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-ca73e7f00f5da0443dfdfe35ae59f668c77b991ca9741dabcac6a360687c6665 | Two players $A$ and $B$ and another 2001 people form a circle, such that $A$ and $B$ are not in consecutive positions. $A$ and $B$ play in alternating turns, starting with $A$. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last man sta... | [
"1000"
] | [
"Two players $A$ and $B$ and another 2001 people form a circle, such that $A$ and $B$ are not in consecutive positions. $A$ and $B$ play in alternating turns, starting with $A$. A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. The winner is the last ma... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0f9aeef11e1df7492448c83d0596ee2c8a9a19af459b1a34a90d4f8bf9096645 | Two players $A$ and $B$ are playing a game with an $m \times n$ chocolate bar divided into a grid of $1 \times 1$ squares. They take turns cutting the chocolate along the grid lines. The player who makes the last cut wins. Determine if there is a winning strategy for either player. | [
"Yes, there is a winning strategy for either player depending on the dimensions \\( m \\) and \\( n \\)."
] | [
"Two players $A$ and $B$ are playing a game with an $m \\times n$ chocolate bar divided into a grid of $1 \\times 1$ squares. They take turns cutting the chocolate along the grid lines. The player who makes the last cut wins. Determine if there is a winning strategy for either player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 9
} | 17 | 0 | 0 |
omr-ws-aa04105de710639f3d6e89dd3403881a8ff27e8a121dfb359f537c97ac7c750d | Two players $A$ and $B$ are playing a game. $B$ has $p$ sheets of paper lying next to each other on a table. On each of the sheets, he writes some of the numbers from $1$ to $n$ (he can write no number, or all the numbers). On the back of each sheet, he writes all those integers from $1$ to $n$ that he did not write in... | [
"\\( \\left\\lfloor \\log_2(n+1) \\right\\rfloor \\)"
] | [
"Two players $A$ and $B$ are playing a game. $B$ has $p$ sheets of paper lying next to each other on a table. On each of the sheets, he writes some of the numbers from $1$ to $n$ (he can write no number, or all the numbers). On the back of each sheet, he writes all those integers from $1$ to $n$ that he did not wri... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-811d308088b4880249b19aa4e6f13ff819104458257110509917922d8461459e | Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighboring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of ... | [
"198"
] | [
"Two players $A$ and $B$ have one stone each on a $100 \\times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighboring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-caa17373c15808b176657e60035f1ab183f9ebf7a8dcb192126c4000c5f90109 | Two players $A$ and $B$ participate in a game where they take turns removing stones from a pile. Initially, there are 2003 stones. On each turn, a player must remove a number of stones that is a divisor of the current number of stones in the pile. The player who removes the last stone loses. Determine the number of sto... | [
"1"
] | [
"Two players $A$ and $B$ participate in a game where they take turns removing stones from a pile. Initially, there are 2003 stones. On each turn, a player must remove a number of stones that is a divisor of the current number of stones in the pile. The player who removes the last stone loses. Determine the number o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efab00f810d9e9e2328aab57ed13c9bd66c009a61c172a4df44f26780dcc8c95 | Two players $A$ and $B$ play a game on a $1 \times 2017$ table. On each turn, a player writes either $+1$ or $-1$ in an empty cell. If a player writes a number in a cell where the adjacent cell has the opposite number, that player loses. Player $A$ starts the game. Determine which player has the winning strategy. | [
"Player $A$ has the winning strategy."
] | [
"Two players $A$ and $B$ play a game on a $1 \\times 2017$ table. On each turn, a player writes either $+1$ or $-1$ in an empty cell. If a player writes a number in a cell where the adjacent cell has the opposite number, that player loses. Player $A$ starts the game. Determine which player has the winning strategy.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-6d18ee7935d067e109f89b5497848d33085c6684f2078fbc2a0235597b644ded | Two players $A$ and $B$ play a game on a $1 \times m$ board using $2012$ pieces numbered from $1$ to $2012$. At each turn, $A$ chooses a piece and $B$ places it in an empty spot. After $k$ turns, if all pieces are placed on the board in increasing order, then $B$ wins; otherwise, $A$ wins. For which values of $(m, k)$ ... | [
"$m \\ge 2012$ or $m < 2012$ and $k < \\log_2(m+2)$"
] | [
"Two players $A$ and $B$ play a game on a $1 \\times m$ board using $2012$ pieces numbered from $1$ to $2012$. At each turn, $A$ chooses a piece and $B$ places it in an empty spot. After $k$ turns, if all pieces are placed on the board in increasing order, then $B$ wins; otherwise, $A$ wins. For which values of $(m... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e06354fa3aa6307e8915c6267dc724347186b885771bda2cbe03ba4aa82f32ba | Two players $A$ and $B$ play a game on a circle with $2n$ cells. $A$ begins and each player alternates turns by choosing a free cell and writing either $O$ or $M$ in it. The player who first creates the sequence $OMO$ in three successive cells wins the game. If no such sequence is formed, the game is a draw. Assuming p... | [
"2n"
] | [
"Two players $A$ and $B$ play a game on a circle with $2n$ cells. $A$ begins and each player alternates turns by choosing a free cell and writing either $O$ or $M$ in it. The player who first creates the sequence $OMO$ in three successive cells wins the game. If no such sequence is formed, the game is a draw. Assum... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-825d68da3583efde80c8ad66cc02708560986e8b1480ab8e26ca8454c3b86e67 | Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially, all the grid cells are empty. In each turn, $A$ places a counter in two adjacent empty hexagons, and $B$ removes any one counter. Define $f(k)$ as the minimum number of turns required for $A$ to ensure that there are $k$ consecutive... | [
"0"
] | [
"Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially, all the grid cells are empty. In each turn, $A$ places a counter in two adjacent empty hexagons, and $B$ removes any one counter. Define $f(k)$ as the minimum number of turns required for $A$ to ensure that there are $k$ consec... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-bd05bc156b5417d52c839069882253d454f8e0750582322b702931dfc7737c48 | Two players $A$ and $B$ play a game where $A$ chooses a point with integer coordinates and colors it green, and $B$ then chooses 10 uncolored points and colors them yellow. The game continues with the same rules. The goal of $A$ is to achieve $111^2$ green points that are the intersections of $111$ horizontal lines and... | [
"A"
] | [
"Two players $A$ and $B$ play a game where $A$ chooses a point with integer coordinates and colors it green, and $B$ then chooses 10 uncolored points and colors them yellow. The game continues with the same rules. The goal of $A$ is to achieve $111^2$ green points that are the intersections of $111$ horizontal line... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-6dc519dff6da34a31bb30921db69b40276c9dc087959430785e4a3f24fce25ab | Two players $A$ and $B$ play a game where $B$ places a stone within a distance of $\leq 2001$ from $A$'s initial position. Player $A$ can see the stone only if within a distance of $\leq 1$ and can move in steps of size $\leq 1$. After each step, $A$ can ask if they are "near" or "far" from the stone, with $B$ respondi... | [
"no_valid_answer_found"
] | [
"Two players $A$ and $B$ play a game where $B$ places a stone within a distance of $\\leq 2001$ from $A$'s initial position. Player $A$ can see the stone only if within a distance of $\\leq 1$ and can move in steps of size $\\leq 1$. After each step, $A$ can ask if they are \"near\" or \"far\" from the stone, with ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac050dc87c32f4568facffb4cebf3142b48b514d500549a147a0b77945e5286a | Two players $A$ and $B$ play a game where they take turns writing one digit on the blackboard, either to the right or left of the existing sequence. $A$ starts with an empty blackboard. $B$ wins if, after any of $B$'s moves, the sequence of digits forms a perfect square. If $A$ follows an optimal strategy, what is the ... | [
"3"
] | [
"Two players $A$ and $B$ play a game where they take turns writing one digit on the blackboard, either to the right or left of the existing sequence. $A$ starts with an empty blackboard. $B$ wins if, after any of $B$'s moves, the sequence of digits forms a perfect square. If $A$ follows an optimal strategy, what is... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba164015a250d44b8a2cdec80cf3e15ae0435764518d09cc7afb4cf084ce9c21 | Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves th... | [
"n-1"
] | [
"Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and mov... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-8dede4444b4aa717e42f73da253dfda1390db6ce9a5c5a113f912a428ba6692a | Two players $A$ and $B$ play a game with a calculator. The calculator has digits (excluding 0) arranged in a grid. Player $A$ starts by pressing a digit and then pressing the "+" button. Player $B$ then presses a digit in the same row or column as the one pressed by $A$, but not the same digit, and also presses "+." Th... | [
"Player $A$ has a winning strategy."
] | [
"Two players $A$ and $B$ play a game with a calculator. The calculator has digits (excluding 0) arranged in a grid. Player $A$ starts by pressing a digit and then pressing the \"+\" button. Player $B$ then presses a digit in the same row or column as the one pressed by $A$, but not the same digit, and also presses ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 20
} | 23 | 0 | 0 |
omr-ws-4c28ccf0209531eb902b66a2c384e44a1febb0434a50f19820cad906e08e417c | Two players $A$ and $B$ play a game with two fair six-sided dice. Player $A$ can choose to either roll one die and double the result or roll both dice and take the sum of the results to get $a$. Player $B$ always rolls both dice and takes the sum of the results to get $b$. If $a=b$, the process is repeated; otherwise, ... | [
"Roll both dice and take the sum of the results."
] | [
"Two players $A$ and $B$ play a game with two fair six-sided dice. Player $A$ can choose to either roll one die and double the result or roll both dice and take the sum of the results to get $a$. Player $B$ always rolls both dice and takes the sum of the results to get $b$. If $a=b$, the process is repeated; otherw... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-bbbb172036d503d492ec292b7fd815a1fcee297813d67bb251ac9919f3145306 | Two players $A$ and $B$ play a game. $A$ writes two different numbers between $1$ and $100$ on two sheets, puts them into two envelopes, and hands them to $B$. $B$ opens one of the envelopes randomly and must guess whether the opened number is larger than the other number. If $B$ uses an optimal strategy, what is the p... | [
"\\dfrac{50}{99}"
] | [
"Two players $A$ and $B$ play a game. $A$ writes two different numbers between $1$ and $100$ on two sheets, puts them into two envelopes, and hands them to $B$. $B$ opens one of the envelopes randomly and must guess whether the opened number is larger than the other number. If $B$ uses an optimal strategy, what is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-dfd4a1f516e79f1ba0f2168052be1e0474b8f27604c2e66ccc692f9a0042c845 | Two players $A$ and $B$ play alternately with respective probabilities of winning each move $a$ and $b$. The game stops when one of the players wins for the first time. Assuming that $A$ goes first, find a relationship between $a$ and $b$ such that the game is fair. | [
"\\( a = \\frac{b}{b+1} \\)"
] | [
"Two players $A$ and $B$ play alternately with respective probabilities of winning each move $a$ and $b$. The game stops when one of the players wins for the first time. Assuming that $A$ goes first, find a relationship between $a$ and $b$ such that the game is fair."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-70326685dcbd76e2b9571d1383accf6440063d109bc2edf051ab3e47ea9e3231 | Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not intersect any previously drawn diagonals. A player loses if after their turn, a quadrilateral is formed such that its two diagonals are not drawn. $A$ starts ... | [
"B"
] | [
"Two players $A$ and $B$ play alternatively in a convex polygon with $n \\geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not intersect any previously drawn diagonals. A player loses if after their turn, a quadrilateral is formed such that its two diagonals are not drawn. $A$ s... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-2b47aa7416f491b3031dd8fde02c466cd24e8488e974e1b5d678cfab08a71e1d | Two players $A$ and $B$ play on a $2N \times 2N$ square. In turn ($A$ starts), they draw a $2 \times 1$ block on the square. Blocks must be placed horizontally or vertically and cannot overlap. Whoever leaves the opponent with no lawful moves wins. Who wins, and what is the winning strategy? | [
"B wins with the strategy of mirroring A's moves with respect to the center of the square."
] | [
"Two players $A$ and $B$ play on a $2N \\times 2N$ square. In turn ($A$ starts), they draw a $2 \\times 1$ block on the square. Blocks must be placed horizontally or vertically and cannot overlap. Whoever leaves the opponent with no lawful moves wins. Who wins, and what is the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-3e701b83317e18f2eb8b85a37b34958970ea9ccee8d5422b0a62b43ac902e812 | Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn, a player chooses a positive integer $n$, erases some primes $p_1, p_2, \dots, p_n$ from the blackboard, and wri... | [
"1"
] | [
"Two players $A$ and $B$ play the following game. Before the game starts, $A$ chooses 1000 not necessarily different odd primes, and then $B$ chooses half of them and writes them on a blackboard. In each turn, a player chooses a positive integer $n$, erases some primes $p_1, p_2, \\dots, p_n$ from the blackboard, a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c006b621d24a534f847fada5e390ab0c9e07da3efe3e186f154b22038a3fff17 | Two players $A$ and $B$ take turns playing a game with a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes that number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones and removes that number of stones, and so on. The player who has to remove the ... | [
"1"
] | [
"Two players $A$ and $B$ take turns playing a game with a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes that number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones and removes that number of stones, and so on. The player who has to remove... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-b22cb2ce97ed4271b7d848835e2dcf4a8d1f360b410140ede45920a2e2a14a89 | Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. Player $A$ starts. On each turn, $B$ can take 1, 2, or 3 stones, while $A$ can take 2, 3, or 4 stones, or 1 stone if it is the last one in the pile. The player who takes the last stone loses. Determine for what values of $N$ player $A$ has a wi... | [
"A \\text{ wins if } N \\not\\equiv 1, 2 \\pmod{5}; B \\text{ wins otherwise}"
] | [
"Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. Player $A$ starts. On each turn, $B$ can take 1, 2, or 3 stones, while $A$ can take 2, 3, or 4 stones, or 1 stone if it is the last one in the pile. The player who takes the last stone loses. Determine for what values of $N$ player $A$ has... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-a62657df8dd55d412a4c366cae2cc98114f7f0d4a8cf1a30dfd6a26dcf60976f | Two players (A and B) play a game with 2n cards, each with a positive integer written on it. The cards are shuffled and laid out in a row. A starts and the players take turns to remove one card from either end of the row, until B picks up the remaining card. Each player's score is the sum of the numbers on their chosen... | [
"n(n+1)"
] | [
"Two players (A and B) play a game with 2n cards, each with a positive integer written on it. The cards are shuffled and laid out in a row. A starts and the players take turns to remove one card from either end of the row, until B picks up the remaining card. Each player's score is the sum of the numbers on their c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-c2bff5dd4d6a3f4242a2098b550f53ec6b9f5b5c7a603573a6ee29f88dbc1282 | Two players (Kate and Adam) play a variant of Nim with 11 sticks. Each player can remove either one or any prime number of sticks. The player who is forced to take the last stick loses. If Kate plays perfectly, she will always win. Give the sum of all the starting moves that lead to a sure win for Kate. | [
"2"
] | [
"Two players (Kate and Adam) play a variant of Nim with 11 sticks. Each player can remove either one or any prime number of sticks. The player who is forced to take the last stick loses. If Kate plays perfectly, she will always win. Give the sum of all the starting moves that lead to a sure win for Kate."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-50d2fb5efaa46d2bd86c41d830b4a2cae284e70fb3e741084c774c54f968ac38 | Two players A and B alternately remove stones from two piles, one with 100 stones and one with 252 stones. On each turn, a player can remove a number of stones from one pile, which is a divisor of the number of stones in the other pile. The player who makes the last move wins. Determine which player can always guarante... | [
"A"
] | [
"Two players A and B alternately remove stones from two piles, one with 100 stones and one with 252 stones. On each turn, a player can remove a number of stones from one pile, which is a divisor of the number of stones in the other pile. The player who makes the last move wins. Determine which player can always gua... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 18
} | 22 | 0 | 0 |
omr-ws-ed6e3c8ee70e91219dad63af52456b81dbe555d1104d012cd5f5d3b3a6d05661 | 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 a multiple of the other pile from a pile. The winner is the one who takes the last chip in one of the piles. For what value of $\alpha$ can player A force a win if initially... | [
"\\( \\alpha = \\frac{\\sqrt{5} + 1}{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 a multiple of the other pile from a pile. The winner is the one who takes the last chip in one of the piles. For what value of $\\alpha$ can player A force a win if ini... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e1188fb1f5857969775887cad722057cd049ab833a0f6a9b669290ff0d45ace2 | Two players A and B are playing a championship series where the winner is the first to win at least 6 games and be ahead by 2 games. The probability of A winning a match is 2/3 and the probability of B winning a match is 1/3. Given that B has won the first four games, find the probability that A wins the championship. | [
"\\( \\frac{1088}{3645} \\)"
] | [
"Two players A and B are playing a championship series where the winner is the first to win at least 6 games and be ahead by 2 games. The probability of A winning a match is 2/3 and the probability of B winning a match is 1/3. Given that B has won the first four games, find the probability that A wins the champions... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-00f08ab0420c0231a52a41b9fed85da8e1ff68d2d6789f75ae961d9a957f784d | Two players A and B play a game consecutively until one loses all their capital. Player A starts with \$a and player B with \$b. The probability of A winning each game is \( p \) and the probability of B winning each game is \( q = 1 - p \). Find the probability of ruin for each player if no limit is set for the number... | [
" \\frac{a}{a + b} "
] | [
"Two players A and B play a game consecutively until one loses all their capital. Player A starts with \\$a and player B with \\$b. The probability of A winning each game is \\( p \\) and the probability of B winning each game is \\( q = 1 - p \\). Find the probability of ruin for each player if no limit is set for... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7b89d8d8131476634652f1656c42a3412b26b532bf66db2a480348d0cc24c2db | Two players A and B play a game of dice. They roll a pair of dice alternately. The player who gets the event (1,1) first wins. If A starts, what is the probability that A wins the game? | [
"\\(\\frac{36}{71}\\)"
] | [
"Two players A and B play a game of dice. They roll a pair of dice alternately. The player who gets the event (1,1) first wins. If A starts, what is the probability that A wins the game?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-04fe57acbd57bedb2db038a63a6511ad76c9c99f1df92c167bdac24d8412a88c | Two players A and B play a game starting with a pile of 1000 stones. Player A first removes any number \( s \) of stones such that \( 0 < s < 1000 \). Thereafter, each player can remove any positive integer number of stones that is a divisor of the number taken in the preceding move. The player who makes the last move ... | [
"Player A can always guarantee victory."
] | [
"Two players A and B play a game starting with a pile of 1000 stones. Player A first removes any number \\( s \\) of stones such that \\( 0 < s < 1000 \\). Thereafter, each player can remove any positive integer number of stones that is a divisor of the number taken in the preceding move. The player who makes the l... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 12
} | 15 | 0 | 0 |
omr-ws-e390ae19c2cf6917b8f06447701427f309b4288add8d4fd976af8d5b9bf2ccd2 | Two players A and B play a game where initially, there are $m$ equal positive integers $n$ written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number $k$ is the smallest among all positive integers on the board, the player repl... | [
"- If \\( m \\) is even, B wins.\n- If \\( m \\) is odd and \\( n \\) is odd, A wins.\n- If \\( m \\) is odd and \\( n \\) is even, B wins."
] | [
"Two players A and B play a game where initially, there are $m$ equal positive integers $n$ written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number $k$ is the smallest among all positive integers on the board, the player... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-0730b689af8935b4da36ba0ef79536096f76dbf20555418c7119efc33b8f5f69 | Two players A and B play a game with a calculator where the figures (except 0) are arranged in a specific table, and the ‘+’ key is also included. Player A starts by pressing a figure and then pressing the + key. Player B then presses a figure in the same row or column as the one pressed by A, but not the same figure, ... | [
"B"
] | [
"Two players A and B play a game with a calculator where the figures (except 0) are arranged in a specific table, and the ‘+’ key is also included. Player A starts by pressing a figure and then pressing the + key. Player B then presses a figure in the same row or column as the one pressed by A, but not the same fig... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-cec78ea320440264957efc4ac8f109a7715d222c3b5a47b4fa663a8be8842c56 | Two players A and B play a series of "2n" games. Each game can result in either a win or a loss for A. Find the total number of ways in which A can win the series of these games. | [
"\\(2^{2n-1} - \\frac{1}{2}\\binom{2n}{n}\\)"
] | [
"Two players A and B play a series of \"2n\" games. Each game can result in either a win or a loss for A. Find the total number of ways in which A can win the series of these games."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-b519b89b8403f0e6e285ecb703899a53c3f5a64e78c05a5f056176c776076085 | Two players A and B play the following game: Suppose on the table is the positive integer \( n \). Player A chooses a positive integer \( x \), and player B will replace \( n \) by either \( n + x \) or \( n - x \) on the table. Find all values of \( k \) such that player A can always ensure that at some point, a power... | [
"\\( k = 2^t + 1 \\) for any non-negative integer \\( t \\)."
] | [
"Two players A and B play the following game: Suppose on the table is the positive integer \\( n \\). Player A chooses a positive integer \\( x \\), and player B will replace \\( n \\) by either \\( n + x \\) or \\( n - x \\) on the table. Find all values of \\( k \\) such that player A can always ensure that at so... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ecd26eb6e78affcd629e271f6f2511ad2554eba067110b2bbb084db090494b1c | Two players A and B take turns drawing a line that joins two points among $n = 2016$ points given in a circle. The first player to form a cycle loses. Who has the winning strategy? | [
"A"
] | [
"Two players A and B take turns drawing a line that joins two points among $n = 2016$ points given in a circle. The first player to form a cycle loses. Who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-e6a866679da6513f428e44c4f4d07c410aeb98dcb60f07b0e1526948b838dd9f | Two players A and B take turns drawing a line that joins two points among $n = 2017$ points given in a circle. The first player to form a cycle loses. Who has the winning strategy? | [
"A"
] | [
"Two players A and B take turns drawing a line that joins two points among $n = 2017$ points given in a circle. The first player to form a cycle loses. Who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-063d5f2283910857e532a30f37b551de0caeddd5a504f41b2466f2b07c9427e6 | Two players A and B take turns replacing asterisks in the matrix $\left({\begin{smallmatrix} * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ \end{smallmatrix}}\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if A goes first? | [
"B has a winning strategy."
] | [
"Two players A and B take turns replacing asterisks in the matrix $\\left({\\begin{smallmatrix}\n* & * & * & * \\\\\n* & * & * & * \\\\\n* & * & * & * \\\\\n* & * & * & * \\\\\n\\end{smallmatrix}}\\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 11
} | 17 | 0 | 0 |
omr-ws-52aa2617da0b8e873fafd7293d724ec8ef4752377d22f3fb2e3d742f06e8d4a3 | Two players A and B take turns replacing asterisks in the matrix $\left({\begin{smallmatrix} * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ * & * & * & * \\ \end{smallmatrix}}\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if B goes first? | [
"A has a winning strategy."
] | [
"Two players A and B take turns replacing asterisks in the matrix $\\left({\\begin{smallmatrix}\n* & * & * & * \\\\\n* & * & * & * \\\\\n* & * & * & * \\\\\n* & * & * & * \\\\\n\\end{smallmatrix}}\\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-bcab2f76f01854e692b4b2df727591fc337bba5b2b4408f159b40e20b98ee8fa | Two players A and B take turns replacing asterisks in the matrix $\left({\begin{smallmatrix} * & * & * \\ * & * & * \\ * & * & * \\ \end{smallmatrix}}\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if A goes first? | [
"A has a winning strategy."
] | [
"Two players A and B take turns replacing asterisks in the matrix $\\left({\\begin{smallmatrix}\n* & * & * \\\\\n* & * & * \\\\\n* & * & * \\\\\n\\end{smallmatrix}}\\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if A goes first?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-ba9119e8235afc51683022298567885caff7159614b5297193b640150bf55c20 | Two players A and B take turns replacing asterisks in the matrix $\left({\begin{smallmatrix} * & * & * \\ * & * & * \\ * & * & * \\ \end{smallmatrix}}\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if B goes first? | [
"A has a winning strategy."
] | [
"Two players A and B take turns replacing asterisks in the matrix $\\left({\\begin{smallmatrix}\n* & * & * \\\\\n* & * & * \\\\\n* & * & * \\\\\n\\end{smallmatrix}}\\right)$ with any numbers. If the resulting matrix is singular, A wins; otherwise, B wins. Who has a winning strategy if B goes first?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f48cea6f2a1fd50a506b75dd6c6d013a0b47f59a21f829f82813f429545fde22 | Two players \( A \) and \( B \) and another 2001 people form a circle, such that \( A \) and \( B \) are not in consecutive positions. \( A \) and \( B \) play in alternating turns, starting with \( A \). A play consists of touching one of the people neighboring you, which such person once touched leaves the circle. Th... | [
"1000"
] | [
"Two players \\( A \\) and \\( B \\) and another 2001 people form a circle, such that \\( A \\) and \\( B \\) are not in consecutive positions. \\( A \\) and \\( B \\) play in alternating turns, starting with \\( A \\). A play consists of touching one of the people neighboring you, which such person once touched le... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-3a6e56008ffb86ec2a827b6fd980ae01dd0013d31914375d53ccbd5e2478373a | Two players \( A \) and \( B \) play a game on a calculator with digits 1 through 9. \( A \) turns on the calculator and presses a digit, then presses the button "+". \( A \) passes the calculator to \( B \), which presses a digit in the same row or column with the one pressed by \( A \) that is not the same as the las... | [
"\\( A \\) has the winning strategy."
] | [
"Two players \\( A \\) and \\( B \\) play a game on a calculator with digits 1 through 9. \\( A \\) turns on the calculator and presses a digit, then presses the button \"+\". \\( A \\) passes the calculator to \\( B \\), which presses a digit in the same row or column with the one pressed by \\( A \\) that is not ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-9b6c44db48b0fe46a9423aee75e0fecc0f0360dc0780da4b3b2ca99852a677c5 | Two players \( A \) and \( B \) play a game where they toss a fair coin. If heads appears, \( A \) gets a point; if tails appears, \( B \) gets a point. The game ends when either \( A \) leads by \( a \) points or \( B \) leads by \( b \) points, where \( a \) and \( b \) are positive integers. What is the expected len... | [
"\\( ab \\)"
] | [
"Two players \\( A \\) and \\( B \\) play a game where they toss a fair coin. If heads appears, \\( A \\) gets a point; if tails appears, \\( B \\) gets a point. The game ends when either \\( A \\) leads by \\( a \\) points or \\( B \\) leads by \\( b \\) points, where \\( a \\) and \\( b \\) are positive integers.... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-f081b28e256b7f5680a5a3ac3d1cfb920fe5f8061fd6e738e360f4e86a277229 | Two players alternate shooting free throws in basketball. For each attempt, player 1 has a \(1/3\) probability of success and player 2 has a \(1/4\) probability of success. What is the probability that player 1 succeeds before player 2? | [
"\\(\\frac{2}{3}\\)"
] | [
"Two players alternate shooting free throws in basketball. For each attempt, player 1 has a \\(1/3\\) probability of success and player 2 has a \\(1/4\\) probability of success. What is the probability that player 1 succeeds before player 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-cd5af0fe8800b47b1bc93c9b41d509a0f207d4c4741fa8f2e051f67c77e044bc | Two players alternate turns changing a number on a blackboard to the difference of that number and one of its divisors. The player who writes zero loses. Starting with the number 2013, which player can guarantee a win? | [
"The first player can guarantee a win."
] | [
"Two players alternate turns changing a number on a blackboard to the difference of that number and one of its divisors. The player who writes zero loses. Starting with the number 2013, which player can guarantee a win?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8b0b36a7cc2d9612e936e83452ee1e3419be1d5a72250f1123b413834138f21b | Two players alternate turns counting consecutive numbers. The first player starts from the number $1$. In the following turns, the next player starts counting from the next number. In a turn, a player can count up to $5$ numbers. The objective of the game is to make your opponent speak the number $21$. Problem 3a: Ther... | [
"First player wins."
] | [
"Two players alternate turns counting consecutive numbers. The first player starts from the number $1$. In the following turns, the next player starts counting from the next number. In a turn, a player can count up to $5$ numbers. The objective of the game is to make your opponent speak the number $21$.\n\nProblem ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-6cc9dd780952eed39134c95de367e5e66383f21a1f0595276f6259d6c13c3acd | Two players alternate writing down integers between $1$ and $1000$ such that no integer which is a divisor of some already written integer is written. Determine who has a winning strategy. | [
"The first player has a winning strategy."
] | [
"Two players alternate writing down integers between $1$ and $1000$ such that no integer which is a divisor of some already written integer is written. Determine who has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 10
} | 28 | 0 | 0 |
omr-ws-129ea8349608b8021b5ec3c90646f1ccbf20d53acaf8c4b77cab2589e099c18a | Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: - No divisor of a previously written number may be written. - The player who writes $328$ loses. Who has a winning strategy, the first player or the second player? | [
"The first player has a winning strategy."
] | [
"Two players alternate writing positive divisors of $328$ on the board, subject to the following rules:\n- No divisor of a previously written number may be written.\n- The player who writes $328$ loses.\nWho has a winning strategy, the first player or the second player?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-a5ffec84e64cd6eb9aad8ea10f7a0e019371c002b5e5a47ee8db0f010e369059 | Two players alternate writing signs and numbers to the right on a board starting with the number 0. The first player always writes either a $+$ or $-$ sign, while the second player writes one of the numbers $1, 2, \ldots, 1993$, each exactly once. The game ends after 1993 moves. The second player wins a score equal to ... | [
"1993"
] | [
"Two players alternate writing signs and numbers to the right on a board starting with the number 0. The first player always writes either a $+$ or $-$ sign, while the second player writes one of the numbers $1, 2, \\ldots, 1993$, each exactly once. The game ends after 1993 moves. The second player wins a score equ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd2a5f8635b9ecbcbfbff4f4cdfb5ecfb24570f0ec9a24ddf727f4d3cf915988 | Two players alternately choose 10 out of 11 boxes and place one ball in each of the chosen boxes. The player who first gets a box with 21 balls wins. Which player has a winning strategy? | [
"The second player has a winning strategy."
] | [
"Two players alternately choose 10 out of 11 boxes and place one ball in each of the chosen boxes. The player who first gets a box with 21 balls wins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-8578351a0d0e6c70f8d34bdc1513c99679204d1e555281788355bfb609953d37 | Two players alternately extract balls (without replacement) from an urn containing $M$ white and $N$ black balls. The winner is the first one to take out a white ball. Find the probability of the first participant winning if $N = 4$ and $M = 1$. | [
"\\(\\frac{3}{5}\\)"
] | [
"Two players alternately extract balls (without replacement) from an urn containing $M$ white and $N$ black balls. The winner is the first one to take out a white ball. Find the probability of the first participant winning if $N = 4$ and $M = 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-a294dc980a700d917f597941880d39670ece62331f7d790f831a674a940aff35 | Two players alternately fold an $m \times n$ checkered sheet of paper along the lines of the cells. The one who cannot make another move loses. Who will win if both play optimally: the beginner or his opponent? | [
"\\text{Second player}"
] | [
"Two players alternately fold an $m \\times n$ checkered sheet of paper along the lines of the cells. The one who cannot make another move loses. Who will win if both play optimally: the beginner or his opponent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b7334656863ff87823cc10f53761cb756c28263d4ef6ced908950f5f3943c6c1 | Two players alternately replace the stars in the expression \( x^{2000} + x^{1999} + \cdots + x + 1 \) by real numbers. The player who makes the last move loses if the resulting polynomial has a real root \( t \) with \( |t| < 1 \), and wins otherwise. Give a winning strategy for one of the players. | [
"\\text{The second player has a winning strategy.}"
] | [
"Two players alternately replace the stars in the expression \\( x^{2000} + x^{1999} + \\cdots + x + 1 \\) by real numbers. The player who makes the last move loses if the resulting polynomial has a real root \\( t \\) with \\( |t| < 1 \\), and wins otherwise. Give a winning strategy for one of the players."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-7c13c842b515e53ac38f901bf598255ac6d4d0c77f2089b024c4acfe0c21dc23 | Two players alternately replace the stars in the expression \[ *x^{2000} + *x^{1999} + \cdots + *x + 1 \] by real numbers. The player who makes the last move loses if the resulting polynomial has a real root \( t \) with \( |t| < 1 \), and wins otherwise. Give a winning strategy for one of the players. | [
"\\text{The second player has a winning strategy.}"
] | [
"Two players alternately replace the stars in the expression\n\\[ *x^{2000} + *x^{1999} + \\cdots + *x + 1 \\]\nby real numbers. The player who makes the last move loses if the resulting polynomial has a real root \\( t \\) with \\( |t| < 1 \\), and wins otherwise. Give a winning strategy for one of the players."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-827989831701816ad3ef97f3242ba31f8bf60444048fbebc3df6a657e2a09479 | Two players alternately replace the stars in the expression \[ *x^{2000} + *x^{1999} + \cdots + *x + 1 \] by real numbers. The player who makes the last move loses if the resulting polynomial has a real root \( t \) with \( |t| < 1 \), and wins otherwise. Provide a winning strategy for one of the players. | [
"The first player (A) has the winning strategy."
] | [
"Two players alternately replace the stars in the expression \n\\[ *x^{2000} + *x^{1999} + \\cdots + *x + 1 \\]\nby real numbers. The player who makes the last move loses if the resulting polynomial has a real root \\( t \\) with \\( |t| < 1 \\), and wins otherwise. Provide a winning strategy for one of the players... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-351bba9c86d9d0b0d851bf757ad6ced1c19c1842bca533e08e6d86856b201400 | Two players alternately replace the stars in the polynomial expression \( *x^{2000} + *x^{1999} + \ldots + *x + 1 \) by real numbers until all stars are replaced. If the resulting polynomial has a real root, the first player wins; otherwise, the second player wins. Who has a winning strategy? | [
"\\text{First player}"
] | [
"Two players alternately replace the stars in the polynomial expression \\( *x^{2000} + *x^{1999} + \\ldots + *x + 1 \\) by real numbers until all stars are replaced. If the resulting polynomial has a real root, the first player wins; otherwise, the second player wins. Who has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-fd5495a888e2304e1b9cabe180813396ec4f539311d7a31ac01a6ff9f49975d0 | Two players alternately roll an $N$-sided die. The player who fails to roll a number greater than the previous roll loses. What is the probability that the first player wins? | [
"1 - \\left( \\frac{N-1}{N} \\right)^N"
] | [
"Two players alternately roll an $N$-sided die. The player who fails to roll a number greater than the previous roll loses. What is the probability that the first player wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f873817d83db607fe4a0e0d625418974056a458d10212c5405268afc0f41f48a | Two players alternately take turns removing 1, 2, 3, 4, or 5 stones from a pile of 2000 stones. A player cannot take the same number of stones as the previous player did on their last turn. The player who cannot make a valid move loses. Determine which player has a winning strategy and describe the strategy. | [
"First player wins by taking 4 stones initially."
] | [
"Two players alternately take turns removing 1, 2, 3, 4, or 5 stones from a pile of 2000 stones. A player cannot take the same number of stones as the previous player did on their last turn. The player who cannot make a valid move loses. Determine which player has a winning strategy and describe the strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 6,
"tir": 15
} | 51 | 0 | 0 |
omr-ws-e79fe1d5ede76d397a7b9c1c9cfef0b7620dc9795d12b380205c266ebaf6be1c | Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an \( n \times n \) grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides wins. What is the outcome of the game given best possible play by both players? Note... | [
"\\text{First player wins}"
] | [
"Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an \\( n \\times n \\) grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides wins. What is the outcome of the game given best possible play by both playe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be12384c3333251916b18edd197241b6dc236429bb2360bc97be3deaa526d9b2 | Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an \( n \times n \) grid. The first player to write four of his/her symbols at the vertices of a square with horizontal and vertical sides is the winner. What is the smallest \( n \) such that the first player has a winn... | [
"5"
] | [
"Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an \\( n \\times n \\) grid. The first player to write four of his/her symbols at the vertices of a square with horizontal and vertical sides is the winner. What is the smallest \\( n \\) such that the first player ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-67096fd78f2f8369faf45476f00d052bc83e4c1a6fcd846dd1d5dead16f1e247 | Two players alternately write divisors of $100!$ on a blackboard, ensuring no number is repeated. The player whose move results in the greatest common divisor (GCD) of all written numbers being $1$ loses the game. Which player has a winning strategy? | [
"The second player has a winning strategy."
] | [
"Two players alternately write divisors of $100!$ on a blackboard, ensuring no number is repeated. The player whose move results in the greatest common divisor (GCD) of all written numbers being $1$ loses the game. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 13
} | 31 | 0 | 0 |
omr-ws-432059e84387a4b4be46425e771e7d935b02d7d3219a44016ef31e5a78a28f20 | Two players alternately write integers on a blackboard as follows: the first player writes \( a_{1} \) arbitrarily, then the second player writes \( a_{2} \) arbitrarily, and thereafter a player writes a number equal to the sum of the two preceding numbers. The player after whose move the obtained sequence contains ter... | [
"The second player has the winning strategy."
] | [
"Two players alternately write integers on a blackboard as follows: the first player writes \\( a_{1} \\) arbitrarily, then the second player writes \\( a_{2} \\) arbitrarily, and thereafter a player writes a number equal to the sum of the two preceding numbers. The player after whose move the obtained sequence con... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-5c28e50faa00235808874684c255383b5d54d1560817d3e413552b3bfb394d8e | Two players and a judge play a game. Each player picks a positive integer \( a \) and \( b \) and tells it to the judge. The judge then discloses \( a+b \) and another number \( x \). The judge takes turns asking both players if they can guess the other's number. Let \( N \) be the smallest number of questions after wh... | [
"2"
] | [
"Two players and a judge play a game. Each player picks a positive integer \\( a \\) and \\( b \\) and tells it to the judge. The judge then discloses \\( a+b \\) and another number \\( x \\). The judge takes turns asking both players if they can guess the other's number. Let \\( N \\) be the smallest number of que... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-b0ba06611c820cc713587ee7c789ebe5a827bc141c876fa15c0ff06857b35b9e | Two players are playing a blackjack machine where they can win or lose $1$ dollar at a time. Both players play once per minute and eventually both win exactly $2N$ dollars. Let $T$ be the length of the shortest time interval during which both players won exactly $N$ dollars. Find the maximum possible value of $T$ in te... | [
"3N"
] | [
"Two players are playing a blackjack machine where they can win or lose $1$ dollar at a time. Both players play once per minute and eventually both win exactly $2N$ dollars. Let $T$ be the length of the shortest time interval during which both players won exactly $N$ dollars. Find the maximum possible value of $T$ ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-d5399b3a462b0e7655b30690263507f51a77b662a776ff1f1c155a9979e5dc56 | Two players are playing a game in which $2n$ numbers are written on a board. Each turn, one player picks one of the numbers on the board, and the other player gets to choose which player to give the number to (the players alternate roles). The game ends when either player has $n$ numbers, at which point all remaining n... | [
"\\left\\lceil \\dfrac{n(2n +1)}{2} \\right\\rceil"
] | [
"Two players are playing a game in which $2n$ numbers are written on a board. Each turn, one player picks one of the numbers on the board, and the other player gets to choose which player to give the number to (the players alternate roles). The game ends when either player has $n$ numbers, at which point all remain... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e5f44af4596dfe8e742ea80f4a804b935e25c9be12a5b9931be3b8c028bcfc2 | Two players are playing a game involving piles of coins. A legal move is to remove a coin or split a pile of coins into two non-empty piles. They are playing with piles of coins where the $k$-th pile contains $k$ coins. Determine which player has the winning strategy. | [
"\\text{First player}"
] | [
"Two players are playing a game involving piles of coins. A legal move is to remove a coin or split a pile of coins into two non-empty piles. They are playing with piles of coins where the $k$-th pile contains $k$ coins. Determine which player has the winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-849a46d857c4c72b23c5a6a2f24df96a6f0adbf363761a727da14fe2880a4a4e | Two players are playing a game on a $1403 \times 1403$ grid with $1403^2$ cells. The first player, on each turn, selects an unchosen cell and draws a vertical line segment from the top to the bottom of the cell. The second player, on each turn, selects an unchosen cell and draws a horizontal line segment from the left ... | [
"Draw"
] | [
"Two players are playing a game on a $1403 \\times 1403$ grid with $1403^2$ cells. The first player, on each turn, selects an unchosen cell and draws a vertical line segment from the top to the bottom of the cell. The second player, on each turn, selects an unchosen cell and draws a horizontal line segment from the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 10
} | 18 | 0 | 0 |
omr-ws-1f77b238aa6729dd0d358629768bc41d22cfd2c489cc986cd97898893f5e60c8 | Two players are playing a game on a $1994 \times 1994$ board with a chess knight piece. Player 1 moves the knight two cells horizontally and one cell vertically, while Player 2 moves the knight two cells vertically and one cell horizontally. No player can move the knight to a cell that has already been used. Determine ... | [
"1988018"
] | [
"Two players are playing a game on a $1994 \\times 1994$ board with a chess knight piece. Player 1 moves the knight two cells horizontally and one cell vertically, while Player 2 moves the knight two cells vertically and one cell horizontally. No player can move the knight to a cell that has already been used. Dete... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-a925060ba8a43921e757574a6fec41c144869aff2eba80db928c8d7e6ac6a472 | Two players are playing a game on a $2018 \times 2018$ board. The first player places blue coins, and the second player places red coins. The first player wins if they can create 5 blue coins in a row or column. Determine the minimum number of blue coins the first player must place to guarantee a win, assuming the seco... | [
"8073"
] | [
"Two players are playing a game on a $2018 \\times 2018$ board. The first player places blue coins, and the second player places red coins. The first player wins if they can create 5 blue coins in a row or column. Determine the minimum number of blue coins the first player must place to guarantee a win, assuming th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-51d5cedc908758f913d55012ad9cd3e90d0bbbb39eb43d0d5ca9206c80def35f | Two players are playing a game on a 3x7 rectangle board $P$ and a set of tetrominoes $S$. Each player alternately chooses a tetromino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the tetromino is precisely above one square on $P$ and does not overlap with any previo... | [
"\\text{Second player}"
] | [
"Two players are playing a game on a 3x7 rectangle board $P$ and a set of tetrominoes $S$. Each player alternately chooses a tetromino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the tetromino is precisely above one square on $P$ and does not overlap with any p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-6e724b8339804797338c7d4c528bfc473df319b66f5f8ce209361e2d9b539f1a | Two players are playing a game on a 4x5 rectangle board $P$ and a set of tetrominoes $S$. Each player alternately chooses a tetromino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the tetromino is precisely above one square on $P$ and does not overlap with any previo... | [
"\\text{Second player}"
] | [
"Two players are playing a game on a 4x5 rectangle board $P$ and a set of tetrominoes $S$. Each player alternately chooses a tetromino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the tetromino is precisely above one square on $P$ and does not overlap with any p... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4ccc25f9ec6748a94bcf05c642bdf8a51618f8ffef33ead26f00fdc04c68c95 | Two players are playing a game on a circle with \( n \) points. On each move, a player chooses a point and connects it with an edge to either an adjacent point or the center of the circle. The winner is the player whose move ensures that every point (including the center) can be reached from any other point via the con... | [
"\\text{Second player}"
] | [
"Two players are playing a game on a circle with \\( n \\) points. On each move, a player chooses a point and connects it with an edge to either an adjacent point or the center of the circle. The winner is the player whose move ensures that every point (including the center) can be reached from any other point via ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e9a996357fe128c9ca5b0fc6dd57b33d38845c9e337918d26ee0927d0d4d8253 | Two players are playing a game on a cross-shaped board $P$ (a central 3x3 square with a 3x3 square attached to each side) and a set of pentominoes $S$. Each player alternately chooses a pentomino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the pentomino is precisel... | [
"\\text{Second player}"
] | [
"Two players are playing a game on a cross-shaped board $P$ (a central 3x3 square with a 3x3 square attached to each side) and a set of pentominoes $S$. Each player alternately chooses a pentomino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the pentomino is pre... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-756409620e64c379dd4889cc0a5a25f0c18bacc6e565e6172325623d0f119863 | Two players are playing a game on an 8x8 square board $P$ and a set of pentominoes $S$. Each player alternately chooses a pentomino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the pentomino is precisely above one square on $P$ and does not overlap with any previous... | [
"\\text{Second player}"
] | [
"Two players are playing a game on an 8x8 square board $P$ and a set of pentominoes $S$. Each player alternately chooses a pentomino from $S$ and places it on $P$ (possibly after rotation and/or reflection) such that each square on the pentomino is precisely above one square on $P$ and does not overlap with any pre... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed9c75b60f5e4b335be59dbeae44ffb3f5371103a61201b8024641a16caab8d8 | Two players are playing a game where on each turn, a player writes a pair of non-negative integer numbers $(a,b)$ such that for each previously written pair $(c,d)$, either $a < c$ or $b < d$. The player who writes the pair $(0,0)$ loses. Determine which player wins if both players play optimally. | [
"\\text{First player}"
] | [
"Two players are playing a game where on each turn, a player writes a pair of non-negative integer numbers $(a,b)$ such that for each previously written pair $(c,d)$, either $a < c$ or $b < d$. The player who writes the pair $(0,0)$ loses. Determine which player wins if both players play optimally."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-0d9354cc87769b8617b3dd99e389a6da887f45e272a0ca6e9405e51709a3f41e | Two players are playing a game where they color the sides of an $n$-gon polygon. Player 1 can color a side if 0 or 2 of the adjacent sides are already colored, and Player 2 can color a side if exactly 1 of the adjacent sides is already colored. The player who cannot color a side loses. Who has the winning strategy? | [
"Player 1 wins for \\( n \\neq 4 \\), Player 2 wins for \\( n = 4 \\)."
] | [
"Two players are playing a game where they color the sides of an $n$-gon polygon. Player 1 can color a side if 0 or 2 of the adjacent sides are already colored, and Player 2 can color a side if exactly 1 of the adjacent sides is already colored. The player who cannot color a side loses. Who has the winning strategy... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 13
} | 15 | 0 | 0 |
omr-ws-c8c360dab58f1403d4201d0299c55fb0bba734a9cc92505ed438bd7e1b3c03c8 | Two players are playing a game where they write down the divisors of the number \(2022! \times 2023!\) except 1 on the blackboard, taking turns. Each player must write a divisor that is different from 1 and all previously written numbers. A player loses if after their turn, there exists a pair of numbers on the blackbo... | [
"Player 1"
] | [
"Two players are playing a game where they write down the divisors of the number \\(2022! \\times 2023!\\) except 1 on the blackboard, taking turns. Each player must write a divisor that is different from 1 and all previously written numbers. A player loses if after their turn, there exists a pair of numbers on the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 10
} | 29 | 0 | 0 |
omr-ws-17b303abb8a4aa4de56c206abe3786291486ed2ecf96560ba37cf0c46ea791d0 | Two players are playing a game with 11 quadratic equations of the form $\star x^2 + \star x + \star = 0$. Each player takes turns replacing one star with a real nonzero number. The first player aims to maximize the number of equations without real roots, while the second player aims to minimize this number. What is the... | [
"6"
] | [
"Two players are playing a game with 11 quadratic equations of the form $\\star x^2 + \\star x + \\star = 0$. Each player takes turns replacing one star with a real nonzero number. The first player aims to maximize the number of equations without real roots, while the second player aims to minimize this number. Wha... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-3a26f5429e68babf16efa5cd4869671ca23676ffef3707bcbdbb9e449b2da094 | Two players are playing a game with an $N \times M$ grid of empty squares. On each turn, a player fills in a single empty square, then fills in all the squares in the same row, and then fills in all the squares in the same column. The players alternate taking turns until all squares are filled in. Let $f(N, M)$ be the ... | [
"0"
] | [
"Two players are playing a game with an $N \\times M$ grid of empty squares. On each turn, a player fills in a single empty square, then fills in all the squares in the same row, and then fills in all the squares in the same column. The players alternate taking turns until all squares are filled in. Let $f(N, M)$ b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa3ec6825ebb4e1efbe7f75e9faaae42784315cc0fbad579d5b098d0df3d196f | Two players are playing a game. Player 1 picks three different non-zero numbers \(a\), \(b\), and \(c\). Player 2 substitutes these numbers into the equation \(x^2 + *x + * = 0\) in any order. Player 2 aims to ensure the resulting quadratic equation has two distinct roots. If Player 2 succeeds, they win. Determine the ... | [
"2"
] | [
"Two players are playing a game. Player 1 picks three different non-zero numbers \\(a\\), \\(b\\), and \\(c\\). Player 2 substitutes these numbers into the equation \\(x^2 + *x + * = 0\\) in any order. Player 2 aims to ensure the resulting quadratic equation has two distinct roots. If Player 2 succeeds, they win. D... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0f59a2c7f0c62669169665f05df0da3e67a8e17c7cb91f531d538f368e48252 | Two players are playing a number subtraction game using two four-digit numbers. The first player calls out a digit between 0 and 9 (these may be repeated), and the second player places it in any free space. The first player tries to make the difference as large as possible, while the second tries to make the difference... | [
"8001"
] | [
"Two players are playing a number subtraction game using two four-digit numbers. The first player calls out a digit between 0 and 9 (these may be repeated), and the second player places it in any free space. The first player tries to make the difference as large as possible, while the second tries to make the diffe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-1d22b25174596f36239149ffdd1dd2f55929ac03db920b24d81af24fdc60d481 | Two players are playing a turn-based game on an $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a ... | [
"4"
] | [
"Two players are playing a turn-based game on an $n \\times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-71396a71cdc735288cf7ad21a6a3cc8ec0f73473077d4c5b6c7fc35f4598a62c | Two players are playing chess, and each player's clock starts and stops as described in the original problem. After each player has made 40 moves, both clocks read exactly 150 minutes. Calculate the minimum difference in seconds that must have occurred between the two clocks at some point during the game. | [
"225"
] | [
"Two players are playing chess, and each player's clock starts and stops as described in the original problem. After each player has made 40 moves, both clocks read exactly 150 minutes. Calculate the minimum difference in seconds that must have occurred between the two clocks at some point during the game."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-269ded1ac432fed5a484a36758845d0116e7dd851ed874e9f7756f4f538b1f73 | Two players are throwing a six-sided die. If the first player rolls a 6, he loses and the game is over. If not, the second player throws the die. If the second player rolls a 6, he loses and the game is over; otherwise, the first player throws again. The game continues until one player rolls a 6 and loses, with the oth... | [
"\\(\\frac{6}{11}\\)"
] | [
"Two players are throwing a six-sided die. If the first player rolls a 6, he loses and the game is over. If not, the second player throws the die. If the second player rolls a 6, he loses and the game is over; otherwise, the first player throws again. The game continues until one player rolls a 6 and loses, with th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4f913b9e1d04112e66cfdc08add396ca4f028204030bccfd9b8e2028789b3958 | Two players compete in a game with N > 2 stacks, each containing one stone. On each turn, a player can combine two stacks such that the total number of stones in the combined stacks is a prime number. The player who cannot make a move loses. Find a winning strategy. | [
"The second player has a winning strategy."
] | [
"Two players compete in a game with N > 2 stacks, each containing one stone. On each turn, a player can combine two stacks such that the total number of stones in the combined stacks is a prime number. The player who cannot make a move loses. Find a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 5,
"tir": 12
} | 40 | 0 | 0 |
omr-ws-f3f5f183803626bc45d78a9f334cb8b5014f41cdbaf2fe47864da5cb4d50099e | Two players continue playing rock-paper-scissors games until one of them wins the third time. Each player selects rock, paper, or scissors at random with equal probability $\frac{1}{3}$ for each choice. Determine the expected value of the number of games needed. | [
"\\dfrac{99}{16}"
] | [
"Two players continue playing rock-paper-scissors games until one of them wins the third time. Each player selects rock, paper, or scissors at random with equal probability $\\frac{1}{3}$ for each choice. Determine the expected value of the number of games needed."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2277cd841b86e58fc943c655eddd90cda38117ef5dbe3f4ef750e8b3766a647e | Two players each have a knight on a band of length $N$. Player 1 rolls a 6-sided die, multiplies the result by 2, and moves their knight that distance. Player 2 rolls the die twice, adds the results, and moves their knight that distance. The winner is the player whose knight passes the band after fewer steps. Assuming ... | [
"Player 1"
] | [
"Two players each have a knight on a band of length $N$. Player 1 rolls a 6-sided die, multiplies the result by 2, and moves their knight that distance. Player 2 rolls the die twice, adds the results, and moves their knight that distance. The winner is the player whose knight passes the band after fewer steps. Assu... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-aafb5be822b18e9d1ab372fad08fbfe731a5d9e3e1aa2fd26e986909ec9f4962 | Two players each have a knight on a chessboard. Each player takes turns moving their knight to an unvisited square. The player who cannot move to a new square loses. Determine which player can win and describe the winning strategy. | [
"\\text{The second player can always win by mirroring the first player's moves across the board's center.}"
] | [
"Two players each have a knight on a chessboard. Each player takes turns moving their knight to an unvisited square. The player who cannot move to a new square loses. Determine which player can win and describe the winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-9ef1bebbc3800854c9930f0837fed8a0f3ee0ad2c846a3174e7d1ab1b2dab94f | Two players each own $n$ dollars and are bidding for an object of value $c > 0$. Each player makes a sealed bid, offering $i$ dollars (where $i$ is an integer from $0$ to $n$) for the sole possession of the object. The higher bidder gets the object and pays the other player the amount they bid. If both bid the same amo... | [
"The payoff matrix has a saddle point at \\( \\lfloor c \\rfloor \\)."
] | [
"Two players each own $n$ dollars and are bidding for an object of value $c > 0$. Each player makes a sealed bid, offering $i$ dollars (where $i$ is an integer from $0$ to $n$) for the sole possession of the object. The higher bidder gets the object and pays the other player the amount they bid. If both bid the sam... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 6
} | 24 | 0 | 0 |
omr-ws-4d15b89e983fdd8c4a09b16b38f87c8fe25b9f2ae32c1cdc3ed470683eb74784 | Two players each roll a pair of standard four-sided dice. The rolls for the first player are \(a_1\) and \(a_2\) with \(a_1 \le a_2\) and the rolls for the second player are \(b_1\) and \(b_2\) with \(b_1 \le b_2\). The probability that \(a_1 > b_1\) and \(a_2 > b_2\) can be expressed as \(\frac{a}{b}\) where \(a\) and... | [
"\\(\\frac{23}{128}\\)"
] | [
"Two players each roll a pair of standard four-sided dice. The rolls for the first player are \\(a_1\\) and \\(a_2\\) with \\(a_1 \\le a_2\\) and the rolls for the second player are \\(b_1\\) and \\(b_2\\) with \\(b_1 \\le b_2\\). The probability that \\(a_1 > b_1\\) and \\(a_2 > b_2\\) can be expressed as \\(\\fra... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-e9f5b0da3c7ec54dcaa9181e72af46c060079b1ab72f05e398fefb9ab4567daf | Two players fill an $n \times n$ square board with $0$ and $1$. After the board is filled, they calculate the sum $r_i$ of numbers in the $i$th row and $c_i$ of numbers in the $i$th column. Let \[ R = \sum_{i=1}^{n} i r_i, \quad C = \sum_{i=1}^{n} i c_i. \] The first player wins if $R < C$, and the second player wins o... | [
"Neither player has a winning strategy."
] | [
"Two players fill an $n \\times n$ square board with $0$ and $1$. After the board is filled, they calculate the sum $r_i$ of numbers in the $i$th row and $c_i$ of numbers in the $i$th column. Let\n\n\\[ R = \\sum_{i=1}^{n} i r_i, \\quad C = \\sum_{i=1}^{n} i c_i. \\]\n\nThe first player wins if $R < C$, and the sec... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d98f84942159c32d7f347bd618bfe6e6cb864e6a8f6ce816f06a68bcbf5040b | Two players have 4 sets of coins with 1, 3, 5, and 7 coins in them. On each turn, a player can select any number of coins but only from one of these 4 sets. The player who must take the last coin loses. Determine if there is a winning strategy for either player. | [
"Second player has a winning strategy."
] | [
"Two players have 4 sets of coins with 1, 3, 5, and 7 coins in them. On each turn, a player can select any number of coins but only from one of these 4 sets. The player who must take the last coin loses. Determine if there is a winning strategy for either player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d11fffc942e7700ccf0baf840f68634b1807d4ead2912bb85e58c379e9a8aa4d | Two players have sets of coins with 1, 3, 5, ..., 2n-1 coins in them. On each turn, a player can select any number of coins but only from one of these sets. The player who must take the last coin loses. Determine if there is a winning strategy for either player. | [
"The first player has a winning strategy if \\( n \\) is not a power of 2; otherwise, the second player has a winning strategy."
] | [
"Two players have sets of coins with 1, 3, 5, ..., 2n-1 coins in them. On each turn, a player can select any number of coins but only from one of these sets. The player who must take the last coin loses. Determine if there is a winning strategy for either player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-cc3f8182a3cc94b06e3dbf95fdcf489324f99bf9ded33f516e1b6525c7694cce | Two players have three piles of rocks, one with 50 rocks, another with 60 rocks, and the last one with 70 rocks. They take turns, and in each turn, a player must split a pile with at least 2 rocks into two piles (not necessarily of equal size). The player who cannot make a move loses. Determine which player has the win... | [
"The first player has the winning strategy."
] | [
"Two players have three piles of rocks, one with 50 rocks, another with 60 rocks, and the last one with 70 rocks. They take turns, and in each turn, a player must split a pile with at least 2 rocks into two piles (not necessarily of equal size). The player who cannot make a move loses. Determine which player has th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f9cf8321f3795fd731c120309067202c1a146efeb608aa8a0c448e928ca8b0b8 | Two players in turn paint cells of a \(7 \times 7\) table each using their own color. A player cannot paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players cannot make their turn. What is the maximal number of cells that can remain unpainted when the g... | [
"25",
"36"
] | [
"Two players in turn paint cells of a \\(7 \\times 7\\) table each using their own color. A player cannot paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players cannot make their turn. What is the maximal number of cells that can remain unpainted wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-b9047a45a87f7e5f72500975c5e1f252aefeb2a22d8a9c4660323e78a8d00949 | Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after both players made five moves, they exchange hats. The player wins if after his move his hat contains one hundred or more coins. Which player has a winning strategy? | [
"The player who places fewer coins has a winning strategy."
] | [
"Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after both players made five moves, they exchange hats. The player wins if after his move his hat contains one hundred or more coins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-a0f6851207774f3c9405bc8a9026ae8becf32d977381d8d85edf44a600a6233f | Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after the second player duplicated the move of the first player, they exchange hats. The player wins if after his move his hat contains one hundred or more coins. Which player has a winning strategy? | [
"Player 1 has a winning strategy."
] | [
"Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after the second player duplicated the move of the first player, they exchange hats. The player wins if after his move his hat contains one hundred or more coins. Which player has a winning strat... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-6e3f98cb2fca191cf523bdcb2c903adf9732b2ff5dd1bde62ef3c47f6043058c | Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ does the second player ha... | [
"n \\equiv 0 \\pmod{4}"
] | [
"Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ does the second play... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-9178c67348c540640cba6f655eb7c9fbf282b4ed5337afd5e367a192fc9c4019 | Two players move alternately, starting with several piles of chips. A move consists of splitting every pile with more than one chip into two piles. The player who makes the last move wins. For what initial conditions does the first player win, and what is his winning strategy? | [
"\\text{The first player wins if and only if there is at least one even-sized pile initially.}"
] | [
"Two players move alternately, starting with several piles of chips. A move consists of splitting every pile with more than one chip into two piles. The player who makes the last move wins. For what initial conditions does the first player win, and what is his winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-c56b9c7dafb5acdc9435dedff050cd3b8327a6f44437a64f31eb758a9db80dbf | Two players play a card game with a deck of $n$ distinct cards. For any two cards, it is known which one beats the other (the relation is not necessarily transitive). The deck is split between the players in an arbitrary manner. In each turn, each player reveals the top card from their deck, and the player whose card b... | [
"45"
] | [
"Two players play a card game with a deck of $n$ distinct cards. For any two cards, it is known which one beats the other (the relation is not necessarily transitive). The deck is split between the players in an arbitrary manner. In each turn, each player reveals the top card from their deck, and the player whose c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
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