeman knows all the gangsters. Two people see each other if they are in the same row or column. Each policeman reports to the sheriff wheth... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-0906edf7f41fbf429106673918ee31c9d239257ec5620ceb269d071629452be4 | At a party, there are $n$ married couples. Nobody knows anyone else except their spouse. To make some new friends, everyone's names are put in a hat and drawn out two at a time (randomly, independently, without replacement) until the hat is empty; a pair of people whose names are drawn together get to know each other (... | [
"\\sum_{k=1}^{n} \\frac{1}{2k - 1}"
] | [
"At a party, there are $n$ married couples. Nobody knows anyone else except their spouse. To make some new friends, everyone's names are put in a hat and drawn out two at a time (randomly, independently, without replacement) until the hat is empty; a pair of people whose names are drawn together get to know each ot... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-85bbe7745122ebe759db559157f0c3a024190342a31a49de1d695fa3ed9ee4c9 | At a party, there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of the women. For which $r$ and $s$ must there be a man and a woman who like each other? | [
"\\( r \\cdot s \\geq n \\)"
] | [
"At a party, there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of the women. For which $r$ and $s$ must there be a man and a woman who like each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c26de2d1cf1c2ee7eeb67b4ebc10d6b3a6824d02b7ac92159b3a1f74a8ef5c90 | At a party, there are 2011 people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules: (a) They do not clink glasses crosswise. (b) At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at le... | [
"2011"
] | [
"At a party, there are 2011 people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules:\n(a) They do not clink glasses crosswise.\n(b) At each point of time, everyone can clink glasses with at most one other person.\nHow many seconds pa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5ac51af4b0176259ab2b0554423b21f713c09ea830724e6afc9ece88fe96010 | At a party, there are 25 boys and some girls. It is known that for any group of at least 10 boys, the number of girls acquainted with at least one of these boys is exactly one more than the number of boys in the group. Find the minimum number of boys that the girl with the most acquaintances among the boys is acquainte... | [
"16"
] | [
"At a party, there are 25 boys and some girls. It is known that for any group of at least 10 boys, the number of girls acquainted with at least one of these boys is exactly one more than the number of boys in the group. Find the minimum number of boys that the girl with the most acquaintances among the boys is acqu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-4718bed16166001ebd79cc2b1f29877a8c67c3ea13475b1ac50caf980e1e60cf | At a party, there are 6 boys and a number of girls. Two of the girls know exactly four boys each, and the remaining girls know exactly two boys each. None of the boys know more than three girls. What is the greatest possible number of girls at the party? | [
"7"
] | [
"At a party, there are 6 boys and a number of girls. Two of the girls know exactly four boys each, and the remaining girls know exactly two boys each. None of the boys know more than three girls. What is the greatest possible number of girls at the party?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ba27aa38874a646f756746623050e4232bc47ae41e73b706f26e4ceeefca956 | At a party, there are five people (Alan, Andrew, Abby, Vincent, and Victoria). There are $4$ kinds of burgers, $2$ of which are vegetarian-friendly. There are two of each kind of burger (and hence $8$ burgers total). How many ways are there to give one burger to each person if Vincent and Victoria are vegetarians (i.e.... | [
"132"
] | [
"At a party, there are five people (Alan, Andrew, Abby, Vincent, and Victoria). There are $4$ kinds of burgers, $2$ of which are vegetarian-friendly. There are two of each kind of burger (and hence $8$ burgers total). How many ways are there to give one burger to each person if Vincent and Victoria are vegetarians ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48f869e85b24bc0cf7ea1353a266e34305da8cdd7424e294bc83e4e14514927f | At a party, there are more than 3 people. Every four of the people have the property that one of the four is acquainted with the other three. Let $n$ be the total number of people at the party, and let $k$ be the number of people who are acquainted with all others except possibly three. If $n = 100$, find the minimum p... | [
"97"
] | [
"At a party, there are more than 3 people. Every four of the people have the property that one of the four is acquainted with the other three. Let $n$ be the total number of people at the party, and let $k$ be the number of people who are acquainted with all others except possibly three. If $n = 100$, find the mini... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-18206e1caf97ba01e859e6f22750b79798ad404ae1537856d172fab5cf859db6 | At a party, there are nine strangers. They wish to acquaint with each other, so they split into groups of three and talk with each other. After talking for an hour, they split into different groups of three and repeat the process. By the end of the party, all nine of them are acquainted with each other. In how many way... | [
"120960"
] | [
"At a party, there are nine strangers. They wish to acquaint with each other, so they split into groups of three and talk with each other. After talking for an hour, they split into different groups of three and repeat the process. By the end of the party, all nine of them are acquainted with each other. In how man... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-977da7462c1813c80a28f21357930e429d0f692c34389990ea53702ccdd75d25 | At a party, there were 12 couples. Each wife drank either 1 bottle of Coca-Cola or none. Each husband drank $2^k a_k$ bottles of Coca-Cola, where $a_k$ is the number of bottles drunk by his wife and $k$ is the pair number. It is known that all husbands together drank 2018 bottles. How many couples did not drink a singl... | [
"5"
] | [
"At a party, there were 12 couples. Each wife drank either 1 bottle of Coca-Cola or none. Each husband drank $2^k a_k$ bottles of Coca-Cola, where $a_k$ is the number of bottles drunk by his wife and $k$ is the pair number. It is known that all husbands together drank 2018 bottles. How many couples did not drink a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f10aa7305129bec4b870cfcc2e33a12207d386cc0412d0e8b911b5ff8c4ff87 | At a quiz show, there are three doors, with a prize hidden behind exactly one of them. You can ask the quizmaster whether the prize is behind the left-hand door or the right-hand door multiple times. The quizmaster can lie at most 10 times. Determine the smallest number of questions you can announce in advance to ensur... | [
"32"
] | [
"At a quiz show, there are three doors, with a prize hidden behind exactly one of them. You can ask the quizmaster whether the prize is behind the left-hand door or the right-hand door multiple times. The quizmaster can lie at most 10 times. Determine the smallest number of questions you can announce in advance to ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-926ba05fedb76a17e1285385a06b212dff19d99bb651ac1a733271e76e31c74f | At a raffle, 300 tickets are sold at $2 each for two prizes of $35 and $25. If you buy a ticket, what is the expected gain? | [
"\\(-1.8\\)"
] | [
"At a raffle, 300 tickets are sold at $2 each for two prizes of $35 and $25. If you buy a ticket, what is the expected gain?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-39fcba6e68e63cd4b6aa7a327cebe56e9da6f899e53cde1245228d9e4f6ff790 | At a raffle, 300 tickets are sold at $2 each for two prizes of $35 and $25. If you buy a ticket, what is the variance of the gain? | [
"6.1267"
] | [
"At a raffle, 300 tickets are sold at $2 each for two prizes of $35 and $25. If you buy a ticket, what is the variance of the gain?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9a09a0db31242d4a7ecee960e9433f9c1186e46fe2419c34d319c7e901f811fc | At a random moment within a period of 20 minutes student $A$ telephones student $B$, waits for 2 minutes and then puts down the receiver. During the same 20 minutes student $B$ arrives home at a random moment, stays for 5 minutes and then leaves. What is the probability that the two will have a talk? | [
"Approx. 0.314"
] | [
"At a random moment within a period of 20 minutes student $A$ telephones student $B$, waits for 2 minutes and then puts down the receiver. During the same 20 minutes student $B$ arrives home at a random moment, stays for 5 minutes and then leaves. What is the probability that the two will have a talk?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdf3a3cd9ec97e791560a36cd19861c906fc54e0724e3dabfceeb03b2d312042 | At a reception desk, every five minutes, three letters arrive and are stacked, and then the top two letters are removed. This process repeats until 36 letters have arrived and the top two letters are removed each time. Once all 36 letters have arrived, the top two letters continue to be removed every five minutes until... | [
"12:25"
] | [
"At a reception desk, every five minutes, three letters arrive and are stacked, and then the top two letters are removed. This process repeats until 36 letters have arrived and the top two letters are removed each time. Once all 36 letters have arrived, the top two letters continue to be removed every five minutes ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 2,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-407173d86a8f26c56c62b97624ca879ad82bad74f1e3828231265d60bee86518 | At a research facility, 97% of bags containing forbidden material trigger an alarm, and 15% of bags without forbidden material also trigger the alarm. If 1 out of every 1000 bags contains forbidden material, what is the probability that a bag triggering the alarm actually contains forbidden material? | [
"0.00640"
] | [
"At a research facility, 97% of bags containing forbidden material trigger an alarm, and 15% of bags without forbidden material also trigger the alarm. If 1 out of every 1000 bags contains forbidden material, what is the probability that a bag triggering the alarm actually contains forbidden material?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 4
} | 13 | 0 | 0 |
omr-ws-e61482db1eb04e5f4b451f4946c0f288c53b768b081838161703cc189ea2a8d8 | At a reunion of 201 people from 5 different countries, it is known that in each group of 6 people, at least two have the same age. Determine the minimum number of people who must have the same country, age, and sex. | [
"5"
] | [
"At a reunion of 201 people from 5 different countries, it is known that in each group of 6 people, at least two have the same age. Determine the minimum number of people who must have the same country, age, and sex."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbada7679575b26893d658d9514985fab6031f8784b88b9a77879dae8880bff3 | At a round table are $1994$ girls playing a deck of $n$ cards. Initially, one girl holds all of the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is holding at most one card. Find the minim... | [
"1994"
] | [
"At a round table are $1994$ girls playing a deck of $n$ cards. Initially, one girl holds all of the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is holding at most one card. Find the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-1ff31134249f46ed22a0967c923caa246a4a1b98014203858d2c487afe4f64cf | At a round table are 1994 girls playing with a deck of $n$ cards. Initially, one girl holds all of the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when each girl is holding at most one card. Let $f(n)$ be the number ... | [
"997"
] | [
"At a round table are 1994 girls playing with a deck of $n$ cards. Initially, one girl holds all of the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when each girl is holding at most one card.\nLet $f(n)$ be the n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-bf944abe594a51dd1ac941c6a00e4caf7aee87c7b297be967918f51d9a4d9428 | At a round table are 2002 people, playing a game with \( n \) cards. Initially, one person holds all the cards. In each turn, if at least one person holds at least 2 cards, one of these people must pass a card to each of her two neighbors. The game ends when each person is holding at most one card. If \( n < 2002 \), f... | [
"(n-1)^2"
] | [
"At a round table are 2002 people, playing a game with \\( n \\) cards. Initially, one person holds all the cards. In each turn, if at least one person holds at least 2 cards, one of these people must pass a card to each of her two neighbors. The game ends when each person is holding at most one card. If \\( n < 20... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a408eccd89f30a166a4d80fdf712ed84a56357501a80407edf81325edda6db69 | At a round table with 36 seats, 12 ghouls, 16 ghosts, and 8 humans are to be seated. Ghouls cannot sit directly adjacent to ghouls, ghosts cannot sit directly adjacent to humans, and humans cannot sit directly adjacent to ghouls nor themselves. What is the maximum number of creatures that can be seated? | [
"31"
] | [
"At a round table with 36 seats, 12 ghouls, 16 ghosts, and 8 humans are to be seated. Ghouls cannot sit directly adjacent to ghouls, ghosts cannot sit directly adjacent to humans, and humans cannot sit directly adjacent to ghouls nor themselves. What is the maximum number of creatures that can be seated?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95781371993f8c514b3626f52c5c115efd3cc5d160e365ddab7eece000d44809 | At a round table with \( n \) forum members, the number of members who have the same sex as their respective right neighbor equals the number of members who do not have the same sex as their respective right neighbor. Let \( n \) be the smallest possible number of forum members for which this condition holds. Calculate... | [
"4"
] | [
"At a round table with \\( n \\) forum members, the number of members who have the same sex as their respective right neighbor equals the number of members who do not have the same sex as their respective right neighbor. Let \\( n \\) be the smallest possible number of forum members for which this condition holds. ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ee671d72abe45d3111fcdf357e3300e2a7f04151ba987f47bf7da4afc3d683e | At a round table, 10 people are sitting, some of whom are knights (who always tell the truth) and the rest are liars (who always lie). There is at least one knight and at least one liar. What is the largest number of those sitting at the table who can say, "Both of my neighbors are knights"? (A statement that is at lea... | [
"9"
] | [
"At a round table, 10 people are sitting, some of whom are knights (who always tell the truth) and the rest are liars (who always lie). There is at least one knight and at least one liar. What is the largest number of those sitting at the table who can say, \"Both of my neighbors are knights\"? (A statement that is... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-4a1916b50eb1b5f6a3a12e116d292b2cb87762c72bbd45395e0ab0087958d147 | At a round table, Anna, Ben, and Chris pass and eat nuts. Initially, Anna has $N$ nuts (where $N > 3$) and divides them equally between Ben and Chris, eating any leftover nuts. Then Ben does the same with his pile, and Chris does the same with his pile. This process repeats. If $N$ is the smallest number such that exac... | [
"7"
] | [
"At a round table, Anna, Ben, and Chris pass and eat nuts. Initially, Anna has $N$ nuts (where $N > 3$) and divides them equally between Ben and Chris, eating any leftover nuts. Then Ben does the same with his pile, and Chris does the same with his pile. This process repeats. If $N$ is the smallest number such that... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-1ee6d404089b98422b10512118f2e428d1e022e86ad1c4f86f13f5b11e94cd01 | At a round table, Anna, Ben, and Chris pass and eat nuts. Initially, Anna has $N$ nuts (where $N > 3$) and divides them equally between Ben and Chris, eating any leftover nuts. Then Ben does the same with his pile, and Chris does the same with his pile. This process repeats. Let $f(N)$ be the minimum number of nuts rem... | [
"99"
] | [
"At a round table, Anna, Ben, and Chris pass and eat nuts. Initially, Anna has $N$ nuts (where $N > 3$) and divides them equally between Ben and Chris, eating any leftover nuts. Then Ben does the same with his pile, and Chris does the same with his pile. This process repeats. Let $f(N)$ be the minimum number of nut... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bd0d0bd6067ae1e14fe9916c2cb8064b1f88aa9f0c8c7308539dfae7f66ae79 | At a round table, there are 10 students. Each student thinks of a number and whispers it to their immediate neighbors. Each student then announces the arithmetic mean of the numbers they heard from their neighbors. The announced arithmetic means are $1, 2, 3, 4, 5, 6, 7, 8, 9,$ and $10$, respectively. Determine the num... | [
"1"
] | [
"At a round table, there are 10 students. Each student thinks of a number and whispers it to their immediate neighbors. Each student then announces the arithmetic mean of the numbers they heard from their neighbors. The announced arithmetic means are $1, 2, 3, 4, 5, 6, 7, 8, 9,$ and $10$, respectively. Determine th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-3dc529cdbf8cf3c0651cf4a51f299afa532b4c730ba9d04a08e0c24401559128 | At a round table, there are 1991 girls playing a game with \( n \) cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when each girl is holding at most one card. Let \( f(n) \) be th... | [
"995"
] | [
"At a round table, there are 1991 girls playing a game with \\( n \\) cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when each girl is holding at most one card. Let \\( f(n) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-64c804c7b45eaeebbf7c11854bb41efea51441f359d99f54bf0e7dff6f46e331 | At a round table, there are 2n Peruvians, 2n Bolivians, and 2n Ecuadorians. People whose neighbors are of the same nationality are asked to stand up. What is the largest possible number of people that can stand up? | [
"\\(6n-4\\)"
] | [
"At a round table, there are 2n Peruvians, 2n Bolivians, and 2n Ecuadorians. People whose neighbors are of the same nationality are asked to stand up. What is the largest possible number of people that can stand up?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c08af051a54421207a46aa872b1d0db34e0175b6406ef8d504366b92620b9597 | At a round table, there are 60 creatures consisting of elves and trolls. Trolls always lie, and elves always speak the truth except for two who made a mistake. Each creature claims to sit between an elf and a troll. Given that exactly two elves made a mistake, how many trolls are there at the table? | [
"20"
] | [
"At a round table, there are 60 creatures consisting of elves and trolls. Trolls always lie, and elves always speak the truth except for two who made a mistake. Each creature claims to sit between an elf and a troll. Given that exactly two elves made a mistake, how many trolls are there at the table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-bc499f49a1a3d06e1eda4621aad14be56a1cf5e10dd0505c9de72f05ee71150e | At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? | [
"\\(\\frac{8}{405 \\pi}\\)"
] | [
"At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-2f36588cbc6904d1e28b388d7a7a96840fcacaf9a8be0895b2c0d1a7f6392676 | At a school, 53.7802% (rounded) of the pupils like mathematics. What is the minimum number of pupils at the school? | [
"1283"
] | [
"At a school, 53.7802% (rounded) of the pupils like mathematics. What is the minimum number of pupils at the school?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 11
} | 14 | 0 | 0 |
omr-ws-1b207fb62524db98f912cb7539c05c877bf3ac1fdb6687b6250ad77cb8e9198a | At a school, 90% of students take maths, 85% take science, and 80% take English. What is the minimum percentage of students who take all three subjects? | [
"27.5%"
] | [
"At a school, 90% of students take maths, 85% take science, and 80% take English. What is the minimum percentage of students who take all three subjects?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-645aab15287d85616d54743a6344b414a33f8a414a736204fa095065a9c28fbf | At a school, all 60 students play on at least one of three teams: Basketball, Soccer, and Mathletics. 8 students play all three sports, half the students play basketball, and the ratio of the size of the math team to the size of the basketball team to the size of the soccer team is 4:3:2. How many students at the schoo... | [
"14"
] | [
"At a school, all 60 students play on at least one of three teams: Basketball, Soccer, and Mathletics. 8 students play all three sports, half the students play basketball, and the ratio of the size of the math team to the size of the basketball team to the size of the soccer team is 4:3:2. How many students at the ... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-4f7154a9378ca25815b9079079fb6e9f069e0ae494c1a77d0ffdebadc8073105 | At a school, each student divides 18 subjects into six disjoint triples such that every possible triple of subjects appears exactly once among the students' triples. Determine the minimum number of students required for this to be possible. | [
"136"
] | [
"At a school, each student divides 18 subjects into six disjoint triples such that every possible triple of subjects appears exactly once among the students' triples. Determine the minimum number of students required for this to be possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b518315b83645341942f1f20d372199f048ed92d978d202a46c0d007cd261e25 | At a school, each student divides 18 subjects into six disjoint triples. If every possible triple of subjects is among the triples of exactly one student, find the total number of students required to achieve this. | [
"136"
] | [
"At a school, each student divides 18 subjects into six disjoint triples. If every possible triple of subjects is among the triples of exactly one student, find the total number of students required to achieve this."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-665451b7f2ddf264f801effecee98b519ae7100d248ed8e16d541f9b03037eef | At a scientific conference with 2020 participants, no three delegates are mutually acquainted. There exists a positive integer \( n \) such that no delegate is acquainted with more than \( n \) other delegates, and there exists an integer \( 1 \leq k < n \) such that at least one delegate is acquainted with exactly \( ... | [
"1010"
] | [
"At a scientific conference with 2020 participants, no three delegates are mutually acquainted. There exists a positive integer \\( n \\) such that no delegate is acquainted with more than \\( n \\) other delegates, and there exists an integer \\( 1 \\leq k < n \\) such that at least one delegate is acquainted with... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 4
} | 8 | 0 | 0 |
omr-ws-fb4ff22540a244ddb305391e490fce6743bc7388267009eace694f05e43f1082 | At a ski race of 10000 m the second skier took the start some time later than the first skier, the speed of the second skier being 1 m/s higher than that of the first. At the moment when the second skier overtook the first, the first skier increased his speed by 2 m/s and the speed of the second skier remained the same... | [
"2 min"
] | [
"At a ski race of 10000 m the second skier took the start some time later than the first skier, the speed of the second skier being 1 m/s higher than that of the first. At the moment when the second skier overtook the first, the first skier increased his speed by 2 m/s and the speed of the second skier remained the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6ba80a0d35e23dc51b16ddb6d86d74c243d0e57dac6551a0154edb4969c3e55 | At a small party with fifteen door prizes, each prize is randomly given to one of the five guests. What is the probability that each of the five guests receives at least one door prize? | [
"\\(\\frac{1001}{5814}\\)"
] | [
"At a small party with fifteen door prizes, each prize is randomly given to one of the five guests. What is the probability that each of the five guests receives at least one door prize?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbc1e5f0a7ea9f50fd9e724d05a0656101b9384d6c22f2939837f4c49e38ed5f | At a socially-distanced Halloween party, two ghosts, Alpha and Demic, decide to scare the guests by playing a game on a geometric cake in the shape of a regular octagon with side length $s = 18\sqrt{2}$ inches. Every $2$nd side is designated as valid, and the two ghosts take turns cutting slices from the cake. The rule... | [
"200"
] | [
"At a socially-distanced Halloween party, two ghosts, Alpha and Demic, decide to scare the guests by playing a game on a geometric cake in the shape of a regular octagon with side length $s = 18\\sqrt{2}$ inches. Every $2$nd side is designated as valid, and the two ghosts take turns cutting slices from the cake. Th... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eaff74d9f3f274e5cfdc66ec4259559856a1b34ab544af086fec6604ede7f6b3 | At a speed of $50$ km/h, Anas' motorcycle can cover $18$ km per liter of gasoline. By lowering the speed from $60$ km/h to $50$ km/h, the distance traveled per liter increases by $3$ km. Determine how many liters of gasoline Anas can save on a trip of $180$ km if he lowers the speed from $60$ km/h to $50$ km/h. | [
"2"
] | [
"At a speed of $50$ km/h, Anas' motorcycle can cover $18$ km per liter of gasoline. By lowering the speed from $60$ km/h to $50$ km/h, the distance traveled per liter increases by $3$ km. Determine how many liters of gasoline Anas can save on a trip of $180$ km if he lowers the speed from $60$ km/h to $50$ km/h."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3ed7f3bb5a6854cfdbe0849a7fdbdead2dbb5ee2d20932219f08925d53dcb406 | At a state contest, 21 Mathletes stay in the same hotel. Each Mathlete gets his/her own room and the room numbers are 1 through 21. When all the Mathletes have arrived, except the ones staying in rooms 12 and 13, what is the median room number of the other 19 Mathletes? | [
"10"
] | [
"At a state contest, 21 Mathletes stay in the same hotel. Each Mathlete gets his/her own room and the room numbers are 1 through 21. When all the Mathletes have arrived, except the ones staying in rooms 12 and 13, what is the median room number of the other 19 Mathletes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3956b42b06ffb1396916e983eecb1b94e8d8300bd3525a205a2b1f7a63dae4df | At a station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, you walk to the station at a random time in the afternoon and wait for the first train in either direction. On av... | [
"74"
] | [
"At a station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. Each day, you walk to the station at a random time in the afternoon and wait for the first train in either direction. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-db1bd13dd6ce9970ae1e4765fc0ca5fcd0675b06c6faba6774f1ff098ae482c8 | At a station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. If you walk to the station at a random time in the afternoon and wait for the first train in either direction, on average, ... | [
"74"
] | [
"At a station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. If you walk to the station at a random time in the afternoon and wait for the first train in either direction, on aver... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6bda3f52a8298203406dda9e487c50959456cdae5fba7b00b6d490c5ac0666d | At a store, pencils cost 10 cents and erasers cost 25 cents. Given that Bob can spend a maximum of five dollars, let $C$ be the number of pencil-eraser combinations he can buy. What is the sum of the digits of $C$? | [
"10"
] | [
"At a store, pencils cost 10 cents and erasers cost 25 cents. Given that Bob can spend a maximum of five dollars, let $C$ be the number of pencil-eraser combinations he can buy. What is the sum of the digits of $C$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-ed82a7476742a766cc396a14445c7d1adebeebf49773b52b91189a76568f23bc | At a summer math program, a contest is held for $n$ teams, each composed of $n$ individuals. Each team is given the same $n$ problems to work on. On each team, there is one person who knows how to solve all $n$ problems, another who knows how to solve $n-1$ of the problems, and so on down to one member who only knows h... | [
"55"
] | [
"At a summer math program, a contest is held for $n$ teams, each composed of $n$ individuals. Each team is given the same $n$ problems to work on. On each team, there is one person who knows how to solve all $n$ problems, another who knows how to solve $n-1$ of the problems, and so on down to one member who only kn... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-a440ccaf7583688f8903d2254b53cc8a05b4abd2b6d5a7693fb1eb6717aeaef7 | At a summer school, there are 7 courses. Each participant was a student in at least one course, and each course was taken by exactly 40 students. It is known that for each pair of courses, there were at most 9 students who took both. Calculate the minimum possible number of students who participated in this summer scho... | [
"124"
] | [
"At a summer school, there are 7 courses. Each participant was a student in at least one course, and each course was taken by exactly 40 students. It is known that for each pair of courses, there were at most 9 students who took both. Calculate the minimum possible number of students who participated in this summer... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cac2bfb41a086458580c135bdb328977e157e69205e0f08b9ae9ab8abea0d6bf | At a table tennis tournament, each one of the $n \geq 2$ participants plays with all the others exactly once. Define a tournament as "circular" if the players can be labeled with the numbers $1, 2, \ldots, n$ such that player $1$ won player $2$, player $2$ won player $3$, ..., player $n-1$ won player $n$, and player $n... | [
"504"
] | [
"At a table tennis tournament, each one of the $n \\geq 2$ participants plays with all the others exactly once. Define a tournament as \"circular\" if the players can be labeled with the numbers $1, 2, \\ldots, n$ such that player $1$ won player $2$, player $2$ won player $3$, ..., player $n-1$ won player $n$, and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84322b32d5f0dcdfd6671d7d60f36f0bc7e81d1db5431830ca2e417b5e65efa2 | At a tennis tournament, there were $2n$ boys and $n$ girls participating. Every player played every other player, and there were no draws. The boys won $\frac{7}{5}$ times as many matches as the girls. Find $n$. | [
"\\( n = 3 \\)"
] | [
"At a tennis tournament, there were $2n$ boys and $n$ girls participating. Every player played every other player, and there were no draws. The boys won $\\frac{7}{5}$ times as many matches as the girls. Find $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27b3d46c2349489f561cee38902d61cad05cb689d01e608b57b8a192e4d3780e | At a tourist camp, each person has at least 50 and at most 100 friends among the other persons at the camp. Suppose t-shirts with at most 1331 different colors are handed out to every person such that any person has 20 friends whose t-shirts all have pairwise different colors. Calculate the minimum number of distinct c... | [
"20"
] | [
"At a tourist camp, each person has at least 50 and at most 100 friends among the other persons at the camp. Suppose t-shirts with at most 1331 different colors are handed out to every person such that any person has 20 friends whose t-shirts all have pairwise different colors. Calculate the minimum number of disti... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c7ec63aaa08c994a1fee6fd345b82bca83b72d7e4ce20dc5687f8b64d38e53f | At a train station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. If you walk to the train station at a random time in the afternoon and wait for the first train in either direction, ... | [
"74"
] | [
"At a train station, northbound trains arrive every three minutes starting at noon and finishing at midnight, while southbound trains arrive every five minutes starting at noon and finishing at midnight. If you walk to the train station at a random time in the afternoon and wait for the first train in either direct... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 64,
"tir": 4
} | 68 | 0 | 0 |
omr-ws-8092cdc9f5a1900dc3e84e9403ef45b20b3c75f047641f26f7ddb0c51aa860d3 | At a university dinner, there are 2017 mathematicians who each order two distinct entrees, with no two mathematicians ordering the same pair of entrees. The cost of each entree is equal to the number of mathematicians who ordered it, and the university pays for each mathematician’s less expensive entree (ties broken ar... | [
"127009"
] | [
"At a university dinner, there are 2017 mathematicians who each order two distinct entrees, with no two mathematicians ordering the same pair of entrees. The cost of each entree is equal to the number of mathematicians who ordered it, and the university pays for each mathematician’s less expensive entree (ties brok... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-117725b58668b471ca2759986d21be11153deb4a70c6f3a958b80f097fc7da86 | At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties broken ar... | [
"127009"
] | [
"At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties brok... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 8,
"tir": 12
} | 43 | 0 | 0 |
omr-ws-f4fb2ba6af8690e6d2f456a3763414d86d34af04c215a54d01b5bb1ff4af20a1 | At a university dinner, there are 2017 mathematicians, each ordering two distinct entrees such that no two mathematicians order the same pair of entrees. The cost of each entree is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entree. What is the m... | [
"127009"
] | [
"At a university dinner, there are 2017 mathematicians, each ordering two distinct entrees such that no two mathematicians order the same pair of entrees. The cost of each entree is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entree. What is ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-1e69499587abc3619f4a9ab7cad0213ac34f4c5b76ff798915576b4c883b2973 | At a university with 1995 students, if two students know each other, they have the same number of acquaintances within the university. Otherwise, they have different numbers of acquaintances. Determine the minimum number of acquaintances a student must have for this condition to hold true. Express your answer as an int... | [
"997"
] | [
"At a university with 1995 students, if two students know each other, they have the same number of acquaintances within the university. Otherwise, they have different numbers of acquaintances. Determine the minimum number of acquaintances a student must have for this condition to hold true. Express your answer as a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-c65f5b4b4647489d4e47ac563594cd58a5875248abc9e2f76b0fa8f332455303 | At a university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science, each with two male and two female professors. A committee of six professors is to be formed such that it contains three men and three women and includes two professors from each departmen... | [
"88"
] | [
"At a university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science, each with two male and two female professors. A committee of six professors is to be formed such that it contains three men and three women and includes two professors from each depa... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-f6f712b3589eb78d96f3154919d122cfe628165791b3501aa7512da0ba2704c7 | At a very crowded theme park, a line of \( n \) people must wait \( 2n - 1 \) seconds for the next person to enter the ride. There are 59 people in a line before I join. Assuming no one else joins the line, how many hours will I have to wait before I enter the ride? | [
"\\(\\frac{3481}{3600}\\)"
] | [
"At a very crowded theme park, a line of \\( n \\) people must wait \\( 2n - 1 \\) seconds for the next person to enter the ride. There are 59 people in a line before I join. Assuming no one else joins the line, how many hours will I have to wait before I enter the ride?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-ad4bef366dac5d35ca6644860736e3f1417feeae9898dcfd63c57d07d8267aed | At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point, and the losing team gets $0$ points. In the final ranking, only one team has the least number of points, and each team, except for the team with the least points, lost exactly one game a... | [
"0"
] | [
"At a volleyball tournament, each team plays exactly once against each other team. Each game has a winning team, which gets $1$ point, and the losing team gets $0$ points. In the final ranking, only one team has the least number of points, and each team, except for the team with the least points, lost exactly one g... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0de21eecd96434d4c8423e080eb3d4696f6dd14c36bf49cd07c642fc8df1fd46 | At a volleyball tournament, every two of the $n$ teams played against each other exactly once. For any two different teams $A$ and $B$, there were exactly $k$ teams that lost against both of them. If $k=5$, find the value of $n$. | [
"23"
] | [
"At a volleyball tournament, every two of the $n$ teams played against each other exactly once. For any two different teams $A$ and $B$, there were exactly $k$ teams that lost against both of them. If $k=5$, find the value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe83d8ef5c27a54891eca44bc6f55014d3cf7642e6c2ddc3f2b4f10bb6dca226 | At a wedding dinner, Mr. Jack invited 7 pairs of guests. Including Mr. Jack and his wife, everyone shook hands with others under the following conditions: a person can shake hands with another person only once, cannot shake hands with their partner, and all guests (including Mr. Jack's wife) gave different answers when... | [
"7"
] | [
"At a wedding dinner, Mr. Jack invited 7 pairs of guests. Including Mr. Jack and his wife, everyone shook hands with others under the following conditions: a person can shake hands with another person only once, cannot shake hands with their partner, and all guests (including Mr. Jack's wife) gave different answers... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-584ededf329f21b50e6326c6f2f804a6c0a1b8b83a4dbfbf2be9e99cdd12d4b1 | At a wedding, every guest who gives a positive integer \( n \) that divides \( 3^n - 3 \) but does not divide \( 2^n - 2 \) receives a prize. Find the sum of all such integers \( n \) that are less than or equal to 100. | [
"163"
] | [
"At a wedding, every guest who gives a positive integer \\( n \\) that divides \\( 3^n - 3 \\) but does not divide \\( 2^n - 2 \\) receives a prize. Find the sum of all such integers \\( n \\) that are less than or equal to 100."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f2e266cc9014004e4b33d9189daf819d4d9131b91d579029054f7a4f8a6ac606 | At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred? | [
"12"
] | [
"At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-956a3580860bb6e9bd54bdc15a473e3e340f3fc3c025e69564fc8cd20c947051 | At a zoo, animals are assigned into many different groups such that each group has at least two animals and every group shares at least one animal with every other group. Determine the number of ways to partition the animals into exactly three "splits" such that each group is represented in at least two splits, and eac... | [
"12"
] | [
"At a zoo, animals are assigned into many different groups such that each group has at least two animals and every group shares at least one animal with every other group. Determine the number of ways to partition the animals into exactly three \"splits\" such that each group is represented in at least two splits, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57d0efdbb0d6b087528c29d479c1158995e8e5d493a7d3bcbae46f0496a9676b | At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. A person is called [i]junior[/i] if they are younger than all their friends, and [i]senior[/i] if they are older than all their friends. A person with no friends is both [i]junior[/i] and [i]senior[/i].... | [
"2"
] | [
"At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. A person is called [i]junior[/i] if they are younger than all their friends, and [i]senior[/i] if they are older than all their friends. A person with no friends is both [i]junior[/i] and [i]senior... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-64827c1334937843f2b9f7752bf349b13e665f65f391c325dbb1106a2876c450 | At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? | [
"\\dfrac{5}{29}"
] | [
"At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1ddaae3e6ea2cafa2074455ae68abb6ac0dc855e080efeefde421e240a372ef2 | At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportio... | [
"\\dfrac{5}{29}"
] | [
"At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What prop... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ad4c32cc73ce5a3d711276a133274b5ba934bae2cadebc06b858db2a9ced4b57 | At an airport, the number of flights arriving at a time follows a Poisson distribution with a mean of 5. What is the probability that the time duration until the third flight arrives is at most two hours? | [
"1"
] | [
"At an airport, the number of flights arriving at a time follows a Poisson distribution with a mean of 5. What is the probability that the time duration until the third flight arrives is at most two hours?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-330806266efed2521ffc679f2e67f9d99d552c3bfe8dd7d7dc5db1611a8b4a79 | At an artisan bakery, French tortes are $52$ dollars, almond tarts are $12$ dollars, and cookies are one dollar each. If Alex has $400$ dollars to purchase exactly $100$ of these items for a party and he buys at least one of each item, how many of each bakery item does he purchase? | [
"\\( f = 2 \\), \\( a = 18 \\), \\( c = 80 \\)"
] | [
"At an artisan bakery, French tortes are $52$ dollars, almond tarts are $12$ dollars, and cookies are one dollar each. If Alex has $400$ dollars to purchase exactly $100$ of these items for a party and he buys at least one of each item, how many of each bakery item does he purchase?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 31
} | 43 | 0 | 0 |
omr-ws-9de53c4b9e9b400cb6f6ad2a4bb236482930eb25f02cfe78e794121a9a6200be | At an award giving ceremony, 36 medals were handed out. The total price of the medals was a 5-digit number with the three middle digits all being 6. If the price of each medal was the same integer, what was the amount spent for each medal? Find the sum of all possible answers. | [
"169992"
] | [
"At an award giving ceremony, 36 medals were handed out. The total price of the medals was a 5-digit number with the three middle digits all being 6. If the price of each medal was the same integer, what was the amount spent for each medal? Find the sum of all possible answers."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb2b5b0a12ce22e1302744a1c948e73e6ae6bec20681090571810ff83747a168 | At an international conference with 4 official languages, each pair of attendees can converse in one of these languages. If the total number of attendees is 100, what is the minimum number of attendees that must speak the most commonly spoken language? | [
"51"
] | [
"At an international conference with 4 official languages, each pair of attendees can converse in one of these languages. If the total number of attendees is 100, what is the minimum number of attendees that must speak the most commonly spoken language?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9046ac939536bd91953fc199cce338e71b6ce7ff809c4c8f7dc561765bdc4a8a | At an international conference with four official languages, any two participants can communicate in one of these languages. If there are 100 participants, find the minimum number of participants who can speak the same language. | [
"51"
] | [
"At an international conference with four official languages, any two participants can communicate in one of these languages. If there are 100 participants, find the minimum number of participants who can speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36ec7d66f209326a35a030485cacd166c49296fd47d1ad782a278a510b9f8f95 | At an international meeting with $1985$ participants, in each set of three participants, there are at least two who speak the same language. Given that no one speaks more than five languages, find the minimum number of participants who speak the same language. | [
"200"
] | [
"At an international meeting with $1985$ participants, in each set of three participants, there are at least two who speak the same language. Given that no one speaks more than five languages, find the minimum number of participants who speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.