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-186ed5f67be72b8deb026ed9c4a0a4f37d9ecb9940c38af060653ba96f284238 | 100,000 sailors are arranged in a circle. Every fourth sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining? | [
"37858"
] | [
"100,000 sailors are arranged in a circle. Every fourth sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-0340690493748d9e50ab4076112a35c36f01fcd7fc4aef6927db4b017670c096 | 100,000 sailors are arranged in a circle. Every other sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining? | [
"68929"
] | [
"100,000 sailors are arranged in a circle. Every other sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-6e8732f8663183de6e48ffa18c12162c7f955db92b0109f8e141039b6b310a40 | 100,000 sailors are arranged in a circle. Every third sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining? | [
"89644"
] | [
"100,000 sailors are arranged in a circle. Every third sailor is shoved overboard until only one remains. Where should you stand to be the last one remaining?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-692e8fadf798a9c14a3319b092da0d53a5226c57aa1a02a3b2eaba72e914e0ef | 1000 name tags, each with a unique number from 1 to 1000, are distributed to 1000 mathematicians sitting randomly around a circular table. Each mathematician replaces the number on their name tag with the floor of the average of the numbers on their neighbors' name tags. Compute the maximum number of full rounds around... | [
"500"
] | [
"1000 name tags, each with a unique number from 1 to 1000, are distributed to 1000 mathematicians sitting randomly around a circular table. Each mathematician replaces the number on their name tag with the floor of the average of the numbers on their neighbors' name tags. Compute the maximum number of full rounds a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-9dac342ba9f37c6c2d76216fa527bb0ac1c0cd0da4860012278924a09f88ec8c | 1000 non-zero numbers are written around a circle, with every other number underlined. Each underlined number is equal to the sum of its two neighbors, and each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be? | [
"375"
] | [
"1000 non-zero numbers are written around a circle, with every other number underlined. Each underlined number is equal to the sum of its two neighbors, and each non-underlined number is equal to the product of its two neighbors. What could the sum of all the numbers written on the circle be?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-acae1211d9bba8a16fa3b8af77501d454946133a5865d4f0859b266f1e8e6287 | 10000 points are marked on a circle and numbered clockwise from 1 to 10000. The points are divided into 5000 pairs, and the points of each pair are joined by a segment such that each segment intersects exactly one other segment. Each segment is labeled with the product of the numbers at its endpoints. Calculate the rem... | [
"0"
] | [
"10000 points are marked on a circle and numbered clockwise from 1 to 10000. The points are divided into 5000 pairs, and the points of each pair are joined by a segment such that each segment intersects exactly one other segment. Each segment is labeled with the product of the numbers at its endpoints. Calculate th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff7167cdde43a783a33360f8e80cdb94ebc0d45c458f70e01a39eef8dc16e1c9 | 10005 children, numbered 1 to 10005, sit around a circle in order. Each child has an integer in hand. The child numbered 1 has integer 1. The child numbered 2 has integer 2. The child numbered 3 has integer 3. Given that the sum of the integers of any 1999 consecutive children is equal to 2018. What is the integer held... | [
"4"
] | [
"10005 children, numbered 1 to 10005, sit around a circle in order. Each child has an integer in hand. The child numbered 1 has integer 1. The child numbered 2 has integer 2. The child numbered 3 has integer 3. Given that the sum of the integers of any 1999 consecutive children is equal to 2018. What is the integer... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-909f16e2eefd22277e9783148e6057a79624e1af0081851732318c7d56b99dbf | 101 cards are marked from 1 to 101, and 2 are drawn at random. Find the probability that both cards drawn are non-composite numbers. | [
"\\(\\frac{351}{5050}\\)"
] | [
"101 cards are marked from 1 to 101, and 2 are drawn at random. Find the probability that both cards drawn are non-composite numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-0815d843e368d98ef89c3c347cfa007b9240cef812fb020a7e9d3af644c8a7f1 | 101 numbers are written on a blackboard: \(1^2, 2^2, 3^2, \cdots, 101^2\). Alex chooses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number. | [
"1"
] | [
"101 numbers are written on a blackboard: \\(1^2, 2^2, 3^2, \\cdots, 101^2\\). Alex chooses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-37b364732c317a4876f7aa0cca68a6edf7989e576503e076057bcd8d6d8bcde7 | 101 wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself.... | [
"50"
] | [
"101 wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion him... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b1412562bbbe7052210bd63b0676609b938e5bf52c47bfccca52cb14cda0ba87 | 1024 Duke students, indexed 1 to 1024, are having a chat. For each \(1 \leq i \leq 1023\), student \(i\) claims that student \(2^{\lfloor \log_2 i \rfloor + 1}\) is lying. Given that exactly 201 people are lying, find the index of the 61st liar (ordered by index from smallest to largest). | [
"119"
] | [
"1024 Duke students, indexed 1 to 1024, are having a chat. For each \\(1 \\leq i \\leq 1023\\), student \\(i\\) claims that student \\(2^{\\lfloor \\log_2 i \\rfloor + 1}\\) is lying. Given that exactly 201 people are lying, find the index of the 61st liar (ordered by index from smallest to largest)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-7a013e01e0c9a3250fc483d0f2c2808fe7a88a8fa74be31f55a9b464bcff2b28 | 11 people will learn 11 languages. The teacher can teach two people two languages in each lesson. What is the minimum number of lessons required for 11 people to learn all 11 languages? (One person can learn the same language several times.) | [
"31"
] | [
"11 people will learn 11 languages. The teacher can teach two people two languages in each lesson. What is the minimum number of lessons required for 11 people to learn all 11 languages? (One person can learn the same language several times.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ad9b5398907a7b4b51a53f51a4b28820294d9754df7aa42127b03c038071c8be | 11 politicians have formed a number of committees. Every committee has exactly 6 members. The first ten politicians are in exactly 7 committees each. If the eleventh politician is in exactly n committees, what is the sum of all possible values of n? | [
"24"
] | [
"11 politicians have formed a number of committees. Every committee has exactly 6 members. The first ten politicians are in exactly 7 committees each. If the eleventh politician is in exactly n committees, what is the sum of all possible values of n?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-3e406018df4dd2d2d0414c40d85e0d4cdb18322428dd78582f82a1c890ac197b | 12 balls labeled with the integers 1 through 12 are placed in a box. Alexandra randomly takes out 3 of them, sets aside the largest, and repeats this procedure (without replacement) until there are no balls left in the box. What is the probability that the 4 balls set aside are labeled 9, 10, 11, and 12? | [
"\\dfrac{9}{55}"
] | [
"12 balls labeled with the integers 1 through 12 are placed in a box. Alexandra randomly takes out 3 of them, sets aside the largest, and repeats this procedure (without replacement) until there are no balls left in the box. What is the probability that the 4 balls set aside are labeled 9, 10, 11, and 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-8017bfccb9370c6cc0478179a107cb0498ea6cfd06ceeafb7521f3854ebf8bd9 | 12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn? | [
"1320"
] | [
"12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-762349d8165f547ec4eeccb4f8b5528a36443cf98d5fa1bb89fbb549d28a236c | 12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible? | [
"1728"
] | [
"12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72c433cc00eea5233b9f8247dcbfe1c0e68861f0a2624e314eb527f51624b63b | 12 knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. 5 knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice? | [
"36"
] | [
"12 knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. 5 knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cfa908a4d984f65d7ed985536458b6c90c99946733b9194e955bfa09d9bf0a06 | 12 people are to be seated on a square table, with three on each side. Two people wish to sit on the North side, two on the East side, and one person insists on occupying the middle seat on any side. Find the number of ways they can be seated. | [
"\\(6 \\cdot 16 \\cdot 7!\\)"
] | [
"12 people are to be seated on a square table, with three on each side. Two people wish to sit on the North side, two on the East side, and one person insists on occupying the middle seat on any side. Find the number of ways they can be seated."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 22
} | 42 | 0 | 0 |
omr-ws-54f142b55aa3ce386ca8a87ab6c5d0c0b2ede6eb78aa7370ed8afb371b83c92e | 12 people, including Alice and Bob, are standing in a circle. Alice leaves the circle first, and each subsequent second, one of the two people who were in the previous second adjacent to the person who just left (chosen at random) leaves, until there is one last person remaining, in which they also leave. If Alice and ... | [
"0"
] | [
"12 people, including Alice and Bob, are standing in a circle. Alice leaves the circle first, and each subsequent second, one of the two people who were in the previous second adjacent to the person who just left (chosen at random) leaves, until there is one last person remaining, in which they also leave. If Alice... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-a7dfc396de963d8b6c6594952e4dd4a8d92be44cba8e4637de45c6c9b17b0da4 | 120 identical balls are tightly stacked in the form of a regular triangular pyramid. How many balls lie at the base of the pyramid? | [
"36"
] | [
"120 identical balls are tightly stacked in the form of a regular triangular pyramid. How many balls lie at the base of the pyramid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef7fcbae372f48a8d51010349c0895d9b6a2d67577cf2ccd9cbe1342dc01a67d | 120 nickels are arranged in a row. Then, every second coin is replaced with a dime, every third coin is replaced with a 20-cent coin, every fourth coin is replaced with a half-dollar, and every fifth coin is replaced with a dollar coin. What is the total value of all the coins, in dollars? | [
"22.00"
] | [
"120 nickels are arranged in a row. Then, every second coin is replaced with a dime, every third coin is replaced with a 20-cent coin, every fourth coin is replaced with a half-dollar, and every fifth coin is replaced with a dollar coin. What is the total value of all the coins, in dollars?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41533a2ee76bcadb9390347be4c93737f8d1e871dbd3a69abc13a6c213db3b3d | 120 pirates distribute 119 gold pieces among themselves. The captain repeatedly checks if any pirate has 15 or more gold pieces. If such a pirate is found, they must give all their gold pieces to other pirates, with no pirate receiving more than one gold piece. Calculate the maximum number of gold pieces any single pir... | [
"14"
] | [
"120 pirates distribute 119 gold pieces among themselves. The captain repeatedly checks if any pirate has 15 or more gold pieces. If such a pirate is found, they must give all their gold pieces to other pirates, with no pirate receiving more than one gold piece. Calculate the maximum number of gold pieces any singl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87349a1869ff44c119abfee2d1952b4d505ca9211a7e4e57ea39212047a9d724 | 14 people named One, Two, ..., Fourteen are seated clockwise in order on a table such that from the viewpoint of Seven, Six is to his right, Eight is to his left, and Ten is directly in front of him. They pass a ball around, with each person passing the ball 3 persons to their left. The game starts with One holding the... | [
"Eleven"
] | [
"14 people named One, Two, ..., Fourteen are seated clockwise in order on a table such that from the viewpoint of Seven, Six is to his right, Eight is to his left, and Ten is directly in front of him. They pass a ball around, with each person passing the ball 3 persons to their left. The game starts with One holdin... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-5118e6a7bdd042d5761b6dab1ae29163d0374a43d036e9436d1ce5c0330c799a | 150 workers were initially engaged in a piece of work. Starting from the second day, 4 workers dropped out each day. It takes 8 more days to complete the work due to the dropouts. Find the total number of days in which the work is completed. | [
"25"
] | [
"150 workers were initially engaged in a piece of work. Starting from the second day, 4 workers dropped out each day. It takes 8 more days to complete the work due to the dropouts. Find the total number of days in which the work is completed."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b401537249cc7b4349a53596ef4b0a80b2c178c26d8fd7fa4affc945735d044d | 155 birds $P_1, \ldots, P_{155}$ are sitting on the boundary of a circle $C$. Two birds $P_i, P_j$ are mutually visible if the angle at the center $m(P_iP_j) \leq 10^\circ$. Find the smallest number of mutually visible pairs of birds, i.e., the minimal set of pairs $\{x,y\}$ of mutually visible pairs of birds with $x,y... | [
"620"
] | [
"155 birds $P_1, \\ldots, P_{155}$ are sitting on the boundary of a circle $C$. Two birds $P_i, P_j$ are mutually visible if the angle at the center $m(P_iP_j) \\leq 10^\\circ$. Find the smallest number of mutually visible pairs of birds, i.e., the minimal set of pairs $\\{x,y\\}$ of mutually visible pairs of birds... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ffc5d8afae643718b23cd3c811984c23faa61563a2b5e44ef00ab2a41cda1d3b | 16 cards are marked from 1 to 16 and 2 are drawn at random. Find the probability that the product of the two cards drawn has an odd number of factors. | [
"\\(\\frac{1}{15}\\)"
] | [
"16 cards are marked from 1 to 16 and 2 are drawn at random. Find the probability that the product of the two cards drawn has an odd number of factors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b62abbc1fc6905c6b8115733234b3c9560ca15131b0b54d4c0865c2544d76b79 | 16 white and 4 red balls that are not identical are distributed randomly into 4 boxes which contain at most 5 balls. What is the probability that each box contains exactly 1 red ball? | [
"\\(\\frac{5^4}{\\binom{20}{4}}\\)"
] | [
"16 white and 4 red balls that are not identical are distributed randomly into 4 boxes which contain at most 5 balls. What is the probability that each box contains exactly 1 red ball?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-008a7cbeaf78c5f44fa9f630de712158e52f26e691fcd117235c27e8131be134 | 17 points are spaced at equal intervals around a unit circle. From one point, segments are drawn to every other point. Find the sum of the squares of the lengths of these segments. | [
"34"
] | [
"17 points are spaced at equal intervals around a unit circle. From one point, segments are drawn to every other point. Find the sum of the squares of the lengths of these segments."
] | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-e280926e274a307258136e3c10d06e8fbfd1cd55cc11ce0ac2cee00d3534f336 | 19 cards are marked from 1 to 19 and 2 are drawn at random. Find the probability that one card drawn is a multiple of 2 and the other is a prime. | [
"\\dfrac{71}{171}"
] | [
"19 cards are marked from 1 to 19 and 2 are drawn at random. Find the probability that one card drawn is a multiple of 2 and the other is a prime."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-773e1c817490207af626efa14dcf02632b6aed36ae7139c7661995a95a8ca551 | 1991 girls are seated at a round table. Initially, one girl holds \( n \) tokens. Each turn, a girl who is holding more than one token passes one token to each of her neighbors. For \( n \leq 1991 \), find the maximum number of turns \( T(n) \) such that the game does not terminate. Calculate \( T(1991) \). | [
"990025"
] | [
"1991 girls are seated at a round table. Initially, one girl holds \\( n \\) tokens. Each turn, a girl who is holding more than one token passes one token to each of her neighbors. For \\( n \\leq 1991 \\), find the maximum number of turns \\( T(n) \\) such that the game does not terminate. Calculate \\( T(1991) \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-6fe5baff0c9de2a2977cf2c128d87e7721dffe197874ed7418c4820d18bc1ab3 | 1994 girls are seated at a round table. Initially, one girl holds \( n \) tokens. Each turn, a girl who is holding more than one token passes one token to each of her neighbors. If \( n = 1994 \), after how many turns will the distribution of tokens first return to the initial state where one girl has all the tokens an... | [
"1994"
] | [
"1994 girls are seated at a round table. Initially, one girl holds \\( n \\) tokens. Each turn, a girl who is holding more than one token passes one token to each of her neighbors. If \\( n = 1994 \\), after how many turns will the distribution of tokens first return to the initial state where one girl has all the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-ccb97e978f19500b43b33199e4542586b6a016087714e124b14504418d7e5fac | 2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon? | [
"\\dfrac{5}{13}"
] | [
"2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e39c69d7b4d6ea4f54822eae8b792a286903613ed397c5ffd2fb455f86accc0 | 2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon? | [
"\\dfrac{14}{39}"
] | [
"2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-276de4aafbe45168758a62b760b7c88716825502fe3a10e5b855a2ddb9187efa | 2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent? | [
"\\frac{2}{7}"
] | [
"2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e94aec99d706119326a07d01034d76936a7c13f51552ec1f8ebcb89bea49b67d | 2 red balls, 2 blue balls, and 6 yellow balls are in a jar. Zion picks 4 balls from the jar at random. What is the probability that Zion picks at least 1 red ball and 1 blue ball? | [
"\\dfrac{17}{42}"
] | [
"2 red balls, 2 blue balls, and 6 yellow balls are in a jar. Zion picks 4 balls from the jar at random. What is the probability that Zion picks at least 1 red ball and 1 blue ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c96be6b6b9358dd17fec20ef39dc398ae97f4d7a4b4012f9a59daa9fee97c340 | 20 chairs are set in a row. 5 people randomly sit in the chairs (no more than one person to a chair). What is the probability that nobody is sitting next to anybody else? | [
"\\(\\frac{91}{323}\\)"
] | [
"20 chairs are set in a row. 5 people randomly sit in the chairs (no more than one person to a chair). What is the probability that nobody is sitting next to anybody else?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-05c0dc906c0e6a89d5c8ba83fa9b938b0293716a1765629306546d75a3f36595 | 20 students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? | [
"\\dfrac{40}{57}"
] | [
"20 students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c321519a1d24488346cbdb5517882e01daa9bd57084c9fa14555b3d853464c5b | 20 tonnes of rice can feed 300 soldiers for 72 days. For how many days can 32 tonnes of rice feed 540 soldiers? | [
"64"
] | [
"20 tonnes of rice can feed 300 soldiers for 72 days. For how many days can 32 tonnes of rice feed 540 soldiers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-271f9fec5860742a4433e683c96ee2d808a54cb0aa962fbe9320ae64acf390b8 | 2006 equilateral triangles are located in a square with side length 1. The sum of their perimeters is equal to 300. Let $N$ be the maximum number of these triangles that can have a common point. Find the minimum possible value of $N$. | [
"3"
] | [
"2006 equilateral triangles are located in a square with side length 1. The sum of their perimeters is equal to 300. Let $N$ be the maximum number of these triangles that can have a common point. Find the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-14c32b70476d5bf92c813245f3f968ae81ee5746d755a484bf4b0ef7c317e9e3 | 2006 points are marked on a circle. Vasya wants to draw two intersecting chords with vertices at these points. In how many ways can he do this? | [
"\\(\\binom{2006}{4}\\)"
] | [
"2006 points are marked on a circle. Vasya wants to draw two intersecting chords with vertices at these points. In how many ways can he do this?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 32
} | 80 | 0 | 0 |
omr-ws-8a09c674593c6ee114f236ba0e00f1b270c455be636c4fe07affe2b700246821 | 2009 unit cubes are glued together to form a solid cube. What is the maximum surface area of the figure? | [
"8038"
] | [
"2009 unit cubes are glued together to form a solid cube. What is the maximum surface area of the figure?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-668db046acfb520984546257843d1ac4277bc59e388aebf598438a3768b8bd39 | 2010 real numbers are written on the circumference of a circle. The sum of all of them is positive. You are allowed to take any triple of consecutive numbers and perform the transformation: \[ (X, Y, Z) \longrightarrow (X + Y, -Y, Z + Y) \] Find the minimum number of transformations required to ensure all the numbers a... | [
"1005"
] | [
"2010 real numbers are written on the circumference of a circle. The sum of all of them is positive. You are allowed to take any triple of consecutive numbers and perform the transformation: \n\\[\n(X, Y, Z) \\longrightarrow (X + Y, -Y, Z + Y)\n\\]\nFind the minimum number of transformations required to ensure all ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-f104880667a55e31c3f382bfead1397ae9e01003e1a50a74eed8ba1baf9a8376 | 2010 ships deliver bananas, lemons, and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Calculate the remainder when... | [
"0"
] | [
"2010 ships deliver bananas, lemons, and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Calculate the remainder... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-706e3dd0f0dace5d007874c33dd929cccdc9f53f648b972f21c000499794a768 | 2011 people line up such that no two people have the same height. Bob is the 27th tallest person. Let \( n \) be the number of ways to arrange these 2011 people in a line such that Bob is shorter than everyone in front of him. Find the product of the 2010 distinct positive integers that uniquely determine \( n \). | [
"\\dfrac{2011!}{1985}"
] | [
"2011 people line up such that no two people have the same height. Bob is the 27th tallest person. Let \\( n \\) be the number of ways to arrange these 2011 people in a line such that Bob is shorter than everyone in front of him. Find the product of the 2010 distinct positive integers that uniquely determine \\( n ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0b5287316e91d84fa3c7b09280f542055f93d8cc8c429241e8afc36bd1a897e0 | 2011 points are evenly spaced in a circle. Let \( P \) be the number of ways of choosing 3 of these points to form an obtuse angle. How many positive divisors does \( P \) have? | [
"128"
] | [
"2011 points are evenly spaced in a circle. Let \\( P \\) be the number of ways of choosing 3 of these points to form an obtuse angle. How many positive divisors does \\( P \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8520865407f215378d64e4e56d69922ae890bd93fd7781584c93e58185517468 | 2020 people stand in a circle, all holding their right hand out in front of them, towards the center of the circle. At once, they all instantly turn to face either the person on the left or the person on their right, each with 50% probability. If two people are now facing each other, they will shake hands. What is the ... | [
"505"
] | [
"2020 people stand in a circle, all holding their right hand out in front of them, towards the center of the circle. At once, they all instantly turn to face either the person on the left or the person on their right, each with 50% probability. If two people are now facing each other, they will shake hands. What is... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0cf1427ee6f4a42bf1034356ba5a2d3485770e3aab7e1680c0ac01d5aff7ca75 | 2023 tokens are placed along a line such that the distance between every two neighboring tokens is 1. In each step, two non-neighboring tokens are moved one unit closer to each other. After a finite number of steps, the procedure stops. Calculate the sum of all possible distinct distances between the leftmost and right... | [
"1023132"
] | [
"2023 tokens are placed along a line such that the distance between every two neighboring tokens is 1. In each step, two non-neighboring tokens are moved one unit closer to each other. After a finite number of steps, the procedure stops. Calculate the sum of all possible distinct distances between the leftmost and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-7809b8b483c16e7cad4c319bbe5c08bd684ec96e2f3a135e381c2c1c0cfc7642 | 204 pirates are going to split 100 coins among themselves. The first pirate proposes a way to split the coins, and the proposal is approved if and only if at least half the pirates vote 'yes'. If the proposal is not approved, the proposing pirate is thrown into the sea. Each pirate wants to stay alive, get as many coin... | [
"0\n\nExplanation:\nThe discussion in the forum indicates that the first pirate needs to secure at least half of the votes to survive. With 204 pirates, he needs 102 votes. To secure these votes, he must bribe 101 other pirates (since he will have his own vote). Given that there are only 100 coins, it is impossible... | [
"204 pirates are going to split 100 coins among themselves. The first pirate proposes a way to split the coins, and the proposal is approved if and only if at least half the pirates vote 'yes'. If the proposal is not approved, the proposing pirate is thrown into the sea. Each pirate wants to stay alive, get as many... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-44fbae962ca9ba825065d89397771497d487e9abff2e1e80dde85413053a75df | 21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on? | [
"6"
] | [
"21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-217a54777068e66a97df5d9b3079f3c08901adf1eec74ea29a0cf369935a7a39 | 22 points are equally spaced around a circle and numbered consecutively. A line is drawn through the center of the circle and passes between points 7 and 8. What is the sum of the other two consecutive numbers that the line passes between? | [
"37"
] | [
"22 points are equally spaced around a circle and numbered consecutively. A line is drawn through the center of the circle and passes between points 7 and 8. What is the sum of the other two consecutive numbers that the line passes between?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-63ada7230465fa2d153cb25e092084408a5013a4d6d1881c16a59c42a8c2bd85 | 23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes? | [
"23"
] | [
"23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-875535f243f7adbfa82b728b78a3ea904f0db4c931657c9e20af2064a80be2e1 | 23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once? | [
"253"
] | [
"23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9c8342fbe00287a58a86827e223787fcf778d0f1313e9346c2985674fc7dacf | 25 knights are seated at a round table, and 3 are chosen at random. Find the probability that at least two of the chosen 3 are sitting next to each other. | [
"\\(\\frac{11}{46}\\)"
] | [
"25 knights are seated at a round table, and 3 are chosen at random. Find the probability that at least two of the chosen 3 are sitting next to each other."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-39f5adfe22315cb98c3385a8f6841d7202873cfef383a8050dc06dd3a786c3d4 | 25 men sit around a circular table. Every hour, there is a vote where each must respond "yes" or "no". Each man's response on the $(n+1)^{\text{th}}$ vote depends on his response and his neighbors' responses on the $n^{\text{th}}$ vote as follows: if his response is the same as at least one of his neighbors, he will re... | [
"12"
] | [
"25 men sit around a circular table. Every hour, there is a vote where each must respond \"yes\" or \"no\". Each man's response on the $(n+1)^{\\text{th}}$ vote depends on his response and his neighbors' responses on the $n^{\\text{th}}$ vote as follows: if his response is the same as at least one of his neighbors,... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-47ca5180efc86aae82cf9ce45d50351e8408dbb7e7d5b7349398b872277296c2 | 25 men sit around a circular table. Every hour, there is a vote, and each must respond "yes" or "no". Each man behaves as follows: on the $n^{\text{th}}$ vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\text{th}}$ vote... | [
"12"
] | [
"25 men sit around a circular table. Every hour, there is a vote, and each must respond \"yes\" or \"no\". Each man behaves as follows: on the $n^{\\text{th}}$ vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{\\text... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97b1c344e72e7e82e9e20e34b6e00ec3aae00fd44450aa0c231562ae93d50ea2 | 25 men sit around a circular table. Every hour, there is a vote, and each must respond 'yes' or 'no'. Each man behaves as follows: on any given vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the next vote; if his response is diff... | [
"12"
] | [
"25 men sit around a circular table. Every hour, there is a vote, and each must respond 'yes' or 'no'. Each man behaves as follows: on any given vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the next vote; if his response is... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-4403c0f4db8d3709d6fe0fa709566bc6eaa8ae2337c27e9c8ecf7aa9696798c3 | 25% of AoPSers thought problem 7 was confusing. 45% of AoPSers thought problem 8 was confusing. 35% thought problem 9 was confusing. 60% of those who thought problem 7 was confusing also thought at least one of problem 8 and/or 9 was confusing, and 80% of those who thought problem 9 was confusing also thought problem 8... | [
"27"
] | [
"25% of AoPSers thought problem 7 was confusing. 45% of AoPSers thought problem 8 was confusing. 35% thought problem 9 was confusing. 60% of those who thought problem 7 was confusing also thought at least one of problem 8 and/or 9 was confusing, and 80% of those who thought problem 9 was confusing also thought prob... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d8e0fe8ec49e912dc47062e8f10f5ee3de92fa7202e5a77a9e5ca14edd504b56 | 26 distinct positive integers consisting of the digits 1 and 2 satisfy: Neither can be obtained by deleting the consecutive digits from the other. Calculate the minimum value of the sum of the digits of these numbers. | [
"185"
] | [
"26 distinct positive integers consisting of the digits 1 and 2 satisfy: Neither can be obtained by deleting the consecutive digits from the other. Calculate the minimum value of the sum of the digits of these numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-0ebce946d3141a8057016eb026b5a371366ef73621db09de8c764da4a3737312 | 26 points are evenly spaced around a circle and are labeled \( A \) through \( Z \) in alphabetical order. Triangle \(\triangle LMT\) is drawn. Three more points, each distinct from \( L, M \), and \( T \), are chosen to form a second triangle. Compute the probability that the two triangles do not overlap. | [
"\\(\\frac{690}{1771}\\)"
] | [
"26 points are evenly spaced around a circle and are labeled \\( A \\) through \\( Z \\) in alphabetical order. Triangle \\(\\triangle LMT\\) is drawn. Three more points, each distinct from \\( L, M \\), and \\( T \\), are chosen to form a second triangle. Compute the probability that the two triangles do not overl... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30d93f1b6f3c77291d55cc2ce0baf268816411bb7b1764dedadabaf177458382 | 27 unit cubes are painted orange such that exactly 4 of its faces are painted and the unpainted faces share a common side. A 3x3x3 cube is formed using these 27 unit cubes. What is the probability that each face of the resulting 3x3x3 cube is entirely orange? | [
"\\dfrac{5^{12}}{2^{34} \\cdot 3^{18}}"
] | [
"27 unit cubes are painted orange such that exactly 4 of its faces are painted and the unpainted faces share a common side. A 3x3x3 cube is formed using these 27 unit cubes. What is the probability that each face of the resulting 3x3x3 cube is entirely orange?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-c8732c6f55b9396cd5338d392c3c470eea5d84360541928e2fe60082e7bd95ec | 28 pirates found a chest with 2015 gold coins in it. The pirates agreed on the following method of dividing the fortune: The most senior pirate suggests a distribution of the coins. Then they vote and if at least 50% of them agree that the distribution is fair, the gold is divided as proposed. Otherwise, the pirate who... | [
"2002"
] | [
"28 pirates found a chest with 2015 gold coins in it. The pirates agreed on the following method of dividing the fortune: The most senior pirate suggests a distribution of the coins. Then they vote and if at least 50% of them agree that the distribution is fair, the gold is divided as proposed. Otherwise, the pirat... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-7973521280b90ec29f6b291e7059aaf3f13e7c7300c347414506b060b5e08f8e | 2n ambassadors are invited to a banquet. Every ambassador has at most \( n-1 \) enemies. Determine the minimum number of distinct seating arrangements around a round table, such that no ambassador sits next to an enemy. (Two seating arrangements are considered the same if one can be obtained by rotating the other.) | [
"1"
] | [
"2n ambassadors are invited to a banquet. Every ambassador has at most \\( n-1 \\) enemies. Determine the minimum number of distinct seating arrangements around a round table, such that no ambassador sits next to an enemy. (Two seating arrangements are considered the same if one can be obtained by rotating the othe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-b039a88105665ad623de02635d5a3e704705f03d6f284a3a3a9f9cc23ea10705 | 3 Elm trees, 4 Dogwood trees, and 5 Oak trees are to be planted in a line in front of a library such that no two Elm trees are next to each other, no Dogwood tree is adjacent to an Oak tree, and all of the trees are planted. How many ways can the trees be situated in this manner? | [
"108"
] | [
"3 Elm trees, 4 Dogwood trees, and 5 Oak trees are to be planted in a line in front of a library such that no two Elm trees are next to each other, no Dogwood tree is adjacent to an Oak tree, and all of the trees are planted. How many ways can the trees be situated in this manner?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-0b58f922944e1a41423094d96bcc00c60841793c627a8fec81fcca899d0c85c7 | 3002 people went to the restroom, including you. 2001 went to the gents' restroom and 1002 went to the ladies' restroom. No one went to both restrooms. Which restroom did you go to? | [
"ladies'"
] | [
"3002 people went to the restroom, including you. 2001 went to the gents' restroom and 1002 went to the ladies' restroom. No one went to both restrooms. Which restroom did you go to?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-7f647a9b945a9af07b995ae7faca19ccbaa41e4e55b335756378b0c20992a47a | 4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) | [
"\\dfrac{27}{128}"
] | [
"4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-029cfa277000e5c2e1cc601fd8112f5bb269ef54c0f1e626473f7587035b8e23 | 40 kg of a salt solution is poured into two vessels so that the second vessel contains 2 kg more pure salt than the first vessel. If 1 kg of salt is added to the second vessel, then it will contain twice the amount of salt than is in the first vessel. Find the mass of the solution in the first vessel. | [
"15 kg"
] | [
"40 kg of a salt solution is poured into two vessels so that the second vessel contains 2 kg more pure salt than the first vessel. If 1 kg of salt is added to the second vessel, then it will contain twice the amount of salt than is in the first vessel. Find the mass of the solution in the first vessel."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2eca4bc5655ab918b1329fca2abdd9af493bbd24029149228fa27161700b2a63 | 40 people vote for 4 candidates. Each voter can choose only one candidate. Find the number of all possible voting results. Express your answer in terms of binomial coefficients. | [
"$\\binom{43}{3}$"
] | [
"40 people vote for 4 candidates. Each voter can choose only one candidate. Find the number of all possible voting results. Express your answer in terms of binomial coefficients."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-9f68d448ce3a2d33d68828b3ce94bfbeed71441ca8f485f0d3360270bca495d8 | 5 cows eat 2 acres of grass in 10 days. 7 cows eat 3 acres of grass in 30 days. The grass grows at a constant rate and each cow eats at a constant rate. How many days will it take 16 cows to eat 7 acres of grass? | [
"70"
] | [
"5 cows eat 2 acres of grass in 10 days. 7 cows eat 3 acres of grass in 30 days. The grass grows at a constant rate and each cow eats at a constant rate. How many days will it take 16 cows to eat 7 acres of grass?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c9183adba2ad171449d3126cf6b48b6fc087f3062223ee6e565211b16b51cda2 | 500 kg of ore contained a certain amount of iron. After removing 200 kg of slag which contains on average 12.5\% by mass of iron, the percentage of iron in the remaining ore increased by 20\%. How much iron in mass percent remained in the ore? | [
"187.5 kg"
] | [
"500 kg of ore contained a certain amount of iron. After removing 200 kg of slag which contains on average 12.5\\% by mass of iron, the percentage of iron in the remaining ore increased by 20\\%. How much iron in mass percent remained in the ore?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afa92c7a5904d45ab8d057d1d271d7be80f406d496a59fce3a49d3da49dfcab9 | 5000 movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie t... | [
"2^{100} \\cdot S(5000, 100)"
] | [
"5000 movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the mo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-847f4e26ecfc43819b4cd434e5a55c51c184fa68ff0b8c9889320c9fd9f13387 | 5N points form an array of 5 rows and \(N\) columns. The points are numbered left to right, top to bottom. Five points, \(P_1, P_2, \ldots, P_5\), are chosen, with \(P_i\) in the \(i\)-th row and having number \(x_i\). The points are renumbered top to bottom, left to right. \(P_i\) now has number \(y_i\). We find that ... | [
"149"
] | [
"5N points form an array of 5 rows and \\(N\\) columns. The points are numbered left to right, top to bottom. Five points, \\(P_1, P_2, \\ldots, P_5\\), are chosen, with \\(P_i\\) in the \\(i\\)-th row and having number \\(x_i\\). The points are renumbered top to bottom, left to right. \\(P_i\\) now has number \\(y... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b61da8b71b102f68b93854badd2af45234c379888896b11a10ae8344c11a3b37 | 6 men and 8 boys can complete a piece of work in 10 days. 26 men and 48 boys can complete the same piece of work in 2 days. Find the time taken by 15 men and 21 boys to complete the work. | [
"\\(\\frac{200}{51}\\)"
] | [
"6 men and 8 boys can complete a piece of work in 10 days. 26 men and 48 boys can complete the same piece of work in 2 days. Find the time taken by 15 men and 21 boys to complete the work."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2f02767079f133a49784fa20f9cbbdf95cb439a099942e9bcef02ab4e5b3ecf4 | 6 people are split into two groups of 3. In each round, each person picks a person from the other group, and any two people who pick each other form a pair and are removed from the game. What is the expected number of rounds needed for all 6 people to be removed from the game? | [
"3"
] | [
"6 people are split into two groups of 3. In each round, each person picks a person from the other group, and any two people who pick each other form a pair and are removed from the game. What is the expected number of rounds needed for all 6 people to be removed from the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e802147b25444a8e14c27d988321f8602175cd7433d8deb4881b48fe9be8d0ba | 6 points are given in the plane, no three of which lie on the same line. Draw 9 segments such that each segment connects two of the given points and you cannot find a triangle with all sides drawn and all vertices located at the given points. | [
"K_{3,3}"
] | [
"6 points are given in the plane, no three of which lie on the same line. Draw 9 segments such that each segment connects two of the given points and you cannot find a triangle with all sides drawn and all vertices located at the given points."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-183332e9d696a3f4a8bff998012b17b38487f8e2771f0e3bce556d25eb88d09c | 65 bugs are placed at different squares of a 9 × 9 square board. Each bug moves to a horizontally or vertically adjacent square, and no bug makes two horizontal or two vertical moves in succession. After a certain number of moves, find the minimum number of bugs that must be in the same square. | [
"2"
] | [
"65 bugs are placed at different squares of a 9 × 9 square board. Each bug moves to a horizontally or vertically adjacent square, and no bug makes two horizontal or two vertical moves in succession. After a certain number of moves, find the minimum number of bugs that must be in the same square."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-0dbedcf4e278c5204cb0098cbce2e72ee140f3c7d9cdd3c28360e59907f4a984 | 8 ants, denoted by $Ant_1$, $Ant_2$, ..., $Ant_8$, must each stand at a vertex of a regular octagon such that each ant stands at exactly one vertex, and the line segments joining $Ant_i$ and $Ant_{i+1}$ for $i = 1, 2, 3, ..., 7$ do not intersect except at the vertices of the octagon. How many ways can they do this? (Ro... | [
"512"
] | [
"8 ants, denoted by $Ant_1$, $Ant_2$, ..., $Ant_8$, must each stand at a vertex of a regular octagon such that each ant stands at exactly one vertex, and the line segments joining $Ant_i$ and $Ant_{i+1}$ for $i = 1, 2, 3, ..., 7$ do not intersect except at the vertices of the octagon. How many ways can they do this... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-a657b24348afbf448d1fd625e6f0ac57d15ae93ded9591bd590ad59d7bd61f66 | 8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them? | [
"\\dfrac{37}{256}"
] | [
"8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-897d38b87ec96bd3b32bf8424b7df72f66bb7595e0443dade6aa76514cd6d740 | 8 computers working for 20 hours can download 150 GB of data. After 2 hours, one computer is damaged. How long will it take the remaining 7 computers to download the remaining data? | [
"\\(\\frac{144}{7}\\)"
] | [
"8 computers working for 20 hours can download 150 GB of data. After 2 hours, one computer is damaged. How long will it take the remaining 7 computers to download the remaining data?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-5a392c0fbad8774746d5afb3eef2521d3dc0660758ad937dec3f0e558e603f1b | 8 computers working for 20 hours can download 150 GB of data. After 2 hours, one computer is damaged. How many new computers should be added so that the remaining data can be downloaded in the remaining 18 hours? | [
"1"
] | [
"8 computers working for 20 hours can download 150 GB of data. After 2 hours, one computer is damaged. How many new computers should be added so that the remaining data can be downloaded in the remaining 18 hours?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbde3801d8a68e26b1e889a268a2c8a1453faf604d0e7b4ddc94b78fa5f053fb | 8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party? | [
"28"
] | [
"8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6edc66f4a0cf135e1ebac0edc825fc1ce23d5a99c2e149188625be952e9a886d | 800 numbers are written to form a rectangular table of 100 rows and 80 columns so that the product of all numbers belonging to any column by the sum of all numbers belonging to, any row is equal to the number standing at the intersection of this row and this column. It is known that the number standing in the right upp... | [
"1"
] | [
"800 numbers are written to form a rectangular table of 100 rows and 80 columns so that the product of all numbers belonging to any column by the sum of all numbers belonging to, any row is equal to the number standing at the intersection of this row and this column. It is known that the number standing in the righ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bcde9ed70a6b5794e6e9ad6ef9da8523c0c631e36b6b9ee139ddb812230ac326 | 93 identical balls move along a line, with 59 moving from left to right at speed v and 34 moving from right to left at speed w. When two balls collide, they exchange their speeds and direction of motion. Determine the total number of collisions that will occur. | [
"1996"
] | [
"93 identical balls move along a line, with 59 moving from left to right at speed v and 34 moving from right to left at speed w. When two balls collide, they exchange their speeds and direction of motion. Determine the total number of collisions that will occur."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5df51e01571460ce2f442b3d8de740e3bb4342cda1e6df5ba664d97638f63959 | 99 Smurfs each have one candy. In each game, each Smurf gives their candy to another Smurf, and each Smurf can receive at most one candy. How many unique games can they play if they do not want to repeat any game configuration and they play one game each day? | [
"\\( !99 \\)"
] | [
"99 Smurfs each have one candy. In each game, each Smurf gives their candy to another Smurf, and each Smurf can receive at most one candy. How many unique games can they play if they do not want to repeat any game configuration and they play one game each day?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-e8617131492944d17d135724f55688ee03d5719b6c13a8d0da7d5f3e059982b7 | 99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats? | [
"33"
] | [
"99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 4,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-78e9253d1e8f81e08c0fb7da1d9ee352c1685f9bb4a32c666a4c71f98efa0772 | 994 cages each contain 2 chickens. Each day, we rearrange the chickens so that the same pair of chickens are never together twice. What is the maximum number of days we can do this? | [
"1987"
] | [
"994 cages each contain 2 chickens. Each day, we rearrange the chickens so that the same pair of chickens are never together twice. What is the maximum number of days we can do this?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39cb4ddee0501bbd0ddec18ff5b31db547ad3e40efb5721c124258eb6ffda4c7 | A "Choombam" is a polyhedron whose faces are only pentagons and hexagons. If a Choombam has $p$ pentagonal faces and $h$ hexagonal faces, find the value of $p + h$ given that the Choombam is topologically equivalent to a soccer ball. | [
"32"
] | [
"A \"Choombam\" is a polyhedron whose faces are only pentagons and hexagons. If a Choombam has $p$ pentagonal faces and $h$ hexagonal faces, find the value of $p + h$ given that the Choombam is topologically equivalent to a soccer ball."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-80e090ddceb3896265905294e9c87e3a94a69038ba34588ef9216a61f853a23e | A "Fibonoid" chain of numbers is constructed by starting with any 2 numbers, adding them to produce a third, adding the 2nd and 3rd to produce a 4th, adding the 3rd and 4th to produce a 5th, etc. The longest possible "Fibonoid" chain of positive integers that ends with the number 2005 is produced. Find the sum of the s... | [
"46"
] | [
"A \"Fibonoid\" chain of numbers is constructed by starting with any 2 numbers, adding them to produce a third, adding the 2nd and 3rd to produce a 4th, adding the 3rd and 4th to produce a 5th, etc. The longest possible \"Fibonoid\" chain of positive integers that ends with the number 2005 is produced. Find the sum... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 6
} | 28 | 0 | 0 |
omr-ws-cf8b05e96d179b2f6ec3a3806e05760eb04a48f380aa43c5144ad2278eba311d | A "Kelvin-magic" square is defined as a $5 \times 5$ grid where each unit square contains an integer from $0$ to $7$ (inclusive). In any row and any column (but not necessarily the diagonals), all nonzero numbers are distinct and their sum is $7$, and at least $2$ entries are zero. Determine the first three digits of t... | [
"144"
] | [
"A \"Kelvin-magic\" square is defined as a $5 \\times 5$ grid where each unit square contains an integer from $0$ to $7$ (inclusive). In any row and any column (but not necessarily the diagonals), all nonzero numbers are distinct and their sum is $7$, and at least $2$ entries are zero. Determine the first three dig... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-05cb26bfdc4e611780aa222a51a3eaec419cab0b05e69b191d3a0cb18b188f0e | A "Remoizaneq" number is defined as a number with at least 3 digits where each digit is unique and the digits form either a geometric sequence or an arithmetic sequence. For example, 123 and 321 are "Remoizaneq" numbers, but 213 and 231 are not. How many such "Remoizaneq" numbers exist? | [
"90"
] | [
"A \"Remoizaneq\" number is defined as a number with at least 3 digits where each digit is unique and the digits form either a geometric sequence or an arithmetic sequence. For example, 123 and 321 are \"Remoizaneq\" numbers, but 213 and 231 are not. How many such \"Remoizaneq\" numbers exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bda1ce6c10f18ed1805ac6259cf5bc97e0ecc94f3aff20fb40ace7d34e7f7e5 | A "backpack" consists of a rectangular prism with a ratio of length to thickness to height of 2:1:3, and a half-cylinder on the top face. What is the difference in volume between a backpack with a thickness of 4 units and a backpack with a thickness of 8 units? | [
"\\( 448(6 + \\frac{\\pi}{4}) \\)"
] | [
"A \"backpack\" consists of a rectangular prism with a ratio of length to thickness to height of 2:1:3, and a half-cylinder on the top face. What is the difference in volume between a backpack with a thickness of 4 units and a backpack with a thickness of 8 units?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b405ec6e5782076fbddb582ea798beb7d3fde3a93cfc02187c61c7dd6f1ff100 | A "concentric cube" is made when an $n \times n \times n$ cube contains a smaller cube with dimensions two less than the one in which it is contained, and this process continues until the dimensions are less than 1. Find the total number of unit cubes in an $n \times n \times n$ concentric cube. Express your answer in ... | [
"\\( n^2 (2n^2 - 1) \\)"
] | [
"A \"concentric cube\" is made when an $n \\times n \\times n$ cube contains a smaller cube with dimensions two less than the one in which it is contained, and this process continues until the dimensions are less than 1. Find the total number of unit cubes in an $n \\times n \\times n$ concentric cube. Express your... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95926197db2c76f0b7142ffdd09854c17eebf10c2ca3981e90f14af339f3bebd | A "dukie" word is a 10-letter word, each letter is one of the four \( D, U, K, E \) such that there are four consecutive letters in that word forming the letter \( DUKE \) in this order. For example, \( DUDKDUKEEK \) is a dukie word, but \( DUEDKUKEDE \) is not. How many different dukie words can we construct in total? | [
"28576"
] | [
"A \"dukie\" word is a 10-letter word, each letter is one of the four \\( D, U, K, E \\) such that there are four consecutive letters in that word forming the letter \\( DUKE \\) in this order. For example, \\( DUDKDUKEEK \\) is a dukie word, but \\( DUEDKUKEDE \\) is not. How many different dukie words can we cons... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eb0419806f1c2135b1072f9fa265a863cde9e4e664e4f001ab8366e2cdfab05 | A "good" number is defined as the sum of a two-digit number with distinct digits and its reverse. How many of these "good" numbers are perfect squares? | [
"1"
] | [
"A \"good\" number is defined as the sum of a two-digit number with distinct digits and its reverse. How many of these \"good\" numbers are perfect squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40a25b048601d5af9579f9d1292034728f13faff7aeab286a485ef881bc41438 | A "lotus design" is defined as a configuration of circles where each circle is externally tangent to two other circles in the configuration, and all circles are contained within a larger enclosing circle. Let $a_n$ represent the number of non-similar lotus designs with exactly $n$ circles. Find the value of $a_5 + a_6$... | [
"2"
] | [
"A \"lotus design\" is defined as a configuration of circles where each circle is externally tangent to two other circles in the configuration, and all circles are contained within a larger enclosing circle. Let $a_n$ represent the number of non-similar lotus designs with exactly $n$ circles. Find the value of $a_5... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-1b26db8686b6e8a3403a7856ea665ba06c3e2562d8c303b4714d8a17f57c3ce6 | A "lotus design" is defined as a configuration of circles where each circle is externally tangent to two other circles in the configuration, forming a closed loop. Let $a_n$ be the number of non-similar lotus designs with exactly $n$ circles. Calculate the value of $a_6$. | [
"1"
] | [
"A \"lotus design\" is defined as a configuration of circles where each circle is externally tangent to two other circles in the configuration, forming a closed loop. Let $a_n$ be the number of non-similar lotus designs with exactly $n$ circles. Calculate the value of $a_6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-d743d45e95a33e8d25e5db80097bab3f14ecc77287965c329520d5f3b7790f92 | A "multiplied number" is a natural number in two-digit form followed by the result of the multiplication. For example, $7\times 8 = 56$, then $7856$ and $8756$ are multiplied numbers. $2\times 3 = 6$, then $236$ and $326$ are multiplied. $2\times 0 = 0$, then $200$ is a multiplied number. The first digit of the number ... | [
"9881"
] | [
"A \"multiplied number\" is a natural number in two-digit form followed by the result of the multiplication. For example, $7\\times 8 = 56$, then $7856$ and $8756$ are multiplied numbers. $2\\times 3 = 6$, then $236$ and $326$ are multiplied. $2\\times 0 = 0$, then $200$ is a multiplied number. The first digit of t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8b07488a0860c652e899b7300fede3f1c3a0cd3628d8a409b09c6b7da11ec4eb | A "pretty" hexagon is defined as a hexagon that can be inscribed in a unit circle. Let $A$ be the area of a "pretty" hexagon. Find the maximum possible value of $100A$. | [
"150\\sqrt{3}"
] | [
"A \"pretty\" hexagon is defined as a hexagon that can be inscribed in a unit circle. Let $A$ be the area of a \"pretty\" hexagon. Find the maximum possible value of $100A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12644082f1a7d69bac59ca3c685e990f881446030cb187901198521723acedd8 | A "pretty" polygon is defined as a polygon with \( n \ge 7 \) sides where each interior angle is equal. For \( n = 10 \), calculate the measure of one interior angle of such a "pretty" polygon. | [
"144"
] | [
"A \"pretty\" polygon is defined as a polygon with \\( n \\ge 7 \\) sides where each interior angle is equal. For \\( n = 10 \\), calculate the measure of one interior angle of such a \"pretty\" polygon."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-778861810b13d1fc7eb2cb0d0d56cb3ab733eb6ed188744cf6037b9f5c9f9ac6 | A "superperfect" number $n$ is defined such that if $\sigma(n)$ denotes the sum of its factors, then $\sigma(\sigma(n)) = 2n$. Find the minimum possible value of $A + B$ where $A$ and $B$ are distinct numbers each having exactly 10 factors and $|A - B| = 1$. | [
"5023"
] | [
"A \"superperfect\" number $n$ is defined such that if $\\sigma(n)$ denotes the sum of its factors, then $\\sigma(\\sigma(n)) = 2n$. Find the minimum possible value of $A + B$ where $A$ and $B$ are distinct numbers each having exactly 10 factors and $|A - B| = 1$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-345b433a9549ff1b99789e1fd49e0b4425af314caa652d4713e5b9d0ed799154 | A "white and nerdy" kid has 27 friends and selects 8 of them randomly for his Myspace top 8 list. This week, he wants to include Caelestor and gfour84, with the condition that gfour84 must come before Caelestor in the list. Let $x$ be the number of possible ordered combinations of the top 8 list under this condition. W... | [
"36"
] | [
"A \"white and nerdy\" kid has 27 friends and selects 8 of them randomly for his Myspace top 8 list. This week, he wants to include Caelestor and gfour84, with the condition that gfour84 must come before Caelestor in the list. Let $x$ be the number of possible ordered combinations of the top 8 list under this condi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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