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-6eb1683eb9c7497cc2f48e136261934a28a744b70dcd5cf9f532e409cc03a08d | Ali and Naqi are playing a game with the polynomial \( P(x) = 1 + x^{1398} \). Naqi starts, and in each turn, a player can choose a natural number \( k \in [0, 1398] \) and add \( x^k \) to the polynomial. Ali's goal is to ensure that \( P(x) \geq 0 \) for all \( x \in \mathbb{R} \). Determine the maximum number of ter... | [
"699"
] | [
"Ali and Naqi are playing a game with the polynomial \\( P(x) = 1 + x^{1398} \\). Naqi starts, and in each turn, a player can choose a natural number \\( k \\in [0, 1398] \\) and add \\( x^k \\) to the polynomial. Ali's goal is to ensure that \\( P(x) \\geq 0 \\) for all \\( x \\in \\mathbb{R} \\). Determine the ma... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6d0242339967fa728a572568e3fa3a0c2a103519d2cb4fd1f25bbdc30a24426f | Ali chooses one of the stones from a group of 2005 stones, marks it, and shuffles the stones. At each move, Betül divides the stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have an equal number of stones, Ali remov... | [
"11"
] | [
"Ali chooses one of the stones from a group of 2005 stones, marks it, and shuffles the stones. At each move, Betül divides the stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have an equal number of stones, Ali ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 5
} | 14 | 0 | 0 |
omr-ws-186a0649e4f3fae62764161240925c5dd8e1b2669cf928f6767d64461c8547e5 | Ali has $2023$ cube-shaped blocks, each with numbers $1$, $2$, and $-3$ on three faces, and each pair of opposite faces has equal numbers. Ali builds a snake by gluing blocks together such that glued faces contain equal numbers. When the snake is finished, the sum of the numbers of all the visible faces is $4046$. Dete... | [
"1012"
] | [
"Ali has $2023$ cube-shaped blocks, each with numbers $1$, $2$, and $-3$ on three faces, and each pair of opposite faces has equal numbers. Ali builds a snake by gluing blocks together such that glued faces contain equal numbers. When the snake is finished, the sum of the numbers of all the visible faces is $4046$.... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-52fcebd9e9a2941f6e403728235694321fef930d982024c4c3d4cc9dd8df9dbf | Ali has 100 cards numbered from 1 to 100. In each step, Ali chooses a card, and Amin decides to either pick it or throw it away. If Amin picks a card, he must throw away the next card chosen by Ali. This process continues until all cards are gone. Amin aims to maximize the sum of the numbers on his cards, while Ali aim... | [
"1717"
] | [
"Ali has 100 cards numbered from 1 to 100. In each step, Ali chooses a card, and Amin decides to either pick it or throw it away. If Amin picks a card, he must throw away the next card chosen by Ali. This process continues until all cards are gone. Amin aims to maximize the sum of the numbers on his cards, while Al... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22d502f045404510a77586e3be94f35754ec58d7ade2eb4d8308faf4a2a4185b | Ali has a collection of 177 snap-together building blocks. Ali has one fourth as many red blocks as blue blocks, half as many white blocks as black blocks, an equal number of black and blue blocks, and 34 miscellaneous blocks of other colors. How many red blocks does Ali have? | [
"13"
] | [
"Ali has a collection of 177 snap-together building blocks. Ali has one fourth as many red blocks as blue blocks, half as many white blocks as black blocks, an equal number of black and blue blocks, and 34 miscellaneous blocks of other colors. How many red blocks does Ali have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fc7949c2f8967451fb641e6a13cf03a74d24c0d7ddd3985bb6a261d14465c46 | Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rock... | [
"18751"
] | [
"Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a \"rocket.\" If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-31b810c366b309b03e0b35b3e0261f47648d4466e41ba07eb9cfbac92a942126 | Ali is situated in the middle square of a $3 \times 3$ square grid. He moves based on the following rules: - If Ali is on the middle square or one of the corner squares, he moves to a horizontally or vertically adjacent square, each with equal probability. - If Ali is on a square horizontally or vertically adjacent to ... | [
"\\(\\frac{1}{2}\\)"
] | [
"Ali is situated in the middle square of a $3 \\times 3$ square grid. He moves based on the following rules:\n- If Ali is on the middle square or one of the corner squares, he moves to a horizontally or vertically adjacent square, each with equal probability.\n- If Ali is on a square horizontally or vertically adja... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-37e28f55b7f694597e7a6c63bca312dd990eb13447db8f02d77917f9e83f4938 | Ali places 5 points on a plane such that no three are collinear. Ramtin then adds a sixth point (also not collinear with any two of the former points). Ali must construct two triangles from these six points where one triangle can be placed inside the other. Let $A$, $B$, and $C$ be the vertices of the outer triangle an... | [
"4"
] | [
"Ali places 5 points on a plane such that no three are collinear. Ramtin then adds a sixth point (also not collinear with any two of the former points). Ali must construct two triangles from these six points where one triangle can be placed inside the other. Let $A$, $B$, and $C$ be the vertices of the outer triang... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-914dce3a061fef2c2abae8e019141b3f9f3e347aa74068aa2cf1d18664e172d7 | Ali presents 7 forbidden $2 \times 2$ tiles. Mohammad attempts to color the cells of an infinite table in black or white to avoid using any of the forbidden tiles. Let $N$ be the number of distinct $2 \times 2$ tiles that Mohammad can use without including any of the forbidden ones. Find $N$. | [
"6"
] | [
"Ali presents 7 forbidden $2 \\times 2$ tiles. Mohammad attempts to color the cells of an infinite table in black or white to avoid using any of the forbidden tiles. Let $N$ be the number of distinct $2 \\times 2$ tiles that Mohammad can use without including any of the forbidden ones. Find $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-cd4da1b8f3c396b9c6c678d5643e1c420dbc5e102e6675cc6639855457751a05 | Ali-Baba and a thief divide a treasure consisting of 100 golden coins split into 10 piles of 10 coins each. Ali-Baba chooses 4 piles, places a mug beside each pile, and puts several coins (not less than 1, but not the whole pile) from the respective pile into each mug. The thief then rearranges the mugs by altering the... | [
"27"
] | [
"Ali-Baba and a thief divide a treasure consisting of 100 golden coins split into 10 piles of 10 coins each. Ali-Baba chooses 4 piles, places a mug beside each pile, and puts several coins (not less than 1, but not the whole pile) from the respective pile into each mug. The thief then rearranges the mugs by alterin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5bef9685c1e37777255f094bcf80b5a27a5e3113916dd6a843133959f4e37136 | Alice and Barbara order a pizza and make three straight cuts through an arbitrary point P, different from the center, such that the cuts pairwise intersect at 60° angles, dividing the pizza into 6 pieces. The center of the pizza is not on the cuts. Alice chooses one piece first, and then the pieces are taken clockwise ... | [
"1"
] | [
"Alice and Barbara order a pizza and make three straight cuts through an arbitrary point P, different from the center, such that the cuts pairwise intersect at 60° angles, dividing the pizza into 6 pieces. The center of the pizza is not on the cuts. Alice chooses one piece first, and then the pieces are taken clock... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-ed86866c7910d09ec676f5a71f158a7e2dbc07b7af9cc2c678ae87b5e3201a6e | Alice and Barbara play a game with a pack of 2n cards, each with a positive integer written on it. The cards are laid out in a row, and the girls take turns removing one card from either end of the row, with Alice starting and Barbara picking up the final card. Each girl's score is the sum of the numbers on her chosen ... | [
"30"
] | [
"Alice and Barbara play a game with a pack of 2n cards, each with a positive integer written on it. The cards are laid out in a row, and the girls take turns removing one card from either end of the row, with Alice starting and Barbara picking up the final card. Each girl's score is the sum of the numbers on her ch... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-b8695b2d0567406ee0ec99789863af0f2f67cb2c129bd127e1d7eff3a9462fd5 | Alice and Barbara play a game with a pack of \(2n\) cards, each with a positive integer written on it. The cards are shuffled and laid out in a row, with the numbers facing upwards. Alice starts, and the girls take turns to remove one card from either end of the row, until Barbara picks up the final card. Each girl's s... | [
"0"
] | [
"Alice and Barbara play a game with a pack of \\(2n\\) cards, each with a positive integer written on it. The cards are shuffled and laid out in a row, with the numbers facing upwards. Alice starts, and the girls take turns to remove one card from either end of the row, until Barbara picks up the final card. Each g... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-345a529eb03e547921de2d99c26f52f8dfd778dcb875cd8a0e184bb2be4d9d47 | Alice and Bernard are playing a game. Alice writes on a blackboard all positive integers from 1 to $N$ inclusive. Each turn, Bernard picks two numbers $x$ and $y$ from the blackboard and performs one of the following operations on $x$ and $y$: (i) $x + y$ (ii) $x + 8y$ (iii) $x - 13y$ He then erases $x$ and $y$ from th... | [
"N \\equiv 0 \\text{ or } 6 \\pmod{7}"
] | [
"Alice and Bernard are playing a game. Alice writes on a blackboard all positive integers from 1 to $N$ inclusive. Each turn, Bernard picks two numbers $x$ and $y$ from the blackboard and performs one of the following operations on $x$ and $y$:\n(i) $x + y$\n(ii) $x + 8y$\n(iii) $x - 13y$\nHe then erases $x$ and $y... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-e31abf781db32df268ff5206224ac62459c4a73e6a564552d796f45836e250e0 | Alice and Bernard play a game with a positive integer \( n \). Starting from an integer \( X \), Alice subtracts an integer \( k \)-th power of \( n \) (possibly \( 0 \)) less than or equal to \( X \). The resulting number \( Y = X - n^k \) is then given to Bernard, who repeats the process. The game continues, and the ... | [
"10924"
] | [
"Alice and Bernard play a game with a positive integer \\( n \\). Starting from an integer \\( X \\), Alice subtracts an integer \\( k \\)-th power of \\( n \\) (possibly \\( 0 \\)) less than or equal to \\( X \\). The resulting number \\( Y = X - n^k \\) is then given to Bernard, who repeats the process. The game ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-6c25406461eadd2e3f9dacf42e69b4ca147416a810824302d45616462c7b5df4 | Alice and Betty are playing a game. First, they pick two positive integers \(k\) and \(n\). They write down the number \(n\) on a whiteboard. On a player's turn, she may replace \(n\) with the number \(n - k^m\) where \(m\) is a nonnegative integer and \(n \ge k^m\). A player loses if she cannot make a move. Let Alice ... | [
"\\(n\\) is a multiple of \\(k\\)."
] | [
"Alice and Betty are playing a game. First, they pick two positive integers \\(k\\) and \\(n\\). They write down the number \\(n\\) on a whiteboard. On a player's turn, she may replace \\(n\\) with the number \\(n - k^m\\) where \\(m\\) is a nonnegative integer and \\(n \\ge k^m\\). A player loses if she cannot mak... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f95e01bbe49cc5c15fce3b79e50450b2804f2fa8af52995e975cb162a01c9a5a | Alice and Bianca have one hundred marbles. At the start of the game, they split these hundred marbles into two piles. A move consists of choosing a pile, then choosing a positive integer not larger than half of the number of marbles in that pile, and finally removing that number of marbles from the chosen pile. The fir... | [
"(50,50), (67,33), (33,67), (95,5), (5,95)"
] | [
"Alice and Bianca have one hundred marbles. At the start of the game, they split these hundred marbles into two piles. A move consists of choosing a pile, then choosing a positive integer not larger than half of the number of marbles in that pile, and finally removing that number of marbles from the chosen pile. Th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 14
} | 25 | 0 | 0 |
omr-ws-61050818d7938e6fe4c5700312d1d5cdf830ccd5a90ac4ebe3793557668149e9 | Alice and Bill are walking in opposite directions along the same route between points A and B. Alice is going from A to B, and Bill from B to A. They start at the same time and pass each other 3 hours later. Alice arrives at B 2.5 hours before Bill arrives at A. How many hours does it take for Bill to go from B to A? | [
"7.5"
] | [
"Alice and Bill are walking in opposite directions along the same route between points A and B. Alice is going from A to B, and Bill from B to A. They start at the same time and pass each other 3 hours later. Alice arrives at B 2.5 hours before Bill arrives at A. How many hours does it take for Bill to go from B to... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-d7595fc972cd6862ee9e0bb1bf32ac08e4cdbcfe7ed6fa9a82de5b820ace4c89 | Alice and Billy are playing a game on a number line starting at $0$. Each turn, Alice has a $\frac{1}{2}$ chance of moving $1$ unit in the positive direction and a $\frac{1}{2}$ chance of moving $1$ unit in the negative direction. Billy has a $\frac{2}{3}$ chance of moving $1$ unit in the positive direction and a $\fra... | [
"\\(\\frac{23}{32}\\)"
] | [
"Alice and Billy are playing a game on a number line starting at $0$. Each turn, Alice has a $\\frac{1}{2}$ chance of moving $1$ unit in the positive direction and a $\\frac{1}{2}$ chance of moving $1$ unit in the negative direction. Billy has a $\\frac{2}{3}$ chance of moving $1$ unit in the positive direction and... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bf64d7812c0823e6c62f4c3359641f4dfe430d3b7b02df0da24cf20daa59a6d | Alice and Bob alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits. The resulting seven-digit number is formed by concatenating the digits in the order selected, with the seventh digit appearing last (i.e., $\overline{A_1B_2A_3B_4A_5B_6A_7}$). Alice wins if and only if ... | [
"Alice"
] | [
"Alice and Bob alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits. The resulting seven-digit number is formed by concatenating the digits in the order selected, with the seventh digit appearing last (i.e., $\\overline{A_1B_2A_3B_4A_5B_6A_7}$). Alice wins if and on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 19
} | 39 | 0 | 0 |
omr-ws-4c890196845028656fc7fdfa70cd1a5272f4d049b49bdcf48d82af061c378eec | Alice and Bob alternately flip 2 coins. The winner is the first one to flip two heads. If Alice begins, what is the probability that Alice wins? | [
"\\(\\frac{16}{27}\\)"
] | [
"Alice and Bob alternately flip 2 coins. The winner is the first one to flip two heads. If Alice begins, what is the probability that Alice wins?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-68c762309b00d04b22156deff1d85061a0cf71a209b39b3b8682871b48531b7f | Alice and Bob are both magically invisible. Alice is trying to find Bob who is randomly placed on a vertex of a $2 \times 2$ grid. Every second Alice guesses where Bob is; if incorrect, Bob moves to a random adjacent vertex and pings his previous location to Alice. If Alice plays optimally, what is the expected number ... | [
"\\dfrac{3}{2}"
] | [
"Alice and Bob are both magically invisible. Alice is trying to find Bob who is randomly placed on a vertex of a $2 \\times 2$ grid. Every second Alice guesses where Bob is; if incorrect, Bob moves to a random adjacent vertex and pings his previous location to Alice. If Alice plays optimally, what is the expected n... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-b7953319914a6b92baa48d6cbc86d79ee4c86be742b1bd53a64a312b8e549188 | Alice and Bob are in a hardware store. The store sells colored sleeves that fit over keys to distinguish them. The following conversation takes place: A: Are you going to cover your keys? B: I would like to, but there are only $7$ colors and I have $8$ keys. A: Yes, but you could always distinguish a key by noticing th... | [
"3"
] | [
"Alice and Bob are in a hardware store. The store sells colored sleeves that fit over keys to distinguish them. The following conversation takes place:\nA: Are you going to cover your keys?\nB: I would like to, but there are only $7$ colors and I have $8$ keys.\nA: Yes, but you could always distinguish a key by not... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-8d69de4473cfda9a3d19f9982804a8e2d52f221d1cc991ec94ed5a3720af65f3 | Alice and Bob are on a coordinate plane at points $A(1, 0)$ and $B(-1, 0)$, respectively. They each pick a direction with uniform random probability and travel at speed $1 \frac{unit}{min}$ in their chosen directions. They each have a $1$-unit radius field of view. What is the probability that they will ever see each o... | [
"\\dfrac{1}{3}"
] | [
"Alice and Bob are on a coordinate plane at points $A(1, 0)$ and $B(-1, 0)$, respectively. They each pick a direction with uniform random probability and travel at speed $1 \\frac{unit}{min}$ in their chosen directions. They each have a $1$-unit radius field of view. What is the probability that they will ever see ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-dcda8b6d2e894b7a9a0e7fd5c4e7c9381b6322b52d0d4fabe89bdd58dc39b89d | Alice and Bob are playing a "factoring game" with the number $270000 (= 2^4 \cdot 3^3 \cdot 5^4)$. Each player picks a number $N$ from the paper, erases $N$, and writes two integers $X$ and $Y$ such that $N = XY$ and $\text{gcd}(X, Y) \neq 1$. Alice goes first, and the player who can no longer make such a factorization... | [
"3"
] | [
"Alice and Bob are playing a \"factoring game\" with the number $270000 (= 2^4 \\cdot 3^3 \\cdot 5^4)$. Each player picks a number $N$ from the paper, erases $N$, and writes two integers $X$ and $Y$ such that $N = XY$ and $\\text{gcd}(X, Y) \\neq 1$. Alice goes first, and the player who can no longer make such a fa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-5cce3e95429605dedc17fbde5d53fc34ca92dd8eba92dc31e5511a0ec7bacb1c | Alice and Bob are playing a game called 'knights of the toroidal table' on a five-by-five square 'chessboard'. Each has a knight, which begins in opposite corners of the board. On each turn, the knight can move as a regular knight, that is, in an L-shape of two steps in one direction, and one step perpendicularly. Howe... | [
"Bob has the winning strategy."
] | [
"Alice and Bob are playing a game called 'knights of the toroidal table' on a five-by-five square 'chessboard'. Each has a knight, which begins in opposite corners of the board. On each turn, the knight can move as a regular knight, that is, in an L-shape of two steps in one direction, and one step perpendicularly.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-d97cd3d639d44f757b767dfebde93300c58d6a91a872f7fdbcbf72c917d6545d | Alice and Bob are playing a game on a $(2n) \times (2n)$ chessboard. Alice starts from the top right corner and Bob from the bottom left corner. Each player moves 1 unit per turn. Alice's goal is to form a closed shape that ends at her starting position without revisiting any previously visited squares by either player... | [
"n^2"
] | [
"Alice and Bob are playing a game on a $(2n) \\times (2n)$ chessboard. Alice starts from the top right corner and Bob from the bottom left corner. Each player moves 1 unit per turn. Alice's goal is to form a closed shape that ends at her starting position without revisiting any previously visited squares by either ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-38ca8390ebcb0814b5704e5be534c5422c78cb97551ce2bf401d9fa2fafbfbd2 | Alice and Bob are playing a game on a 64×64 chessboard. Alice starts by placing a 1×2 domino, and then Bob places an L-shaped tromino. They alternate turns, placing one piece per turn, with the pieces being able to be rotated but not overlapping. The player who cannot place a piece loses. Determine which player has a w... | [
"Bob has the winning strategy."
] | [
"Alice and Bob are playing a game on a 64×64 chessboard. Alice starts by placing a 1×2 domino, and then Bob places an L-shaped tromino. They alternate turns, placing one piece per turn, with the pieces being able to be rotated but not overlapping. The player who cannot place a piece loses. Determine which player ha... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-86598da27c50be907ca9c21ad4af216c41c796f71389b092b5a2280ac687ca89 | Alice and Bob are playing a game on a graph with $n \ge 3$ vertices, initially a complete graph. At each turn, Alice must choose two vertices such that the graph remains connected even if one of them (along with its incident edges) is removed. Bob then removes one edge, ensuring that any two vertices in the graph are s... | [
"n - 2"
] | [
"Alice and Bob are playing a game on a graph with $n \\ge 3$ vertices, initially a complete graph. At each turn, Alice must choose two vertices such that the graph remains connected even if one of them (along with its incident edges) is removed. Bob then removes one edge, ensuring that any two vertices in the graph... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-3bad6f594b0b10810952e109ff374897a592a3a8f6cd059ce36f85a613e8e819 | Alice and Bob are playing a game on a tree where each point has a certain number of coins. They take turns selecting a point, and the point must be adjacent to the point selected by the other player in the previous turn. Alice goes first. If the tree is a straight line with 10 points and the number of coins at each poi... | [
"60"
] | [
"Alice and Bob are playing a game on a tree where each point has a certain number of coins. They take turns selecting a point, and the point must be adjacent to the point selected by the other player in the previous turn. Alice goes first. If the tree is a straight line with 10 points and the number of coins at eac... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-e84fe9bc0d02d612ee91e5cb5439a6d65881c88a14cd53520f93fe9ffc1c0192 | Alice and Bob are playing a game on an $8 \times 8$ chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black, or he does nothing. Alice may decide to continue the game or not. Alice aims to maximize the... | [
"16"
] | [
"Alice and Bob are playing a game on an $8 \\times 8$ chessboard. Initially, all grids are white. Each round, Alice chooses a white grid and paints it black. Then Bob chooses one of the neighbors of that grid and paints it black, or he does nothing. Alice may decide to continue the game or not. Alice aims to maximi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-c221200a3781454117130df4b52117f58797457fed3938c10f40ac64fd1efac5 | Alice and Bob are playing a game on the grid $[2024]^2$, where $[2024]:=\{1,2,\ldots,2024\}$. They take turns choosing a point that has not been chosen before, and the first person to get 2024 collinear points wins. If the game is played optimally by both players, calculate the maximum number of points that can be chos... | [
"2024 \\times 2023"
] | [
"Alice and Bob are playing a game on the grid $[2024]^2$, where $[2024]:=\\{1,2,\\ldots,2024\\}$. They take turns choosing a point that has not been chosen before, and the first person to get 2024 collinear points wins. If the game is played optimally by both players, calculate the maximum number of points that can... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-e92702156d8dd83f5b58933cc5e54426703f1d0421da31c79610f921cf26b313 | Alice and Bob are playing a game where Bob chooses a secret fixed point $B$ in the unit square. Alice then chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After each point $P_k$ (for $k \geq 1$), Bob tells Alice whether $P_k$ is closer to $B$ than $P_{k-1}$ ("warmer") or not ("colder"). After choosin... | [
"22"
] | [
"Alice and Bob are playing a game where Bob chooses a secret fixed point $B$ in the unit square. Alice then chooses a sequence of points $P_0, P_1, \\ldots, P_N$ in the plane. After each point $P_k$ (for $k \\geq 1$), Bob tells Alice whether $P_k$ is closer to $B$ than $P_{k-1}$ (\"warmer\") or not (\"colder\"). Af... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-007f7927a35f716f1ffa19921a1c0c7cbfcf055f927182eca4064ebe6c962605 | Alice and Bob are playing a game where Bob writes down a polynomial \( f(x) \) with the following properties: - For any integer \( n \), \( f(n) \) is an integer. - The degree of \( f(x) \) is less than 187. Alice knows these properties but does not know the polynomial. In each turn, Alice picks a number \( k \) from t... | [
"63"
] | [
"Alice and Bob are playing a game where Bob writes down a polynomial \\( f(x) \\) with the following properties:\n- For any integer \\( n \\), \\( f(n) \\) is an integer.\n- The degree of \\( f(x) \\) is less than 187.\n\nAlice knows these properties but does not know the polynomial. In each turn, Alice picks a num... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16819f701b06bb02078fffe5d025e446bc3f8150f26fe6c85f84a676cce6af1a | Alice and Bob are playing a game where they initially have $a$ and $b$ coins respectively. At each round, they toss a fair coin. If it is heads, Alice wins and takes $\min(x, y)$ coins from Bob, where $x$ and $y$ are the number of coins Alice and Bob had before the round. If it is tails, Bob wins and takes $\min(x, y)$... | [
"\\dfrac{5}{12}"
] | [
"Alice and Bob are playing a game where they initially have $a$ and $b$ coins respectively. At each round, they toss a fair coin. If it is heads, Alice wins and takes $\\min(x, y)$ coins from Bob, where $x$ and $y$ are the number of coins Alice and Bob had before the round. If it is tails, Bob wins and takes $\\min... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-774f7dfcd663d3c00c40d5cf2045912b505ae2036bc40622fe7d37ac3d84903b | Alice and Bob are playing a game where they take turns writing natural numbers not exceeding $2018$ on the board, with no number being written twice. Alice begins. A player wins if, after their move, there are three numbers on the board that form an arithmetic progression. Which player has a winning strategy? | [
"Alice has the winning strategy."
] | [
"Alice and Bob are playing a game where they take turns writing natural numbers not exceeding $2018$ on the board, with no number being written twice. Alice begins. A player wins if, after their move, there are three numbers on the board that form an arithmetic progression. Which player has a winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-8946c67e3abc228e9b10b4aec3b30977ff389bd8b778c7ea48c57698c65ba73c | Alice and Bob are playing a game with 100 coins of various denominations lined up in a row. Alice starts by taking either the first or the last coin, and then Bob takes either the first or the last coin from the remaining row of 99 coins. They alternate taking one coin from either end of the remaining row of coins unti... | [
"1"
] | [
"Alice and Bob are playing a game with 100 coins of various denominations lined up in a row. Alice starts by taking either the first or the last coin, and then Bob takes either the first or the last coin from the remaining row of 99 coins. They alternate taking one coin from either end of the remaining row of coins... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-7a48a4b63683d5b0369fc8002614554108822a4103aa372fa6e202c8904ad4ec | Alice and Bob are playing a game with \( n \) distinct Lego bricks labeled from \( 1 \) through \( n \) inclusive, where \( n \) is a positive integer. Alice randomly arranges the bricks into a line, and Bob can take the leftmost brick and place it in any other position each move. Bob's goal is to rearrange the bricks ... | [
"\\( n - \\sum_{k=1}^n \\frac{1}{k!} \\)"
] | [
"Alice and Bob are playing a game with \\( n \\) distinct Lego bricks labeled from \\( 1 \\) through \\( n \\) inclusive, where \\( n \\) is a positive integer. Alice randomly arranges the bricks into a line, and Bob can take the leftmost brick and place it in any other position each move. Bob's goal is to rearrang... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-405b657836570571a89155e67ebd8f5be48dee2fb8ba0a6fe4b34d372e40c23a | Alice and Bob are playing a game with a cursed coin. They take turns flipping the coin, and the winner is the first person to get heads. Initially, the probability of the coin coming up heads is $\frac{1}{2}$. After each flip, the probability of getting heads is halved. If Alice flips first, what is the probability of ... | [
"5"
] | [
"Alice and Bob are playing a game with a cursed coin. They take turns flipping the coin, and the winner is the first person to get heads. Initially, the probability of the coin coming up heads is $\\frac{1}{2}$. After each flip, the probability of getting heads is halved. If Alice flips first, what is the probabili... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e77b6f456b06ecc4c4f8a9a46f5521762b1c1abf510ffc7259cbf6100754784d | Alice and Bob are playing a game with a cursed coin. They take turns flipping the coin, and the winner is the first person to get heads. The probability of the coin coming up heads is initially $\frac{1}{2}$, but after every flip, this probability is halved. If Alice flips first, what is the probability that Alice wins... | [
"45"
] | [
"Alice and Bob are playing a game with a cursed coin. They take turns flipping the coin, and the winner is the first person to get heads. The probability of the coin coming up heads is initially $\\frac{1}{2}$, but after every flip, this probability is halved. If Alice flips first, what is the probability that Alic... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-324d9f2785fdfc61284fde0c1a33ff85ade3bcca9001fc991f0c26f33a7ea897 | Alice and Bob are playing a game with a pile of 2014 stones. Alice goes first and can take a number of stones that is a power of 2 (i.e., 1, 2, 4, ...), while Bob can take a number of stones that is a power of 4 (i.e., 1, 4, 16, ...). The player who takes the last stone wins. Determine the minimum number of stones Alic... | [
"2"
] | [
"Alice and Bob are playing a game with a pile of 2014 stones. Alice goes first and can take a number of stones that is a power of 2 (i.e., 1, 2, 4, ...), while Bob can take a number of stones that is a power of 4 (i.e., 1, 4, 16, ...). The player who takes the last stone wins. Determine the minimum number of stones... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-bce8156713ff63eb1acabe57debdd6e4b53b6c1155191c3fa52d34f605372925 | Alice and Bob are playing a game. Alice starts by picking an integer between 1 and 2013 (inclusive). On each turn, Bob tries to guess Alice's number. Alice then tells Bob if his guess is too high, too low, or correct. If Bob's guess is incorrect, Alice adds or subtracts 1 from her number (always remaining in the range ... | [
"63"
] | [
"Alice and Bob are playing a game. Alice starts by picking an integer between 1 and 2013 (inclusive). On each turn, Bob tries to guess Alice's number. Alice then tells Bob if his guess is too high, too low, or correct. If Bob's guess is incorrect, Alice adds or subtracts 1 from her number (always remaining in the r... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-8fc2f7f8e401efc2e6fbcae7907f0250da949d3b7e737abb4f711c766ca38dc3 | Alice and Bob are playing a game. Alice starts with \$5, and Bob starts with \$10. Alice flips a coin. If it is heads, Bob loses \$1, and Alice repeats. If it is tails, Alice gives the coin to Bob. If Bob gets the coin, Bob flips the coin. If it is heads, Alice loses \$1, and Bob repeats. If it is tails, Bob gives the ... | [
"\\(\\frac{1}{3}\\)"
] | [
"Alice and Bob are playing a game. Alice starts with \\$5, and Bob starts with \\$10. Alice flips a coin. If it is heads, Bob loses \\$1, and Alice repeats. If it is tails, Alice gives the coin to Bob. If Bob gets the coin, Bob flips the coin. If it is heads, Alice loses \\$1, and Bob repeats. If it is tails, Bob g... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-d45e7ed0b781cba77a9d2013b3a1504760ef63ec041b37b83f69a782bc5f8a65 | Alice and Bob are playing a game. Finitely many points are hidden in the plane, whose exact locations are known only to Alice. Alice has enumerated the points and Bob can refer to them by these numbers. Bob's goal is to determine the maximum number of points among the given points that lie on a line. The only type of i... | [
"10"
] | [
"Alice and Bob are playing a game. Finitely many points are hidden in the plane, whose exact locations are known only to Alice. Alice has enumerated the points and Bob can refer to them by these numbers. Bob's goal is to determine the maximum number of points among the given points that lie on a line. The only type... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-d8f3b9d89875709df4270db0467066cc1a02e0e5f212c7aa1ae2c35041379600 | Alice and Bob are playing on the grid-plane. Alice starts by placing one X in an empty square, and then Bob places 100 Os in empty squares. This process repeats with Alice placing one X and Bob placing 100 Os. Let $N$ be the minimum number of turns Alice needs to guarantee the existence of a square with vertices at Xs ... | [
"201"
] | [
"Alice and Bob are playing on the grid-plane. Alice starts by placing one X in an empty square, and then Bob places 100 Os in empty squares. This process repeats with Alice placing one X and Bob placing 100 Os. Let $N$ be the minimum number of turns Alice needs to guarantee the existence of a square with vertices a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-91063d7e19390ac9ccada1af0ca3eb6b42ea623ff4bd0343e0de76e8cc85fa47 | Alice and Bob are playing the Smallest Positive Integer Game. Alice says, "My number is 24." Bob says, "What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number." What is the smallest possible number that Bob could have? (Remember that Bob's number has to be a po... | [
"6"
] | [
"Alice and Bob are playing the Smallest Positive Integer Game. Alice says, \"My number is 24.\" Bob says, \"What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number.\"\n\nWhat is the smallest possible number that Bob could have? (Remember that Bob's number ha... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e05034861ee4de0b8ee68a292d08ec7445caaddcdc4d9b0007d499cbfc76e586 | Alice and Bob are running a race with 10 laps around a 400-meter circular track. Every time Alice passes the starting line, Alice’s speed slows down to \(\frac{9}{10}\) of her current speed, and every time Bob passes the starting line, Bob’s speed slows down to \(\frac{8}{10}\) of his current speed. If both runners sta... | [
"65.765 m/s\n\nGiven the unusual value, it's important to note that this might need further verification. However, based on the forum discussions, the answer provided is:"
] | [
"Alice and Bob are running a race with 10 laps around a 400-meter circular track. Every time Alice passes the starting line, Alice’s speed slows down to \\(\\frac{9}{10}\\) of her current speed, and every time Bob passes the starting line, Bob’s speed slows down to \\(\\frac{8}{10}\\) of his current speed. If both ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-52403e1ea1c5cf613b0a6c57018ceef076cf0d0ac7e7afed7078b9b56797575a | Alice and Bob are running around a circular track at a constant speed. After 8 minutes, Alice ran 46 complete laps. After 7 minutes, Bob ran 41 complete laps. After 29 minutes, Alice and Bob both ran $x$ complete laps. What is the sum of all possible integer values of $x$? | [
"0"
] | [
"Alice and Bob are running around a circular track at a constant speed. After 8 minutes, Alice ran 46 complete laps. After 7 minutes, Bob ran 41 complete laps. After 29 minutes, Alice and Bob both ran $x$ complete laps. What is the sum of all possible integer values of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-8ac4062ab7285f5eca52e69b622f5a28852fb5e526a7b0258effdc6302de659c | Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200-meter side and run in the same direction, with Alice running twice as fast as Bob. After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the build... | [
"\\(\\frac{1}{6}\\)"
] | [
"Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200-meter side and run in the same direction, with Alice running twice as fast as Bob. After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-e81b6c84ac17234a9ba0e06998a3ccde1751be0a3206771567dfc549b660faaa | Alice and Bob are running on a circular track with a circumference of 2 km. Alice runs at 5 km/h, and Bob runs at 15 km/h. Both start running clockwise from random, uniformly chosen points on the track at a random, uniformly chosen time between 10 a.m. and 11 a.m. What is the probability that Alice and Bob will meet be... | [
"\\(\\frac{61}{75}\\)"
] | [
"Alice and Bob are running on a circular track with a circumference of 2 km. Alice runs at 5 km/h, and Bob runs at 15 km/h. Both start running clockwise from random, uniformly chosen points on the track at a random, uniformly chosen time between 10 a.m. and 11 a.m. What is the probability that Alice and Bob will me... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-a051abe6e1eb725448697c3850e15975f46cb5ecb55a75bb36f7f8b8abd211a4 | Alice and Bob are teleported to random positions on a 1 km long train. They can see each other's facing direction, total walking distance, and whether they are at the front or end of the train. They can only discuss a strategy before the game starts. Determine the least value \( x \) such that they are guaranteed to me... | [
"1.5"
] | [
"Alice and Bob are teleported to random positions on a 1 km long train. They can see each other's facing direction, total walking distance, and whether they are at the front or end of the train. They can only discuss a strategy before the game starts. Determine the least value \\( x \\) such that they are guarantee... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c38d313b92c18504f2e20a41507ab756bf5379d340409c11da9da30f39d5f7a5 | Alice and Bob are trying to figure out a 4-digit password for Cathy’s bitcoin wallet. They know the following: - The first digit is 5. - The number is a multiple of 9. - Alice knows the second and third digits. - Bob knows the third and fourth digits. - After sharing this information, Alice and Bob both say they do not... | [
"5940"
] | [
"Alice and Bob are trying to figure out a 4-digit password for Cathy’s bitcoin wallet. They know the following:\n- The first digit is 5.\n- The number is a multiple of 9.\n- Alice knows the second and third digits.\n- Bob knows the third and fourth digits.\n- After sharing this information, Alice and Bob both say t... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-08d807e248ceedb7733a9e3fbf27ee14cd62d5bb499d3cff8d014c0e374de708 | Alice and Bob both flip a fair coin. Alice’s coin is fair, but Bob’s has a probability of 1/3 of landing on heads. Each stops after flipping their first head. What is the probability that after they stopped, Alice will have flipped more times than Bob? | [
"\\(\\frac{1}{4}\\)"
] | [
"Alice and Bob both flip a fair coin. Alice’s coin is fair, but Bob’s has a probability of 1/3 of landing on heads. Each stops after flipping their first head. What is the probability that after they stopped, Alice will have flipped more times than Bob?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-3d956114680e4514a3a29d5f6e81904e1b90a8b026b956a847e1154c148a1013 | Alice and Bob both randomly choose two integers \( m \) and \( n \) with \( m < n \) from the set \(\{1, 2, \ldots, 20\}\) in different ways: - Alice first chooses \( m \) uniformly at random from the set \(\{1, 2, \ldots, 19\}\) and then chooses \( n \) uniformly at random from the set \(\{m+1, m+2, \ldots, 20\}\). - ... | [
"\\(\\frac{9}{2}\\)"
] | [
"Alice and Bob both randomly choose two integers \\( m \\) and \\( n \\) with \\( m < n \\) from the set \\(\\{1, 2, \\ldots, 20\\}\\) in different ways:\n\n- Alice first chooses \\( m \\) uniformly at random from the set \\(\\{1, 2, \\ldots, 19\\}\\) and then chooses \\( n \\) uniformly at random from the set \\(\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-90c7b2964fca28decbcf4ba1e9b95907ae4b696c7201a0acc78682bab9165ae4 | Alice and Bob choose random nonnegative real numbers independently. The distribution of Alice's number is an exponential distribution with a mean of 1. The distribution of the sum of the two numbers is an exponential distribution with a median of 1. What is the distribution of Bob's number? | [
"\\text{Bob's number is } 0 \\text{ with probability } \\ln 2 \\text{ and exponential with parameter } \\ln 2 \\text{ otherwise.}"
] | [
"Alice and Bob choose random nonnegative real numbers independently. The distribution of Alice's number is an exponential distribution with a mean of 1. The distribution of the sum of the two numbers is an exponential distribution with a median of 1. What is the distribution of Bob's number?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8ce640286eb2e9a56da7c027b9e7ce745b3d8893e414636d100b77fc91679ec2 | Alice and Bob determine a number with 2018 digits in the decimal system by choosing digits from left to right, with Alice starting and each digit differing from the previously chosen digit modulo 3. Bob will make the last move. If Alice starts with the digit 1, what is the smallest digit Bob can choose on his last move... | [
"2"
] | [
"Alice and Bob determine a number with 2018 digits in the decimal system by choosing digits from left to right, with Alice starting and each digit differing from the previously chosen digit modulo 3. Bob will make the last move. If Alice starts with the digit 1, what is the smallest digit Bob can choose on his last... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99c583624804c93787bc8287826fce866846ab4df395bed98a5318e157b380b0 | Alice and Bob draw 2565 different points on the circumference of a circle and play a game in turns. A straight line is drawn connecting two points on the circle, one line at each turn, with the following rules: "Do not redraw a link that has been drawn, but you may use the point that was previously connected." The game... | [
"Alice"
] | [
"Alice and Bob draw 2565 different points on the circumference of a circle and play a game in turns. A straight line is drawn connecting two points on the circle, one line at each turn, with the following rules: \"Do not redraw a link that has been drawn, but you may use the point that was previously connected.\" T... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bebb8e17bd53a3b22531dc19f85df978c68cf429d9b0e367dba3cb3e92d6e0b7 | Alice and Bob each arrive at a party at a random time between 1:00 and 2:00. If Alice arrives after Bob, what is the probability that Bob arrived before 1:30? | [
"\\frac{3}{4}"
] | [
"Alice and Bob each arrive at a party at a random time between 1:00 and 2:00. If Alice arrives after Bob, what is the probability that Bob arrived before 1:30?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fc359700c0dfac2c4b4ba6723adba5031b5ddcb45aed40c83e510863931ba9d | Alice and Bob each choose a number between 16 and 25. Alice guesses Bob's number from the remaining choices. Bob is 1.5 times as likely to guess Alice's number correctly compared to Alice guessing Bob's number. What is the probability that Alice wins, assuming she goes first? | [
"\\dfrac{3}{7}"
] | [
"Alice and Bob each choose a number between 16 and 25. Alice guesses Bob's number from the remaining choices. Bob is 1.5 times as likely to guess Alice's number correctly compared to Alice guessing Bob's number. What is the probability that Alice wins, assuming she goes first?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-295d169210ceb383b5b418c40df9d0113066294e569207369fab1b246967f2ce | Alice and Bob each choose a number between 16 and 25. Alice guesses Bob's number from the remaining choices. Bob is 1.5 times as likely to guess Alice's number correctly compared to Alice guessing Bob's number. What is the probability that the game has to go up to the 10th round, assuming Bob goes first? | [
"\\dfrac{1}{100}"
] | [
"Alice and Bob each choose a number between 16 and 25. Alice guesses Bob's number from the remaining choices. Bob is 1.5 times as likely to guess Alice's number correctly compared to Alice guessing Bob's number. What is the probability that the game has to go up to the 10th round, assuming Bob goes first?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-3b1de488ce6e56f666119e82d436315b86f7ed985b2b7194c3242e89ebf3fdca | Alice and Bob each divide a 20 × 21 board into 1×3 trominos. After dividing, they compare the results and compute how many trominos are the same in both splittings. What is the maximal amount Bob may guarantee to himself no matter how Alice plays? | [
"70"
] | [
"Alice and Bob each divide a 20 × 21 board into 1×3 trominos. After dividing, they compare the results and compute how many trominos are the same in both splittings. What is the maximal amount Bob may guarantee to himself no matter how Alice plays?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b475efeda544c3dd92135c269512ee369ce052566c4a42ca094d9b2b53c30b5 | Alice and Bob each have a bag of 9 balls numbered from 1 to 9. Each removes one ball uniformly at random from their own bag. Let \( a \) be the sum of the numbers on the balls remaining in Alice’s bag, and let \( b \) be the sum of the numbers on the balls remaining in Bob’s bag. Determine the probability that \( a \) ... | [
"\\(\\frac{10}{81}\\)"
] | [
"Alice and Bob each have a bag of 9 balls numbered from 1 to 9. Each removes one ball uniformly at random from their own bag. Let \\( a \\) be the sum of the numbers on the balls remaining in Alice’s bag, and let \\( b \\) be the sum of the numbers on the balls remaining in Bob’s bag. Determine the probability that... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76c03238e9e2552fc130ca6dc1e7a16377640c80d54b91612df9c9ccecf69a09 | Alice and Bob each have a countably infinite collection of boxes labelled by $\mathbb{Z}^{+}$. For some unknown sequence $(a_k)$ of real numbers, Alice and Bob’s $k$th box each contains the number $a_k$. They agree on a strategy where each will open some boxes and predict the value of another box. Let $S$ be the set of... | [
"1"
] | [
"Alice and Bob each have a countably infinite collection of boxes labelled by $\\mathbb{Z}^{+}$. For some unknown sequence $(a_k)$ of real numbers, Alice and Bob’s $k$th box each contains the number $a_k$. They agree on a strategy where each will open some boxes and predict the value of another box. Let $S$ be the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-566d559148b82332f725c257968b1a63ab4d5f0ba0f99a8a3087bb4417bdefd0 | Alice and Bob each pick 3 distinct numbers randomly from the set $\{1, 2, 3, \ldots, 9\}$ and arrange them in ascending order to form a three-digit number. If the numbers formed by Alice and Bob are $a$ and $b$ respectively, find the probability that $a < b$. If this probability equals $\frac{m}{n}$ in reduced form, wr... | [
"83"
] | [
"Alice and Bob each pick 3 distinct numbers randomly from the set $\\{1, 2, 3, \\ldots, 9\\}$ and arrange them in ascending order to form a three-digit number. If the numbers formed by Alice and Bob are $a$ and $b$ respectively, find the probability that $a < b$. If this probability equals $\\frac{m}{n}$ in reduced... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-b6a9ed0c59ce520624eb4872b6fb52f2471e5e7dc08a6d499aee58aead3d749f | Alice and Bob each roll a die six times and take the sums of their outcomes. The player with the higher sum wins, and if the sums are equal, nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \( \frac{a}{b} \)... | [
"3895"
] | [
"Alice and Bob each roll a die six times and take the sums of their outcomes. The player with the higher sum wins, and if the sums are equal, nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \\( \\frac{a... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-00ae3d566a9d4b0a7c3672ffd7fb10e6939be667e8bb6e2fedf6c2a1eae521a6 | Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? | [
"\\dfrac{5}{12}"
] | [
"Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1e8967b84a3dab54bcff7c2d2d92fda208f77ad4b593e43f7eabc9227621da53 | Alice and Bob have 100 bricks, 50 white and 50 black. They play a game where initially, each brick is a separate tower. In each turn, a player must place one tower on top of another tower such that no two adjacent bricks in the resulting tower are of the same color. Alice starts first. The player unable to make a move ... | [
"Alice"
] | [
"Alice and Bob have 100 bricks, 50 white and 50 black. They play a game where initially, each brick is a separate tower. In each turn, a player must place one tower on top of another tower such that no two adjacent bricks in the resulting tower are of the same color. Alice starts first. The player unable to make a ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 8
} | 19 | 0 | 0 |
omr-ws-668f2ab89146b183a7537f6ba88bf2a6f08504ab65f90773dd5fdc5bf72e9ee8 | Alice and Bob live 10 miles apart. Alice looks due north and sees an airplane at an angle of elevation of $30^\circ$. Bob looks due west and sees the same airplane at an angle of elevation of $60^\circ$. What is the closest altitude of the airplane in miles? | [
"5.5"
] | [
"Alice and Bob live 10 miles apart. Alice looks due north and sees an airplane at an angle of elevation of $30^\\circ$. Bob looks due west and sees the same airplane at an angle of elevation of $60^\\circ$. What is the closest altitude of the airplane in miles?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-c790b1de44078dc5a46bc3abd2018a94596395df8ea80cf23742b3c2ed45c685 | Alice and Bob play a game in which they take turns choosing integers from 1 to $n$. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of th... | [
"All positive integers \\( n \\)"
] | [
"Alice and Bob play a game in which they take turns choosing integers from 1 to $n$. Before any integers are chosen, Bob selects a goal of \"odd\" or \"even\". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-bed315e39acec656720e6b497d0475597682b927cdcbc109f301941d4828e77a | Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Find the smallest $n > 100$ such that Bob has a winning stra... | [
"104"
] | [
"Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Find the smallest $n > 100$ such that Bob has a winning... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-8054bcf182eba3173ddb222ab5788074e27e75461dd49622ffe4af4da28799d7 | Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? | [
"35"
] | [
"Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e93be7a249d2e4d77a09c157fff045e0eba4f275895bcb23a1208ec7ac1af92f | Alice and Bob play a game on a $1 \times m$ board using 2012 cards numbered from 1 through 2012. At each step, Alice chooses a card and Bob places it on an empty square of the board. Bob wins the game if the numbers on the cards on the board are in increasing order after $k$ steps, where $1 \le k \le 2012$. Find all pa... | [
"\\((k, m) \\text{ such that } m \\geq 2^k - 1 \\text{ for } k \\leq 10 \\text{ and } m \\geq 2012 \\text{ for } k > 10\\)"
] | [
"Alice and Bob play a game on a $1 \\times m$ board using 2012 cards numbered from 1 through 2012. At each step, Alice chooses a card and Bob places it on an empty square of the board. Bob wins the game if the numbers on the cards on the board are in increasing order after $k$ steps, where $1 \\le k \\le 2012$. Fin... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7fb902265ae072cbb8e769047e1fa4304a0efb2e49cc97adacf206780a7a8b5 | Alice and Bob play a game on a $2024 \times 2024$ grid. On the $k$-th turn, the player selects at most $n_k$ unit squares, where $n_k$ is the least integer such that $n_k \geq \frac{k}{2024}$, and covers them with stickers bearing their name. If a square already has a sticker, it is replaced. A player wins if they can ... | [
"Yes"
] | [
"Alice and Bob play a game on a $2024 \\times 2024$ grid. On the $k$-th turn, the player selects at most $n_k$ unit squares, where $n_k$ is the least integer such that $n_k \\geq \\frac{k}{2024}$, and covers them with stickers bearing their name. If a square already has a sticker, it is replaced. A player wins if t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aded37ab418254ef18bde7c0ac6d284abb92011124d599a4571b33b277450228 | Alice and Bob play a game on a 5 x 5 chessboard where the first player places a king on an empty board, and then they alternate moving the king to a square that has not been previously occupied. The player who cannot move loses. Which player has a winning strategy? | [
"2nd player has a winning strategy."
] | [
"Alice and Bob play a game on a 5 x 5 chessboard where the first player places a king on an empty board, and then they alternate moving the king to a square that has not been previously occupied. The player who cannot move loses. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-ef7b3ace623eff514f1e63bab782e117d684952a29f53dd892d96ebf928160cf | Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number ... | [
"3"
] | [
"Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest nu... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-894a256eae8ec07afd794fe08b7d8d37d6074682d3869e18241920b80f00da86 | Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number ... | [
"15"
] | [
"Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest nu... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 2
} | 2 | 0 | 0 |
omr-ws-0d3476ba969ef4bbbc5da4f9881cf2fd0b090fead5d1cf475913cb54c30cfc93 | Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. A tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's ... | [
"290"
] | [
"Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. A tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Ali... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05784aba52e84e877b51d7d8ab6b256aa29f474901e3a2fad681b7c49808cdbb | Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move, Alice directs one undirected edge of $G$. At each move, Bob chooses a positive integer $m$, where $1 \le m \le 1000$, and directs $m$ undirected edges of $G$. The game ends when all edges ... | [
"no_valid_answer_found"
] | [
"Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move, Alice directs one undirected edge of $G$. At each move, Bob chooses a positive integer $m$, where $1 \\le m \\le 1000$, and directs $m$ undirected edges of $G$. The game ends when all... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59ec1758eee0501cff14b2261aa73927efa7b020cf22895690c0b9693da1ce66 | Alice and Bob play a game on a numbered row of $n \ge 5$ squares. At the beginning, a pebble is placed on the first square. During a move, a player can choose to move the pebble one square forward, four squares forward, or two squares backwards, provided the pebble stays within the borders of the square row. The player... | [
"Alice wins for odd \\( n \\), Bob wins for even \\( n \\)."
] | [
"Alice and Bob play a game on a numbered row of $n \\ge 5$ squares. At the beginning, a pebble is placed on the first square. During a move, a player can choose to move the pebble one square forward, four squares forward, or two squares backwards, provided the pebble stays within the borders of the square row. The ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-6b152d5514f0bcddb648b7cbcd99c3eacf9490b6f8350f6e6d6db4d87105c1b1 | Alice and Bob play a game on a regular $n$-gon ($n \geq 3$). Starting with Alice, they take turns choosing a vertex that does not have a number yet and writes either $1$ or $0$ on that vertex. Alice wins if, after her turn, there are three consecutive vertices such that the sum of the numbers on those vertices is divis... | [
"\\( n \\geq 5 \\) and \\( n \\) is odd."
] | [
"Alice and Bob play a game on a regular $n$-gon ($n \\geq 3$). Starting with Alice, they take turns choosing a vertex that does not have a number yet and writes either $1$ or $0$ on that vertex. Alice wins if, after her turn, there are three consecutive vertices such that the sum of the numbers on those vertices is... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-49f3c4b4b0378bdab701f71ad624e4da1f1ebff68a01cd0a1fe9726e76fba398 | Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. In each move, a player advances their own game piece by one or two squares in the direction of the opponent’s piece, landing on an empty square ... | [
"n \\not\\equiv 2 \\pmod{3}"
] | [
"Alice and Bob play a game on a strip of $n \\ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. In each move, a player advances their own game piece by one or two squares in the direction of the opponent’s piece, landing on an empty s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-f48a4a591eb203803790592be2fd3286430f71c39a69c294953d20fe3b21188e | Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. In each move, a player advances their own game piece by one or two squares in the direction of the opponent’s piece, landing on an empty square ... | [
"n \\equiv 2 \\pmod{3}"
] | [
"Alice and Bob play a game on a strip of $n \\ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. In each move, a player advances their own game piece by one or two squares in the direction of the opponent’s piece, landing on an empty s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-7dd930610c892ff312c6ee0f7a4149c195dc2669ff296db8ff888d79d4e73e73 | Alice and Bob play a game on an $8 \times 8$ board, alternating moves, with Alice starting. Alice marks an unmarked and uncovered cell in her turn. Bob covers two adjacent uncovered cells with a $1 \times 2$ domino, provided there are an even number of marked cells ($0$ or $2$) among the two cells. The player who canno... | [
"Alice"
] | [
"Alice and Bob play a game on an $8 \\times 8$ board, alternating moves, with Alice starting. Alice marks an unmarked and uncovered cell in her turn. Bob covers two adjacent uncovered cells with a $1 \\times 2$ domino, provided there are an even number of marked cells ($0$ or $2$) among the two cells. The player wh... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-86b7fb6dd445f2ecccfde7d974bf85228cd486387491d57185640052a2fed1e6 | Alice and Bob play a game on an 8 x 8 chessboard where the first player places a king on an empty board, and then they alternate moving the king to a square that has not been previously occupied. The player who cannot move loses. Which player has a winning strategy? | [
"First player has a winning strategy."
] | [
"Alice and Bob play a game on an 8 x 8 chessboard where the first player places a king on an empty board, and then they alternate moving the king to a square that has not been previously occupied. The player who cannot move loses. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d066482be5c57f3a670b7a4423ef512258e0078c8ce0244f6ec30776df9ce84a | Alice and Bob play a game on an infinite checkered strip where cells are numbered consecutively with integers. Alice places one cross in any free cell on her turn, and Bob places zeros in any 2020 free cells on his turn. Alice wins if she can get four cells marked with crosses such that their corresponding cell numbers... | [
"2021"
] | [
"Alice and Bob play a game on an infinite checkered strip where cells are numbered consecutively with integers. Alice places one cross in any free cell on her turn, and Bob places zeros in any 2020 free cells on his turn. Alice wins if she can get four cells marked with crosses such that their corresponding cell nu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-ffa18851a30dc6a19ecb3825fcc4453cf444551a81cd27163581c3c057682b96 | Alice and Bob play a game on the complete graph $K_{100}$. In each round, Alice colors an unmarked edge red, and Bob colors an unmarked edge blue. After 124 rounds, let $N$ be the maximum number of disjoint cycles of length 4 (each with all edges colored red) that Alice can achieve. Find the minimum value of $N$ that B... | [
"15"
] | [
"Alice and Bob play a game on the complete graph $K_{100}$. In each round, Alice colors an unmarked edge red, and Bob colors an unmarked edge blue. After 124 rounds, let $N$ be the maximum number of disjoint cycles of length 4 (each with all edges colored red) that Alice can achieve. Find the minimum value of $N$ t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-5cea0b00deba4756761cbc6f28a7012202c85266dc1ae52ad126d00e0c1e5f7e | Alice and Bob play a game starting from a positive integer $N$. On each turn, a player must count down a number of steps equal to a Fibonacci number. Alice starts first, and the player who counts to 1 wins. Define a "Bob-winning" number as a value of $N$ for which Bob has a winning strategy regardless of Alice's moves.... | [
"555"
] | [
"Alice and Bob play a game starting from a positive integer $N$. On each turn, a player must count down a number of steps equal to a Fibonacci number. Alice starts first, and the player who counts to 1 wins. Define a \"Bob-winning\" number as a value of $N$ for which Bob has a winning strategy regardless of Alice's... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cbd5b20a09505bee091c1854acfc9f480ae70a8d955d9c0c50427d2c72a6fb2 | Alice and Bob play a game starting with a number from the set $\{9, 10\}$. On each turn, the player adds either $19$ or $20$ to the current number, with Alice going first. The game ends when the number reaches or exceeds $2019$. Let $N$ be the number of starting numbers from the set $\{9, 10\}$ for which Bob cannot for... | [
"1"
] | [
"Alice and Bob play a game starting with a number from the set $\\{9, 10\\}$. On each turn, the player adds either $19$ or $20$ to the current number, with Alice going first. The game ends when the number reaches or exceeds $2019$. Let $N$ be the number of starting numbers from the set $\\{9, 10\\}$ for which Bob c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fe9f3865f495f5878998461f8ea0f60acc7003ba25cff21bcd4b9bcbcd13c69 | Alice and Bob play a game starting with a positive integer. On each move, a player may subtract 1 or divide by 2 (rounding up if necessary) to get an integer. The player who first reaches 0 wins. Given the starting values 512 and 513, which of these two values leads to a winning strategy for Alice? | [
"513"
] | [
"Alice and Bob play a game starting with a positive integer. On each move, a player may subtract 1 or divide by 2 (rounding up if necessary) to get an integer. The player who first reaches 0 wins. Given the starting values 512 and 513, which of these two values leads to a winning strategy for Alice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-bb59faeca58ba9cbc2a1c6e41d0f550e108dde9826278ecf4e78c1463d0246ab | Alice and Bob play a game starting with an empty set $S$. Alice picks a positive integer $m$ and adds it to $S$. Bob then picks a positive integer which is not a multiple of $m$ and adds it to $S$. On each subsequent turn, a player must pick a positive integer $k$ that cannot be obtained by adding together elements alr... | [
"Alice has a winning strategy."
] | [
"Alice and Bob play a game starting with an empty set $S$. Alice picks a positive integer $m$ and adds it to $S$. Bob then picks a positive integer which is not a multiple of $m$ and adds it to $S$. On each subsequent turn, a player must pick a positive integer $k$ that cannot be obtained by adding together element... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 3
} | 12 | 0 | 0 |
omr-ws-039bb491cdcec9868a3d4c84e8a38954d9be018fa8f05fbc8817380c298df9f3 | Alice and Bob play a game starting with the number $2$ on a blackboard. Each player, in turn, changes the current number $n$ to $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate turns. The game is lost by the player who is forced to write a number greater than $\underbrace{22...2}... | [
"Alice"
] | [
"Alice and Bob play a game starting with the number $2$ on a blackboard. Each player, in turn, changes the current number $n$ to $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate turns. The game is lost by the player who is forced to write a number greater than $\\underbrace{2... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-dcc2ece37a7abaf6344ddb2d00f4a7cbd5d58fa84a014e105d657e6a7a3b5b7d | Alice and Bob play a game starting with the numbers 47 and 2016 on the blackboard. In turns, they choose two numbers \(a\) and \(b\) with \(a > b\) whose difference \(a - b\) is not yet written on the blackboard and write this difference on the board. The game ends when no further move is possible. Let \(N\) be the num... | [
"14"
] | [
"Alice and Bob play a game starting with the numbers 47 and 2016 on the blackboard. In turns, they choose two numbers \\(a\\) and \\(b\\) with \\(a > b\\) whose difference \\(a - b\\) is not yet written on the blackboard and write this difference on the board. The game ends when no further move is possible. Let \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b8def3cf171b3b6d820fc8e46893876e7e4f12bf310fd55df1da4ba0cb09e763 | Alice and Bob play a game where Alice arranges the numbers $1, 2, \ldots, n$ in some order in a row. Bob then places a pebble on one of the numbers. A player's turn consists of moving the pebble to an adjacent number, with the restriction that the pebble can be placed on the number $k$ at most $k$ times. The players al... | [
"Alice wins if \\( n \\equiv 0, 3 \\pmod{4} \\); Bob wins if \\( n \\equiv 1, 2 \\pmod{4} \\)."
] | [
"Alice and Bob play a game where Alice arranges the numbers $1, 2, \\ldots, n$ in some order in a row. Bob then places a pebble on one of the numbers. A player's turn consists of moving the pebble to an adjacent number, with the restriction that the pebble can be placed on the number $k$ at most $k$ times. The play... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-f97404157af06bb9bf1528e3208d41f6f1b65d6ac6d403b4893cb099a6c8daaf | Alice and Bob play a game where Alice arranges the numbers \(1, 2, \ldots, n\) in some order in a row and Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number \(k\) ... | [
"Bob"
] | [
"Alice and Bob play a game where Alice arranges the numbers \\(1, 2, \\ldots, n\\) in some order in a row and Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the numbe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4d7e365cfc2ceddaaea7f9eb2db73d6433bea90257d36bc45b4bfbdcf61a1bb3 | Alice and Bob play a game where Alice chooses a non-zero real number \( a \), Bob chooses a non-zero real number \( b \), and Alice chooses a non-zero real number \( c \). Bob then forms a quadratic polynomial using \( a \), \( b \), and \( c \) in some order. Bob wins if the quadratic polynomial has a real root, and A... | [
"Bob has a winning strategy."
] | [
"Alice and Bob play a game where Alice chooses a non-zero real number \\( a \\), Bob chooses a non-zero real number \\( b \\), and Alice chooses a non-zero real number \\( c \\). Bob then forms a quadratic polynomial using \\( a \\), \\( b \\), and \\( c \\) in some order. Bob wins if the quadratic polynomial has a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-76d6f14c80f17c98c527f03a09bab2bbae39d310ede07d1205fd60de659638ba | Alice and Bob play a game where Alice chooses a non-zero real number \(a\), Bob chooses a non-zero real number \(b\), and Alice chooses a non-zero real number \(c\). Bob then forms a quadratic polynomial using \(a\), \(b\), and \(c\) in some order. Alice wins if the quadratic polynomial has a real root, and Bob wins ot... | [
"Bob has a winning strategy."
] | [
"Alice and Bob play a game where Alice chooses a non-zero real number \\(a\\), Bob chooses a non-zero real number \\(b\\), and Alice chooses a non-zero real number \\(c\\). Bob then forms a quadratic polynomial using \\(a\\), \\(b\\), and \\(c\\) in some order. Alice wins if the quadratic polynomial has a real root... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-efe7bcd5b965dab5e14e522a872c17957edd2b44a4e3ca5af1390e097a464939 | Alice and Bob play a game where Alice chooses a non-zero real number \(a\), Bob chooses a non-zero real number \(b\), and Alice chooses a non-zero real number \(c\). Bob then forms a quadratic polynomial using \(a\), \(b\), and \(c\) in some order. Bob wins if the quadratic polynomial has a real root, and Alice wins ot... | [
"Bob has a winning strategy."
] | [
"Alice and Bob play a game where Alice chooses a non-zero real number \\(a\\), Bob chooses a non-zero real number \\(b\\), and Alice chooses a non-zero real number \\(c\\). Bob then forms a quadratic polynomial using \\(a\\), \\(b\\), and \\(c\\) in some order. Bob wins if the quadratic polynomial has a real root, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-785db550359e88d8e858a9e20671992b2ae47ab6dab1131d30503d6102a87a0f | Alice and Bob play a game where Alice draws \( n \) distinct circles in the plane, creating several bounded regions. Bob then colors each bounded region either red or blue. Alice can choose to change the color of a single region. Alice wins if there is an even number of red regions within each circle; otherwise, Bob wi... | [
"3"
] | [
"Alice and Bob play a game where Alice draws \\( n \\) distinct circles in the plane, creating several bounded regions. Bob then colors each bounded region either red or blue. Alice can choose to change the color of a single region. Alice wins if there is an even number of red regions within each circle; otherwise,... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-79bb388ef62725aa2740f8af5651d0eb194680d94b865d258aa45fe4317e9c96 | Alice and Bob play a game where Alice picks a set \( A = \{1, 2, \ldots, n\} \) for some natural number \( n \ge 2 \). Bob starts by choosing any number from \( A \), and then they take turns choosing distinct numbers from \( A \) such that each chosen number differs by 1 from an already chosen number. The game ends wh... | [
"Alice has a winning strategy when \\( n = 8 \\)."
] | [
"Alice and Bob play a game where Alice picks a set \\( A = \\{1, 2, \\ldots, n\\} \\) for some natural number \\( n \\ge 2 \\). Bob starts by choosing any number from \\( A \\), and then they take turns choosing distinct numbers from \\( A \\) such that each chosen number differs by 1 from an already chosen number.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9fb3a77f44cfc66e55b2ca365deaf13d57a4fd75fd728c569327c446477ec2d | Alice and Bob play a game where each starts with an ordered $n$-tuple $(1, 1, 1, \dots, 1)$, and each $n$-tuple is an element of $\mathbb{Z}_m^n$. They take turns making a move, which involves selecting an element from their tuple and adding it to an element of the other player's tuple, updating the other player's tupl... | [
"Bob"
] | [
"Alice and Bob play a game where each starts with an ordered $n$-tuple $(1, 1, 1, \\dots, 1)$, and each $n$-tuple is an element of $\\mathbb{Z}_m^n$. They take turns making a move, which involves selecting an element from their tuple and adding it to an element of the other player's tuple, updating the other player... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
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