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-0d1fa150c8e588653e6c9440d59d2917306d692eeb949885b3b4176bbe903a66 | Two people take turns saying numbers, with each person able to say 1, 2, or 3 numbers at a time. The person who says the number 21 loses. Determine the winning strategy for the second player. | [
"The second player should always say a number that makes the total sum a multiple of 4."
] | [
"Two people take turns saying numbers, with each person able to say 1, 2, or 3 numbers at a time. The person who says the number 21 loses. Determine the winning strategy for the second player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-bf2e5135620f1e0ce01e462dea3d8d4a58c443efd5d49bf157c31fc39b6b9d56 | Two people toss a fair coin until one of the following patterns appears: HHT (selected by A) or HTT (selected by B). The person whose pattern appears first wins. Find the probability that A wins. | [
"\\(\\frac{2}{3}\\)"
] | [
"Two people toss a fair coin until one of the following patterns appears: HHT (selected by A) or HTT (selected by B). The person whose pattern appears first wins. Find the probability that A wins."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-17ddce51236f49017a6b8280500a01225107bd02e3f5da7ced6a3b45396b935a | Two people went from point A to point B. The first one walked along the highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first person arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. | [
"10"
] | [
"Two people went from point A to point B. The first one walked along the highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first person arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8dd46f2a61b7f05dda46fd110759ae35339622f381ad3b0b4e25c8b4bc35afc | Two people will travel to a place 20 miles away from their house. They both will start at the same time and walk at a constant velocity of 4 miles/hour. They own a horse that can carry one person at a time and can run at 10 miles/hour, regardless of whether someone is riding or not. Find the minimum time required for b... | [
"\\dfrac{34}{11}"
] | [
"Two people will travel to a place 20 miles away from their house. They both will start at the same time and walk at a constant velocity of 4 miles/hour. They own a horse that can carry one person at a time and can run at 10 miles/hour, regardless of whether someone is riding or not. Find the minimum time required ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-999e5179c0f541009c0e882dcbafe1ea012c9c37e8791e6a35871e280e618c22 | Two people, $A$ and $B$, are given consecutive natural numbers $n$ and $n+1$, respectively. They do not know each other's numbers but know that their numbers are consecutive. They are asked in turns whether they know the other's number. Both can hear each other's responses. Do they ever respond "yes"? If so, who respon... | [
"$A$ will always respond \"yes\" first, after $\\frac{n-1}{2}$ \"no\" responses if $n$ is odd, and $\\frac{n}{2}$ \"no\" responses if $n$ is even."
] | [
"Two people, $A$ and $B$, are given consecutive natural numbers $n$ and $n+1$, respectively. They do not know each other's numbers but know that their numbers are consecutive. They are asked in turns whether they know the other's number. Both can hear each other's responses. Do they ever respond \"yes\"? If so, who... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f6ab8c420dfd32aca38389ae0ef7e1578eedb550a5c6a9625d3ee66d3da851d8 | Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $... | [
"20"
] | [
"Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03780c39848f67d39d45e8660e47e5893d520aa56b988bca84f31ca7e2f17d3d | Two people, $A$ and $B$, each roll dice and compute a sum. $A$ rolls two regular dice and adds the results to get a value from $2$ to $12$. $B$ rolls $2019$ regular dice, selects the minimum and maximum values, and adds them to get a value from $2$ to $12$. Who has a higher probability of obtaining a larger result? | [
"B"
] | [
"Two people, $A$ and $B$, each roll dice and compute a sum. $A$ rolls two regular dice and adds the results to get a value from $2$ to $12$. $B$ rolls $2019$ regular dice, selects the minimum and maximum values, and adds them to get a value from $2$ to $12$. Who has a higher probability of obtaining a larger result... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-789303a3fd5cb21bd9133de201e60a7f9fb17dcc4d91c02727b084b0d0cdb6b5 | Two people, $A$ and $B$, play a game where $A$ starts by choosing a positive integer $n$. Each player, on their turn, modifies the last number said according to the following rules: - If the last number is odd, the player adds 7 to it. - If the last number is even, the player divides it by 2. The winner is the player w... | [
"22"
] | [
"Two people, $A$ and $B$, play a game where $A$ starts by choosing a positive integer $n$. Each player, on their turn, modifies the last number said according to the following rules:\n- If the last number is odd, the player adds 7 to it.\n- If the last number is even, the player divides it by 2.\n\nThe winner is th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b22106b8ff9f1c4fe32bb76dd986a8fb6e45445c380cc3849b987de4559992a | Two people, $A$ and $B$, play a game where they take turns blowing up a balloon. The balloon will explode when its volume exceeds 2014 mL. $A$ starts by blowing in 1 mL, and each subsequent turn, a player must blow in a volume that is at least as much as the previous turn but no more than twice the previous turn's volu... | [
"Player \\( A \\) has a winning strategy."
] | [
"Two people, $A$ and $B$, play a game where they take turns blowing up a balloon. The balloon will explode when its volume exceeds 2014 mL. $A$ starts by blowing in 1 mL, and each subsequent turn, a player must blow in a volume that is at least as much as the previous turn but no more than twice the previous turn's... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 5
} | 9 | 0 | 0 |
omr-ws-5d94148786cdd04faf04dcd50dad50610cf9965892aa0fd052dc5c6ccd128bb7 | Two people, $A$ and $B$, play a game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. The person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy? | [
"The second player will win."
] | [
"Two people, $A$ and $B$, play a game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. The person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3e09b562f0b12d0f0e7049dae1efcb104f11deee02209fe216b210be8d2ef44 | Two people, $A$ and $B$, play a game. On the board are written numbers $0, 1, 2, \dots, 1024$. $A$ erases $512$ numbers, $B$ erases $256$ remaining numbers, then $A$ erases $128$... In the $10^{th}$ move, $B$ erases $1$ number, leaving two numbers $x$ and $y$. After this, $B$ pays $|x-y|$ to $A$. How much will $B$ pay ... | [
"32"
] | [
"Two people, $A$ and $B$, play a game. On the board are written numbers $0, 1, 2, \\dots, 1024$. $A$ erases $512$ numbers, $B$ erases $256$ remaining numbers, then $A$ erases $128$... In the $10^{th}$ move, $B$ erases $1$ number, leaving two numbers $x$ and $y$. After this, $B$ pays $|x-y|$ to $A$. How much will $B... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-90ffa8e8536c2cb197fba2545b3e4b01d5f7fb29ec1312f18f05636c4348c375 | Two people, A and B, and an umpire U play a game. A chooses a positive integer \( m \), and B chooses a positive integer \( n \). U writes two integers on a blackboard, one of which is \( m + n \). U asks A if he knows \( n \). If A does not know, U asks B if he knows \( m \). If B does not know, U asks A again, and so... | [
"3"
] | [
"Two people, A and B, and an umpire U play a game. A chooses a positive integer \\( m \\), and B chooses a positive integer \\( n \\). U writes two integers on a blackboard, one of which is \\( m + n \\). U asks A if he knows \\( n \\). If A does not know, U asks B if he knows \\( m \\). If B does not know, U asks ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-d76438e45d29bbd4c489111dd3dbff103d22396af9e91cefd03c8479ee23cdba | Two people, A and B, are discussing the ages of B’s four children. B states that the product of their ages is 72. A states that this is not enough information. B then states that the sum of their ages equals A's eldest daughter's age. A still states that this is not enough information. B then states that the oldest chi... | [
"1, 3, 3, 8"
] | [
"Two people, A and B, are discussing the ages of B’s four children. B states that the product of their ages is 72. A states that this is not enough information. B then states that the sum of their ages equals A's eldest daughter's age. A still states that this is not enough information. B then states that the oldes... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-97b69951454b849fcbc0d6d87fd729e8447706af0d8a48f6b38f949faf6ac40f | Two people, A and B, are playing a game where they take turns drawing a diagonal across a regular 14-sided polygon. The diagonals cannot cross each other but can start or end at the same point. The person who is unable to draw a diagonal loses the game. Person A starts the game. Assuming both players play optimally, ca... | [
"11"
] | [
"Two people, A and B, are playing a game where they take turns drawing a diagonal across a regular 14-sided polygon. The diagonals cannot cross each other but can start or end at the same point. The person who is unable to draw a diagonal loses the game. Person A starts the game. Assuming both players play optimall... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-9dee56353a74046ad4bdb03bbf2f0574cada94ae90c4fc2bd9cfb505936efa73 | Two people, A and B, are playing a game. They each have a white card and a black card. At the same time, they put either their white or black card on the table. If there are 2 black cards, A gets 5 dollars. If there are 2 white cards, A gets 1 dollar. If there is a white card and a black card, B gets 2 dollars. What is... | [
"\\( \\frac{1}{2} \\) black and \\( \\frac{1}{2} \\) white."
] | [
"Two people, A and B, are playing a game. They each have a white card and a black card. At the same time, they put either their white or black card on the table. If there are 2 black cards, A gets 5 dollars. If there are 2 white cards, A gets 1 dollar. If there is a white card and a black card, B gets 2 dollars. Wh... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-61ba1d857fd3b083aa275c0203b6fcd7e1f031b7d0343bb96867d40872cb8b30 | Two people, A and B, must travel from point M to point N, located 15 km from M. On foot, they can move at a speed of 6 km/h. They also have a bicycle that can be ridden at a speed of 15 km/h. A starts walking, and B rides the bicycle until meeting pedestrian C, who is walking from N to M. Then B walks and C rides the b... | [
"\\dfrac{3}{11}"
] | [
"Two people, A and B, must travel from point M to point N, located 15 km from M. On foot, they can move at a speed of 6 km/h. They also have a bicycle that can be ridden at a speed of 15 km/h. A starts walking, and B rides the bicycle until meeting pedestrian C, who is walking from N to M. Then B walks and C rides ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-aa52c7445ac80d511ecc935d7e4b9f66b35f231b889af767b1bb6121761f5472 | Two people, A and B, play a game on a 100x100 chessboard. A thinks of a position (xA, yA). B can ask how many moves at least are required for a king to get from any chosen position (x, y) to (xA, yA). The king can move one step to any of the 8 neighboring squares. What is the minimal number of questions B needs to dete... | [
"14"
] | [
"Two people, A and B, play a game on a 100x100 chessboard. A thinks of a position (xA, yA). B can ask how many moves at least are required for a king to get from any chosen position (x, y) to (xA, yA). The king can move one step to any of the 8 neighboring squares. What is the minimal number of questions B needs to... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1134529c3072edc3da7eeec8219bba857b61be6b3d1fc7f32a39896e1c06546 | Two people, A and B, play a game where they can either remove a coin from a pile or split a pile into two non-empty piles. The last person to remove a coin loses. The game starts with A1 piles of single coins and A2 piles of two coins. Determine the strategy for losing based on the sequence (A1, A2, A3, ...). | [
"Ensure the state is a [b]0[/b]-state after each move."
] | [
"Two people, A and B, play a game where they can either remove a coin from a pile or split a pile into two non-empty piles. The last person to remove a coin loses. The game starts with A1 piles of single coins and A2 piles of two coins. Determine the strategy for losing based on the sequence (A1, A2, A3, ...)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee523ef5df712adab149404055bd1f6f17d6d6bba2d5cc522a5c121512c7120f | Two people, A and B, play a game with a deck of 32 cards. A starts, and thereafter the players alternate, each taking either 1 card or a prime number of cards. The player who cannot pick up any cards loses. Who will win the game if both follow optimal strategy? | [
"B will win."
] | [
"Two people, A and B, play a game with a deck of 32 cards. A starts, and thereafter the players alternate, each taking either 1 card or a prime number of cards. The player who cannot pick up any cards loses. Who will win the game if both follow optimal strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9c4424b8f954ecf883484dccc7981c2f766caa494842cb2a0af7d02a0312cba5 | Two people, A and B, play a game with a positive integer \( n \). A gives \( n \) distinct real numbers one by one. After each number, B must predict its position in a decreasingly sorted list of A's numbers by giving an integer between 1 and \( n \) inclusive. B's score is the number of correct predictions. Can B achi... | [
"n \\geq 3"
] | [
"Two people, A and B, play a game with a positive integer \\( n \\). A gives \\( n \\) distinct real numbers one by one. After each number, B must predict its position in a decreasingly sorted list of A's numbers by giving an integer between 1 and \\( n \\) inclusive. B's score is the number of correct predictions.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-f8b739735b15191560bc913eab084d19888a5fa0b78cbf9fb89aedcf464ff488 | Two people, A and B, start from the same point on the equator. Person A moves 40 km north and then 40 km east. Person B moves 40 km east and then 40 km south. What is the distance between A and B? | [
"\\( 40\\sqrt{2} \\) km"
] | [
"Two people, A and B, start from the same point on the equator. Person A moves 40 km north and then 40 km east. Person B moves 40 km east and then 40 km south. What is the distance between A and B?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c2c5396b366fd18a552cd773d16997f02d029227d9ceff2403b8b9ae7ce0948 | Two people, Ana and Banana, are having a tournament where the first person to get at least 4 points wins (1 point for a win, 0.5 points for a tie, and 0 points for a loss). Ana wins each game with a probability of $\frac{3}{4}$, loses with a probability of $\frac{1}{5}$, and ties with the remaining probability. If Bana... | [
"207"
] | [
"Two people, Ana and Banana, are having a tournament where the first person to get at least 4 points wins (1 point for a win, 0.5 points for a tie, and 0 points for a loss). Ana wins each game with a probability of $\\frac{3}{4}$, loses with a probability of $\\frac{1}{5}$, and ties with the remaining probability. ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-84405188593f527f6193f3d12a76bad2c1764aaebfc108458d9261f2422cd1ab | Two people, Bert and Sara, are flipping three coins. Bert guesses a possible sequence of the coins (such as HHH). Sara, after hearing Bert's guess, guesses a different sequence. They then flip the three coins. The first one to guess the exact sequence correctly wins the game. What is the probability that Sara wins the ... | [
"\\dfrac{1}{2}"
] | [
"Two people, Bert and Sara, are flipping three coins. Bert guesses a possible sequence of the coins (such as HHH). Sara, after hearing Bert's guess, guesses a different sequence. They then flip the three coins. The first one to guess the exact sequence correctly wins the game. What is the probability that Sara wins... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-84bb7d372423f81ecfd5550342874887a3a1bc41948045df8f7d0fe29b6b7065 | Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probability... | [
"1"
] | [
"Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probab... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-eddc22f5d2806acdc8a4cd63ad53d197c19547cd082963e683d25539b88f7fd8 | Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probability... | [
"\\dfrac{49}{80}"
] | [
"Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probab... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e3e63508b3252bcd61ea2cabf14ce2ffaeedf8eb4a853d524e5f5d81b4d3ff6b | Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probability... | [
"\\(\\frac{1}{2}\\)"
] | [
"Two people, E and O, play a game where each round they simultaneously choose either a 1 or a 2. If the sum of their numbers is even, player E takes that sum of points from player O; if the sum is odd, player O takes that sum of points from player E. Both players choose a 1 with probability 7/12 and a 2 with probab... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe373adf95e01b822ad83872142683c5b47c63a9d029af38fbcab869c44eaf66 | Two people, X and Y, play a game with an acute-angled triangle \( \triangle ABC \). X chooses a point \( P \) on \( AB \), Y chooses a point \( Q \) on \( BC \), and X then chooses a point \( R \) on \( CA \). The objective is for X to maximize the area of \( \triangle PQR \) and for Y to minimize it. What are their be... | [
"X chooses the midpoint of \\( AB \\) and Y chooses \\( Q \\) such that \\( PP' \\parallel AC \\) and \\( P' \\) is the corresponding point on \\( BC \\)."
] | [
"Two people, X and Y, play a game with an acute-angled triangle \\( \\triangle ABC \\). X chooses a point \\( P \\) on \\( AB \\), Y chooses a point \\( Q \\) on \\( BC \\), and X then chooses a point \\( R \\) on \\( CA \\). The objective is for X to maximize the area of \\( \\triangle PQR \\) and for Y to minimiz... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29836a6d9922b3398d369057386277cadaf12fec6fcf35bff4e89198065759c0 | Two perfect logicians, S and P, are told that integers \( x \) and \( y \) have been chosen such that \( 1 < x < y \) and \( x + y < 100 \). S is given the value \( x + y \) and P is given the value \( xy \). They then have the following conversation: P: I cannot determine the two numbers. S: I knew that. P: Now I can ... | [
"4, 13",
"\\( x = 4 \\), \\( y = 13 \\)"
] | [
"Two perfect logicians, S and P, are told that integers \\( x \\) and \\( y \\) have been chosen such that \\( 1 < x < y \\) and \\( x + y < 100 \\). S is given the value \\( x + y \\) and P is given the value \\( xy \\). They then have the following conversation:\n\nP: I cannot determine the two numbers.\nS: I kne... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 33
} | 53 | 0 | 0 |
omr-ws-85f7a3c474e00bb24b53abe11c50285f7c7d6ce4b22a65a8a8db8325e37e27b5 | Two perfect logicians, \( S \) and \( P \), are told that integers \( x \) and \( y \) have been chosen such that \( 1 < x < y \) and \( x + y < 100 \). \( S \) is given the value \( x + y \) and \( P \) is given the value \( x \cdot y \). They then have the following conversation: - \( P \): I cannot determine the two... | [
"4 \\text{ and } 13"
] | [
"Two perfect logicians, \\( S \\) and \\( P \\), are told that integers \\( x \\) and \\( y \\) have been chosen such that \\( 1 < x < y \\) and \\( x + y < 100 \\). \\( S \\) is given the value \\( x + y \\) and \\( P \\) is given the value \\( x \\cdot y \\). They then have the following conversation:\n\n- \\( P ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-2e72f72f8acd1ec2a5c4a4ebc52f198a4ec9de3ddac2a13726c5cf65323e116f | Two perpendicular chords $AM$ and $BN$ of a circle intersect at point $K$, defining four arcs with pairwise different lengths, with $AB$ being the smallest. Chords $AD$ and $BC$ are drawn such that $AD \parallel BC$ and $C, D$ are different from $N, M$. If $L$ is the intersection point of $DN$ and $MC$, and $T$ is the ... | [
"no_valid_answer_found"
] | [
"Two perpendicular chords $AM$ and $BN$ of a circle intersect at point $K$, defining four arcs with pairwise different lengths, with $AB$ being the smallest. Chords $AD$ and $BC$ are drawn such that $AD \\parallel BC$ and $C, D$ are different from $N, M$. If $L$ is the intersection point of $DN$ and $MC$, and $T$ i... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b3ba3904409445f46c1be53f3c9a50633775051ee1ec21638bff77aa8db10dc | Two perpendicular chords \(AM\) and \(BN\) of a circle intersect at point \(K\), defining four arcs with pairwise different lengths, with \(AB\) being the smallest of them. We draw chords \(AD\) and \(BC\) such that \(AD \parallel BC\) and \(C, D\) are different from \(N, M\). If \(L\) is the point of intersection of \... | [
"\\dfrac{1}{9}"
] | [
"Two perpendicular chords \\(AM\\) and \\(BN\\) of a circle intersect at point \\(K\\), defining four arcs with pairwise different lengths, with \\(AB\\) being the smallest of them. We draw chords \\(AD\\) and \\(BC\\) such that \\(AD \\parallel BC\\) and \\(C, D\\) are different from \\(N, M\\). If \\(L\\) is the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-301daa4261ce865fbb0f668c0a37efeb3bf335c95fdd65958c833d42161f2cb7 | Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R$. The distance from $P$ to the center of the circle is $kR$. Let $S$ be the sum of the lengths of these two chords. Find the value of $S^2$ when $k = \frac{1}{2}$. | [
"14R^2"
] | [
"Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R$. The distance from $P$ to the center of the circle is $kR$. Let $S$ be the sum of the lengths of these two chords. Find the value of $S^2$ when $k = \\frac{1}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5b552bae194530d5dad51494f8d324b05fb8d8dcdb4ff8fbb2cbdf6347921788 | Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle. | [
"\\(\\sqrt{65}\\)"
] | [
"Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-5379a76fb4d8dba8fe8a857f48093c22ebb9fc0e193dd858ec8c005c42544c9a | Two perpendicular chords intersect in a circle. The segments of one chord are $3$ and $4$, and the segments of the other are $6$ and $2$. Compute the diameter of the circle. | [
"\\(\\frac{\\sqrt{65}}{2}\\)"
] | [
"Two perpendicular chords intersect in a circle. The segments of one chord are $3$ and $4$, and the segments of the other are $6$ and $2$. Compute the diameter of the circle."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a0f8cdfcf30b208612b839be83a44337dc5ff2fb0c717740c23d4b6643dc53b | Two perpendicular chords of a circle $\omega$ have lengths $8$ and $10$ and intersect at a point $P$. The distance from $P$ to the center of the circle is $\sqrt{19}$. If the area of the circle is $A\pi$, what is $A$? | [
"\\( 60 \\)"
] | [
"Two perpendicular chords of a circle $\\omega$ have lengths $8$ and $10$ and intersect at a point $P$. The distance from $P$ to the center of the circle is $\\sqrt{19}$. If the area of the circle is $A\\pi$, what is $A$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4adf7be70a9a713e6d2a1dd4f68ed4c6a33b74abe13c621afb7f35b47311fcd | Two perpendicular generatrices of a right circular cone divide the circle of the base in the ratio $1 : 2$. Find the volume of the cone if its altitude is $h$. | [
"$\\frac{2\\pi }{3}h^3$"
] | [
"Two perpendicular generatrices of a right circular cone divide the circle of the base in the ratio $1 : 2$. Find the volume of the cone if its altitude is $h$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ba9b09bf0acfe1c5002635d984b0191cfd2c28696c769eb5456c2182d680083 | Two perpendicular intersecting lines \(a\) and \(b\) intersect three different parallel planes at points \(A_1\) and \(B_1\), \(A_2\) and \(B_2\), \(A_3\) and \(B_3\). Spheres are constructed with segments \(A_1B_1\), \(A_2B_2\), and \(A_3B_3\) as diameters. Let the radius of the circle formed by the intersection of th... | [
"no_valid_answer_found"
] | [
"Two perpendicular intersecting lines \\(a\\) and \\(b\\) intersect three different parallel planes at points \\(A_1\\) and \\(B_1\\), \\(A_2\\) and \\(B_2\\), \\(A_3\\) and \\(B_3\\). Spheres are constructed with segments \\(A_1B_1\\), \\(A_2B_2\\), and \\(A_3B_3\\) as diameters. Let the radius of the circle forme... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a7e5e4fb5193f38c8c80e0c1a7fd57ca4eb1187e3ffcc41a4cc53eecef427d8 | Two perpendicular lines \( l_1 \) and \( l_2 \) intersect at \( O \). A circle \((K, r)\) has its center on \( l_1 \) and \( S \) is a point on \((K)\). Another circle with center \( L \) and diameter \( OS \) intersects \((K)\) again at \( P \) and \( l_2 \) at \( Q \). If \( PQ \) intersects \( l_1 \) at \( T \) and ... | [
"90"
] | [
"Two perpendicular lines \\( l_1 \\) and \\( l_2 \\) intersect at \\( O \\). A circle \\((K, r)\\) has its center on \\( l_1 \\) and \\( S \\) is a point on \\((K)\\). Another circle with center \\( L \\) and diameter \\( OS \\) intersects \\((K)\\) again at \\( P \\) and \\( l_2 \\) at \\( Q \\). If \\( PQ \\) int... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a49aa28fa4396078a61e679e5615ab2b2bbb8bdaf636cad273b88a9674f95892 | Two perpendicular lines and a circle \( C \) pass through a common point. Three line segments \( AB \), \( CD \), and \( EF \), with endpoints on the two perpendicular lines, are tangent to \( C \) at their midpoints. If the lengths of \( AB \), \( CD \), and \( EF \) are \( 10 \), \( x \), and \( y \) respectively, an... | [
"10"
] | [
"Two perpendicular lines and a circle \\( C \\) pass through a common point. Three line segments \\( AB \\), \\( CD \\), and \\( EF \\), with endpoints on the two perpendicular lines, are tangent to \\( C \\) at their midpoints. If the lengths of \\( AB \\), \\( CD \\), and \\( EF \\) are \\( 10 \\), \\( x \\), and... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-698af99a66c8867f9f9d79dc08ef3cb9c629ad4f02666e2e9a11f421011b30a6 | Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ$ and $BZ$ are parallel to $HX$ and $HY$, respectively. If $AX = 12$, $AY = 16$, and $AB = 20$, find the length of $XY$. | [
"20"
] | [
"Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ$ and $BZ$ are parallel to $HX$ and $HY$, respectively. If $AX = 12$, $AY = 16$, and $AB = 20$, find the length of $XY$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-2e38af0ca051d0280bcd90e686cd21330ca604fea7e99cea397c2d208c24e191 | Two perpendicular lines are drawn through the orthocenter \( H \) of triangle \( ABC \). These lines intersect \( AB \) at points \( D \) and \( E \), \( AC \) at points \( G \) and \( H \), and \( BC \) at points \( P \) and \( Q \). Let \( M \), \( N \), and \( O \) be the midpoints of segments \( DE \), \( GH \), an... | [
"-\\dfrac{3}{4}"
] | [
"Two perpendicular lines are drawn through the orthocenter \\( H \\) of triangle \\( ABC \\). These lines intersect \\( AB \\) at points \\( D \\) and \\( E \\), \\( AC \\) at points \\( G \\) and \\( H \\), and \\( BC \\) at points \\( P \\) and \\( Q \\). Let \\( M \\), \\( N \\), and \\( O \\) be the midpoints o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff9a76897b31a8614be1bc8cbfa4c199f4d9ca5e3d9c68dbe18829f5613cd2a8 | Two perpendicular lines divide a square into four parts. Three of these parts have area $1$. If the side length of the square is $s$, find the value of $s^2$. | [
"4"
] | [
"Two perpendicular lines divide a square into four parts. Three of these parts have area $1$. If the side length of the square is $s$, find the value of $s^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fcde1cbfcd088d6e829ad369e9129af266f371593d8ef90d53eee2d464977f33 | Two perpendicular lines pass through the point $F(1,1)$ on the coordinate plane. One of these lines intersects the hyperbola $y=\frac{1}{2x}$ at points $A$ and $C$ ($C_x > A_x$), and the other line intersects the left part of the hyperbola at point $B$ and the right part at point $D$. Let $m = (C_x - A_x)(D_x - B_x)$. ... | [
"\\(\\frac{m^2}{2}\\)"
] | [
"Two perpendicular lines pass through the point $F(1,1)$ on the coordinate plane. One of these lines intersects the hyperbola $y=\\frac{1}{2x}$ at points $A$ and $C$ ($C_x > A_x$), and the other line intersects the left part of the hyperbola at point $B$ and the right part at point $D$. Let $m = (C_x - A_x)(D_x - B... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-72d557ed41e3f84476028eb60604a4c61791dfcab560f6fab920e819893c41c7 | Two perpendicular lines with $x$-intercepts $-2$ and $8$ intersect at $(0, b)$. Determine all possible values of $b$. | [
"\\( b = \\pm 4 \\)"
] | [
"Two perpendicular lines with $x$-intercepts $-2$ and $8$ intersect at $(0, b)$. Determine all possible values of $b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 18
} | 22 | 0 | 0 |
omr-ws-931c6cfb26079eccb9f66419395af06a169c372bbc49aeee27cbfd3adcc46f1f | Two perpendicular lines with x-intercepts -2 and 8 intersect at (0, b). Determine both possible values for b. | [
"\\( b = 4 \\) or \\( b = -4 \\)"
] | [
"Two perpendicular lines with x-intercepts -2 and 8 intersect at (0, b). Determine both possible values for b."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-7999197727a57cb87002664bdffaa846b8cd1ff4bd132d88c6c004ae99a96680 | Two perpendicular planes \( P_1 \) and \( P_2 \) intersect in line \( l \). Plane \( P_3 \) passes through point \( A \) on line \( l \) and is perpendicular to line \( l \). Points in space which lie on two of the planes \( P_1 \), \( P_2 \), and \( P_3 \) and are less than one inch from point \( A \) or are between 3... | [
"12"
] | [
"Two perpendicular planes \\( P_1 \\) and \\( P_2 \\) intersect in line \\( l \\). Plane \\( P_3 \\) passes through point \\( A \\) on line \\( l \\) and is perpendicular to line \\( l \\). Points in space which lie on two of the planes \\( P_1 \\), \\( P_2 \\), and \\( P_3 \\) and are less than one inch from point... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6c3cabfaa1a4a1cdbf74f55af28f1157b6bbb5a76c7d8a7c3060366453fd747 | Two perpendicular planes intersect a sphere in two circles with radii 54 and 66. These circles intersect in two points, $A$ and $B$, such that $AB = 42$. Find $R^2$, where $R$ is the radius of the sphere. | [
"4518"
] | [
"Two perpendicular planes intersect a sphere in two circles with radii 54 and 66. These circles intersect in two points, $A$ and $B$, such that $AB = 42$. Find $R^2$, where $R$ is the radius of the sphere."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de51c3e78288ffa2804dfff65025ac57c17b8f5c46935cec255256322c20ff13 | Two perpendicular planes intersect a sphere in two circles with radii 54 and 66. These circles intersect in two points, $A$ and $B$, such that $AB=42$. Find $R^2$, where $R$ is the radius of the sphere. | [
"6831"
] | [
"Two perpendicular planes intersect a sphere in two circles with radii 54 and 66. These circles intersect in two points, $A$ and $B$, such that $AB=42$. Find $R^2$, where $R$ is the radius of the sphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-e1b59f398c9f027a16763d280c17626be714e4876822ef084132a4445e031296 | Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$. | [
"\\( DC^2 = \\frac{2R^2 - PO^2}{4} \\)"
] | [
"Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-4de72101724ce42fa3deac20681f0f46c804fb9bdd630235523cdbc4b722a30b | Two persons $A$ and $B$ are playing Tic Tac Toe. It is known that $A$ loses $p$ times while $B$ loses $q$ times. The total number of wins in the game is $r$ times. How many times do $A$ and $B$ play the game? | [
"\\(\\frac{1}{2}(p + q + r)\\)"
] | [
"Two persons $A$ and $B$ are playing Tic Tac Toe. It is known that $A$ loses $p$ times while $B$ loses $q$ times. The total number of wins in the game is $r$ times. How many times do $A$ and $B$ play the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-4e8cd1d006a2d7378cc1e1c9238af1ab346131ae5eed683d72731e34d733b4c8 | Two persons $A$ and $B$ each pick $5$ positive integers (not necessarily distinct) from the set $\{1, 2, \dots, 5\}$. Let $S$ be the sum of all possible equal subset sums that can be formed by taking some nonempty subset of $A$'s integers and some nonempty subset of $B$'s integers. Find the maximum possible value of $S... | [
"120"
] | [
"Two persons $A$ and $B$ each pick $5$ positive integers (not necessarily distinct) from the set $\\{1, 2, \\dots, 5\\}$. Let $S$ be the sum of all possible equal subset sums that can be formed by taking some nonempty subset of $A$'s integers and some nonempty subset of $B$'s integers. Find the maximum possible val... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-71c22e0b9a66989929fd9cb7007d7aade977c9e3a22347c5f3f4de682a478145 | Two persons $A$ and $B$ play a game repeatedly. The probability that $A$ wins any given game is $p$. Let $q$ be the probability that $A$ wins at least one game before $B$ wins two games. Given that $q = \frac{1}{2}$, find the value of $p$. | [
"\\( \\frac{\\sqrt{2} - 1}{\\sqrt{2}} \\)"
] | [
"Two persons $A$ and $B$ play a game repeatedly. The probability that $A$ wins any given game is $p$. Let $q$ be the probability that $A$ wins at least one game before $B$ wins two games. Given that $q = \\frac{1}{2}$, find the value of $p$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-e247b9019d5437493fae52f9be32db0af45cd67b84216edbdb2a48ce77f1291b | Two persons $A$ and $B$ play a game with two decks of 26 cards each, numbered from 1 to 26. The cards are distributed randomly between the two players, and each player arranges their 26 cards in a deck. At each turn, they reveal the top card of their deck, and the player with the higher number gets a point. If player $... | [
"226"
] | [
"Two persons $A$ and $B$ play a game with two decks of 26 cards each, numbered from 1 to 26. The cards are distributed randomly between the two players, and each player arranges their 26 cards in a deck. At each turn, they reveal the top card of their deck, and the player with the higher number gets a point. If pla... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a87944a1033b8b117878b5e37116b1349a1d5a11ed145dfae5995cece8532e4 | Two persons A and B start moving from point P towards Q, which are 1400 km apart. The speed of A is 50 km/hr and that of B is 20 km/hr. How far is A from Q when he meets B for the 22nd time? | [
"0 km"
] | [
"Two persons A and B start moving from point P towards Q, which are 1400 km apart. The speed of A is 50 km/hr and that of B is 20 km/hr. How far is A from Q when he meets B for the 22nd time?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44884d510fc5afd5e480eee754d0a4585f2b4863586f35d80fe9df8fa88ccc15 | Two persons are awarded pensions in proportion to the square root of the number of years they have served. One has served 9 years longer than the other and receives a pension greater by $50. If the length of service of the first had exceeded that of the second by 4 1/4 years, their pensions would have been in the propo... | [
"200"
] | [
"Two persons are awarded pensions in proportion to the square root of the number of years they have served. One has served 9 years longer than the other and receives a pension greater by $50. If the length of service of the first had exceeded that of the second by 4 1/4 years, their pensions would have been in the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-a3c1b86227f1acaa6cd2a2de1fb5c955a43bc22fa50af582e4983a12fdcf3808 | Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, and they continue this process until there are 5 pieces. After that, the first person chooses one of the pieces, then the second person chooses one of the remaining pieces, and... | [
"\\dfrac{3}{8}"
] | [
"Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, and they continue this process until there are 5 pieces. After that, the first person chooses one of the pieces, then the second person chooses one of the remaining pieces... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-ae2be480680e627ffc0139bc3715330a17d31d55777e96ce53078e820b0a1189 | Two persons are playing a game on an $n \times m$ table with drawn lines. Person $\#1$ starts the game. Each person, on their turn, folds the table along one of its lines. The player who cannot make a fold loses the game. Determine which player has a winning strategy. | [
"The first player has a winning strategy if and only if \\( m \\) and \\( n \\) are not both of the form \\( 2^k - 1 \\)."
] | [
"Two persons are playing a game on an $n \\times m$ table with drawn lines. Person $\\#1$ starts the game. Each person, on their turn, folds the table along one of its lines. The player who cannot make a fold loses the game. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 7
} | 16 | 0 | 0 |
omr-ws-b5852be84af2f298ab1dc05d53543d3a677bc0b45e58da6fc4ff8eb1735c084a | Two persons are playing a game with a pile of stones. The first player can remove either 1 or 10 stones on their turn, and the second player can remove either m or n stones on their turn. The player who cannot make a move loses. Given that the first player can always win regardless of the second player's moves, what ar... | [
"\\( |m - n| \\neq 9 \\) and \\( m, n \\notin \\{1, 2, 3, 4, 5, 6, 7, 8\\} \\)"
] | [
"Two persons are playing a game with a pile of stones. The first player can remove either 1 or 10 stones on their turn, and the second player can remove either m or n stones on their turn. The player who cannot make a move loses. Given that the first player can always win regardless of the second player's moves, wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-0de2a4e4486a7405e0d8727b11ac6d3ebc4d7df5d24da9caf76f97d74fbddc59 | Two persons deposited equal sums of money in a savings bank. One of them withdrew his money after $m$ months and received $p$ roubles, and the other withdrew the money after $n$ months and received $q$ roubles. How much money did either person deposit and what interest does the savings bank pay? | [
"x=\\frac{pn-qm}{n-m}, y=\\frac{1200(p-q)}{qm-pn}\\%"
] | [
"Two persons deposited equal sums of money in a savings bank. One of them withdrew his money after $m$ months and received $p$ roubles, and the other withdrew the money after $n$ months and received $q$ roubles. How much money did either person deposit and what interest does the savings bank pay?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d297235185bbb6b83e3fe1eec0c5cf6e2d68c672cb6607ac3d7eb2959bc74ba | Two persons play a game in which they call an integer \( m \) (\( 1 \leq m \leq n \)) alternately. Note that you may not call the number which has already been said. The game is over when no one can call numbers. If the sum of the numbers that the lead player has said is divisible by 3, then the lead player wins; other... | [
"n \\not\\equiv 2 \\pmod{3} \\text{ or } n \\text{ is even}"
] | [
"Two persons play a game in which they call an integer \\( m \\) (\\( 1 \\leq m \\leq n \\)) alternately. Note that you may not call the number which has already been said. The game is over when no one can call numbers. If the sum of the numbers that the lead player has said is divisible by 3, then the lead player ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-3e10ad1a7280c49fdf9cad70c5c35663c42d4fb946680a2fbda4a99bdf589529 | Two persons play a game on a $3 \times 100$ board. The first player places tiles of size $1 \times 2$ lengthwise (along the long axis of the board), and the second player places tiles of size $1 \times 2$ in the perpendicular direction. The loser is the one who cannot make a move. Which player can always win, and what ... | [
"First player wins."
] | [
"Two persons play a game on a $3 \\times 100$ board. The first player places tiles of size $1 \\times 2$ lengthwise (along the long axis of the board), and the second player places tiles of size $1 \\times 2$ in the perpendicular direction. The loser is the one who cannot make a move. Which player can always win, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-3ec31e29f863349fd424685e24aba51fe5f54fecc9c0e81ddb74bb0ed074c89c | Two persons play a game starting with 2011 balls. On each turn, a player can remove $2^k$ balls where $k \in \mathbf{N}$. The player who takes the last ball wins. Assuming both players play optimally, who will win, the first or second player? | [
"The first player will win."
] | [
"Two persons play a game starting with 2011 balls. On each turn, a player can remove $2^k$ balls where $k \\in \\mathbf{N}$. The player who takes the last ball wins. Assuming both players play optimally, who will win, the first or second player?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 28
} | 38 | 0 | 0 |
omr-ws-42819ef185687b9eacb25593ea84ce8f888b6996c523913e3cda56423f04433c | Two persons play a game starting with one pile of $n \geq 4$ nuts. During a move, a player takes one of the piles and splits it into two nonempty sets. If a player cannot make a move, they lose. For which values of $n$ does the first player have a winning strategy? | [
"\\( n \\neq 4k+3 \\)"
] | [
"Two persons play a game starting with one pile of $n \\geq 4$ nuts. During a move, a player takes one of the piles and splits it into two nonempty sets. If a player cannot make a move, they lose. For which values of $n$ does the first player have a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30d8fd8c30d351915a50b9d1413dd02c21ab560aab08e0c23765e873cde363e8 | Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtracting an integer between $1$ and $2010$ inclusive, or dividing by $2011$, rounding up if the fractional part of the result is more than $\frac{1}{2}$ and rounding down otherwise... | [
"First player wins."
] | [
"Two persons play the following game with integers. The initial number is $2011^{2011}$. The players move in turns. Each move consists of subtracting an integer between $1$ and $2010$ inclusive, or dividing by $2011$, rounding up if the fractional part of the result is more than $\\frac{1}{2}$ and rounding down oth... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0dd72567909271d9e220f020673ac829f2ce4ef35b522f1d6ebdfacb657ebd65 | Two persons simultaneously leave cities $A$ and $B$ and travel towards each other. The first person travels 2 km/h faster than does the second and arrives in $B$ one hour before the second arrives in $A$. $A$ and $B$ are 24 km apart. How many kilometers does each person make in one hour? | [
"8 km/h; 6 km/h"
] | [
"Two persons simultaneously leave cities $A$ and $B$ and travel towards each other. The first person travels 2 km/h faster than does the second and arrives in $B$ one hour before the second arrives in $A$. $A$ and $B$ are 24 km apart. How many kilometers does each person make in one hour?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eeb5375f65a9b877db670d777a7b5b88f9116c79d528938a234bd0afda871439 | Two persons start from points A and B, which are 400 km apart, at the same time. The speed of the person starting from A is 10 km/h greater than the speed of the person starting from B. After some time, the distance between them is 10 km. Determine the speeds of the two persons. The given answer is 70 km/h and 60 km/h,... | [
"70 \\text{ km/h and } 60 \\text{ km/h}"
] | [
"Two persons start from points A and B, which are 400 km apart, at the same time. The speed of the person starting from A is 10 km/h greater than the speed of the person starting from B. After some time, the distance between them is 10 km. Determine the speeds of the two persons. The given answer is 70 km/h and 60 ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-d8cbe3662ef59becdc18df106300929f97ca3ccebb2a7f7f1eac47325fe3c479 | Two persons start from the same point and travel in opposite directions. After each has gone 3 meters, they turn left and continue for 4 meters. How far are they from the point of departure? | [
"5 meters"
] | [
"Two persons start from the same point and travel in opposite directions. After each has gone 3 meters, they turn left and continue for 4 meters. How far are they from the point of departure?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-9843a5349df230bbfad2bd8d97519f82c1ddef98ee9b91b180c5454a18656686 | Two persons start running from opposite corners A and C of a square field with side length 'a' in the clockwise direction with speeds v1 and v2, respectively. Initially, the distance between them is √2a. Find the time t in terms of a, v1, and v2 when the distance between them is √2a again. | [
"\\( t = \\frac{4a}{v_1 - v_2} \\)"
] | [
"Two persons start running from opposite corners A and C of a square field with side length 'a' in the clockwise direction with speeds v1 and v2, respectively. Initially, the distance between them is √2a. Find the time t in terms of a, v1, and v2 when the distance between them is √2a again."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-249319a6ad8401a69f320e75f13acd3b6a386715d922b70aa2a7e9aaab9ac2fc | Two persons take turns throwing a coin. The first player to get two consecutive heads wins. Find the probability that the first player wins. | [
"\\(\\frac{14}{25}\\)"
] | [
"Two persons take turns throwing a coin. The first player to get two consecutive heads wins. Find the probability that the first player wins."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4b0c85a9c976c3a0050afde494122f927eb0f0c60d13dbda2f09cf13e0a3a35f | Two persons, $A$ and $B$, each tell a positive integer to $C$ individually. $C$ writes the sum of those two numbers and another randomly chosen positive integer on a board. $C$ then asks $A$ whether he knows $B$'s number; if $A$ says no, $C$ will ask $B$ whether he knows $A$'s number; and so on. Let $N$ be the maximum ... | [
"199"
] | [
"Two persons, $A$ and $B$, each tell a positive integer to $C$ individually. $C$ writes the sum of those two numbers and another randomly chosen positive integer on a board. $C$ then asks $A$ whether he knows $B$'s number; if $A$ says no, $C$ will ask $B$ whether he knows $A$'s number; and so on. Let $N$ be the max... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b94460bb411bfd1115aa77bbbe7e8e8983c257bf0c4024fae849d7625e10c49f | Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $X$ winning a match in an unlimited number of games. | [
"\\(\\frac{5}{21}\\)"
] | [
"Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $X$ winning a match in an unlimited number of games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 6
} | 54 | 0 | 0 |
omr-ws-79aab8ffa39e124e0ca38983216c4d3b7768b04d2468f15679faa6d9fc87687a | Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $X$ winning a match within $5$ games. | [
"\\(\\frac{55}{243}\\)"
] | [
"Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $X$ winning a match within $5$ games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 14
} | 53 | 0 | 0 |
omr-ws-9b911b7aacecbdec1ffcd1c36804b165c0873b152fe19c8f7e747ee1c3f1dfa4 | Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $Y$ winning a match in an unlimited number of games. | [
"\\(\\frac{16}{21}\\)"
] | [
"Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $Y$ winning a match in an unlimited number of games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 6,
"tir": 5
} | 49 | 0 | 0 |
omr-ws-f0abffcec93d124d589d326bb642a9805d5272993006771fa68db4d668982ea5 | Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $Y$ winning a match within $5$ games. | [
"\\(\\frac{176}{243}\\)"
] | [
"Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. Determine the probability of $Y$ winning a match within $5$ games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 42,
"tir": 12
} | 54 | 0 | 0 |
omr-ws-fcd4ac8446f4b8397f63c840757ebade7da648d09951182f3ba931e665231d07 | Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair? | [
"\\( \\frac{16}{5} \\) or \\( 3.2 \\)"
] | [
"Two persons, $X$ and $Y$, play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 8,
"tir": 4
} | 46 | 0 | 0 |
omr-ws-02ceaa8d601cfd6601e2c82de67789019fb94b38c455b90b27f0f766e178722e | Two persons, A and B, play a game. A has 20 coins and B has 50 coins. They roll a dice. If the result is a multiple of 3, A gives B one coin; otherwise, B gives A one coin. They continue this until one person has 70 coins. What is the probability of A winning the game? | [
"\\(\\frac{2^{70}-2^{50}}{2^{70}-1}\\)"
] | [
"Two persons, A and B, play a game. A has 20 coins and B has 50 coins. They roll a dice. If the result is a multiple of 3, A gives B one coin; otherwise, B gives A one coin. They continue this until one person has 70 coins. What is the probability of A winning the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7d6b58d2541f14aff56862147ff84b2861bccb3defdc2d29c62dfc404f37dec8 | Two persons, A and B, play a game. A has 20 coins and B has 50 coins. They roll a die. If the result is a multiple of 3, A gives B one coin; otherwise, B gives A one coin. They continue this until one person has 70 coins. What is the probability of A winning the game? | [
"\\(\\frac{1154047404513689600}{1154048505100108801}\\)",
"\\frac{\\left( \\frac{1}{2} \\right)^{20} - 1}{\\left( \\frac{1}{2} \\right)^{70} - 1}"
] | [
"Two persons, A and B, play a game. A has 20 coins and B has 50 coins. They roll a die. If the result is a multiple of 3, A gives B one coin; otherwise, B gives A one coin. They continue this until one person has 70 coins. What is the probability of A winning the game?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-41d0002779a365678dad8ec62eedd5995c4a1a298792f0b140909cc47828c518 | Two persons, A and B, set up an incantation contest with the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii) The f... | [
"B will win."
] | [
"Two persons, A and B, set up an incantation contest with the following rules:\ni) Any incantation can appear no more than once;\nii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;\nii... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-cf450b3d54be3eae133215402db9dc42d195afb78e48d1ea9026cf9a39df397f | Two persons, \( A \) and \( B \), each think of a positive integer. You know both numbers but \( A \) and \( B \) do not know each other's numbers. On a board, you write the sum of the numbers and another number larger than the sum. Both \( A \) and \( B \) can see the board. You repeatedly ask each person if they know... | [
"40"
] | [
"Two persons, \\( A \\) and \\( B \\), each think of a positive integer. You know both numbers but \\( A \\) and \\( B \\) do not know each other's numbers. On a board, you write the sum of the numbers and another number larger than the sum. Both \\( A \\) and \\( B \\) can see the board. You repeatedly ask each pe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9220f30dc8d71620ea65dd554cecb254bd2bae53d2170c7a251fd064be03e13a | Two persons, \(A\) and \(B\), take turns tossing a fair coin. Person \(A\) goes first. The first person to toss heads \(n\) times in total wins. Find the probability \(a_n\) that \(A\) wins, and then study the limit \(\lim_{n\to\infty} a_n\). | [
"\\(\\frac{1}{2}\\)"
] | [
"Two persons, \\(A\\) and \\(B\\), take turns tossing a fair coin. Person \\(A\\) goes first. The first person to toss heads \\(n\\) times in total wins. Find the probability \\(a_n\\) that \\(A\\) wins, and then study the limit \\(\\lim_{n\\to\\infty} a_n\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 16
} | 24 | 0 | 0 |
omr-ws-72b43fa74eaa695f86c7dbea3759b505a8c536eb0548edfea33db76dbaec3e3d | Two persons, both of height \( h \), are standing at a distance of \( h \) from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is \( h \), find the height of the lamp-post in terms... | [
"\\( 2h \\)"
] | [
"Two persons, both of height \\( h \\), are standing at a distance of \\( h \\) from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is \\( h \\), find the height of the lamp-po... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d519c38baedb14574e7352c130b75f88d11d58c76292e255c1c5b707987e8597 | Two persons, both of height \( h \), are standing at a distance of \( h \) from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is \( h \), what is the height of the lamp post? | [
"\\( h \\)"
] | [
"Two persons, both of height \\( h \\), are standing at a distance of \\( h \\) from each other. The shadow of one person cast by a vertical lamp-post placed between the two persons is double the length of the shadow of the other. If the sum of the lengths of the shadows is \\( h \\), what is the height of the lamp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-559ef29aa4b85e3f464b2e523e4d680209556d35486240be654835deb1777f7f | Two piles contain 2000 and 2017 coins respectively. Ann and Bob take alternate turns, where each player on their turn picks a pile with at least two coins, removes $t$ coins (where $2 \leq t \leq 4$), and adds 1 coin to the other pile. The player who cannot make a move loses. If Ann plays first, determine which player ... | [
"Ann",
"Bob has a winning strategy."
] | [
"Two piles contain 2000 and 2017 coins respectively. Ann and Bob take alternate turns, where each player on their turn picks a pile with at least two coins, removes $t$ coins (where $2 \\leq t \\leq 4$), and adds 1 coin to the other pile. The player who cannot make a move loses. If Ann plays first, determine which ... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-ba5a02943e3e2ac2f0d1205aca31c0b30a6b0ec7284639b04f6902c98c94f35d | Two piles contain 2000 and 2018 coins respectively. Anna and Bob take alternate turns, where each player on their turn must remove 2, 3, or 4 coins from one pile and add 1 coin to the other pile. The player who cannot make a move loses. If Anna plays first, determine which player has a winning strategy. | [
"Anna"
] | [
"Two piles contain 2000 and 2018 coins respectively. Anna and Bob take alternate turns, where each player on their turn must remove 2, 3, or 4 coins from one pile and add 1 coin to the other pile. The player who cannot make a move loses. If Anna plays first, determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-52dc32cdc363956b1e30c7c39f0d5a55e4dd0f03f71593c0647be0e9a1816b32 | Two piles of coins each have 2010 coins. In a turn, a player can take as many coins as they want from one pile or take one coin from each pile. Players A and B play optimally, with A going first. Who wins the game, where the last player to take a coin is the winner? | [
"B"
] | [
"Two piles of coins each have 2010 coins. In a turn, a player can take as many coins as they want from one pile or take one coin from each pile. Players A and B play optimally, with A going first. Who wins the game, where the last player to take a coin is the winner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-80bcb50d4fa460ec702b35a0fe64fe63c5789317db73c4e468b2f1f4d2c19648 | Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number $k$, not exceeding the number of coins in either pile, the sum of the weights of the $k$ heaviest coins in the first pile is not more than that of the second pile. If each coin ... | [
"0"
] | [
"Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number $k$, not exceeding the number of coins in either pile, the sum of the weights of the $k$ heaviest coins in the first pile is not more than that of the second pile. If each ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc240c5391f3b902de5ba9eef3ad30acd04bbf5241807f264cf40068e8862de3 | Two piles of stones are placed on a table with equal total mass. For every integer \( k \) not exceeding the number of stones in any pile, the \( k \) heaviest stones in the second pile are heavier than the \( k \) heaviest stones in the first pile. Determine the smallest positive number \( x \) such that if every ston... | [
"no_valid_answer_found"
] | [
"Two piles of stones are placed on a table with equal total mass. For every integer \\( k \\) not exceeding the number of stones in any pile, the \\( k \\) heaviest stones in the second pile are heavier than the \\( k \\) heaviest stones in the first pile. Determine the smallest positive number \\( x \\) such that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b827281414c79dd7c138ca02c0846167675efa9489b683ad017e260688eb3df | Two pipelines operating together fill a pool $3/4$ full of water. If one pipeline fills one-fourth of the pool first, and then the second (the first is then switched off) brings the volume of water up to $3/4$ the capacity of the pool, then this will require 2.5 hours. Now if the first pipeline is in operation for one ... | [
"The first pipe will fill the pool in 2 hours, the second in 4 hours"
] | [
"Two pipelines operating together fill a pool $3/4$ full of water. If one pipeline fills one-fourth of the pool first, and then the second (the first is then switched off) brings the volume of water up to $3/4$ the capacity of the pool, then this will require 2.5 hours. Now if the first pipeline is in operation for... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-527c34511076557fd3ccfcc62dd8c15fda0a1331439325e11df64d5d9de0bcac | Two pipes are connected to a reservoir with a volume of 24 m$^3$. The outlet pipe only drains the water at the rate of 2 m$^3$/h, and the inlet pipe only fills the reservoir. When the two valves were opened, the reservoir was empty. When the reservoir was half full, the first pipe was closed and the second pipe kept fi... | [
"5/2 m$^3$"
] | [
"Two pipes are connected to a reservoir with a volume of 24 m$^3$. The outlet pipe only drains the water at the rate of 2 m$^3$/h, and the inlet pipe only fills the reservoir. When the two valves were opened, the reservoir was empty. When the reservoir was half full, the first pipe was closed and the second pipe ke... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-430f147db270291b813feaa2f36a390bbd3926042671f90f61adfb7acd17e118 | Two pipes can fill a tank in 6 minutes and 7 minutes respectively. If both pipes are opened alternatively for 1 minute each, in what time will they fill the tank? | [
"$\\frac{45}{7}$"
] | [
"Two pipes can fill a tank in 6 minutes and 7 minutes respectively. If both pipes are opened alternatively for 1 minute each, in what time will they fill the tank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-066739c7a4ddbaae95ce7f5bd12cd9f7699e3a881445c9a527fdf849d1f918c9 | Two pipes running together fill a tank in \( \frac{20}{3} \) minutes. After adjusting their rates, one pipe running alone takes 1 minute less to fill the tank, and the other pipe running alone takes 2 minutes more to fill the tank. Together, they now fill the tank in 7 minutes. Find the time it takes for each pipe to f... | [
"14 minutes"
] | [
"Two pipes running together fill a tank in \\( \\frac{20}{3} \\) minutes. After adjusting their rates, one pipe running alone takes 1 minute less to fill the tank, and the other pipe running alone takes 2 minutes more to fill the tank. Together, they now fill the tank in 7 minutes. Find the time it takes for each p... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-23614bfff8d10783465c2a85c2c3915457474f90be918d8890c4812503f60ddb | Two pipes, operating simultaneously, feed 100 liters of a liquid per minute into a tank. There are two acid solutions, one strong and the other weak. If we mix 10 liters of each solution and 20 liters of water, we get 40 liters of a 20\% solution. It is also known that if we feed the weak solution into an empty tank th... | [
"50\\%"
] | [
"Two pipes, operating simultaneously, feed 100 liters of a liquid per minute into a tank. There are two acid solutions, one strong and the other weak. If we mix 10 liters of each solution and 20 liters of water, we get 40 liters of a 20\\% solution. It is also known that if we feed the weak solution into an empty t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85bed50b8f6ff748ebb345f8d5cdd3c405969a4d690face12128233c6a42836b | Two pirates, Adam and Bob, found a treasure box containing 100 golden coins. They decide to separate the coins by taking turns, with Adam going first. In each turn, a pirate takes a pile of coins, and the other pirate decides who this pile belongs to. If a pirate receives 9 piles of coins, the game ends and the remaini... | [
"Adam should take 6 coins each time."
] | [
"Two pirates, Adam and Bob, found a treasure box containing 100 golden coins. They decide to separate the coins by taking turns, with Adam going first. In each turn, a pirate takes a pile of coins, and the other pirate decides who this pile belongs to. If a pirate receives 9 piles of coins, the game ends and the re... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-a62f2c24a6822f05ad8a5a031dc8f2109847290ee2ef79f5098d42dd0747d579 | Two planes are drawn through a point taken on a lateral edge of a regular triangular prism with the side of the base $a$. One of them passes through a side of the lower base of the prism at an angle $\alpha $ to the base, the other, through the parallel side of the upper base and at an angle $\beta $ to it. Determine t... | [
"$V=\\frac{3a^3\\sin(\\alpha +\\beta )}{8\\cos \\alpha \\cos \\beta }$; $S+S'=\\frac{a^2\\sqrt 3}{4}\\cdot \\frac{\\cos \\alpha +\\cos\\beta }{\\cos\\alpha \\cos\\beta } =\\frac{a^2\\sqrt 3\\cos\\frac{\\alpha +\\beta }{2}\\cos\\frac{\\alpha -\\beta }{2}} {2\\cos \\alpha \\cos \\beta }$"
] | [
"Two planes are drawn through a point taken on a lateral edge of a regular triangular prism with the side of the base $a$. One of them passes through a side of the lower base of the prism at an angle $\\alpha $ to the base, the other, through the parallel side of the upper base and at an angle $\\beta $ to it. Dete... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44bfb372f1355b64b8b8f71c0d808266a1d4afa487447a9bef17b4d98ced9397 | Two planes are drawn through the vertex of a cone. One of them makes an angle $\alpha $ with the plane of the base of the cone and cuts the base along a chord $a$ long, and the other makes an angle $\beta $ with the plane of the base and cuts the base along a chord $b$ long. Find the volume of the cone. | [
"$\\frac{\\pi \\sqrt {b^2-a^2}(b^2\\cot^2 \\alpha -a^2\\cot^2 \\beta )} {24(\\cot^2 \\alpha -\\cot^2 \\beta )^{3/2}}$"
] | [
"Two planes are drawn through the vertex of a cone. One of them makes an angle $\\alpha $ with the plane of the base of the cone and cuts the base along a chord $a$ long, and the other makes an angle $\\beta $ with the plane of the base and cuts the base along a chord $b$ long. Find the volume of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88ff51b065d2b8009475e8b9a8ba9988d6b780ec01c58b59b57cb5504d27a577 | Two planes are flying along two coplanar lines at (possibly different) constant speeds. At $12:00$ PM, they are five miles apart. At $12:04$ PM, they are three miles apart, and at $12:07$ PM the planes are six miles apart. Let $d$ be the shortest distance between the two planes at any point in time. Find $d^2$. | [
"\\( 9 \\)"
] | [
"Two planes are flying along two coplanar lines at (possibly different) constant speeds. At $12:00$ PM, they are five miles apart. At $12:04$ PM, they are three miles apart, and at $12:07$ PM the planes are six miles apart. Let $d$ be the shortest distance between the two planes at any point in time. Find $d^2$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-827bed8590d8f59996f61d77cf192a12d2415f6946f0a5dbbc34a3ab09ff5abf | Two planes, \( \alpha \) and \( \beta \), form a dihedral angle of \( 30^\circ \), and their intersection is the line \( d \). A point \( A \) situated at the exterior of this angle projects itself in \( P \not\in d \) on \( \alpha \), and in \( Q \not\in d \) on \( \beta \) such that \( AQ < AP \). Name \( B \) the pr... | [
"Yes, \\( A, B, P, Q \\) are coplanar."
] | [
"Two planes, \\( \\alpha \\) and \\( \\beta \\), form a dihedral angle of \\( 30^\\circ \\), and their intersection is the line \\( d \\). A point \\( A \\) situated at the exterior of this angle projects itself in \\( P \\not\\in d \\) on \\( \\alpha \\), and in \\( Q \\not\\in d \\) on \\( \\beta \\) such that \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-4773a06703a30e6eaa6454715cde9abf3d44a4095abe6d92500b25a969ddfaef | Two planes, \( \alpha \) and \( \beta \), form a dihedral angle of \( 30^\circ \), and their intersection is the line \( d \). A point \( A \) situated at the exterior of this angle projects itself in \( P \not\in d \) on \( \alpha \), and in \( Q \not\in d \) on \( \beta \) such that \( AQ < AP \). Name \( B \) the pr... | [
"4"
] | [
"Two planes, \\( \\alpha \\) and \\( \\beta \\), form a dihedral angle of \\( 30^\\circ \\), and their intersection is the line \\( d \\). A point \\( A \\) situated at the exterior of this angle projects itself in \\( P \\not\\in d \\) on \\( \\alpha \\), and in \\( Q \\not\\in d \\) on \\( \\beta \\) such that \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68b455b7858fbd9ec18fc5aeca4717b39a4b90e4b270b09852df0d8770a36509 | Two players $A$ and $B$ alternately take chips from two piles with $a$ and $b$ chips respectively, where initially $a > b$. A move consists of taking from one pile a multiple of the number of chips in the other pile. The winner is the one who takes the last chip from one of the piles. For what value of $\alpha$ can $A$... | [
"\\(\\alpha = 2\\)"
] | [
"Two players $A$ and $B$ alternately take chips from two piles with $a$ and $b$ chips respectively, where initially $a > b$. A move consists of taking from one pile a multiple of the number of chips in the other pile. The winner is the one who takes the last chip from one of the piles. For what value of $\\alpha$ c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5c7272928e8f0af38108f0d9697432cbf605c3195b288a2d51e040ca0c55442 | Two players $A$ and $B$ alternately take chips from two piles with $a$ and $b$ chips respectively, where initially $a > b$. A move consists of taking from one pile a multiple of the number of chips in the other pile. The winner is the one who takes the last chip from one of the piles. If $a > 2b$, find the minimum numb... | [
"3"
] | [
"Two players $A$ and $B$ alternately take chips from two piles with $a$ and $b$ chips respectively, where initially $a > b$. A move consists of taking from one pile a multiple of the number of chips in the other pile. The winner is the one who takes the last chip from one of the piles. If $a > 2b$, find the minimum... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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