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-a2c3135f8b14c3db92280244a507cac8dd8d728e1fcb3646a3ef63c6db80453a | Two players play a cooperative game. They can discuss a strategy prior to the game but cannot communicate during the game. The game master chooses one of the players in each round. The chosen player must guess the number of the current round. Players keep track of the number of rounds they were chosen but have no infor... | [
"200"
] | [
"Two players play a cooperative game. They can discuss a strategy prior to the game but cannot communicate during the game. The game master chooses one of the players in each round. The chosen player must guess the number of the current round. Players keep track of the number of rounds they were chosen but have no ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-7d8fda4522840b3299beca141e4b046df8d039d7148850b116d4c4cb51cbfd19 | Two players play a game as follows. The first player chooses two non-zero integers \( A \) and \( B \). The second player forms a quadratic equation with \( A \), \( B \), and 1998 as coefficients (in any order). The first player wins if the quadratic equation has two distinct rational roots. Find the sum of all possib... | [
"no_valid_answer_found"
] | [
"Two players play a game as follows. The first player chooses two non-zero integers \\( A \\) and \\( B \\). The second player forms a quadratic equation with \\( A \\), \\( B \\), and 1998 as coefficients (in any order). The first player wins if the quadratic equation has two distinct rational roots. Find the sum ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce8d6d48c61dffbcc4e961feb0955ca44ec97f6b8049d78b64d429a17375e298 | Two players play a game on a $1 \times n$ table. In each move, a player can eat two consecutive cells. A player who cannot make a move loses. Determine which player has a winning strategy, depending on the value of $n$. | [
"\\text{Second player wins if } n \\equiv 1 \\pmod{4}; \\text{ otherwise, first player wins}"
] | [
"Two players play a game on a $1 \\times n$ table. In each move, a player can eat two consecutive cells. A player who cannot make a move loses. Determine which player has a winning strategy, depending on the value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-47c10bd509e3443e45343e9a78764d07aab9c9b9bdb43daee0faf16e81c04614 | Two players play a game on a $100 \times 100$ board. The first player marks a free square, and the second player places a domino (a $2 \times 1$ rectangle) covering two free squares, one of which is marked. The first player wins if the entire board is covered; otherwise, the second player wins. Which player has a winni... | [
"The second player has a winning strategy."
] | [
"Two players play a game on a $100 \\times 100$ board. The first player marks a free square, and the second player places a domino (a $2 \\times 1$ rectangle) covering two free squares, one of which is marked. The first player wins if the entire board is covered; otherwise, the second player wins. Which player has ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-5b69a6da6b21e25f35768ca1bd10a3b42c1213f4ed82293a47693ebdc4b16fa8 | Two players play a game on a $100 \times 100$ board. The first player marks a free square, then the second player places a $1 \times 2$ domino covering two free squares, one of which is marked. This continues until one player is unable to move. The first player wins if the entire board is covered; otherwise, the second... | [
"\\text{First player}"
] | [
"Two players play a game on a $100 \\times 100$ board. The first player marks a free square, then the second player places a $1 \\times 2$ domino covering two free squares, one of which is marked. This continues until one player is unable to move. The first player wins if the entire board is covered; otherwise, the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97744c7fffc8284e75d4d26bd5ae99196cbc2d68eea53eb18a3dcb9b96fd5f36 | Two players play a game on a 10x10 board with all squares initially colored white. The players take turns to either: A. Change a white square to a black square or B. Choose a row or column containing strictly more white squares than black squares, and change the color of every square in that row or column. Assuming bot... | [
"10"
] | [
"Two players play a game on a 10x10 board with all squares initially colored white. The players take turns to either:\nA. Change a white square to a black square or\nB. Choose a row or column containing strictly more white squares than black squares, and change the color of every square in that row or column.\nAssu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-9e0ccd943584566e6e81ac82e6d1f4d8f8f00d64fc5b86b4ed767e8859561e74 | Two players play a game on a 10x10 board with tokens numbered 1 to 100 (even and odd tokens split between the two players). Players take turns placing one token per turn on any empty cell. The sum of the numbers on any two adjacent cells (vertically or horizontally) must always be a prime number. If a player cannot pla... | [
"Yes, the second player has a winning strategy."
] | [
"Two players play a game on a 10x10 board with tokens numbered 1 to 100 (even and odd tokens split between the two players). Players take turns placing one token per turn on any empty cell. The sum of the numbers on any two adjacent cells (vertically or horizontally) must always be a prime number. If a player canno... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-255adca91d3fa4301c84015d16141794613c9db03a36e635bdf1540287913a6d | Two players play a game on a 1x2003 strip of squares. They alternately write either S or O into an empty square. The first player to make three consecutive squares spell "S O S" wins. Does either player have a winning strategy? | [
"The first player has a winning strategy."
] | [
"Two players play a game on a 1x2003 strip of squares. They alternately write either S or O into an empty square. The first player to make three consecutive squares spell \"S O S\" wins. Does either player have a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-ca399eaa44dd955efb646e0b3d6a5e790e0ec753cec6207d67619f069b960811 | Two players play a game on a blackjack machine where one can win or lose one dollar at a time. Both players eventually win exactly $2N$ dollars. Let $T$ be the total number of minutes played until both players have won exactly $2N$ dollars. Find the minimum possible value of $T$ such that there exists a time interval d... | [
"2N"
] | [
"Two players play a game on a blackjack machine where one can win or lose one dollar at a time. Both players eventually win exactly $2N$ dollars. Let $T$ be the total number of minutes played until both players have won exactly $2N$ dollars. Find the minimum possible value of $T$ such that there exists a time inter... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f9aa519e81ac6646626ae8bb1ccccfa014bea8a8aba34d87d70fedfa522b3ccd | Two players play a game on a board starting with the numbers $1, 2, 3$. A move consists of writing a new number on the board which equals the sum of the last written number and one of the other numbers. A player who writes a number greater than or equal to 1000 wins. Who has a winning strategy, the player who moves fir... | [
"The player who moves first has a winning strategy."
] | [
"Two players play a game on a board starting with the numbers $1, 2, 3$. A move consists of writing a new number on the board which equals the sum of the last written number and one of the other numbers. A player who writes a number greater than or equal to 1000 wins. Who has a winning strategy, the player who move... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-e4bad363603d7b3a4a766668c3553e51831e7122c09435fc0085df3bdbd18c3c | Two players play a game on a regular $102$-gon. The first player marks one of the vertices. Then, starting with the second player, they take turns moving a chip from its current vertex to another vertex along either the main diagonal or the shortest diagonal, with the restriction that the chip cannot be moved back to t... | [
"51"
] | [
"Two players play a game on a regular $102$-gon. The first player marks one of the vertices. Then, starting with the second player, they take turns moving a chip from its current vertex to another vertex along either the main diagonal or the shortest diagonal, with the restriction that the chip cannot be moved back... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-4a969cd322d43b252d315ba155d8301c3ae5530ceebb503ed01d733a9fb08717 | Two players play a game on a regular $n$-gon where the vertices are marked. On each turn, a player selects a vertex and connects it by a segment to either an adjacent vertex or the center of the $n$-gon. The winner is the player who, after their move, makes it possible to get from any vertex to any other vertex by movi... | [
"First player wins if $n$ is odd; Second player wins if $n$ is even."
] | [
"Two players play a game on a regular $n$-gon where the vertices are marked. On each turn, a player selects a vertex and connects it by a segment to either an adjacent vertex or the center of the $n$-gon. The winner is the player who, after their move, makes it possible to get from any vertex to any other vertex by... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 12
} | 25 | 0 | 0 |
omr-ws-be10bb13afb255c7f97ca8642d1e22f466985677487a337224f88a94af29d8b2 | Two players play a game on a regular $n$-gon with its vertices and center marked. On each turn, a player selects a vertex and connects it by a segment to either an adjacent vertex or the center. The winner is the player who, after their move, makes it possible to get from any marked point to any other marked point movi... | [
"\\text{Second player}"
] | [
"Two players play a game on a regular $n$-gon with its vertices and center marked. On each turn, a player selects a vertex and connects it by a segment to either an adjacent vertex or the center. The winner is the player who, after their move, makes it possible to get from any marked point to any other marked point... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c67422a75633c45cea95c30281e5c0fc8ac27cf70cc134087deb4f2a47d3a59e | Two players play a game on an $N \times N$ grid by coloring one square at a time, ensuring no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy? | [
"\\( N \\) is odd."
] | [
"Two players play a game on an $N \\times N$ grid by coloring one square at a time, ensuring no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 11
} | 28 | 0 | 0 |
omr-ws-920286aa9e27b1c139d123d3bd9e8a8f3e05a3143a61fdddb8e1b784e0609fb5 | Two players play a game on an $n \times n$ board where each field contains a disc with a red and a blue face. The players move alternately, and on each turn, the player can choose one of the following options: - (i) Turn a blue disc over. - (ii) Turn a blue disc over and simultaneously turn over all discs to the right ... | [
"\\text{At least one disc is blue}"
] | [
"Two players play a game on an $n \\times n$ board where each field contains a disc with a red and a blue face. The players move alternately, and on each turn, the player can choose one of the following options:\n- (i) Turn a blue disc over.\n- (ii) Turn a blue disc over and simultaneously turn over all discs to th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-bb40e2235f110c7c1312e9357bb526ecfb5359668294cb26b0b696bc390bf386 | Two players play a game on an infinite board that consists of unit squares. Player I chooses a square and marks it with O. Then player II chooses another square and marks it with X. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no p... | [
"4"
] | [
"Two players play a game on an infinite board that consists of unit squares. Player I chooses a square and marks it with O. Then player II chooses another square and marks it with X. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53e40bd0bf2fa9952836bbbf9bc89b189b040dddb2fbc145f5340a672f17de15 | Two players play a game on four piles of pebbles labeled with the numbers $1, 2, 3, 4$ respectively. On each turn, a player selects integers $m$ and $n$ with $1 \leq m < n \leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1, n-2, \dots, n-m$. A player loses if they cannot make a ... | [
"(b \\bmod 2) \\oplus (c \\bmod 3) \\oplus (d \\bmod 4) \\neq 0"
] | [
"Two players play a game on four piles of pebbles labeled with the numbers $1, 2, 3, 4$ respectively. On each turn, a player selects integers $m$ and $n$ with $1 \\leq m < n \\leq 4$, removes $m$ pebbles from pile $n$, and places one pebble in each of the piles $n-1, n-2, \\dots, n-m$. A player loses if they cannot... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-ca4ba75c7cb2f70b39564ad131672d7ab10631172d9d367ed7d4925219d26055 | Two players play a game starting with 2013 candies. On each turn, a player can take any amount of candies that is a power of 2 (i.e., $2^k$ where $k \geq 0$). The player who eats the last candy wins. Assuming both players play optimally, which player has a winning strategy? | [
"Second player has a winning strategy."
] | [
"Two players play a game starting with 2013 candies. On each turn, a player can take any amount of candies that is a power of 2 (i.e., $2^k$ where $k \\geq 0$). The player who eats the last candy wins. Assuming both players play optimally, which player has a winning strategy?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-52788500b14688598b561b1e2673c67131a50025d1d29170f05f24d17e2000f3 | Two players play a game starting with a number $N$ on the board. In each turn, a player can replace $N$ with either $N-1$ or $\left\lfloor \frac{N}{2} \right\rfloor$. The player who first writes the number $0$ on the board wins. Determine which player has the winning strategy if $N = 2012$. | [
"The first player has the winning strategy. \n\nSince the question asks for a numerical value or a mathematical expression and the answer is a statement about the winning strategy, the closest numerical representation would be to indicate the winning player as 1 (first player)."
] | [
"Two players play a game starting with a number $N$ on the board. In each turn, a player can replace $N$ with either $N-1$ or $\\left\\lfloor \\frac{N}{2} \\right\\rfloor$. The player who first writes the number $0$ on the board wins. Determine which player has the winning strategy if $N = 2012$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 18
} | 40 | 0 | 0 |
omr-ws-15a867cfc577f9a7e004e963b1fcb8545e2785ab06e42e874d11d20063e99a68 | Two players play a game starting with one pile of at least two stones. On each turn, a player chooses one pile and divides it into two or three nonempty piles. The player who cannot make a legal move loses. Which player has a winning strategy? | [
"The first player has a winning strategy."
] | [
"Two players play a game starting with one pile of at least two stones. On each turn, a player chooses one pile and divides it into two or three nonempty piles. The player who cannot make a legal move loses. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1eb0f85d53d8613fc4b7e61259bef6c971a78565a5cc556a90feae44f33ab5a3 | Two players play a game starting with the integer 2020. On each turn, a player can replace the current integer $N$ with either $\left \lfloor{N/2}\right \rfloor$ or $N-1$. The player who writes down 0 wins. Which player has a winning strategy? | [
"The first player has a winning strategy."
] | [
"Two players play a game starting with the integer 2020. On each turn, a player can replace the current integer $N$ with either $\\left \\lfloor{N/2}\\right \\rfloor$ or $N-1$. The player who writes down 0 wins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-c5c266514fa6b49152e54a2558c2e4c403b8e7d75cfdd13282041b6a2cae3186 | Two players play a game starting with the number $k=2$. Each player can either replace $k$ by $k+1$ or by $2k$. The player who is forced to write a number greater than a fixed integer $n > 1$ loses the game. Determine for which values of $n$ one of the players has a winning strategy. | [
"Player 2 has a winning strategy if \\( n = 2^{2k+1} + 2c \\) where \\( k \\) is a nonnegative integer and \\( c < 2^k \\); otherwise, Player 1 has a winning strategy."
] | [
"Two players play a game starting with the number $k=2$. Each player can either replace $k$ by $k+1$ or by $2k$. The player who is forced to write a number greater than a fixed integer $n > 1$ loses the game. Determine for which values of $n$ one of the players has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7d8929175f650d58860dde013978cf28d4e3959b6cf0e48a864a14321b46d972 | Two players play a game starting with the number 1234 on the board. On each turn, a player can subtract a nonzero digit of the current number from the number itself, replacing the old number with the new difference. The player who reaches 0 wins. Which player has a winning strategy? | [
"First"
] | [
"Two players play a game starting with the number 1234 on the board. On each turn, a player can subtract a nonzero digit of the current number from the number itself, replacing the old number with the new difference. The player who reaches 0 wins. Which player has a winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea237ac7efd6fbc0a4f0e7e0528073313a14205954a302f6ca77fbde3f91ced0 | Two players play a game starting with the number \( 2004! \) on the blackboard. On each turn, a player chooses a number \( A \) with no more than 20 distinct prime divisors and \( A < B \), where \( B \) is the current number on the blackboard. The number \( B \) is replaced by \( B - A \). The player who writes 0 on t... | [
"The first player has the winning strategy."
] | [
"Two players play a game starting with the number \\( 2004! \\) on the blackboard. On each turn, a player chooses a number \\( A \\) with no more than 20 distinct prime divisors and \\( A < B \\), where \\( B \\) is the current number on the blackboard. The number \\( B \\) is replaced by \\( B - A \\). The player ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 12
} | 22 | 0 | 0 |
omr-ws-c4c9a80e928a19b9d278445ea9c3499e2fd67780551c7a22a696351b0f8721b0 | Two players play a game starting with the number \( k = 2 \). Each player can either replace \( k \) with \( k + 1 \) or \( 2k \). The player who is forced to write a number greater than \( n \) loses the game. Determine for which values of \( n \) each player has a winning strategy. | [
"The second player has a winning strategy if and only if \\( n \\) is an odd power of 2."
] | [
"Two players play a game starting with the number \\( k = 2 \\). Each player can either replace \\( k \\) with \\( k + 1 \\) or \\( 2k \\). The player who is forced to write a number greater than \\( n \\) loses the game. Determine for which values of \\( n \\) each player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-7c8ac7ca150d94df5bcb4e2cc65cea3e9346d9ff0630742ca8dfdd16e1d48554 | Two players play a game starting with the polynomial \(2013x^2 + 2012x + 2011\). On each turn, a player subtracts one of the polynomials \(x^2\), \(x\), \(x^2 - x + 1\), or \(x^2 + x - 1\) from the current polynomial. If the resulting polynomial has an integer root, the player who made the move loses. Determine which p... | [
"\\text{Second player}"
] | [
"Two players play a game starting with the polynomial \\(2013x^2 + 2012x + 2011\\). On each turn, a player subtracts one of the polynomials \\(x^2\\), \\(x\\), \\(x^2 - x + 1\\), or \\(x^2 + x - 1\\) from the current polynomial. If the resulting polynomial has an integer root, the player who made the move loses. De... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-bee4a79458b4575ced5d09f502b57a094ae03c5a564a56143a9c1f68abbcd353 | Two players play a game until one has won a total of five games. If player A wins each individual game with probability 0.6, what is the probability that player A will win the match? | [
"0.7334"
] | [
"Two players play a game until one has won a total of five games. If player A wins each individual game with probability 0.6, what is the probability that player A will win the match?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-a50308d5c58fd84c71d85f447ea87397b812be480f2a549e900ef424139f9bcd | Two players play a game using $923^k$ balls. Each player can remove either 2, 3, or 9 balls on their turn. The player who cannot make a move loses. Which player has a winning strategy, and what is it? | [
"First player wins if \\( k \\) is odd; second player wins if \\( k \\) is even."
] | [
"Two players play a game using $923^k$ balls. Each player can remove either 2, 3, or 9 balls on their turn. The player who cannot make a move loses. Which player has a winning strategy, and what is it?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 11
} | 17 | 0 | 0 |
omr-ws-dad51f05696daeafffcaff5858044c4f53a1915a263d7745c2d7345d49d36cbf | Two players play a game where Player 1 chooses a number between 1 and \( n \). Player 2 has \( q \) questions to determine the number. In Stage 1, Player 2 asks questions of the form "Is the number less than \( k \)?" and Player 1 answers truthfully. Stage 1 ends when Player 2 uses all \( q \) questions or when Player ... | [
"\\( \\frac{1}{\\lceil n/(q+1) \\rceil} \\)"
] | [
"Two players play a game where Player 1 chooses a number between 1 and \\( n \\). Player 2 has \\( q \\) questions to determine the number. In Stage 1, Player 2 asks questions of the form \"Is the number less than \\( k \\)?\" and Player 1 answers truthfully. Stage 1 ends when Player 2 uses all \\( q \\) questions ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-743a5fb326222c263af2deee0d294f77d0b8a7387288f35d64d91237580148ba | Two players play a game where the First Player has cards with numbers $2, 4, \ldots, 2000$ and the Second Player has cards with numbers $1, 3, \ldots, 2001$. In each turn, a player chooses one of their cards and places it on the table; the opponent then places one of their cards next to it. The player with the larger n... | [
"First Player: 499, Second Player: 501"
] | [
"Two players play a game where the First Player has cards with numbers $2, 4, \\ldots, 2000$ and the Second Player has cards with numbers $1, 3, \\ldots, 2001$. In each turn, a player chooses one of their cards and places it on the table; the opponent then places one of their cards next to it. The player with the l... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a883908315552b797fc753f3aa74c1c4d8d8c062b00d52c1d5916a0b996bae2b | Two players play a game where the first player chooses a decimal digit, and the second player substitutes it for one of the stars in the subtraction **** - ****. The first player aims to maximize the result, while the second player aims to minimize it. If both players play optimally, what is the guaranteed result of th... | [
"0"
] | [
"Two players play a game where the first player chooses a decimal digit, and the second player substitutes it for one of the stars in the subtraction **** - ****. The first player aims to maximize the result, while the second player aims to minimize it. If both players play optimally, what is the guaranteed result ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-e51d8230ecb2c2c6dc168aaf91eaa1df4a133c76ae052b21258d774c6b5b18cc | Two players play a game where the first player writes down one digit, and then the second player writes one digit to the left or right of the previously written digit. The first player then writes one digit to the left or right of the previously written number, and so on. The first player's strategy is to ensure that a... | [
"619"
] | [
"Two players play a game where the first player writes down one digit, and then the second player writes one digit to the left or right of the previously written digit. The first player then writes one digit to the left or right of the previously written number, and so on. The first player's strategy is to ensure t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-6d503230cd5a4434f6fb8dd26d2a8d6021cf58953c0f11c58263c1b8ea88d7a2 | Two players play a game where the first to win 6 coin tosses gets 64,000 dollars. The game stops after 8 tosses with a score of 5:3. Propose different ways to share the money. | [
"40000, 24000 or 48000, 16000 or 32000, 32000"
] | [
"Two players play a game where the first to win 6 coin tosses gets 64,000 dollars. The game stops after 8 tosses with a score of 5:3. Propose different ways to share the money."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 19
} | 21 | 0 | 0 |
omr-ws-24e9d730cdcde82aa5b605d8bf9bf99c8b21374dd80605c3379127d455b0046e | Two players play a game where there are $n$ chips on the table for integer $n$. They alternate taking away a positive integer number of chips. The first player can take between 1 and $n-1$ chips, and subsequent players can only take between 1 and three times the number of chips the previous player took away. A player w... | [
"\\text{First player}"
] | [
"Two players play a game where there are $n$ chips on the table for integer $n$. They alternate taking away a positive integer number of chips. The first player can take between 1 and $n-1$ chips, and subsequent players can only take between 1 and three times the number of chips the previous player took away. A pla... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-f168b9f3815dcbd38f0017e8f4faba5cd585c3790205300e9af0dc1be7658859 | Two players play a game where they alternate taking a positive integer \(N\) and decreasing it by some divisor \(n\) of \(N\) such that \(n < N\). A player loses if they are given \(N = 1\). For how many of the first 2015 positive integers is the player who moves first guaranteed to win, given optimal play from both pl... | [
"1007"
] | [
"Two players play a game where they alternate taking a positive integer \\(N\\) and decreasing it by some divisor \\(n\\) of \\(N\\) such that \\(n < N\\). A player loses if they are given \\(N = 1\\). For how many of the first 2015 positive integers is the player who moves first guaranteed to win, given optimal pl... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fcd5f9656b1b36665d3c17d6902c9b1e16cc5d9ba860632ac6078bb3f391022 | Two players play a game with $n > 2$ piles, each containing $n^{10} + 1$ stones. A move consists of removing all but one pile and dividing the remaining pile into $n$ nonempty piles. The player who cannot make a move loses. Determine which player has a winning strategy: the first player or the second player. | [
"Second player has a winning strategy."
] | [
"Two players play a game with $n > 2$ piles, each containing $n^{10} + 1$ stones. A move consists of removing all but one pile and dividing the remaining pile into $n$ nonempty piles. The player who cannot make a move loses. Determine which player has a winning strategy: the first player or the second player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 17,
"genselect": 4,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-9d32c77f45dcaa2f2d74abfba257287f16160bb748854d13a633c4e7bc2b8c59 | Two players play a game with 2 6-sided dice. Before they roll the dice, they have to guess the sum of the 2 dice. They can pick any number between 2 and 12, but they cannot be the same. The player whose guess is closer to the actual sum of the 2 dice wins the game. If you go first, what number should you guess to maxim... | [
"7, \\(\\frac{127}{198}\\)"
] | [
"Two players play a game with 2 6-sided dice. Before they roll the dice, they have to guess the sum of the 2 dice. They can pick any number between 2 and 12, but they cannot be the same. The player whose guess is closer to the actual sum of the 2 dice wins the game. If you go first, what number should you guess to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f94c4838044a2ffe1592d2aa2fe3847aec576e0a20c57c3288d11a98a89b256 | Two players play a game with 2017 cards arranged in a row, each having one red side and one white side, initially all showing red. Each move consists of choosing a block of 50 consecutive cards, the leftmost of which is showing red, and turning them all over. The last player who can make a legal move wins. Determine th... | [
"2^{49}"
] | [
"Two players play a game with 2017 cards arranged in a row, each having one red side and one white side, initially all showing red. Each move consists of choosing a block of 50 consecutive cards, the leftmost of which is showing red, and turning them all over. The last player who can make a legal move wins. Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-8277bf3d11cb3e187b1a4404caff26774c4cba52481ccc50ad5f410b4262d689 | Two players play a game with a chocolate bar of dimension \( m \times n \) partitioned into \( 1 \times 1 \) squares. A move consists of choosing a square \((i, j)\) and taking all the chocolate from the squares \((k, l)\) with \( k \ge i \) and \( l \ge j \). The chocolate at position \((1,1)\) is poisoned, and the pl... | [
"3"
] | [
"Two players play a game with a chocolate bar of dimension \\( m \\times n \\) partitioned into \\( 1 \\times 1 \\) squares. A move consists of choosing a square \\((i, j)\\) and taking all the chocolate from the squares \\((k, l)\\) with \\( k \\ge i \\) and \\( l \\ge j \\). The chocolate at position \\((1,1)\\) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b5253ed9bcc5c6ab96ba3d69eb343555bb8f5cdcb4e5dad32c2989779cd8c32 | Two players play a game with a knight on an $8 \times 8$ chessboard. The first player places the knight on any square of the board. Each player in turn must then move the knight by a normal knight's move to a square not previously occupied by the knight. Which player has a winning strategy? | [
"The second player has a winning strategy."
] | [
"Two players play a game with a knight on an $8 \\times 8$ chessboard. The first player places the knight on any square of the board. Each player in turn must then move the knight by a normal knight's move to a square not previously occupied by the knight. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 11
} | 31 | 0 | 0 |
omr-ws-55bd11a0ac22f60137819500ea3e126e2247b618c3e9006e67d837aa48fe4ab3 | Two players play a game with a pile of 2024 chips. Each player, in turn, removes at least one chip but no more than half of the chips still on the table. The player who leaves only one chip on the table loses. Determine which player has a winning strategy and describe the strategy. | [
"\\text{First player}"
] | [
"Two players play a game with a pile of 2024 chips. Each player, in turn, removes at least one chip but no more than half of the chips still on the table. The player who leaves only one chip on the table loses. Determine which player has a winning strategy and describe the strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00d23ab6a23e4f68d6f3a90f585e2bccf0a176cc53de3f660aca307b96b1b9cb | Two players play a game with numbers $1, 2, 3, \ldots, 2004$ written on a board. Each turn, a player selects two numbers $a$ and $b$, deletes them, and replaces them with $a^b$. The game ends when only one number remains on the board. The first player wins if the final number ends with $2, 3, 7,$ or $8$, and loses if i... | [
"The second player has a winning strategy."
] | [
"Two players play a game with numbers $1, 2, 3, \\ldots, 2004$ written on a board. Each turn, a player selects two numbers $a$ and $b$, deletes them, and replaces them with $a^b$. The game ends when only one number remains on the board. The first player wins if the final number ends with $2, 3, 7,$ or $8$, and lose... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 10
} | 13 | 0 | 0 |
omr-ws-de1e2780bc892526954b18bdb1834ef9a79f55be475e7d8a7479d9ba9cd0ec52 | Two players play a game with the positive integers from 1 to n. In each step, a player can delete one number and all its divisors. The player who cannot make a move loses. Determine which player has a winning strategy, the first player or the second player, and describe the winning strategy. | [
"\\text{First player}"
] | [
"Two players play a game with the positive integers from 1 to n. In each step, a player can delete one number and all its divisors. The player who cannot make a move loses. Determine which player has a winning strategy, the first player or the second player, and describe the winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-88aa26c9d49ac285b7fc4b31c1921e7f52117ed6a34036d7992e5947420a7dad | Two players play a game with two piles of stones, initially sized 6 and 6. On each turn, a player can either remove 4 or 5 stones from any pile or add 2 stones to any nonempty pile. The player who takes the last stone wins. Determine who wins the game. | [
"2"
] | [
"Two players play a game with two piles of stones, initially sized 6 and 6. On each turn, a player can either remove 4 or 5 stones from any pile or add 2 stones to any nonempty pile. The player who takes the last stone wins. Determine who wins the game."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 20
} | 70 | 0 | 0 |
omr-ws-b3cbfada835d4f23db1fab59baaadf4dc39487ce37f973673aef1393e2a9c6df | Two players play a game with two piles of stones, initially sized 6 and 9. On each turn, a player can either remove 4 or 5 stones from any pile or add 2 stones to any nonempty pile. The player who takes the last stone wins. Determine who wins the game. | [
"O"
] | [
"Two players play a game with two piles of stones, initially sized 6 and 9. On each turn, a player can either remove 4 or 5 stones from any pile or add 2 stones to any nonempty pile. The player who takes the last stone wins. Determine who wins the game."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d82420cb1825ba6f2e979fa6312d3e9e6480de37415e41b5043dbb01360fb82f | Two players play a game with two piles of tokens, initially containing 10,000 and 20,000 tokens, respectively. A move consists of removing any positive number of tokens from a single pile or removing \( x > 0 \) tokens from one pile and \( y > 0 \) tokens from the other, where \( x + y \) is divisible by 2015. The play... | [
"First player has a winning strategy.",
"\\text{First player}"
] | [
"Two players play a game with two piles of tokens, initially containing 10,000 and 20,000 tokens, respectively. A move consists of removing any positive number of tokens from a single pile or removing \\( x > 0 \\) tokens from one pile and \\( y > 0 \\) tokens from the other, where \\( x + y \\) is divisible by 201... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 22
} | 54 | 0 | 0 |
omr-ws-2f601e9134ef94d8197a743bf5c31b25a7f196d360a14760af6f04d49e3c326b | Two players play a game. Initially, 1 million is written on the whiteboard. In each player's turn, the player chooses two numbers that multiply to make the number on the board and replaces the original number with the difference of the two chosen numbers. Player 1 goes first, then alternating between player 1 and 2. If... | [
"1000000"
] | [
"Two players play a game. Initially, 1 million is written on the whiteboard. In each player's turn, the player chooses two numbers that multiply to make the number on the board and replaces the original number with the difference of the two chosen numbers. Player 1 goes first, then alternating between player 1 and ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f84c206d9017e7c1b6112c9605649f70ae35efd36ad3f01a05aea6cd967f07d | Two players play a game. The first player writes ten positive real numbers on a board. The second player then writes another ten distinct positive real numbers. The first player then arranges the twenty numbers into 10 ordered pairs (a, b). The first player wins if the ten quadratic equations \(x^2 + ax + b\) have 11 d... | [
"Second player wins."
] | [
"Two players play a game. The first player writes ten positive real numbers on a board. The second player then writes another ten distinct positive real numbers. The first player then arranges the twenty numbers into 10 ordered pairs (a, b). The first player wins if the ten quadratic equations \\(x^2 + ax + b\\) ha... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 21
} | 31 | 0 | 0 |
omr-ws-635a129afc3002b05bd2f6cf348a3844410a1837f44bd119d32cb8cf519438fd | Two players play alternately on a $5 \times 5$ board. The first player always enters a $1$ into an empty square and the second player always enters a $0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $3 \times 3$ squares is calculated and the first player's score is the larges... | [
"6"
] | [
"Two players play alternately on a $5 \\times 5$ board. The first player always enters a $1$ into an empty square and the second player always enters a $0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $3 \\times 3$ squares is calculated and the first player's score is the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bb4349645c6937b726eeb3107058a5ed765e32d0500fe68c79338b608cd95653 | Two players play alternately on a 5 x 5 board. The first player always enters a 1 into an empty square and the second player always enters a 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3 x 3 squares is calculated, and the first player's score is the largest such sum. What ... | [
"6"
] | [
"Two players play alternately on a 5 x 5 board. The first player always enters a 1 into an empty square and the second player always enters a 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3 x 3 squares is calculated, and the first player's score is the largest such sum. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bae7ec5742a9c9afb985bad0ea402bedca87baac6e8618755cf63a0b326b1c24 | Two players play alternately. The first player is given a pair of positive integers \((x_1, y_1)\). Each player must replace the pair \((x_n, y_n)\) with a pair of non-negative integers \((x_{n+1}, y_{n+1})\) such that \(x_{n+1} = \min(x_n, y_n)\) and \(y_{n+1} = \max(x_n, y_n) - k \cdot x_{n+1}\) for some positive int... | [
"\\frac{x_1}{y_1} \\notin \\left\\{ \\frac{F_n}{F_{n+1}} \\mid n \\geq 1 \\right\\}"
] | [
"Two players play alternately. The first player is given a pair of positive integers \\((x_1, y_1)\\). Each player must replace the pair \\((x_n, y_n)\\) with a pair of non-negative integers \\((x_{n+1}, y_{n+1})\\) such that \\(x_{n+1} = \\min(x_n, y_n)\\) and \\(y_{n+1} = \\max(x_n, y_n) - k \\cdot x_{n+1}\\) for... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-b2342b80c93029c7d32cae4b229d6adce35373cee060ff452e6fcf2b48e7c6ac | Two players play alternatively on an infinite square grid. The first player places an $X$ in an empty cell and the second player places an $O$ in an empty cell. The first player wins if they get $n$ adjacent $X$'s in a line—horizontally, vertically, or diagonally. Determine the smallest value of $n$ for which the first... | [
"2"
] | [
"Two players play alternatively on an infinite square grid. The first player places an $X$ in an empty cell and the second player places an $O$ in an empty cell. The first player wins if they get $n$ adjacent $X$'s in a line—horizontally, vertically, or diagonally. Determine the smallest value of $n$ for which the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-53266a3f913a129d79a26acd9b325da8958fcded809660079ceefba0ed89922c | Two players play alternatively on an infinite square grid. The first player places an X in an empty cell, and the second player places an O in an empty cell. The first player wins if they get 11 adjacent Xs in a line, either horizontally, vertically, or diagonally. Let $N$ be the minimum number of O's the second player... | [
"20"
] | [
"Two players play alternatively on an infinite square grid. The first player places an X in an empty cell, and the second player places an O in an empty cell. The first player wins if they get 11 adjacent Xs in a line, either horizontally, vertically, or diagonally. Let $N$ be the minimum number of O's the second p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-5fdaed5a4107886d27cfedd843ef7f47eab49a8fcd1758287c710d3954973899 | Two players play on a 1×2000 table. In each turn, a player places either 'S' or 'O' in an empty cell. The game ends when the consecutive triple "SOS" appears for the first time. Let $N$ be the number of ways the second player can ensure a win by placing their first move in the $k$-th cell, where $1 \leq k \leq 2000$. C... | [
"2001"
] | [
"Two players play on a 1×2000 table. In each turn, a player places either 'S' or 'O' in an empty cell. The game ends when the consecutive triple \"SOS\" appears for the first time. Let $N$ be the number of ways the second player can ensure a win by placing their first move in the $k$-th cell, where $1 \\leq k \\leq... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-18d3853fd040619050a2353956e1abfd391f0e5e1955caaaf7bb9d18c41fcb6e | Two players play on a 5x5 grid. Player 1 enters a 1 on an empty square, and Player 2 enters a 0 on an empty square. When the board is full, the sum of the numbers in all 25 squares is calculated, and Player 1 scores this sum. What is the largest score Player 1 can make, regardless of Player 2's strategy? | [
"13"
] | [
"Two players play on a 5x5 grid. Player 1 enters a 1 on an empty square, and Player 2 enters a 0 on an empty square. When the board is full, the sum of the numbers in all 25 squares is calculated, and Player 1 scores this sum. What is the largest score Player 1 can make, regardless of Player 2's strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72124ed8b113ae93fd79220c02ed898a9a0a5320f18430e096e19927ebe47daf | Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the ... | [
"\\dfrac{2}{3}"
] | [
"Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-c4959ad1c2affca082397642ca5d7ad54ef3a64d8c70e6df035babefdc107931 | Two players play the following game on an $8 \times 8$ chessboard. The first player moves a rook from the bottom-left square to any square to the right or above. The second player then moves the rook similarly. The player who moves the rook to the top-right square wins. Who has a winning strategy? | [
"Second player wins."
] | [
"Two players play the following game on an $8 \\times 8$ chessboard. The first player moves a rook from the bottom-left square to any square to the right or above. The second player then moves the rook similarly. The player who moves the rook to the top-right square wins. Who has a winning strategy?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-2276f343d99d37eb88e8aaaf1702786ca614e373b658e422a89d06f773380e7f | Two players play the following game. The first player chooses a 10-digit number $\overline{a_9a_8\dots a_0}$. The second player, on each move, selects a subset $S \subset \{0, 1, \dots, 9\}$ and asks about the multiset $\{a_s \mid s \in S\}$ (i.e., which digits are in the corresponding positions, without specifying the... | [
"10"
] | [
"Two players play the following game. The first player chooses a 10-digit number $\\overline{a_9a_8\\dots a_0}$. The second player, on each move, selects a subset $S \\subset \\{0, 1, \\dots, 9\\}$ and asks about the multiset $\\{a_s \\mid s \\in S\\}$ (i.e., which digits are in the corresponding positions, without... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-3429122dd9bb5b2bdf7c238e08f47775b158b8a1b3430fdfce58676e68e86597 | Two players play the following game. They alternately write divisors of \(100!\) on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals \(1\) loses the game. Which player has a winning strategy? | [
"The first player has a winning strategy."
] | [
"Two players play the following game. They alternately write divisors of \\(100!\\) on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals \\(1\\) loses the game. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 10
} | 17 | 0 | 0 |
omr-ws-8e72b90ebe69ad20b8b09f5d6a2082adc8a42089a60c447f300270c6e6374774 | Two players play the following game: there are \( n \) stones. A player can take \( 2^m \) stones with \( m \in \mathbb{N} \). For which values of \( n \) does the first player win? | [
"\\( n \\not\\equiv 0 \\pmod{3} \\)"
] | [
"Two players play the following game: there are \\( n \\) stones. A player can take \\( 2^m \\) stones with \\( m \\in \\mathbb{N} \\). For which values of \\( n \\) does the first player win?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 31
} | 49 | 0 | 0 |
omr-ws-27a15efbecc0310012ae2143862c172989d7897f1b1455e04f6a0787acc6ffa6 | Two players replace in turn the hearts in the formula \[ M=\frac{1\cdot\heartsuit+2\cdot\heartsuit+\ldots+1001\cdot\heartsuit}{1+2+\ldots+1001} \] starting from the leftmost heart and ending with the rightmost one by numbers $1, 2, \ldots, 1001$ in some order (each number can be used only once). When all the hearts are... | [
"1"
] | [
"Two players replace in turn the hearts in the formula\n\\[ M=\\frac{1\\cdot\\heartsuit+2\\cdot\\heartsuit+\\ldots+1001\\cdot\\heartsuit}{1+2+\\ldots+1001} \\]\nstarting from the leftmost heart and ending with the rightmost one by numbers $1, 2, \\ldots, 1001$ in some order (each number can be used only once). When... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-e596ec52d9225fb7814dfdec666593143177c1e83a74fc9cdffbd94c1af5546a | Two players select cards numbered 1, 2, 3, 4. The game stops and the player who drew the last card wins if the sum of the numbers on all the cards that have appeared is a multiple of 3. Suppose player A goes first. What is the probability that player A wins? (For example, player A could draw 3 and win immediately). The... | [
"$\\frac{13}{23}$"
] | [
"Two players select cards numbered 1, 2, 3, 4. The game stops and the player who drew the last card wins if the sum of the numbers on all the cards that have appeared is a multiple of 3. Suppose player A goes first. What is the probability that player A wins? (For example, player A could draw 3 and win immediately)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 22
} | 69 | 0 | 0 |
omr-ws-11d927e925d88dd72231068c33fe979b50d1253bc5cb86605842cf16099716a3 | Two players share an unbiased coin and take turns playing "The Race". On Player 1's turn, he tosses the coin once: if it comes up Heads, he scores one point; if it comes up Tails, he scores nothing. On Player 2's turn, she chooses a positive integer \( T \) and tosses the coin \( T \) times: if it comes up all Heads, s... | [
"0.66666667"
] | [
"Two players share an unbiased coin and take turns playing \"The Race\". On Player 1's turn, he tosses the coin once: if it comes up Heads, he scores one point; if it comes up Tails, he scores nothing. On Player 2's turn, she chooses a positive integer \\( T \\) and tosses the coin \\( T \\) times: if it comes up a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0c38831b3bfff2fa2370859e8bae768981aef7d994e99d639efbfe2f5cd19347 | Two players start with two coins each. They take turns tossing a coin; if it comes down heads, the player keeps it, and if tails, they give it to the other player. The game ends when one player has all four coins. If the first player always goes first, what is the expected number of games that the first player would wi... | [
"360"
] | [
"Two players start with two coins each. They take turns tossing a coin; if it comes down heads, the player keeps it, and if tails, they give it to the other player. The game ends when one player has all four coins. If the first player always goes first, what is the expected number of games that the first player wou... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-082fa3885495fb345fd3df42ad48957f15f1657bdecca944d614a0fd66d37d9d | Two players start with two coins each. They take turns tossing a coin; if it comes up heads, the player keeps it, and if it comes up tails, the player gives it to the other player. The game ends when one player has all four coins. If the game is played 840 times with the first player always going first, what is the exp... | [
"360"
] | [
"Two players start with two coins each. They take turns tossing a coin; if it comes up heads, the player keeps it, and if it comes up tails, the player gives it to the other player. The game ends when one player has all four coins. If the game is played 840 times with the first player always going first, what is th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b50efbc2b9f2abaff60627a9653d5848882111948557eeb7b7f5a2341d14aa78 | Two players take turns alternatively and remove a number from the set $\{1, 2, \dots, 1000\}$. A player cannot remove a number that differs by 1 from a number already removed, nor can they remove a number that sums to 1001 with another removed number. The player who cannot make a move loses. Determine the winner. | [
"The first player wins."
] | [
"Two players take turns alternatively and remove a number from the set $\\{1, 2, \\dots, 1000\\}$. A player cannot remove a number that differs by 1 from a number already removed, nor can they remove a number that sums to 1001 with another removed number. The player who cannot make a move loses. Determine the winne... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-d2c5d50be60d07461a3c2c76a3532b2f6123eee59252530085a87d2f9f62e4eb | Two players take turns choosing any two of four heaps containing 38, 45, 61, and 70 matches respectively. Each player can take some non-zero number of matches from each of the two chosen heaps. The player who cannot make a move loses. Which player has a winning strategy? | [
"First player has a winning strategy.",
"The first player has a winning strategy."
] | [
"Two players take turns choosing any two of four heaps containing 38, 45, 61, and 70 matches respectively. Each player can take some non-zero number of matches from each of the two chosen heaps. The player who cannot make a move loses. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 4,
"tir": 43
} | 76 | 0 | 0 |
omr-ws-aabf7dce4d997a24e59b8baf87cf6df5e37415dd8cc0f3dee9d6b71a2d40bd6a | Two players take turns coloring cells in a $2009 \times 2009$ table. The first player colors any cell. Subsequently, each player must color a cell that is at most 7 cells away (horizontally, vertically, or diagonally) from a cell already colored by the other player. The player who cannot make a move loses. Does either ... | [
"The first player has a winning strategy."
] | [
"Two players take turns coloring cells in a $2009 \\times 2009$ table. The first player colors any cell. Subsequently, each player must color a cell that is at most 7 cells away (horizontally, vertically, or diagonally) from a cell already colored by the other player. The player who cannot make a move loses. Does e... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-98ce804f7603465e401231d44ff5f0eea559ee899298604125aaa1293e89ffa5 | Two players take turns coloring cells in a 2009 × 2009 table. The first player colors any cell. Subsequently, each player must color a cell that is at most 7 cells away (horizontally, vertically, or diagonally) from a cell already colored by the other player. The player who cannot make a move loses. Does either player ... | [
"The first player has a winning strategy."
] | [
"Two players take turns coloring cells in a 2009 × 2009 table. The first player colors any cell. Subsequently, each player must color a cell that is at most 7 cells away (horizontally, vertically, or diagonally) from a cell already colored by the other player. The player who cannot make a move loses. Does either pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-fa4078125b20db5b4b2bd1c5b362be4da287237cec277223aa4e62433e0f9092 | Two players take turns coloring 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 values of $n$ does the se... | [
"\\( n = 4 \\)"
] | [
"Two players take turns coloring 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 values of $n$ does t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-c1be8b96907eb138b387e3a46f8b6a93bcc48b2a5b5d4eed0dda9a774a3436aa | Two players take turns coloring the squares of a $4 \times 4$ grid, one square at a time. The player who completes a $2 \times 2$ square loses. Determine which player has the winning strategy, the First player or the Second player. | [
"Second"
] | [
"Two players take turns coloring the squares of a $4 \\times 4$ grid, one square at a time. The player who completes a $2 \\times 2$ square loses. Determine which player has the winning strategy, the First player or the Second player."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2f5804550823169c146c5aa3c75916874cc19c08df3a1ed7fee0f1b42a5273fe | Two players take turns coloring the squares of a $4 \times 4$ grid, one square at a time. The player who completes a $2 \times 2$ square loses. Which player has the winning strategy, the first or the second? | [
"\\text{Second}"
] | [
"Two players take turns coloring the squares of a $4 \\times 4$ grid, one square at a time. The player who completes a $2 \\times 2$ square loses. Which player has the winning strategy, the first or the second?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3769b7522dd70e8d1225d20e45ff255b71e8fea3c77abef07d0be69ae6169c12 | Two players take turns coloring the ten line segments (five sides and five diagonals) of a regular pentagon, with the first player coloring in blue and the second player in red. A player wins if they are the first to create a triangle in their own color, whose three vertices are also vertices of the regular pentagon. T... | [
"Neither player can force a win."
] | [
"Two players take turns coloring the ten line segments (five sides and five diagonals) of a regular pentagon, with the first player coloring in blue and the second player in red. A player wins if they are the first to create a triangle in their own color, whose three vertices are also vertices of the regular pentag... | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-15a55a8641330535c7798b8392c0e56710c8147d58157236f249b3a2bee174ef | Two players take turns cutting a cubic cardboard box along one of its edges. The winner is the one after whose turn the box can be opened by cutting along three edges of one face. Who can secure the victory - the beginner or the second player? | [
"The beginner can secure the victory."
] | [
"Two players take turns cutting a cubic cardboard box along one of its edges. The winner is the one after whose turn the box can be opened by cutting along three edges of one face. Who can secure the victory - the beginner or the second player?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-980961fdc939626b4b3de2a52f805a1e806f15e8aff393c140935688145a88c8 | Two players take turns cutting a regular $n$-gonal prism along one of its edges. The winner is the one after whose turn the box can be opened by cutting along all edges of one face except for one edge. Who can secure the victory - the beginner or the second player? | [
"\\text{Second player}"
] | [
"Two players take turns cutting a regular $n$-gonal prism along one of its edges. The winner is the one after whose turn the box can be opened by cutting along all edges of one face except for one edge. Who can secure the victory - the beginner or the second player?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-5e883b3c52dc4a386adbe7b5a08bc69bb46e5a30ec7a202378934c337642479f | Two players take turns drawing diagonals in a regular $1999$-gon. After each move, the player pays the opponent a number of rubles equal to the number of crossed diagonals. Which of the players can get more money regardless of the opponent's game? | [
"\\text{First player}"
] | [
"Two players take turns drawing diagonals in a regular $1999$-gon. After each move, the player pays the opponent a number of rubles equal to the number of crossed diagonals. Which of the players can get more money regardless of the opponent's game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-9a0f6ebe7aa1b8017ae7c41637c55c4a30cf22922d1f05d7d824ae5f0c90a8d0 | Two players take turns drawing diagonals in a regular 1995-gon. It is forbidden to draw a diagonal if it intersects with already drawn ones. The player who forms a quadrilateral with diagonals not yet drawn loses. Who can win no matter how the opponent plays? | [
"\\text{Second player}"
] | [
"Two players take turns drawing diagonals in a regular 1995-gon. It is forbidden to draw a diagonal if it intersects with already drawn ones. The player who forms a quadrilateral with diagonals not yet drawn loses. Who can win no matter how the opponent plays?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-98be617917b2ac400d5d92a49ae8fee2b612a69e33730070c3f65c3198724e86 | Two players take turns drawing lines through pairs of 'free' points of the same parity on a circle divided into $n$ congruent arcs, labeled consecutively from $1$ to $n$. The lines must not intersect any previously drawn lines. The player who makes the last move wins. Assuming both players use optimal strategies, who w... | [
"\\text{Second player}"
] | [
"Two players take turns drawing lines through pairs of 'free' points of the same parity on a circle divided into $n$ congruent arcs, labeled consecutively from $1$ to $n$. The lines must not intersect any previously drawn lines. The player who makes the last move wins. Assuming both players use optimal strategies, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f502ee5a206a9e6938a0194b09e73648440666b794353c204f6e0c039d02b0c0 | Two players take turns eating up a chocolate bar in the shape of an equilateral triangle with side length $n$, consisting of triangular chips with side length 1. Each player breaks off a triangular piece along one of the lines and eats it, passing the leftovers to the other player. A player who has no move or leaves ex... | [
"\\text{Second player}"
] | [
"Two players take turns eating up a chocolate bar in the shape of an equilateral triangle with side length $n$, consisting of triangular chips with side length 1. Each player breaks off a triangular piece along one of the lines and eats it, passing the leftovers to the other player. A player who has no move or leav... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d25095075a04a78f44780d377710940238a46874e0c0b2237b2ac87004c902f5 | Two players take turns entering a symbol in an empty cell of a $1 \times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, w... | [
"Betty"
] | [
"Two players take turns entering a symbol in an empty cell of a $1 \\times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes fi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f42977bc78fdc8e04c1397de9a89cfa010f6cc68e98a7f1af1cab2b4f11e964d | Two players take turns entering a symbol in an empty cell of an 1 x n chessboard, where n is an integer greater than 1. Aaron always enters the symbol X and Betty always enters the symbol O. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player ha... | [
"Betty"
] | [
"Two players take turns entering a symbol in an empty cell of an 1 x n chessboard, where n is an integer greater than 1. Aaron always enters the symbol X and Betty always enters the symbol O. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which play... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-34a31fe1deed5b8eef105d7a137b4b9dee1280f9ce4f201b4876316613427bdf | Two players take turns marking the vertices of a regular $2012$-gon. Each player marks any previously unmarked vertex that is neither adjacent to nor opposite any marked vertex. The game stops when a player cannot make a move. If the first player starts the game, what is the maximum number of vertices that the first pl... | [
"503"
] | [
"Two players take turns marking the vertices of a regular $2012$-gon. Each player marks any previously unmarked vertex that is neither adjacent to nor opposite any marked vertex. The game stops when a player cannot make a move. If the first player starts the game, what is the maximum number of vertices that the fir... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6f8a7ef0309477ec7a6e87cf0ad5ae8c93b6e879f79ef7159dcaa3fc57d3fe67 | Two players take turns modifying an integer starting from $2011^{2011}$. Each move consists of either subtracting an integer between $1$ and $2010$ inclusive, or dividing by $2011$ and rounding down to the nearest integer. The player who first obtains a non-positive integer wins. Which player has a winning strategy? | [
"The second player has a winning strategy. \n\nSince the problem asks for a player with a winning strategy and not a numerical value or mathematical expression, the answer is stated in words."
] | [
"Two players take turns modifying an integer starting from $2011^{2011}$. Each move consists of either subtracting an integer between $1$ and $2010$ inclusive, or dividing by $2011$ and rounding down to the nearest integer. The player who first obtains a non-positive integer wins. Which player has a winning strateg... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-424f84a30311669247708d69eeab4f11061036338d07eafa105b18cae7d8908e | Two players take turns moving a chip between vertices of a regular $n$-gon, drawing a segment between the starting and ending points of each move. If two points are already connected by a segment, that move is prohibited. A player who cannot make a move loses. Which player can guarantee a victory? | [
"The first player can guarantee a victory."
] | [
"Two players take turns moving a chip between vertices of a regular $n$-gon, drawing a segment between the starting and ending points of each move. If two points are already connected by a segment, that move is prohibited. A player who cannot make a move loses. Which player can guarantee a victory?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-4bddbd88f632dffe48af85cf2608798d5f3d4c4d3730b2913b2fd8a2461cdfb4 | Two players take turns on a strip of 12 cells. On each turn, a player can either place a checker on any empty cell or move an already placed checker one cell to the right. The player who places a checker on the last free cell wins. Who has a winning strategy? | [
"\\text{Second player}"
] | [
"Two players take turns on a strip of 12 cells. On each turn, a player can either place a checker on any empty cell or move an already placed checker one cell to the right. The player who places a checker on the last free cell wins. Who has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-3b04abcbe9b06e571c442757799692428e7b93b97f8508bce72a92d9b9eb18b6 | Two players take turns painting cells of a $7 \times 7$ table, each using their own color. A player cannot paint a cell if its row or column contains a cell painted by the other player. The game stops when one player cannot make a move. What is the maximal number of cells that can remain unpainted when the game stops? | [
"42"
] | [
"Two players take turns painting cells of a $7 \\times 7$ table, each using their own color. A player cannot paint a cell if its row or column contains a cell painted by the other player. The game stops when one player cannot make a move. What is the maximal number of cells that can remain unpainted when the game s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-26ea3d005693ba22105546f27a537bb54c80e4453f06afeab949762ebb71d86f | Two players take turns painting the circles on a given picture with their respective colors. At the end, the remaining area of each small triangle is painted by the color of the majority of its vertices. The winner is the player who ends up with a larger area of their color (including the area of the circles). Assuming... | [
"3"
] | [
"Two players take turns painting the circles on a given picture with their respective colors. At the end, the remaining area of each small triangle is painted by the color of the majority of its vertices. The winner is the player who ends up with a larger area of their color (including the area of the circles). Ass... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-63291d94490b1c3e21f65316aba87789a6529ab365bb4250580acbbed2a5ca39 | Two players take turns placing $X$'s and $Y$'s on an infinite chessboard. For each $X$ placed by the first player, the second player responds by placing 100 $Y$'s. The first player aims to form the vertices of a square with sides parallel to the lines of the chessboard using their $X$'s. Determine the minimum number of... | [
"101"
] | [
"Two players take turns placing $X$'s and $Y$'s on an infinite chessboard. For each $X$ placed by the first player, the second player responds by placing 100 $Y$'s. The first player aims to form the vertices of a square with sides parallel to the lines of the chessboard using their $X$'s. Determine the minimum numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-2520d8b8e1d2b27aa98e6d347a5311feb51c5ebb42edf7ea23159180de2e60d2 | Two players take turns placing a rectangle of size $1 \times 1$, $1 \times 10$, or $1 \times 11$ on a $1 \times 100$ table. The player who places a rectangle covering the 100th square wins. Which player has a winning strategy? | [
"The second player has a winning strategy."
] | [
"Two players take turns placing a rectangle of size $1 \\times 1$, $1 \\times 10$, or $1 \\times 11$ on a $1 \\times 100$ table. The player who places a rectangle covering the 100th square wins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 6,
"tir": 5
} | 50 | 0 | 0 |
omr-ws-a170acc75d35dbd0e57fa4477a30a2a0cc85e675975b39f52d4c9e4b50fdb058 | Two players take turns placing an unused number from \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) into one of the empty squares in a 2 by 2 array. At the end, if the product of the top row is greater than the product of the bottom row, the first player wins. Otherwise, the second player wins. What is the winning strategy for each pl... | [
"\\text{Second player can always win}"
] | [
"Two players take turns placing an unused number from \\(\\{1, 2, 3, 4, 5, 6, 7, 8\\}\\) into one of the empty squares in a 2 by 2 array. At the end, if the product of the top row is greater than the product of the bottom row, the first player wins. Otherwise, the second player wins. What is the winning strategy fo... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0a69dbab6a3416c824dc5d1ec360ed253a1c467503386f158162dca5a581f812 | Two players take turns placing cards with numbers from 1 to 5 into a pile and summing the numbers. The player who makes the total sum equal to 14 wins. What is the optimal strategy for the players? | [
"Start with 2 and aim to make the sum a multiple of 6 after each of your turns."
] | [
"Two players take turns placing cards with numbers from 1 to 5 into a pile and summing the numbers. The player who makes the total sum equal to 14 wins. What is the optimal strategy for the players?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-3ef355c5359460fb93ba7c1d38860b50abf891ac7c6f6ad079c134feca6595aa | Two players take turns placing cards with numbers from 1 to 5 into a pile and summing the numbers. The player who makes the total sum equal to 15 wins. What is the optimal strategy for the players? | [
"The optimal strategy is for the first player to start with a card numbered 3 and then always play a card that makes the total sum 9 after their turn, eventually reaching 15."
] | [
"Two players take turns placing cards with numbers from 1 to 5 into a pile and summing the numbers. The player who makes the total sum equal to 15 wins. What is the optimal strategy for the players?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 5,
"tir": 19
} | 54 | 0 | 0 |
omr-ws-7dcc2b25eca31f086859e7102651ca28b94dcf3ed6bc82aa2cd5a923f2b989fc | Two players take turns placing either two or three coins into their own hats, which are initially empty. After the second player duplicates the first player's move, they exchange hats. The player wins if, after their move, their hat contains one hundred or more coins. Which player has a winning strategy? | [
"Player 1 has a winning strategy."
] | [
"Two players take turns placing either two or three coins into their own hats, which are initially empty. After the second player duplicates the first player's move, they exchange hats. The player wins if, after their move, their hat contains one hundred or more coins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7f2955d89e73b7ee28256c81c3d6c6824679ee92d6e5dafa2df6c1c4b4a4a558 | Two players take turns placing either two or three coins into their own hats. After each player has made five moves, they exchange hats. The player wins if, after their move, their hat contains one hundred or more coins. Which player has a winning strategy? | [
"\\text{First player}"
] | [
"Two players take turns placing either two or three coins into their own hats. After each player has made five moves, they exchange hats. The player wins if, after their move, their hat contains one hundred or more coins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-eb6e98bde7a967f95f02517d2efcc93eec42b5abe0dda63ff646e6a108dd77f3 | Two players take turns placing kings on the squares of a 9 x 9 chessboard, so that no king can capture another. The player who is unable to place a king loses. Define a strategy for one of the players so that they can always win. | [
"The first player can always win by placing a king at the center of the board and then playing diametrically opposite to the second player's moves."
] | [
"Two players take turns placing kings on the squares of a 9 x 9 chessboard, so that no king can capture another. The player who is unable to place a king loses. Define a strategy for one of the players so that they can always win."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-2f78f335cf80c39bf2e5a138659bd0db6d8b1500be689e5f1696a491dfc854f7 | Two players take turns playing on a $3 \times 1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allo... | [
"\\text{First player}"
] | [
"Two players take turns playing on a $3 \\times 1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy tha... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-c5a247ad0cadc66cf4f400136062c8af32dc012f29f52da6e0e104a45698e1b7 | Two players take turns removing a ball from a jar that initially contains $m$ white and $n$ black balls. The first player to remove a white ball wins. Develop a recursive formula to compute the probability that the starting player wins. | [
"\\( p_{n,m} = 1 - \\frac{n}{n+m} p_{n-1,m} \\) with \\( p_{1,m} = \\frac{m}{m+1} \\)"
] | [
"Two players take turns removing a ball from a jar that initially contains $m$ white and $n$ black balls. The first player to remove a white ball wins. Develop a recursive formula to compute the probability that the starting player wins."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 15
} | 28 | 0 | 0 |
omr-ws-63c21bdc694f9b6f49bd0996b22c723fc4c7228c3fd40f23109578999d84bfa0 | Two players take turns removing a prime number of coins from a pile, initially with $n$ coins. The player unable to make a move loses. Let $\ell(n)$ denote the number of losing positions from $0$ to $n$, and define $\tau(n) = \frac{\ell(n)}{n}$. Given $\tau(100) = 0.12$, $\tau(10^4) \approx 0.1325$, and $\tau(10^5) \ap... | [
"0"
] | [
"Two players take turns removing a prime number of coins from a pile, initially with $n$ coins. The player unable to make a move loses. Let $\\ell(n)$ denote the number of losing positions from $0$ to $n$, and define $\\tau(n) = \\frac{\\ell(n)}{n}$. Given $\\tau(100) = 0.12$, $\\tau(10^4) \\approx 0.1325$, and $\\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-1b8faa435488a9ab11de3916ba06956bde753c75a0226b0a9d16171a8e630e13 | Two players take turns removing any non-zero number of stones from 100 heaps on a table, with the condition that at least one heap remains untouched after each move. The player who cannot make a move loses. Determine which player, the first or the second, has a winning strategy for any initial configuration of the heap... | [
"The second player has a winning strategy if and only if all 100 heaps have the same number of stones."
] | [
"Two players take turns removing any non-zero number of stones from 100 heaps on a table, with the condition that at least one heap remains untouched after each move. The player who cannot make a move loses. Determine which player, the first or the second, has a winning strategy for any initial configuration of the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8a6f8a5ba86aa8ef05c4fbdf351d19378edf183fd882763d5dfb67c11cfc6aa | Two players take turns removing coins from two heaps containing 100 and 99 coins, respectively. On each turn, a player can remove between 1 and 10 coins from the first heap or between 1 and 9 coins from the second heap. The player who takes the last coin wins. Determine the winning strategy for this game. | [
"Move from (100, 99) to (100, 91) and then play the \"opposite\" to 11 and 10 in the respective heaps."
] | [
"Two players take turns removing coins from two heaps containing 100 and 99 coins, respectively. On each turn, a player can remove between 1 and 10 coins from the first heap or between 1 and 9 coins from the second heap. The player who takes the last coin wins. Determine the winning strategy for this game."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 25
} | 39 | 0 | 0 |
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