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-bb0421f7b7107e4fab5d4d4871c1ec48cf55e6076f05ff87fe27d5f88a06b163 | Two players take turns removing either one coin or two touching coins from a circle of 13 coins. The player who takes the last coin wins. Formulate a strategy that will ensure a victory for the second player. | [
"Ensure symmetry by picking 2 coins opposite if the first player picks 1, or respond symmetrically if the first player picks 2."
] | [
"Two players take turns removing either one coin or two touching coins from a circle of 13 coins. The player who takes the last coin wins. Formulate a strategy that will ensure a victory for the second player."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-329526f5294f3cbc9f994b89d42693c1b48b3f9f0e35af7f5af45e4f9f3074c6 | Two players take turns rolling a die, with the first player going first. The game ends when a player rolls a 1, and that player wins. What is the probability that the player who goes first wins the game? | [
"\\(\\frac{6}{11}\\)"
] | [
"Two players take turns rolling a die, with the first player going first. The game ends when a player rolls a 1, and that player wins. What is the probability that the player who goes first wins the game?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4ee6d44cbbe544a9c3b40e08fa9c182f1b63ca3af89f64496985b661b71c2110 | Two players take turns saying a number between 1 and 10 inclusive, with player 1 going first. The cumulative sum of the numbers said is kept track of. The person who says the number that makes the sum reach or exceed 50 wins. If a player cannot say a number such that its sum with the last number said equals 11, which p... | [
"1"
] | [
"Two players take turns saying a number between 1 and 10 inclusive, with player 1 going first. The cumulative sum of the numbers said is kept track of. The person who says the number that makes the sum reach or exceed 50 wins. If a player cannot say a number such that its sum with the last number said equals 11, wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-37e730e5c1c9ad520866a63519476e8f9e363893d5163ed1cbfeafbdd3bee471 | Two players take turns saying a number from the set {1, 2, 4}, with player 1 going first. The cumulative sum of the numbers said is kept track of. The person who says the number that makes the sum reach or exceed 50 wins. Which player has the winning strategy? | [
"Second player"
] | [
"Two players take turns saying a number from the set {1, 2, 4}, with player 1 going first. The cumulative sum of the numbers said is kept track of. The person who says the number that makes the sum reach or exceed 50 wins. Which player has the winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-6a8daf849f81f5428dc7b80aa3d7ab1d2353b3121bc2e5de20bb0e347857307b | Two players take turns splitting a 4x6x9 rectangular parallelepiped, made up of 216 unit cubes, into two smaller rectangular parallelepipeds using a plane. The player who makes the last split wins. Which player has the winning strategy? | [
"The first player has the winning strategy."
] | [
"Two players take turns splitting a 4x6x9 rectangular parallelepiped, made up of 216 unit cubes, into two smaller rectangular parallelepipeds using a plane. The player who makes the last split wins. Which player has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-d462c1ba272b18aee0922f2fc9492ef4af6b444b7f3ddb867b383c3ae879cd7a | Two players take turns subtracting 1 from one of the coefficients of the quadratic function \( f(x) = x^2 + 25x + 425 \). The player who makes the function have real roots wins. Determine which player has a winning strategy. | [
"The second player has a winning strategy."
] | [
"Two players take turns subtracting 1 from one of the coefficients of the quadratic function \\( f(x) = x^2 + 25x + 425 \\). The player who makes the function have real roots wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 14
} | 33 | 0 | 0 |
omr-ws-2e7d4f0e9e121b15e47301cf9e14ac067b990f6f98c80cd381a91188f4c7c894 | Two players take turns taking 1 or 2 stones from one pile, randomly selected from a given set of 3 ordered piles, initially containing 100, 200, and 300 stones, in order from left to right. It is forbidden to make a move that results in the left pile having more stones than the right pile. If after a player's move, no ... | [
"\\text{Second player}"
] | [
"Two players take turns taking 1 or 2 stones from one pile, randomly selected from a given set of 3 ordered piles, initially containing 100, 200, and 300 stones, in order from left to right. It is forbidden to make a move that results in the left pile having more stones than the right pile. If after a player's move... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-41a4ecb9619bfe3019c52f1c0c6de25a70f4bdcbe21ef3396cc3112842ed2aac | Two players take turns taking 1, 2, or 3 stones from a given set of 3 piles, initially containing 11, 22, and 33 stones. If after a player's move, any two piles have the same number of stones, that player wins. Who will win with the right game of both players? | [
"\\text{Second player}"
] | [
"Two players take turns taking 1, 2, or 3 stones from a given set of 3 piles, initially containing 11, 22, and 33 stones. If after a player's move, any two piles have the same number of stones, that player wins. Who will win with the right game of both players?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-7ff20053c619938d8703deed5b8ae4c388c9789eeccab90cde71b9e73db1582c | Two players take turns taking stones from a pile of 10 million stones. On every move, a player must take \( p^n \) stones, where \( p \) is any prime number and \( n \) is any non-negative integer (chosen by the player making the move). The person who takes the last stone wins. Who has a winning strategy and what is it... | [
"First player has a winning strategy."
] | [
"Two players take turns taking stones from a pile of 10 million stones. On every move, a player must take \\( p^n \\) stones, where \\( p \\) is any prime number and \\( n \\) is any non-negative integer (chosen by the player making the move). The person who takes the last stone wins. Who has a winning strategy and... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b51dd530b1f15f3e8c5acafa0026f51368494ea673b38322c6f2b4c05e6b9092 | Two players take turns writing down all proper non-decreasing fractions with denominators from $1$ to $1999$ and placing a "$+$" sign before each fraction. After all such fractions are written out, their sum is found. If this sum is an integer, the player who made the last entry wins; otherwise, the opponent wins. Who ... | [
"The player who makes the last entry can secure a win."
] | [
"Two players take turns writing down all proper non-decreasing fractions with denominators from $1$ to $1999$ and placing a \"$+$\" sign before each fraction. After all such fractions are written out, their sum is found. If this sum is an integer, the player who made the last entry wins; otherwise, the opponent win... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1d85f0719430d85189a4839926118f088ee53b7f23fb7ad1e004bee4deb10d68 | Two players take turns writing natural numbers on a board, with the rules that numbers cannot be greater than $p$ and cannot be divisors of previously written numbers. The player who cannot make a move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy. | [
"The first player has a winning strategy."
] | [
"Two players take turns writing natural numbers on a board, with the rules that numbers cannot be greater than $p$ and cannot be divisors of previously written numbers. The player who cannot make a move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 17
} | 37 | 0 | 0 |
omr-ws-45c31d281b98073da5689bd29239758f3b35b718d545c9ea006dfc860c6ee630 | Two players take turns writing on a board starting with the number "0". The first player writes a "+" or "-" sign, and the second player writes a natural number from 1 to 1993, each number used exactly once. After 1993 moves by each player, the second player receives a prize equal to the absolute value of the arithmeti... | [
"993010"
] | [
"Two players take turns writing on a board starting with the number \"0\". The first player writes a \"+\" or \"-\" sign, and the second player writes a natural number from 1 to 1993, each number used exactly once. After 1993 moves by each player, the second player receives a prize equal to the absolute value of th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f402209142d4b56e1d704d0dced37dbcbce296e9e9abfd09efc3880d8a028e4e | Two players write integers on a blackboard according to the following rules: - The first player writes $1$. - At each subsequent turn, a player must write either $a+1$ or $2a$, where $a$ is any number already written on the blackboard and $2a \leq 1000$. - No number can be written more than once. - The player who write... | [
"\\text{Second player}"
] | [
"Two players write integers on a blackboard according to the following rules:\n- The first player writes $1$.\n- At each subsequent turn, a player must write either $a+1$ or $2a$, where $a$ is any number already written on the blackboard and $2a \\leq 1000$.\n- No number can be written more than once.\n- The player... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-9a0ee6424439dff8a4f3915be24d1ce4d91450e2c2061765bcde96670179495c | Two players write integers on a blackboard according to the following rules: - The first player writes $1$. - At each subsequent turn, the player writes either $a+1$ or $2a$, where $a$ is any number already written on the blackboard and $2a \leq 1000$. - No number can be written more than once. - The player who writes ... | [
"The first player loses."
] | [
"Two players write integers on a blackboard according to the following rules:\n- The first player writes $1$.\n- At each subsequent turn, the player writes either $a+1$ or $2a$, where $a$ is any number already written on the blackboard and $2a \\leq 1000$.\n- No number can be written more than once.\n- The player w... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-6290ee5ade490ca64e2e3c5a1c0e815d0b6454ff67e5eb18d9ac6e70e68dfc3f | Two players, $A$ and $B$, alternatively take stones from a pile of $n \geq 2$ stones. $A$ plays first and must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as their opponent took in the previous move. The player who takes the last stone wins.... | [
"Player $B$ has a winning strategy if $n$ is a power of 2; otherwise, player $A$ has a winning strategy."
] | [
"Two players, $A$ and $B$, alternatively take stones from a pile of $n \\geq 2$ stones. $A$ plays first and must take at least one stone and at most $n-1$ stones. Then each player must take at least one stone and at most as many stones as their opponent took in the previous move. The player who takes the last stone... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 26
} | 27 | 0 | 0 |
omr-ws-cb605191d94da4658109cddfba14a957d368797ee7b5d6f00221b8e4da78331a | Two players, $A$ and $B$, are playing a game on a plane with 2020 points. They take turns adding segments between these points, with $A$ going first. The rule is that no cycle of length 4 can be formed. The first player who cannot add a segment loses. Calculate the maximum number of segments that can be added to the pl... | [
"45450"
] | [
"Two players, $A$ and $B$, are playing a game on a plane with 2020 points. They take turns adding segments between these points, with $A$ going first. The rule is that no cycle of length 4 can be formed. The first player who cannot add a segment loses. Calculate the maximum number of segments that can be added to t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-1395f1cc1fd39212c2259ac7499ae3d8f373624a590f2a6567b6814a1c968f40 | Two players, $A$ and $B$, consecutively make moves on a blackboard with natural numbers $1, 2, \ldots, 500$. Each move consists of choosing two numbers $n$ and $2n$ and erasing them. If a player cannot perform a valid move, he loses. Which player can guarantee a win? | [
"A"
] | [
"Two players, $A$ and $B$, consecutively make moves on a blackboard with natural numbers $1, 2, \\ldots, 500$. Each move consists of choosing two numbers $n$ and $2n$ and erasing them. If a player cannot perform a valid move, he loses. Which player can guarantee a win?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f6f5dbd95e25b88f37b09735b10b2b6970cf5ea83f5ac247d41242052cc3e6e1 | Two players, $A$ and $B$, each have two piles of $n$ coins, where $n$ is a positive integer with at least 2 distinct prime factors and is a perfect square of a composite number. Each player can remove or add a number of coins from/to one of their piles, which is a divisor of the number of coins already in that pile (ex... | [
"Yes"
] | [
"Two players, $A$ and $B$, each have two piles of $n$ coins, where $n$ is a positive integer with at least 2 distinct prime factors and is a perfect square of a composite number. Each player can remove or add a number of coins from/to one of their piles, which is a divisor of the number of coins already in that pil... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-65d4b36069055e1adb250f5903b9c211ec32a61ef0e3bbb95097d4ef9cd6abe3 | Two players, $A$ and $B$, play a game on a rhombus-shaped board of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles. The tokens are initially placed in cells containing opposite $60^{\circ}$ corners. Players take turns moving their token to a neighboring cell (sharing a s... | [
"A has a winning strategy."
] | [
"Two players, $A$ and $B$, play a game on a rhombus-shaped board of side $n$ and angles of $60^{\\circ}$ and $120^{\\circ}$, divided into $2n^2$ equilateral triangles. The tokens are initially placed in cells containing opposite $60^{\\circ}$ corners. Players take turns moving their token to a neighboring cell (sha... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 4
} | 24 | 0 | 0 |
omr-ws-02fcf97384c85d47a2534a54566fb1e52e40fbf4bdf317c536d7a85bf07142c4 | Two players, $A$ and $B$, play a game on an $n \times n$ board. Each player has a rook on opposite corners. In a move, a player can move their rook horizontally or vertically through any number of squares, provided all squares passed through are previously unvisited. After the move, all passed squares are considered vi... | [
"B"
] | [
"Two players, $A$ and $B$, play a game on an $n \\times n$ board. Each player has a rook on opposite corners. In a move, a player can move their rook horizontally or vertically through any number of squares, provided all squares passed through are previously unvisited. After the move, all passed squares are conside... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-786bd092fc334509ca96d00d47468ac0512e48d9b56ea5dd8831d41bd6bc15e6 | Two players, $A$ and $B$, play a game on an $n \times n$ empty square table. $A$ starts the game and they take turns. On each move, a player places one marble on a cell if all neighboring cells are empty. Two cells are neighbors if they share a vertex. The player who cannot make a move loses. Determine the winner for e... | [
"A"
] | [
"Two players, $A$ and $B$, play a game on an $n \\times n$ empty square table. $A$ starts the game and they take turns. On each move, a player places one marble on a cell if all neighboring cells are empty. Two cells are neighbors if they share a vertex. The player who cannot make a move loses. Determine the winner... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-4cf10a272d7ef611c60dc4593eb991dff9a4c33b6cf4a8bd43c5b4022d26f78f | Two players, $A$ and $B$, play a game starting with a positive integer $n$. On each turn, a player can subtract one of the following from $n$: - $1$ - the number of digits of $n$ - the sum of the digits of $n$ The player who reduces $n$ to $0$ wins. If $A$ goes first, determine who will be the winner given the initial ... | [
"B"
] | [
"Two players, $A$ and $B$, play a game starting with a positive integer $n$. On each turn, a player can subtract one of the following from $n$: \n- $1$\n- the number of digits of $n$\n- the sum of the digits of $n$\n\nThe player who reduces $n$ to $0$ wins. If $A$ goes first, determine who will be the winner given ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-b48f0ce1271d0daac1c1b7754530eccd4ab4301f640ec627fecf3a0a10f98c58 | Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\left\lfloor \frac{n_k}{3} \right\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine which player... | [
"B"
] | [
"Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\\left\\lfloor \\frac{n_k}{3} \\right\\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-b920c9c6a10b87b23bd9df7da75ee5371abde6bf38e34dc73d0a837480a2c262 | Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\left\lfloor \frac{n_k}{3} \right\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine which player... | [
"B"
] | [
"Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\\left\\lfloor \\frac{n_k}{3} \\right\\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 5,
"tir": 17
} | 69 | 0 | 0 |
omr-ws-8c4941c44789058a3c875fba4e3e624815a79aa61aa8ed417a231351d1f800ab | Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\left\lfloor \frac{n_k}{3} \right\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine which player... | [
"A"
] | [
"Two players, $A$ and $B$, play a game starting with a positive integer $n_0$ on a blackboard. On each turn, a player deletes the current number $n_k$ and writes either $n_k-1$ or $\\left\\lfloor \\frac{n_k}{3} \\right\\rfloor$. The game ends when the number becomes 0, and the player who writes 0 wins. Determine wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 13
} | 36 | 0 | 0 |
omr-ws-18456032619bd00591aed862553bdd17cf88be261d9ce9b0178913a183eb35bf | Two players, $A$ and $B$, play a game starting with the number $1234567890$ on the blackboard. In each move, a player erases the current number $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$, and writes the result. The first player to reduce the number to $0$ wins. Determine whi... | [
"Player \\(A\\) has the winning strategy.",
"\\( A \\) has the winning strategy."
] | [
"Two players, $A$ and $B$, play a game starting with the number $1234567890$ on the blackboard. In each move, a player erases the current number $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$, and writes the result. The first player to reduce the number to $0$ wins. Determin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 3
} | 11 | 0 | 0 |
omr-ws-b35b20500b3d7ac88f696ab65313df984c1d36555c2efc49af82b1ed0df75977 | Two players, $A$ and $B$, play a game starting with the number $2024$ on the board. They take turns reducing the number by a natural number that does not exceed the number of the move (with the first move $A$ must reduce by $1$, $B$ by $1$ or $2$, then $A$ by $1$, $2$, or $3$, and so on). The player who first reduces t... | [
"B has the winning strategy."
] | [
"Two players, $A$ and $B$, play a game starting with the number $2024$ on the board. They take turns reducing the number by a natural number that does not exceed the number of the move (with the first move $A$ must reduce by $1$, $B$ by $1$ or $2$, then $A$ by $1$, $2$, or $3$, and so on). The player who first redu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 10
} | 30 | 0 | 0 |
omr-ws-2b8ca93cb413d4139e4c21ca3a24de0e4c08ec1291911add6e516eede12aee92 | Two players, $A$ and $B$, play a game where $A$ chooses a positive integer $k$ and $B$ chooses a positive integer $n$. They alternately mark points in the plane until both have marked $n$ points. $B$ wins if the number of lines passing through at least two marked points is divisible by $k$. Otherwise, $A$ wins. Which p... | [
"B"
] | [
"Two players, $A$ and $B$, play a game where $A$ chooses a positive integer $k$ and $B$ chooses a positive integer $n$. They alternately mark points in the plane until both have marked $n$ points. $B$ wins if the number of lines passing through at least two marked points is divisible by $k$. Otherwise, $A$ wins. Wh... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-0f38473b93398049e8741dfaa15335e48f3838c408212c71e8d00f52e2e0afb3 | Two players, $A$ and $B$, play a game with a pile of $N$ stones where $2 \leq N$. The players alternate turns, with $A$ going first. On the first turn, $A$ must remove at least $1$ and at most $N-1$ stones. If a player removes $k$ stones on their turn, the other player must remove at least $1$ and at most $2k-1$ stones... | [
"\\( A \\) wins if \\( N \\) is not a power of 2, and \\( B \\) wins if \\( N \\) is a power of 2."
] | [
"Two players, $A$ and $B$, play a game with a pile of $N$ stones where $2 \\leq N$. The players alternate turns, with $A$ going first. On the first turn, $A$ must remove at least $1$ and at most $N-1$ stones. If a player removes $k$ stones on their turn, the other player must remove at least $1$ and at most $2k-1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-6b137f2b5c02d218828fc4e4154b4f0125c3219771847c60dd8ae45af8b31c2b | Two players, $A$ and $B$, play a game with a positive integer $n$. At each turn, starting with player $A$, a player chooses 2 out of $n$ sets and removes the same number of elements from both sets, where each removed element appears in both sets and in those 2 sets only, and the number of elements removed is greater th... | [
"n \\equiv 0 \\text{ or } 1 \\pmod{4}"
] | [
"Two players, $A$ and $B$, play a game with a positive integer $n$. At each turn, starting with player $A$, a player chooses 2 out of $n$ sets and removes the same number of elements from both sets, where each removed element appears in both sets and in those 2 sets only, and the number of elements removed is great... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-349d7d0ec85cf891168e7ece2165f25abd326695b266052eb8e8b8c30e8f0b5f | Two players, $A$ and $B$, play a game with the expression $\pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box$. Player $A$ moves first by replacing one of the $\Box$ with an arbitrary natural number. After all $\Box$ are replaced, $A$ replaces each $\pm$ with a plus or a minus. $A$ wins if the resulting expression's... | [
"A"
] | [
"Two players, $A$ and $B$, play a game with the expression $\\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box$. Player $A$ moves first by replacing one of the $\\Box$ with an arbitrary natural number. After all $\\Box$ are replaced, $A$ replaces each $\\pm$ with a plus or a minus. $A$ wins if the re... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-a60f5e377958856b6bf0b4ce66719627d1fe204af79565d4eac325d00eceed5e | Two players, $A$ and $B$, play a game with the expression \[ \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box \pm \Box. \] Player $A$ takes the first turn, and in each turn, the player replaces a symbol $\Box$ with a positive integer. After all $\Box$ symbols are replaced, player $A$ replaces each $\pm$ with either $+$ or ... | [
"A"
] | [
"Two players, $A$ and $B$, play a game with the expression\n\\[ \\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box \\pm \\Box. \\]\nPlayer $A$ takes the first turn, and in each turn, the player replaces a symbol $\\Box$ with a positive integer. After all $\\Box$ symbols are replaced, player $A$ replaces each $\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-73125a770925a5ccabbdae99c2d7a772194f8a9a6b503314ca18aded3e0b6183 | Two players, $A$ and $B$, play a game with the following rules: - $B$ starts by writing a positive integer $N$ on the blackboard. - Players alternate turns, with $A$ taking the first turn. - On each turn, a player can either "mirror" or "mimic" (if the previous player mirrored): - To "mirror," the player writes a posit... | [
"A \\text{ wins if } N \\text{ is prime; otherwise, } B \\text{ wins}"
] | [
"Two players, $A$ and $B$, play a game with the following rules:\n- $B$ starts by writing a positive integer $N$ on the blackboard.\n- Players alternate turns, with $A$ taking the first turn.\n- On each turn, a player can either \"mirror\" or \"mimic\" (if the previous player mirrored):\n - To \"mirror,\" the play... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-f3a3d76287e614648fb188a0657eb36b6ff76e2feb61e81ca7d145fdd83df6d7 | Two players, $A$ and $B$, play a game with the following rules: - The set $S$ contains all perfect squares: $S = \{1^2, 2^2, 3^2, \ldots\}$. - Initially, $M = 0$. - Players take turns choosing an element $X$ from $S$ such that $M + X \leq 2013$. - The chosen element $X$ is added to $M$, updating $M$ to $M + X$. - The p... | [
"B"
] | [
"Two players, $A$ and $B$, play a game with the following rules:\n- The set $S$ contains all perfect squares: $S = \\{1^2, 2^2, 3^2, \\ldots\\}$.\n- Initially, $M = 0$.\n- Players take turns choosing an element $X$ from $S$ such that $M + X \\leq 2013$.\n- The chosen element $X$ is added to $M$, updating $M$ to $M ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-c01d07d044702f1387dab97c6ddf8483e425afead39da68b320e6fc3c9db6e93 | Two players, $A$ and $B$, play a game with the polynomial $x^{30} + *x^{29} + \ldots + *x + 8 = 0$. Player $A$ starts by replacing all asterisks with the natural numbers from $1$ to $30$ (each used exactly once). Player $B$ then replaces some of the $+$ signs with $-$ signs. Player $A$ aims to create a polynomial with ... | [
"B"
] | [
"Two players, $A$ and $B$, play a game with the polynomial $x^{30} + *x^{29} + \\ldots + *x + 8 = 0$. Player $A$ starts by replacing all asterisks with the natural numbers from $1$ to $30$ (each used exactly once). Player $B$ then replaces some of the $+$ signs with $-$ signs. Player $A$ aims to create a polynomial... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34a130c4d20d513d53261c2f6a59140f43357b18187b484ce8598a4fbd9303f0 | Two players, $A$ and $B$, play a game with the positive divisors of a positive integer $N$ written on a blackboard. In the first move, player $A$ erases $N$. If the last erased number is $d$, the next player must erase either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all ... | [
"\\( N \\) must be a perfect square."
] | [
"Two players, $A$ and $B$, play a game with the positive divisors of a positive integer $N$ written on a blackboard. In the first move, player $A$ erases $N$. If the last erased number is $d$, the next player must erase either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-c6f01b4e41018c5838270e183c444bacd5c167e5bd84b31520bbe125156d8698 | Two players, $A$ and $B$, play a series of games where the probability of $A$ winning a game is $\frac{1}{3}$ and the probability of $B$ winning a game is $\frac{2}{3}$. The series is won by $A$ if $A$ wins 2 consecutive games, and by $B$ if $B$ wins 4 consecutive games. Find the probability that $B$ wins the series. | [
"\\(\\frac{64}{129}\\)"
] | [
"Two players, $A$ and $B$, play a series of games where the probability of $A$ winning a game is $\\frac{1}{3}$ and the probability of $B$ winning a game is $\\frac{2}{3}$. The series is won by $A$ if $A$ wins 2 consecutive games, and by $B$ if $B$ wins 4 consecutive games. Find the probability that $B$ wins the se... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-3151b5da32177c68e76b60820231f30a289f803627f7a2bfd289d6822425dd3c | Two players, $A$ and $B$, play alternately, starting with $A$. There are 2009 boxes numbered from 1 to 2009, with 2009 stones in box 2 and the others empty. A move consists in selecting a non-empty box $i$, taking one or more stones from that box and putting them in box $i + 1$. If $i = 2009$, the selected stones are e... | [
"Player $B$ has a winning strategy."
] | [
"Two players, $A$ and $B$, play alternately, starting with $A$. There are 2009 boxes numbered from 1 to 2009, with 2009 stones in box 2 and the others empty. A move consists in selecting a non-empty box $i$, taking one or more stones from that box and putting them in box $i + 1$. If $i = 2009$, the selected stones ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 5,
"tir": 14
} | 45 | 0 | 0 |
omr-ws-cd7e55fafcef34a7be02136e7f23e6ce812f1be8017ad5df26e37453e3168d4f | Two players, $A$ and $B$, play alternately, starting with $A$. There are 2009 boxes numbered from 1 to 2009, with exactly one stone in each box. A move consists in selecting a non-empty box $i$, taking one or more stones from that box and putting them in box $i + 1$. If $i = 2009$, the selected stones are eliminated. T... | [
"Player \\( A \\) has a winning strategy."
] | [
"Two players, $A$ and $B$, play alternately, starting with $A$. There are 2009 boxes numbered from 1 to 2009, with exactly one stone in each box. A move consists in selecting a non-empty box $i$, taking one or more stones from that box and putting them in box $i + 1$. If $i = 2009$, the selected stones are eliminat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 12
} | 30 | 0 | 0 |
omr-ws-c099f5e31726bc0e8ddb9ecf1c7d7d5bbd9df7cbd0583145475426fe45a921d4 | Two players, $A$ and $B$, take turns drawing pieces of paper with numbers $1$ through $9$ written on them. The winner is the first player to have three pieces of paper whose numbers sum to $15$. If $A$ goes first, determine who has a winning strategy. | [
"A"
] | [
"Two players, $A$ and $B$, take turns drawing pieces of paper with numbers $1$ through $9$ written on them. The winner is the first player to have three pieces of paper whose numbers sum to $15$. If $A$ goes first, determine who has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-180eff04d1a143add3ee3977bb454592ff8f0489ab0df31cb105674d67960ef5 | Two players, $A$ and $B$, take turns picking up cards numbered from $1$ to $2002$. The player who ends up with the sum of their cards having a larger last digit than their opponent wins. $A$ goes first. Who has a winning strategy, and what is the strategy to win? | [
"Player \\( A \\) has a winning strategy."
] | [
"Two players, $A$ and $B$, take turns picking up cards numbered from $1$ to $2002$. The player who ends up with the sum of their cards having a larger last digit than their opponent wins. $A$ goes first. Who has a winning strategy, and what is the strategy to win?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-07149165e929a9ecb7d84b78ab04f0e80686f1b81d62a50794a34724f0725a36 | Two players, $A$ and $B$, take turns saying an integer greater than 1. A number cannot be said if it can be written as the sum of positive multiples of numbers already said. The player who cannot say a number loses. If $A$ starts, who has a winning strategy? | [
"A has a winning strategy."
] | [
"Two players, $A$ and $B$, take turns saying an integer greater than 1. A number cannot be said if it can be written as the sum of positive multiples of numbers already said. The player who cannot say a number loses. If $A$ starts, who has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-549980373d17492d9afcf342399a09d70d558cb21c0411b3f995de88121d935a | Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$ 's turn, $B$ selects one white unit square and colors it blue. On $R$ 's turn, $R$ selects two white unit squares and colors them red. The players alternat... | [
"4"
] | [
"Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$ 's turn, $B$ selects one white unit square and colors it blue. On $R$ 's turn, $R$ selects two white unit squares and colors them red. The players alt... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d3a8e559d17f0dfd1a3674466a9ceccbc4e473d7022813e3f946d3fdf7dc536 | Two players, A and B, are playing a game starting with a cubic polynomial. During the game, the players can perform the following operations: 1. Replace the polynomial \( f(x) \) with \( f(nx) \), \( f(x+n) \), or \( f(x) + n \) where \( n \) is a natural number. 2. Change the sign of the coefficient of \( x^2 \) only.... | [
"f(x) = ax^3"
] | [
"Two players, A and B, are playing a game starting with a cubic polynomial. During the game, the players can perform the following operations:\n1. Replace the polynomial \\( f(x) \\) with \\( f(nx) \\), \\( f(x+n) \\), or \\( f(x) + n \\) where \\( n \\) is a natural number.\n2. Change the sign of the coefficient o... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-f12438d537d91f42378b541c07ba8d6784310b42f9bdc1e18e386c90480f50d9 | Two players, A and B, are playing a game starting with a number \( n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} \). The players take turns dividing \( n \) by \( p^k \) where \( k \leq x \) and \( p \in \{p_1, p_2, \ldots, p_k\} \) such that \( p^k \mid n \). The player who reduces \( n \) to 1 wins. Who has... | [
"Player B"
] | [
"Two players, A and B, are playing a game starting with a number \\( n = p_1^{\\alpha_1} p_2^{\\alpha_2} \\cdots p_k^{\\alpha_k} \\). The players take turns dividing \\( n \\) by \\( p^k \\) where \\( k \\leq x \\) and \\( p \\in \\{p_1, p_2, \\ldots, p_k\\} \\) such that \\( p^k \\mid n \\). The player who reduces... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d2af834129dda4dd075bc7bd0d60c19c8876b24e1ee121d339f55d6dde759d4d | Two players, A and B, are playing a game with 4 boxes of candies. The first box contains 9 candies, the second contains 18 candies, and the third contains 34 candies. Player B can choose the number of candies in the fourth box. Each player, on their turn, can take any number of candies from exactly one box. The player ... | [
"57"
] | [
"Two players, A and B, are playing a game with 4 boxes of candies. The first box contains 9 candies, the second contains 18 candies, and the third contains 34 candies. Player B can choose the number of candies in the fourth box. Each player, on their turn, can take any number of candies from exactly one box. The pl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-3eda2a494f721197cface8f699d78e69729b4936f013fff3835c11785b57b8c6 | Two players, A and B, are playing a sequence of matches where the first to reach 6 points wins. Each match results in one player winning 1 point and the other losing. The game is interrupted when the score is 4:3 in favor of player A. Assuming both players have approximately equal performance, what should be the ratio ... | [
"1:1"
] | [
"Two players, A and B, are playing a sequence of matches where the first to reach 6 points wins. Each match results in one player winning 1 point and the other losing. The game is interrupted when the score is 4:3 in favor of player A. Assuming both players have approximately equal performance, what should be the r... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-93ebb6c35f84570b579e8edea5d9ea2f729ef9c2a5d9524308608fefa052aad8 | Two players, A and B, are throwing two fair dice. Player A wins if the sum of the dice is 7, and player B wins if the sum is 6 or 8. Player A goes first. What is the probability that player A will win after n throws? | [
"\\dfrac{18}{43} \\left(1 - \\left(\\dfrac{65}{108}\\right)^{\\lceil n/2 \\rceil}\\right)"
] | [
"Two players, A and B, are throwing two fair dice. Player A wins if the sum of the dice is 7, and player B wins if the sum is 6 or 8. Player A goes first. What is the probability that player A will win after n throws?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-6b60593472bb4b55c1c1f4d3a0b35eb7e1c1b4b994af7ab50c8dcf0f5fd8d210 | Two players, A and B, play a game on a circle with 99 equally spaced points. On each turn, a player selects an unused number from 1 to 99 and assigns it to one of the points. The player who first completes an arithmetic sequence of three consecutive numbers around the circle wins. Player A goes first. Determine which p... | [
"A"
] | [
"Two players, A and B, play a game on a circle with 99 equally spaced points. On each turn, a player selects an unused number from 1 to 99 and assigns it to one of the points. The player who first completes an arithmetic sequence of three consecutive numbers around the circle wins. Player A goes first. Determine wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78c344d4e4bc43f097c51cfe9ea00c334dbb708a6eaababed60f772f288bef43 | Two players, A and B, play a game on a graph G with vertices labeled $P_1, P_2, \ldots, P_n$. Positive real numbers $x_{i \to j}$ are given for each edge. Player A starts by selecting a vertex, writing 1 on it, and coloring it red. Player B then selects a vertex, writes a number on it, and colors it blue. Players alter... | [
"A"
] | [
"Two players, A and B, play a game on a graph G with vertices labeled $P_1, P_2, \\ldots, P_n$. Positive real numbers $x_{i \\to j}$ are given for each edge. Player A starts by selecting a vertex, writing 1 on it, and coloring it red. Player B then selects a vertex, writes a number on it, and colors it blue. Player... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-efb084732b69a3c177aba0a5463ede1eb58304fc592c16f6ca949200aba98361 | Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns moving the coin either 53 squares forward or 2 squares backward, without going beyond the bounds of the row. The player who places the coin on square 2017 wins. Determine which... | [
"A"
] | [
"Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns moving the coin either 53 squares forward or 2 squares backward, without going beyond the bounds of the row. The player who places the coin on square 2017 wins. Determine ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-f8f45700e44a56168b095c06b02ce49fc210cbb8def455c22087fc66f8ca7338 | Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns, starting with A, and on each turn, a player can either move the coin forward by 53 squares or backward by 2 squares. The coin cannot move to a number less than 0 or greater th... | [
"A"
] | [
"Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns, starting with A, and on each turn, a player can either move the coin forward by 53 squares or backward by 2 squares. The coin cannot move to a number less than 0 or great... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-2214b24d4732733ae219f0ab201597c96f8bf853af053c52120bca441215685e | Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns, starting with A, and on each turn, a player can either move the coin forward by 53 squares or backward by 2 squares. The coin cannot move to a position less than 0 or greater ... | [
"Player A has the winning strategy."
] | [
"Two players, A and B, play a game on a row of 2018 squares numbered from 0 to 2017. Initially, a coin is placed on square 0. Players take turns, starting with A, and on each turn, a player can either move the coin forward by 53 squares or backward by 2 squares. The coin cannot move to a position less than 0 or gre... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-4deb48cb1517b18f71ddec47ac08ef95d1a20f26f52f70955016845daba03f0d | Two players, A and B, play a game over $n > 1$ rounds with a fixed integer $d \geq 1$. In the first round, A picks a positive integer $m_1$, and B picks a positive integer $n_1 \neq m_1$. In each subsequent round $k$ (for $k = 2, \ldots, n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$, and B pi... | [
"B always has a winning strategy."
] | [
"Two players, A and B, play a game over $n > 1$ rounds with a fixed integer $d \\geq 1$. In the first round, A picks a positive integer $m_1$, and B picks a positive integer $n_1 \\neq m_1$. In each subsequent round $k$ (for $k = 2, \\ldots, n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \\leq m_{k-1} + d$,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-e39486bdbc90807867c4d87ad63e7ca69129f2a575dfdb246693e0f8d40b99d3 | Two players, A and B, play a game starting with a positive integer \( N \). On each turn, a player can either subtract 1 from the number or, if the number is even, halve the number. A goes first, followed by B, and they alternate turns. The player who reduces the number to 0 wins. Assuming both players play optimally, ... | [
"A can win when the starting number is 1, 3, or any even number greater than 2."
] | [
"Two players, A and B, play a game starting with a positive integer \\( N \\). On each turn, a player can either subtract 1 from the number or, if the number is even, halve the number. A goes first, followed by B, and they alternate turns. The player who reduces the number to 0 wins. Assuming both players play opti... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 19
} | 41 | 0 | 0 |
omr-ws-8e9744ac8aa28f7eaf0422d3f6d69c3d7d4a2f0d4981532e955feb96dc7082ec | Two players, A and B, play a game starting with a positive integer \( N_0 \) on the blackboard. A erases \( N_0 \) and writes \( N_1 \in \{N_0 - 1, \lfloor N_0 / 3 \rfloor\} \). B erases \( N_1 \) and writes \( N_2 \in \{N_1 - 1, \lfloor N_1 / 3 \rfloor\} \). They continue until 0 appears on the blackboard. The player ... | [
"B"
] | [
"Two players, A and B, play a game starting with a positive integer \\( N_0 \\) on the blackboard. A erases \\( N_0 \\) and writes \\( N_1 \\in \\{N_0 - 1, \\lfloor N_0 / 3 \\rfloor\\} \\). B erases \\( N_1 \\) and writes \\( N_2 \\in \\{N_1 - 1, \\lfloor N_1 / 3 \\rfloor\\} \\). They continue until 0 appears on th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9239e97126a704687d816f9c10361aa4f7636ce95f35b6dd9a781d4ee172de59 | Two players, A and B, play a game starting with a positive integer \( N_0 \) on the blackboard. A erases \( N_0 \) and writes \( N_1 \in \{N_0 - 1, \lfloor N_0 / 3 \rfloor\} \). B erases \( N_1 \) and writes \( N_2 \in \{N_1 - 1, \lfloor N_1 / 3 \rfloor\} \). They continue until 0 appears on the blackboard. The player ... | [
"B"
] | [
"Two players, A and B, play a game starting with a positive integer \\( N_0 \\) on the blackboard. A erases \\( N_0 \\) and writes \\( N_1 \\in \\{N_0 - 1, \\lfloor N_0 / 3 \\rfloor\\} \\). B erases \\( N_1 \\) and writes \\( N_2 \\in \\{N_1 - 1, \\lfloor N_1 / 3 \\rfloor\\} \\). They continue until 0 appears on th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-41a4d28db358cdc2577e5af54facabba2d813409ab2bff3b6f0636fd0879eb62 | Two players, A and B, play a game starting with a positive integer \( N_0 \) on the blackboard. A erases \( N_0 \) and writes \( N_1 \in \{N_0 - 1, \lfloor N_0 / 3 \rfloor\} \). B erases \( N_1 \) and writes \( N_2 \in \{N_1 - 1, \lfloor N_1 / 3 \rfloor\} \). They continue until 0 appears on the blackboard. The player ... | [
"A"
] | [
"Two players, A and B, play a game starting with a positive integer \\( N_0 \\) on the blackboard. A erases \\( N_0 \\) and writes \\( N_1 \\in \\{N_0 - 1, \\lfloor N_0 / 3 \\rfloor\\} \\). B erases \\( N_1 \\) and writes \\( N_2 \\in \\{N_1 - 1, \\lfloor N_1 / 3 \\rfloor\\} \\). They continue until 0 appears on th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a2ac153555454b5da72db33221b9ad6cecfe60fe5ae8c870ef2dfe80f3a7d9d6 | Two players, A and B, play a game starting with the number \(2017^{2017}\). On each turn, a player can either subtract an integer between 1 and 2016 (inclusive) or divide the number by 2017, rounding down to the nearest integer. The player who first obtains a non-positive integer wins. Which player has a winning strate... | [
"A"
] | [
"Two players, A and B, play a game starting with the number \\(2017^{2017}\\). On each turn, a player can either subtract an integer between 1 and 2016 (inclusive) or divide the number by 2017, rounding down to the nearest integer. The player who first obtains a non-positive integer wins. Which player has a winning... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-74cc9fe07f1270ad830144ed87dcbee2628f75aa3f596bb4f290a5aa318c51c7 | Two players, A and B, play a game where they repeatedly flip a coin. If the coin lands on heads (with probability \( p \)), B gives A one unit. If the coin lands on tails (with probability \( 1 - p \)), A gives B one unit. Initially, A has \( i \) units and B has \( N - i \) units. What is the probability that A wins (... | [
"\\( \\frac{(1-p)^N - p^{N-i}(1-p)^i}{(1-p)^N - p^N} \\)"
] | [
"Two players, A and B, play a game where they repeatedly flip a coin. If the coin lands on heads (with probability \\( p \\)), B gives A one unit. If the coin lands on tails (with probability \\( 1 - p \\)), A gives B one unit. Initially, A has \\( i \\) units and B has \\( N - i \\) units. What is the probability ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 10
} | 13 | 0 | 0 |
omr-ws-43150b67058b552463675a799bba10d82827a03490ff9d6bae6ea1f93a078ec3 | Two players, A and B, play a game with 101 numbers on a board (1 through 101). In each turn, a player erases 9 numbers. After 11 turns (with A playing 6 turns and B playing 5 turns), two numbers remain. Player A aims to maximize the difference between these two numbers, while player B aims to minimize it. Determine the... | [
"55"
] | [
"Two players, A and B, play a game with 101 numbers on a board (1 through 101). In each turn, a player erases 9 numbers. After 11 turns (with A playing 6 turns and B playing 5 turns), two numbers remain. Player A aims to maximize the difference between these two numbers, while player B aims to minimize it. Determin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 35,
"genselect": 7
} | 42 | 0 | 0 |
omr-ws-e672a631685ea2afdf6347cb833f129fab69eac365911eb883c92832d6d4b109 | Two players, A and B, play a game with \( n \) piles of stones. Player A starts by choosing 2 piles, then player B takes an arbitrary (but nonzero) number of stones from one of the 2 chosen piles. Then player B selects 2 piles, and player A takes some stones from one of them. The player who removes the last stone wins.... | [
"Player A has a winning strategy if there is an even number of smallest piles; otherwise, player B has a winning strategy."
] | [
"Two players, A and B, play a game with \\( n \\) piles of stones. Player A starts by choosing 2 piles, then player B takes an arbitrary (but nonzero) number of stones from one of the 2 chosen piles. Then player B selects 2 piles, and player A takes some stones from one of them. The player who removes the last ston... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60cd4592c58ad327511aaa11ef4dd8d894169245a07308ea71ed9920dd936f7b | Two players, A and B, play a game with a pile of 2006 coins. Players take turns removing between 1 and 7 coins each turn. A player can also choose to pass their turn but must pay 7 coins from their previously collected coins to a separate box. The player who takes the last coin wins, and A starts the game. Determine wh... | [
"Player A can win for sure."
] | [
"Two players, A and B, play a game with a pile of 2006 coins. Players take turns removing between 1 and 7 coins each turn. A player can also choose to pass their turn but must pay 7 coins from their previously collected coins to a separate box. The player who takes the last coin wins, and A starts the game. Determi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 16
} | 25 | 0 | 0 |
omr-ws-573ecb91f31fc462ad9b1c64aa9a4c8203402d7c651d97c6cacfbac239689756 | Two players, A and B, play a game with a regular convex polygon with $2n+1$ edges, where $n > 1$. Player A starts and each player takes turns coloring one uncolored vertex. The player who makes it impossible for the other to form an acute triangle with the remaining uncolored vertices wins. Determine which player has t... | [
"A"
] | [
"Two players, A and B, play a game with a regular convex polygon with $2n+1$ edges, where $n > 1$. Player A starts and each player takes turns coloring one uncolored vertex. The player who makes it impossible for the other to form an acute triangle with the remaining uncolored vertices wins. Determine which player ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-afd65ae7d347298b38b009a2a1750d8c18375ab8cf38e8d88e537424eaec0b89 | Two players, A and B, play a game with five jugs numbered 0 through 4. Jug 0 is empty, and each of the other jugs contains some number of coins. On each turn, a player can move any number of coins (at least one) from any jug to the jug numbered one less. Player A starts. The player who makes the last move, resulting in... | [
"B"
] | [
"Two players, A and B, play a game with five jugs numbered 0 through 4. Jug 0 is empty, and each of the other jugs contains some number of coins. On each turn, a player can move any number of coins (at least one) from any jug to the jug numbered one less. Player A starts. The player who makes the last move, resulti... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-cab12e8689742aa83728aa819834e25f71ad984b6003c4ac4e7dd09ca1b248dd | Two players, A and B, play a game with two piles of 2010 coins each. Players take turns, with A going first. On each turn, a player can take one or more coins from one pile or exactly one coin from each pile. The player who takes the last coin wins. Which player will win if both play optimally? | [
"B wins."
] | [
"Two players, A and B, play a game with two piles of 2010 coins each. Players take turns, with A going first. On each turn, a player can take one or more coins from one pile or exactly one coin from each pile. The player who takes the last coin wins. Which player will win if both play optimally?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-3a597fbba15576e0d8a818127737d6e9e217f04a3637d5742a33e4107a904eca | Two players, A and B, play a game with two piles of stones. In each round, the current player chooses a positive integer \( k \) and removes \( k \) stones from one of the piles (if \( k \) exceeds the number of stones in the chosen pile, the player takes all the stones). The player who takes the last stone loses. Dete... | [
"A has a winning strategy if adding \\(a-1\\) and \\(b-1\\) in binary involves no carry-forwards. The strategy for A is to ensure that after each of their moves, adding the updated pile sizes minus one involves a carry-forward."
] | [
"Two players, A and B, play a game with two piles of stones. In each round, the current player chooses a positive integer \\( k \\) and removes \\( k \\) stones from one of the piles (if \\( k \\) exceeds the number of stones in the chosen pile, the player takes all the stones). The player who takes the last stone ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5780bdea683f8e2174054b87bedd99aa48f8910a798913adb7b241024dc2ff24 | Two players, A and B, start with the number 60 on a board. In each move, a player subtracts any positive divisor (including 1 and the number itself) from the current number and replaces it with the difference. Player A starts first. The player who reaches 0 loses. Determine who will win and provide a winning strategy f... | [
"Player A wins. Strategy: Always subtract an odd divisor."
] | [
"Two players, A and B, start with the number 60 on a board. In each move, a player subtracts any positive divisor (including 1 and the number itself) from the current number and replaces it with the difference. Player A starts first. The player who reaches 0 loses. Determine who will win and provide a winning strat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 13
} | 60 | 0 | 0 |
omr-ws-e8f8ecf499fe26fb98ddc35a3de69a314b5cccb20594f5f5761173d3ad2552d5 | Two players, A and B, take turns filling in the coefficients of the polynomial \( f(X) = X^{10} + ?X^9 + ?X^8 + \ldots + 1 \). If the resulting polynomial \( f(X) \) has no real roots, A wins. If \( f(X) \) has real roots, B wins. A goes first. Determine who has a winning strategy and describe the strategy. | [
"A has a winning strategy."
] | [
"Two players, A and B, take turns filling in the coefficients of the polynomial \\( f(X) = X^{10} + ?X^9 + ?X^8 + \\ldots + 1 \\). If the resulting polynomial \\( f(X) \\) has no real roots, A wins. If \\( f(X) \\) has real roots, B wins. A goes first. Determine who has a winning strategy and describe the strategy.... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-f13ce5e1535c383fcd04063af51241ffb9470d841124b8f8aafb2c8965b09e90 | Two players, A and B, take turns removing chips from a pile that initially contains \( n \) chips. Player A cannot remove all chips on his first turn and must remove at least one chip. Thereafter, a player can remove up to twice the number of chips removed by the opponent on the previous turn. The player who removes th... | [
"n - f"
] | [
"Two players, A and B, take turns removing chips from a pile that initially contains \\( n \\) chips. Player A cannot remove all chips on his first turn and must remove at least one chip. Thereafter, a player can remove up to twice the number of chips removed by the opponent on the previous turn. The player who rem... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-fb91a438dafebea29c9194714859274637c008729f0b7883939804b598d5d17c | Two players, A and B, take turns removing chips from a pile that initially contains \( n \) chips. Player A cannot remove all chips on his first turn and must remove at least one chip. Thereafter, a player cannot remove more chips than the opponent removed on the previous turn. The player who removes the last chip wins... | [
"A wins if \\( n \\) is not a power of 2."
] | [
"Two players, A and B, take turns removing chips from a pile that initially contains \\( n \\) chips. Player A cannot remove all chips on his first turn and must remove at least one chip. Thereafter, a player cannot remove more chips than the opponent removed on the previous turn. The player who removes the last ch... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-1eb89087189f677f6febb41691ddc60eae853a7e6e946771b481ccc0ed27e12f | Two players, A and B, take turns removing stones from a heap with at least 2 stones. Player A starts and can take between 1 and n-1 stones. Each subsequent player must take between 1 and the number of stones the previous player took. The player who takes the last stone wins. Determine which player has a winning strateg... | [
"Player B has a winning strategy if \\( n \\) is a power of 2; otherwise, player A has a winning strategy."
] | [
"Two players, A and B, take turns removing stones from a heap with at least 2 stones. Player A starts and can take between 1 and n-1 stones. Each subsequent player must take between 1 and the number of stones the previous player took. The player who takes the last stone wins. Determine which player has a winning st... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 23
} | 30 | 0 | 0 |
omr-ws-f709db514388b756e1205f95605b63674f5bb3de36bffe0757151d5828b06eb5 | Two players, A and B, take turns writing composite divisors of \( N = p_1 p_2 \cdots p_n \), where \( p_1, p_2, \ldots, p_n \) are distinct primes. The rule is that no two numbers written should be such that one divides the other. Player A starts the game. Determine who can win the game. | [
"A wins."
] | [
"Two players, A and B, take turns writing composite divisors of \\( N = p_1 p_2 \\cdots p_n \\), where \\( p_1, p_2, \\ldots, p_n \\) are distinct primes. The rule is that no two numbers written should be such that one divides the other. Player A starts the game. Determine who can win the game."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-35a86ad7c557b352d1361f596744820ed5a8dca70871700565305dc480b3c244 | Two players, Alice and Bob, play a game on a \( 10 \times 10 \) grid. Initially, all cells are empty. Alice and Bob take turns placing a single coin in an empty cell, with Alice starting first. The game ends when a player cannot make a move. The player who cannot make a move loses. Determine which player has a winning ... | [
"Bob has a winning strategy."
] | [
"Two players, Alice and Bob, play a game on a \\( 10 \\times 10 \\) grid. Initially, all cells are empty. Alice and Bob take turns placing a single coin in an empty cell, with Alice starting first. The game ends when a player cannot make a move. The player who cannot make a move loses. Determine which player has a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-05bbe6ed44d1972e63b7e98fc95f8a4c629edc783cb04c164afda7f7d6cfe7d9 | Two players, Alice and Bob, play a game starting with the number $10^{2018}$ on a blackboard. The players take turns performing one of two actions: 1. Divide a number $x$ on the board into integers $a$ and $b$ such that $ab = x$. 2. If a number $x$ occurs twice, erase one or both instances. The player who cannot make a... | [
"Alice has a winning strategy."
] | [
"Two players, Alice and Bob, play a game starting with the number $10^{2018}$ on a blackboard. The players take turns performing one of two actions:\n1. Divide a number $x$ on the board into integers $a$ and $b$ such that $ab = x$.\n2. If a number $x$ occurs twice, erase one or both instances.\n\nThe player who can... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-3cb868c5bb12f507c8ffc71d61b346640bcfe6071e4586656d58eafc2c04e4b7 | Two players, Ana and Ben, are playing a game on a 4x4 grid of points with integer coordinates $(m,n)$ where $1 \leq m, n \leq 4$. Ana starts by coloring one point red, then Ben colors one of the remaining points yellow, and they alternate turns. The first player to create a rectangle with vertices of the same color and... | [
"Ben"
] | [
"Two players, Ana and Ben, are playing a game on a 4x4 grid of points with integer coordinates $(m,n)$ where $1 \\leq m, n \\leq 4$. Ana starts by coloring one point red, then Ben colors one of the remaining points yellow, and they alternate turns. The first player to create a rectangle with vertices of the same co... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-b599b72e7b2338ec16138020aae3d50d700f0417f7e201c66f99a92c215e48d9 | Two players, Ana and Bogdan, play a game with 2017 pebbles on a table. Ana starts by taking one pebble. On each subsequent turn, the player can take between 1 and \(2^{(j-1)}\) pebbles, where \(j\) is the turn number. The player who takes the last pebble wins. Determine which player has a winning strategy. | [
"Ana"
] | [
"Two players, Ana and Bogdan, play a game with 2017 pebbles on a table. Ana starts by taking one pebble. On each subsequent turn, the player can take between 1 and \\(2^{(j-1)}\\) pebbles, where \\(j\\) is the turn number. The player who takes the last pebble wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-b4c4bcd4f7d03dcffca5eb23a436c89a4bc54d83c091755c9d028d00fb6599d1 | Two players, Andria and Bob, are playing a game with two piles of stones containing $x$ and $y$ stones respectively. Andria moves first. On each turn, a player can remove $k$ stones from one pile, where $k$ must be divisible by the number of stones in the other pile. The player who takes the last stone wins. Find a win... | [
"\\text{The first player wins if the number of steps in the Euclidean algorithm for gcd}(x, y) \\text{ is odd.}"
] | [
"Two players, Andria and Bob, are playing a game with two piles of stones containing $x$ and $y$ stones respectively. Andria moves first. On each turn, a player can remove $k$ stones from one pile, where $k$ must be divisible by the number of stones in the other pile. The player who takes the last stone wins. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-49058cefc89892e25b2de3f5ee24d5e0edb6a4fa0d8715d35cad9be37456f6c1 | Two players, Andriy and Olesya, play a game with a round cake cut into $2n$ ($n>1$) sectors of distinct weights. Olesya takes 1 piece first, then Andriy takes 2 pieces such that the remaining pieces form a sector. They continue taking 2 pieces each turn, ensuring the remaining pieces form a sector, until one player tak... | [
"\\( n \\neq 2, 4 \\)"
] | [
"Two players, Andriy and Olesya, play a game with a round cake cut into $2n$ ($n>1$) sectors of distinct weights. Olesya takes 1 piece first, then Andriy takes 2 pieces such that the remaining pieces form a sector. They continue taking 2 pieces each turn, ensuring the remaining pieces form a sector, until one playe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5e5ac946821ee8820cf101b46cf56c9417e377a75875e615fd84fcfe1ded9638 | Two players, Anne and Berit, play a game starting with the number $10^{2007}$ on the blackboard. On each turn, a player can either replace a number $x$ with two integers $a, b > 1$ such that $ab = x$, or strike off one or both of two equal numbers on the blackboard. The player who cannot perform any operation loses. Wh... | [
"Anne has the winning strategy."
] | [
"Two players, Anne and Berit, play a game starting with the number $10^{2007}$ on the blackboard. On each turn, a player can either replace a number $x$ with two integers $a, b > 1$ such that $ab = x$, or strike off one or both of two equal numbers on the blackboard. The player who cannot perform any operation lose... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 8
} | 15 | 0 | 0 |
omr-ws-64fddcc5c79c2198dc31e6ae97e7e47472071ce2ed629a31feeee9721bd16cb0 | Two players, Arnaldo and Betania, play alternately with Arnaldo starting first. Initially, there are two piles of stones containing $x$ and $y$ stones respectively. In each turn, a player can either: 1. Choose two non-empty piles and take one stone from each pile. 2. Choose a pile with an odd amount of stones, take one... | [
"Arnaldo wins if $x \\equiv y \\pmod{2}$ and both are odd, or if $x \\equiv 1 \\pmod{4}$ and $y$ is even, or if $x \\equiv 3 \\pmod{4}$ and $y \\equiv 0 \\pmod{4}$. Betania wins if $x \\equiv y \\pmod{2}$ and both are even, or if $x \\equiv 3 \\pmod{4}$ and $y \\equiv 2 \\pmod{4}$."
] | [
"Two players, Arnaldo and Betania, play alternately with Arnaldo starting first. Initially, there are two piles of stones containing $x$ and $y$ stones respectively. In each turn, a player can either:\n1. Choose two non-empty piles and take one stone from each pile.\n2. Choose a pile with an odd amount of stones, t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-873f0f1731f8c51372cb4fdb0184869643ffa18890ccad365c1aa431804af3db | Two players, Arnould and Betty, play alternately with Arnould starting first. Initially, there are two piles of stones containing \( x \) and \( y \) stones, respectively. In each move, a player can: 1. Choose two non-empty piles and take one stone from each. 2. Choose a pile with an odd number of stones, take one ston... | [
"Arnould"
] | [
"Two players, Arnould and Betty, play alternately with Arnould starting first. Initially, there are two piles of stones containing \\( x \\) and \\( y \\) stones, respectively. In each move, a player can:\n1. Choose two non-empty piles and take one stone from each.\n2. Choose a pile with an odd number of stones, ta... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-83341a85aa865acc21881a18fd29cb60fccd2528b83f4f9d36350074ee8e44bf | Two players, Aryan and Anurag, alternately move a marble on a coordinate plane starting from the origin \(O(0,0)\). The marble must stay within the circle with diameter \(OB\), where \(B\) is at \((5,0)\). On each turn, the marble must be moved to a new position \((c,d)\) such that the distance from the current positio... | [
"Aryan has the winning strategy."
] | [
"Two players, Aryan and Anurag, alternately move a marble on a coordinate plane starting from the origin \\(O(0,0)\\). The marble must stay within the circle with diameter \\(OB\\), where \\(B\\) is at \\((5,0)\\). On each turn, the marble must be moved to a new position \\((c,d)\\) such that the distance from the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 8
} | 20 | 0 | 0 |
omr-ws-7ae0b50c3aac4e733a46d24214cd2c350bfddf77589a80751bde164d7f43ef7b | Two players, Diego and Pablo, play a game with a ladder sequence $1, 2, \dots, 2020$. On each turn, a player replaces an entry $a_i$ with a smaller non-negative integer $a_i'$ such that the sequence remains non-decreasing. The player who first achieves the sequence $(0, 0, \dots, 0)$ wins. Determine which player has a ... | [
"Pablo has a winning strategy."
] | [
"Two players, Diego and Pablo, play a game with a ladder sequence $1, 2, \\dots, 2020$. On each turn, a player replaces an entry $a_i$ with a smaller non-negative integer $a_i'$ such that the sequence remains non-decreasing. The player who first achieves the sequence $(0, 0, \\dots, 0)$ wins. Determine which player... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 5,
"tir": 11
} | 35 | 0 | 0 |
omr-ws-ffc37bfc19c38694edf300a03ee7ef212d301ee285228ff817c1514339ec8e54 | Two players, F and S, are playing a game with a fair coin. Player F wins if there are two tails in a row, and player S wins if there are three heads in a row. What is the probability that player F will win the game? | [
"\\(\\frac{7}{10}\\)",
"\\dfrac{7}{10}"
] | [
"Two players, F and S, are playing a game with a fair coin. Player F wins if there are two tails in a row, and player S wins if there are three heads in a row. What is the probability that player F will win the game?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 22
} | 69 | 0 | 0 |
omr-ws-12b0cced15282ac258f3e6e0023393674ab2680d803566390103561ca175a777 | Two players, Jacob and David, play a game in a convex polygon with $n \geq 5$ sides. On each turn, they draw a diagonal which does not intersect any previously drawn diagonal. The player that creates a quadrilateral (with no diagonal yet drawn) loses. If Jacob plays first, for which values of $n$ can he win? | [
"\\( n \\) is even."
] | [
"Two players, Jacob and David, play a game in a convex polygon with $n \\geq 5$ sides. On each turn, they draw a diagonal which does not intersect any previously drawn diagonal. The player that creates a quadrilateral (with no diagonal yet drawn) loses. If Jacob plays first, for which values of $n$ can he win?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 3,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-9e4588262265aae52db9d4a3f65f934c508d882b2dd21d8802bbce808457bdab | Two players, Mathinity and Golden Math, take turns filling in the blanks with real numbers in the polynomial \(x^{69} + \_\_\_\_ x^{68} + \_\_\_\_ x^{67} + \ldots + \_\_\_\_ x + \_\_\_\_\). Mathinity goes first. If the resulting polynomial has at least 3 distinct real roots, Mathinity wins; otherwise, Golden Math wins.... | [
"Mathinity"
] | [
"Two players, Mathinity and Golden Math, take turns filling in the blanks with real numbers in the polynomial \\(x^{69} + \\_\\_\\_\\_ x^{68} + \\_\\_\\_\\_ x^{67} + \\ldots + \\_\\_\\_\\_ x + \\_\\_\\_\\_\\). Mathinity goes first. If the resulting polynomial has at least 3 distinct real roots, Mathinity wins; othe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-f68e181d74350346a4f05410ce29cd86bbe663da2fd621bd1d782a27305ecf5c | Two players, Red and Blue, play on a 10x10 board in alternating turns. Blue goes first. In each turn, a player picks an unchosen row or column and colors all its squares with their own color, overwriting any previous colors. The game ends after 20 turns when all rows and columns are chosen. Red wins if the number of re... | [
"Red"
] | [
"Two players, Red and Blue, play on a 10x10 board in alternating turns. Blue goes first. In each turn, a player picks an unchosen row or column and colors all its squares with their own color, overwriting any previous colors. The game ends after 20 turns when all rows and columns are chosen. Red wins if the number ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-e574d244a3d253e8e62736a7e3028845abb04ccebf311367949d0dabd6221b82 | Two players, S and G, play a game on a $2017 \times n$ board. The first player, S, picks a square of size $k \times k$ and colors it. The next player, G, then picks another uncolored $k \times k$ square and colors it. They take turns coloring squares until one player cannot make a move, at which point the other player ... | [
"All integers \\( n \\geq 2017 \\)"
] | [
"Two players, S and G, play a game on a $2017 \\times n$ board. The first player, S, picks a square of size $k \\times k$ and colors it. The next player, G, then picks another uncolored $k \\times k$ square and colors it. They take turns coloring squares until one player cannot make a move, at which point the other... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-9052814422dd81b383369eefc11086c17401e0216503a972ecaaf8486506ad77 | Two players, SB and RH, play a game starting with the number \(10^{2007}\) on the board. The players take turns performing one of the following actions: 1. Choose a number \(x\) on the board and replace it with two numbers \(a\) and \(b\) such that \(ab = x\). 2. If there are two identical numbers on the board, erase o... | [
"SB wins."
] | [
"Two players, SB and RH, play a game starting with the number \\(10^{2007}\\) on the board. The players take turns performing one of the following actions:\n1. Choose a number \\(x\\) on the board and replace it with two numbers \\(a\\) and \\(b\\) such that \\(ab = x\\).\n2. If there are two identical numbers on t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 1
} | 6 | 0 | 0 |
omr-ws-c8447cffc479d0b7ccc41debd73ccd89ea05e50adc2cc3d6c454921f1ef2807a | Two players, Tom and Sid, play a game with two piles of rocks. The first pile has 10 rocks, and the second pile has 12 rocks. Each player, on their turn, can take any amount of rocks from one pile or the same amount of rocks from both piles. The player who takes the last rock wins. Who wins if Tom starts the game? | [
"Tom wins."
] | [
"Two players, Tom and Sid, play a game with two piles of rocks. The first pile has 10 rocks, and the second pile has 12 rocks. Each player, on their turn, can take any amount of rocks from one pile or the same amount of rocks from both piles. The player who takes the last rock wins. Who wins if Tom starts the game?... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-b09d67b2672433b5bbe7d555b732540288d8bb41c63df47ce2dd3e86fd0287de | Two players, Wester Olfram and Peter Enis, are playing a game. Initially, there are $2021$ times the integer $69$ written on a blackboard. Wester begins the game and the players move alternately. The player who is currently making a move chooses one of the non-zero numbers on the board. If this number $n$ is the smalle... | [
"Wester Olfram"
] | [
"Two players, Wester Olfram and Peter Enis, are playing a game. Initially, there are $2021$ times the integer $69$ written on a blackboard. Wester begins the game and the players move alternately. The player who is currently making a move chooses one of the non-zero numbers on the board. If this number $n$ is the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-34b680a23bb53977f23173b97bccf85423f3d541263047023981a7814654595a | Two players, X and Y, play a game consisting of multiple matches. The probability that X wins a match is $\frac{1}{2}$, the probability that Y wins a match is $\frac{1}{4}$, and the probability of a draw is $\frac{1}{4}$. They continue playing until either X or Y wins two consecutive matches. Let $P(X)$ be the probabil... | [
"\\( a = \\frac{16}{21} \\), \\( b = \\frac{5}{21} \\)"
] | [
"Two players, X and Y, play a game consisting of multiple matches. The probability that X wins a match is $\\frac{1}{2}$, the probability that Y wins a match is $\\frac{1}{4}$, and the probability of a draw is $\\frac{1}{4}$. They continue playing until either X or Y wins two consecutive matches. Let $P(X)$ be the ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-159f0d0a71dde1a069bffa50f5e596a1fbffe40e4363c8bf161a2114d47617e1 | Two players, \( A \) and \( B \), play a game on a circle with 10 cells. Player \( A \) starts, and they take turns choosing a free cell and writing either \( O \) or \( M \) in it. The player who first creates the sequence \( OMO \) in three consecutive cells wins. If no such sequence is formed, the game is a draw. As... | [
"0"
] | [
"Two players, \\( A \\) and \\( B \\), play a game on a circle with 10 cells. Player \\( A \\) starts, and they take turns choosing a free cell and writing either \\( O \\) or \\( M \\) in it. The player who first creates the sequence \\( OMO \\) in three consecutive cells wins. If no such sequence is formed, the g... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-b513f8a36adde523d6936c1dd0340e7ec4bb4d3144b89915ad11850b9f3c3c80 | Two players, \(A\) (first player) and \(B\), take alternate turns in playing a game using 2016 chips. On each turn, a player must remove \(s\) chips from the remaining pile, where \(s \in \{ 2, 4, 5 \}\). The player who is unable to make a move loses the game. Determine which player can force a win. | [
"Player \\(B\\) can force a win."
] | [
"Two players, \\(A\\) (first player) and \\(B\\), take alternate turns in playing a game using 2016 chips. On each turn, a player must remove \\(s\\) chips from the remaining pile, where \\(s \\in \\{ 2, 4, 5 \\}\\). The player who is unable to make a move loses the game. Determine which player can force a win."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-ec4b264f33ab319590869b18adae1d0d105ac0d9bc9ed9797292e76a28cf7cf7 | Two players, \(A\) and \(B\), play a game starting with the number \(10^{2007}\) on the blackboard. On each turn, a player can either: - Replace a number \(x\) on the blackboard by two integers \(a\) and \(b\) greater than 1 such that \(ab = x\), or - Strike off one or both of two equal numbers on the blackboard. The p... | [
"A will win."
] | [
"Two players, \\(A\\) and \\(B\\), play a game starting with the number \\(10^{2007}\\) on the blackboard. On each turn, a player can either:\n- Replace a number \\(x\\) on the blackboard by two integers \\(a\\) and \\(b\\) greater than 1 such that \\(ab = x\\), or\n- Strike off one or both of two equal numbers on ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-93bde43ef2843818718b4ca7bf590253536694940f4f806e5796e60ed96cd17f | Two players, \(A\) and \(B\), play a game with 400 numbers written on a board. On each turn, a player erases three numbers that can form the sides of a triangle. Player \(A\) goes first. The player who cannot make a move loses. Determine which player has a winning strategy. | [
"A"
] | [
"Two players, \\(A\\) and \\(B\\), play a game with 400 numbers written on a board. On each turn, a player erases three numbers that can form the sides of a triangle. Player \\(A\\) goes first. The player who cannot make a move loses. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-632d6056f6d088c844f4402cffd43e5d5d159d2b990c1285195d9fae3663f723 | Two players, \(A\) and \(B\), play a game with \(2n\) cards. Each card has a positive integer written on it. The cards are shuffled and laid out in a row with the numbers facing upwards. \(A\) starts and the players take turns to remove one card from either end of the row until \(B\) picks up the remaining card. Each p... | [
"dp[0][2n-1]"
] | [
"Two players, \\(A\\) and \\(B\\), play a game with \\(2n\\) cards. Each card has a positive integer written on it. The cards are shuffled and laid out in a row with the numbers facing upwards. \\(A\\) starts and the players take turns to remove one card from either end of the row until \\(B\\) picks up the remaini... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-273e54d37fc004a36b699972144eff5c8a41c4488f2cfeafe5dfe6a6373719e5 | Two players, \(A\) and \(B\), play a game with seven cards numbered \(1, 2, 3, 4, 5, 6, 7\). Player \(A\) picks a card first, writes the number on the card down, and returns it to the deck. If the sum of the numbers written is a multiple of 3, the game ends and the player who went last wins. Compute the probability tha... | [
"\\(\\frac{43}{75}\\)"
] | [
"Two players, \\(A\\) and \\(B\\), play a game with seven cards numbered \\(1, 2, 3, 4, 5, 6, 7\\). Player \\(A\\) picks a card first, writes the number on the card down, and returns it to the deck. If the sum of the numbers written is a multiple of 3, the game ends and the player who went last wins. Compute the pr... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 23
} | 43 | 0 | 0 |
omr-ws-a0b222f099a8d6c5f0125047867541e4a23a88ba336593da98a8c1ef2467931d | Two players, \(A\) and \(B\), play a game. At the beginning there is the number \(2\). \(A\) goes first. In each turn, the player chooses a proper divisor of the current number, and adds it to the current number. The first player who, after his/her turn, makes the number exceed a certain integer \(k\), loses. Suppose t... | [
"A wins for both \\( k = 2011^{2011} \\) and \\( k = 2012^{2012} \\)."
] | [
"Two players, \\(A\\) and \\(B\\), play a game. At the beginning there is the number \\(2\\). \\(A\\) goes first. In each turn, the player chooses a proper divisor of the current number, and adds it to the current number. The first player who, after his/her turn, makes the number exceed a certain integer \\(k\\), l... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 11
} | 18 | 0 | 0 |
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