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-3eb6808cf57126dc679c0e4885bb5cc09000873739f4b04606549fa3ab2a8e1e | Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the figure already drawn, such that one of its sides is a complete side of the figure already drawn. A player wins if she completes a rectangle whose ar... | [
"Beatriz"
] | [
"Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the figure already drawn, such that one of its sides is a complete side of the figure already drawn. A player wins if she completes a rectangle who... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-01a378e0e24686bd1f393a90fd589db37778aa3a5a3ba2c3ca6a9952fa459e01 | Ana and Benito play a game consisting of 2020 turns. Initially, there are 2020 cards on the table, numbered from 1 to 2020, and Ana possesses an extra card with number 0. In the $k$-th turn, the player that doesn't possess card $k-1$ chooses whether to take the card with number $k$ or to give it to the other player. Th... | [
"Both players can force a tie."
] | [
"Ana and Benito play a game consisting of 2020 turns. Initially, there are 2020 cards on the table, numbered from 1 to 2020, and Ana possesses an extra card with number 0. In the $k$-th turn, the player that doesn't possess card $k-1$ chooses whether to take the card with number $k$ or to give it to the other playe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-7653de08f365ca3ca9d73dc27b9db50972cb39936a2a403e41c47a19ee748808 | Ana and Bernardo take turns playing a game starting with the number 3 on a board. If the number on the board is \( n \), the player must replace it with an integer \( m \) such that \( m \) is coprime with \( n \) and \( n < m < n^2 \). The first player to reach a number greater than or equal to 2016 loses. Determine w... | [
"Ana has the winning strategy."
] | [
"Ana and Bernardo take turns playing a game starting with the number 3 on a board. If the number on the board is \\( n \\), the player must replace it with an integer \\( m \\) such that \\( m \\) is coprime with \\( n \\) and \\( n < m < n^2 \\). The first player to reach a number greater than or equal to 2016 los... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 19
} | 30 | 0 | 0 |
omr-ws-5b3e770d875d4c9f338f79928cb09d9da8f79c4d63cd7ed6edc86d05fa7aa6bf | Ana and Beth decide to play a math game. Starting with Ana, they take turns saying an integer between 1 and 9 inclusive that has not yet been said by either of them. The first player to have said three numbers that sum to 15 is the winner. If all numbers have been said and neither player has won, then the game ends in ... | [
"Neither"
] | [
"Ana and Beth decide to play a math game. Starting with Ana, they take turns saying an integer between 1 and 9 inclusive that has not yet been said by either of them. The first player to have said three numbers that sum to 15 is the winner. If all numbers have been said and neither player has won, then the game end... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-de57ce63c44860dceac238adf9eeef8e80c867e82561ab4c94d189a1e3be2613 | Ana and Beto have 13 stones, each with a distinct color and weights from 1 to 13 grams. Ana knows the coloring of the stones, but Beto does not. An operation consists of placing stones on a weight balance until it is balanced. Determine the minimum number of operations Ana must perform so that Beto can identify the col... | [
"2"
] | [
"Ana and Beto have 13 stones, each with a distinct color and weights from 1 to 13 grams. Ana knows the coloring of the stones, but Beto does not. An operation consists of placing stones on a weight balance until it is balanced. Determine the minimum number of operations Ana must perform so that Beto can identify th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1d77d29394bf5e941f5c80d4f6907f42cc152ef5da13d4b803b9dfcc2507e0a | Ana and Beto play a game where Ana chooses a non-negative integer $N$ and announces it to Beto. Beto then writes a sequence of $2016$ numbers, consisting of $1008$ ones and $1008$ negative ones. Ana must partition this sequence into several blocks of consecutive terms, calculate the sum of the numbers in each block, an... | [
"Any even number from 0 to 2016."
] | [
"Ana and Beto play a game where Ana chooses a non-negative integer $N$ and announces it to Beto. Beto then writes a sequence of $2016$ numbers, consisting of $1008$ ones and $1008$ negative ones. Ana must partition this sequence into several blocks of consecutive terms, calculate the sum of the numbers in each bloc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-ba13ba04cc9e7f07d5f12bb18c12e2bc30b088ff8e7fb1be1783ad1d1125df46 | Ana and Beto play a game with a stick of length $15$. Ana starts by cutting the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Dete... | [
"Ana has the winning strategy."
] | [
"Ana and Beto play a game with a stick of length $15$. Ana starts by cutting the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses.... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 9
} | 31 | 0 | 0 |
omr-ws-83629a0fa9077a59f36904bbff4b8a668e8cb3aba06da9f6c33973de0067a30c | Ana and Beto play a turn-based game where each player draws a segment of length 1 on the plane. Ana starts by drawing the first segment, and each subsequent segment must start at the endpoint of the previous segment. The player who draws a segment that intersects another segment at a point other than its endpoints lose... | [
"4"
] | [
"Ana and Beto play a turn-based game where each player draws a segment of length 1 on the plane. Ana starts by drawing the first segment, and each subsequent segment must start at the endpoint of the previous segment. The player who draws a segment that intersects another segment at a point other than its endpoints... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4a5c7c724eeee7ca16089d539b8e79ab39e873aef19a8e76536598ac7eeeb7a6 | Ana and Beto play on a blackboard where all integers from 1 to 2024 (inclusive) are written. On each turn, Ana chooses three numbers \(a, b, c\) and Beto erases them and writes one of the following numbers: \(a+b-c\), \(a-b+c\), or \(-a+b+c\). The game ends when only two numbers are left on the board and Ana cannot pla... | [
"Beto"
] | [
"Ana and Beto play on a blackboard where all integers from 1 to 2024 (inclusive) are written. On each turn, Ana chooses three numbers \\(a, b, c\\) and Beto erases them and writes one of the following numbers: \\(a+b-c\\), \\(a-b+c\\), or \\(-a+b+c\\). The game ends when only two numbers are left on the board and A... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1018c5631dade5456c9ecef8c4b33238dde9afc9b7a8c874cea06eb507a6df63 | Ana and Beto play on a grid of $2022 \times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Bo... | [
"Beto has a winning strategy."
] | [
"Ana and Beto play on a grid of $2022 \\times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments.... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-b91db39d4fb8edacd06588699f0297e7757e91d56e86ebd646b7bb3d907274f0 | Ana and Bety play a game alternating turns. Initially, Ana chooses an odd composite integer \( n \) such that \( 2^j < n < 2^{j+1} \) with \( 2 < j \). In her first turn, Bety chooses an odd composite integer \( n_1 \) such that \[ n_1 \leq \frac{1^n + 2^n + \dots + (n-1)^n}{2(n-1)^{n-1}}. \] Then, on her turn, Ana cho... | [
"Ana"
] | [
"Ana and Bety play a game alternating turns. Initially, Ana chooses an odd composite integer \\( n \\) such that \\( 2^j < n < 2^{j+1} \\) with \\( 2 < j \\). In her first turn, Bety chooses an odd composite integer \\( n_1 \\) such that\n\\[ n_1 \\leq \\frac{1^n + 2^n + \\dots + (n-1)^n}{2(n-1)^{n-1}}. \\]\nThen, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 24
} | 42 | 0 | 0 |
omr-ws-6d794dff84e951358abfd68c6848a1b651179882efaaa46d1b5659f48f90b279 | Ana and Bob take turns writing $0$s and $1$s on a board, with Ana starting first. Each writes a total of $2021$ digits. Ana wins if the resulting binary number can be expressed as the sum of two squares; otherwise, Bob wins. Determine who has a winning strategy. | [
"Bob has a winning strategy."
] | [
"Ana and Bob take turns writing $0$s and $1$s on a board, with Ana starting first. Each writes a total of $2021$ digits. Ana wins if the resulting binary number can be expressed as the sum of two squares; otherwise, Bob wins. Determine who has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-879ba4f3722b94da11710e2a66331da643053dbddd677e1edced87ed53c6d033 | Ana and Bojan are playing a game where Ana chooses positive integers $a$ and $b$. Ana gets pieces of paper with numbers $a+1, a+2, \ldots, a+2016$ and Bojan gets pieces with numbers $b+1, b+2, \ldots, b+2016$. They write the number $a+b$ on the board. In each move, Ana gives one of her pieces to Bojan, who then writes ... | [
"Bojan"
] | [
"Ana and Bojan are playing a game where Ana chooses positive integers $a$ and $b$. Ana gets pieces of paper with numbers $a+1, a+2, \\ldots, a+2016$ and Bojan gets pieces with numbers $b+1, b+2, \\ldots, b+2016$. They write the number $a+b$ on the board. In each move, Ana gives one of her pieces to Bojan, who then ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 14
} | 27 | 0 | 0 |
omr-ws-4054190fc7b13461da28a69941a7d4c13fd58d422aa729389357a15f7d072814 | Ana and Bora each have a paper strip with the letter $A$ and $B$ respectively. Every minute, one of them writes the word on the other's strip either to the left or right of their own word. After 10 minutes, let $N$ be the number of ways the word on Ana's strip can be cut into two parts and the parts exchanged to form a... | [
"248"
] | [
"Ana and Bora each have a paper strip with the letter $A$ and $B$ respectively. Every minute, one of them writes the word on the other's strip either to the left or right of their own word. After 10 minutes, let $N$ be the number of ways the word on Ana's strip can be cut into two parts and the parts exchanged to f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-bf9970a284e29a3748bfa25756e8238e65bd2178bafe597b92eca3dadd2b4102 | Ana and Celia sell various objects and obtain as many euros as objects they sold. The money obtained consists of some $10$ euro bills and less than $10$ coins of $1$ euro. They decide to distribute the money as follows: Ana takes a $10$ euro bill, then Celia, and so on, until Ana takes the last $10$ euro note, and Celi... | [
"\\{ -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \\}"
] | [
"Ana and Celia sell various objects and obtain as many euros as objects they sold. The money obtained consists of some $10$ euro bills and less than $10$ coins of $1$ euro. They decide to distribute the money as follows: Ana takes a $10$ euro bill, then Celia, and so on, until Ana takes the last $10$ euro note, and... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-cbe023c0a52d5e1e449eefa7d4a21051d08d48bab42246aa8cd6df9011ac956b | Ana and Natalia alternately play on an $n \times n$ board (Ana rolls first and $n > 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), witho... | [
"8"
] | [
"Ana and Natalia alternately play on an $n \\times n$ board (Ana rolls first and $n > 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally),... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-2cdc86249e7b34580095083227d2eb942db9a22d635d17d592d051e16f0d4671 | Ana draws three numbers from the set $\{1, 2, \ldots, 9\}$, none of which are prime, and two of which are consecutive. She finds that their product is 4 times their sum. What are the three numbers Ana drew? | [
"(2, 6, 7)"
] | [
"Ana draws three numbers from the set $\\{1, 2, \\ldots, 9\\}$, none of which are prime, and two of which are consecutive. She finds that their product is 4 times their sum. What are the three numbers Ana drew?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18cd9906c384501aa95f9e7e96bdc64899dbbc4823390445366c6f16fb6dd72b | Ana has 22 coins. She can take either 6 coins or 18 coins from her friends, or she can give 12 coins to her friends. She can perform these operations any number of times. Find the least number of coins Ana can have. | [
"4"
] | [
"Ana has 22 coins. She can take either 6 coins or 18 coins from her friends, or she can give 12 coins to her friends. She can perform these operations any number of times. Find the least number of coins Ana can have."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-40ee4e94d6f9026e1c2fcad202f61d39413e556002d68fd1979b83fd12c5a7c0 | Ana has a scale that shows which side weighs more or if both sides are equal. She has 4 weights that look the same but weigh 1001g, 1002g, 1004g, and 1005g, respectively. Is it possible for Ana to determine the weight of each weight with only 4 measurements? | [
"Yes"
] | [
"Ana has a scale that shows which side weighs more or if both sides are equal. She has 4 weights that look the same but weigh 1001g, 1002g, 1004g, and 1005g, respectively. Is it possible for Ana to determine the weight of each weight with only 4 measurements?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9dcbf939e7944b2eb81bdb40e3fc47b40f17e7c2e39780902f46293b682fdde9 | Ana has an $8 \times 8$ lattice grid of points, and Banana gives her the integers $a_1, a_2, \ldots, a_k$ which sum to exactly $64$. Banana challenges Ana to partition the $64$ points in the lattice grid into sets $S_1, S_2, \ldots, S_k$ so that for all $i \in \{1, 2, \ldots, k\}$: (i) $|S_i| = a_i$, and (ii) the set $... | [
"2080"
] | [
"Ana has an $8 \\times 8$ lattice grid of points, and Banana gives her the integers $a_1, a_2, \\ldots, a_k$ which sum to exactly $64$. Banana challenges Ana to partition the $64$ points in the lattice grid into sets $S_1, S_2, \\ldots, S_k$ so that for all $i \\in \\{1, 2, \\ldots, k\\}$:\n(i) $|S_i| = a_i$, and\n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-5a2076a722249d495cad040b5194872f66364babdef4ea66707d7f5543351fd4 | Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighing scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the ... | [
"2020"
] | [
"Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighing scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-298e88b3413379092e0aef44ace7d1490a807dbafe1a771b1760c0459558c290 | Ana has cards $1, 3, 5, 7, \ldots, 2n-1$ and Bruno has cards $2, 4, 6, 8, \ldots, 2n$. During odd turns, Bruno chooses a card first and Ana chooses second; during even turns, Ana chooses a card first and Bruno chooses second. The player with the higher card earns a point. Each card can only be used once. The game ends ... | [
"\\(\\left\\lfloor \\frac{n}{2} \\right\\rfloor\\)"
] | [
"Ana has cards $1, 3, 5, 7, \\ldots, 2n-1$ and Bruno has cards $2, 4, 6, 8, \\ldots, 2n$. During odd turns, Bruno chooses a card first and Ana chooses second; during even turns, Ana chooses a card first and Bruno chooses second. The player with the higher card earns a point. Each card can only be used once. The gam... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-cf20a4079a8d3ae7e7938b5b5f9fc44988936f9c46084e49ec474947c8840eb3 | Ana has decided to go out for a walk of exactly one kilometer each day. She lives in a city grid that is $5$ km $\times 5$ km, with each block measuring $100$ m. Her house is on a center corner. For how many days can she take different tours, if she always begins and ends at her house without passing through the same p... | [
"56"
] | [
"Ana has decided to go out for a walk of exactly one kilometer each day. She lives in a city grid that is $5$ km $\\times 5$ km, with each block measuring $100$ m. Her house is on a center corner. For how many days can she take different tours, if she always begins and ends at her house without passing through the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-001ec4274ef720c3df5c5c4a22ffd4c86288bc4669748b497a28b33b1f7fde42 | Ana is taking a 30-item exam and answers each item randomly with a $\frac{5}{11}$ probability of being correct. She gets exactly 12 items correct, and these correct answers are all consecutive. Find the probability that this happens. | [
"\\(\\frac{19 \\times 5^{12} \\times 6^{18}}{11^{30}}\\)"
] | [
"Ana is taking a 30-item exam and answers each item randomly with a $\\frac{5}{11}$ probability of being correct. She gets exactly 12 items correct, and these correct answers are all consecutive. Find the probability that this happens."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-aa2d19ad73c492184a517a36bbafac254167267761a937f125177461ef866069 | Ana places white pawns in every square of the bottom row and black pawns in every square of the top row on an $8 \times 8$ chessboard. A pawn can move upward and capture a pawn of the opposite color in the upward vertical square. Determine the minimum number of pawns that can remain on the board after all possible capt... | [
"0"
] | [
"Ana places white pawns in every square of the bottom row and black pawns in every square of the top row on an $8 \\times 8$ chessboard. A pawn can move upward and capture a pawn of the opposite color in the upward vertical square. Determine the minimum number of pawns that can remain on the board after all possibl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bbccafccae32f52a9eae4916d670ad364a01c3e64a6a7bc8974b32b2c81c21f | Ana plays a game on a $100 \times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, with no other pawns anywhere else. Each white pawn moves toward the top row and each black pawn moves toward the bottom row according to the following rule... | [
"2"
] | [
"Ana plays a game on a $100 \\times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, with no other pawns anywhere else. Each white pawn moves toward the top row and each black pawn moves toward the bottom row according to the followin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-4060295f8695de87bd962524376e85a125a5d2bf59f7198d8be34b4911627b6c | Ana selected some cells from a board of $50 \times 50$ and placed a chip in each one. Beto wants to place chips in some empty cells so that each row and each column of the board contains an even number of chips. What is the maximum number of additional chips Beto might need to place to achieve this, assuming Ana's sele... | [
"50"
] | [
"Ana selected some cells from a board of $50 \\times 50$ and placed a chip in each one. Beto wants to place chips in some empty cells so that each row and each column of the board contains an even number of chips. What is the maximum number of additional chips Beto might need to place to achieve this, assuming Ana'... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-14810c7b48ad764ec00223116d53f44e6d96e60d2df4abceb2dceb132fbe6a36 | Ana wants to cut a cubical cake with exactly four straight cuts, where each cut is parallel to the edges of the cube and passes through the interior of the cube. Compute the sum of the possible numbers of pieces of cake Ana can end up with. | [
"34"
] | [
"Ana wants to cut a cubical cake with exactly four straight cuts, where each cut is parallel to the edges of the cube and passes through the interior of the cube. Compute the sum of the possible numbers of pieces of cake Ana can end up with."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-ca6c6e923da1a270aadbc01d60af611ec8bf8ca1618df6a66b75ff908050a40e | Ana writes 7 positive integers in a circle such that the sum of the numbers is 36 and the absolute value of the difference between any two neighboring numbers is 2 or 3. Determine the maximum value of the largest number in this circle. | [
"10"
] | [
"Ana writes 7 positive integers in a circle such that the sum of the numbers is 36 and the absolute value of the difference between any two neighboring numbers is 2 or 3. Determine the maximum value of the largest number in this circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2f04cbb058194597a68d8d53b09c424ec240aba4695f536e512721d2f6747f2 | Ana writes a whole number on the board. Beto can either add 2 or subtract 3 from the number as many times as he wants. Beto wins if he can make the number equal to 2021 after a finite number of moves; otherwise, Ana wins. Which player has a winning strategy? | [
"Beto has a winning strategy."
] | [
"Ana writes a whole number on the board. Beto can either add 2 or subtract 3 from the number as many times as he wants. Beto wins if he can make the number equal to 2021 after a finite number of moves; otherwise, Ana wins. Which player has a winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-b0e8b9ba5ee3202711531b13f25942b7e19da8513bbfeb7fe1e0eabb3ec8b7d2 | Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs 5 minutes to cook, Bob's food needs 7 minutes, Cho's needs 1 minute, Dan's needs 12 minutes, and Eve's needs 5 minutes. What is the expected nu... | [
"\\(\\frac{37}{2}\\)"
] | [
"Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs 5 minutes to cook, Bob's food needs 7 minutes, Cho's needs 1 minute, Dan's needs 12 minutes, and Eve's needs 5 minutes. What is the expect... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-85331d0d32da14103fe5135e04ae0a2f00f458d71977f16341c6aa8f3b07d9f0 | Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs 5 minutes to cook, Bob's food needs 7 minutes, Cho's needs 1 minute, Dan's needs 12 minutes, and Eve's needs 5 minutes. What is the expected nu... | [
"\\(\\frac{37}{2}\\)"
] | [
"Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs 5 minutes to cook, Bob's food needs 7 minutes, Cho's needs 1 minute, Dan's needs 12 minutes, and Eve's needs 5 minutes. What is the expect... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-2ee1753ab2c8592df2ecba428e18e638c50c5c084ccb7cf05b00a663a6a1c009 | Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading... | [
"061"
] | [
"Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially he... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 13
} | 61 | 0 | 0 |
omr-ws-2c0253f8aba00949cab152055c929c485534dc505ef4f6b6bb6148e5ffd94d50 | Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along t... | [
"61"
] | [
"Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field. Ana starts biking along the edge of the field, initially heading west, Bob starts biking al... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 3,
"tir": 4
} | 28 | 0 | 0 |
omr-ws-f788cc7d51e460fc4fd3530f04d24bbc0279324fc3a766ca9e485693dd5001be | Ana, Bruno, and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played 12 games, Bruno played 21 games, and Carolina rested for 8 games. Who rested in the last game? | [
"Ana"
] | [
"Ana, Bruno, and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played 12 games, Bruno played 21 games, and Carolina rested for 8 games. Who rested in the last gam... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-8474a4d8ff0be67d1beba84143a575b7f90b6d835138d3b37c435d4cf498ae2a | Analyse the singularity at \( z = 0 \) for \( e^{\sin(1/z)} \). | [
"\\text{Essential singularity}"
] | [
"Analyse the singularity at \\( z = 0 \\) for \\( e^{\\sin(1/z)} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e29b25bc78b165d14aa31e20b50b820293c59a477a7cf977f1807299ae03dfb3 | Analytically determine \( t^* \), the optimal period a truck should be retained in the fleet to minimize the average cost of owning a truck. | [
"7.32"
] | [
"Analytically determine \\( t^* \\), the optimal period a truck should be retained in the fleet to minimize the average cost of owning a truck."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2116091f22f93a7f3ba5cb0e7665baac66b60104630f10b45041553f04845766 | Analyze the behavior of the following sum for large \( N \): \[ \sum_{m=0}^\infty \frac{N^{(1-\alpha)m}}{m!} \frac{(N-1)!}{(N+m)!} \sum_{n=1}^N \frac{(n+m)!}{(n-1)!} \] where \( \alpha \in (0,1) \). | [
"\\(\\frac{N}{2}\\)"
] | [
"Analyze the behavior of the following sum for large \\( N \\):\n\\[\n\\sum_{m=0}^\\infty \\frac{N^{(1-\\alpha)m}}{m!} \\frac{(N-1)!}{(N+m)!} \\sum_{n=1}^N \\frac{(n+m)!}{(n-1)!}\n\\]\nwhere \\( \\alpha \\in (0,1) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b7d71ca5137aba11d0d255b77b0625227f12fc7b330e3e3afa8ba458815dfba | Analyze the behavior of the function $\frac{2(b^{n}-1)}{b^{n}(b-1)n}$ as $b$ changes while $n$ remains constant, where $b, n \in \mathbb{N}$ and $b, n > 1$. | [
"0"
] | [
"Analyze the behavior of the function $\\frac{2(b^{n}-1)}{b^{n}(b-1)n}$ as $b$ changes while $n$ remains constant, where $b, n \\in \\mathbb{N}$ and $b, n > 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-bb6936829af007515dd20f8a676b5b1d00b02970b403d18d24af92d260e0409c | Analyze the behavior of the function $\frac{2(b^{n}-1)}{b^{n}(b-1)n}$ as $n$ changes while $b$ remains constant, where $b, n \in \mathbb{N}$ and $b, n > 1$. | [
"The function \\(\\frac{2(b^{n}-1)}{b^{n}(b-1)n}\\) decreases and approaches 0 as \\(n\\) increases while \\(b\\) remains constant."
] | [
"Analyze the behavior of the function $\\frac{2(b^{n}-1)}{b^{n}(b-1)n}$ as $n$ changes while $b$ remains constant, where $b, n \\in \\mathbb{N}$ and $b, n > 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-ad7a5f1d907b00edd1d2e4bee3d653501b4dc8467e40e748fd2ce223cff06154 | Analyze the behavior of the function $f(x,y) = x^2 + y^3$ at the origin and classify the type of critical point it is. | [
"\\text{saddle point}"
] | [
"Analyze the behavior of the function $f(x,y) = x^2 + y^3$ at the origin and classify the type of critical point it is."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdde50431b518730300cda0dbbc7c14626f7bd23cea32611910e86e4c3c01fdd | Analyze the behavior of the function $g(x,y) = x^2 - 2xy + y^2$ at the origin and classify the type of critical point it is. | [
"\\text{Local Minimum}"
] | [
"Analyze the behavior of the function $g(x,y) = x^2 - 2xy + y^2$ at the origin and classify the type of critical point it is."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99794ea1ec609d2a37f6dd9d981b1e2ff973581a096c8954268080b3bd989e04 | Analyze the behavior of the function $i(x,y) = x^2 + 11xy + \frac{y^2}{2} + x^4$ at the origin and classify the type of critical point it is. | [
"saddle point"
] | [
"Analyze the behavior of the function $i(x,y) = x^2 + 11xy + \\frac{y^2}{2} + x^4$ at the origin and classify the type of critical point it is."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a4fdba0a6c2ed14e8828a2fb37a9f71a82c6c0995877e7e504400832ede1a433 | Analyze the behavior of the solution to the differential equation $\sqrt{4-x^{2}} \frac{dy}{dx} = \sqrt{4-y^{2}}$ for the following intervals: (a) $-2 < x \leq -1$ (b) $x \geq 2$ (c) $x \leq -2$ | [
"(a) $y = -2$, (b) Not defined, (c) Not defined"
] | [
"Analyze the behavior of the solution to the differential equation $\\sqrt{4-x^{2}} \\frac{dy}{dx} = \\sqrt{4-y^{2}}$ for the following intervals:\n(a) $-2 < x \\leq -1$\n(b) $x \\geq 2$\n(c) $x \\leq -2$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-ec4c3de1903f5abccc0646c06316402d74bb9f801ac0f8b7dd930036ab4e7cb9 | Analyze the convergence (or divergence) of the limit \(\lim_{x \to \infty} \frac{e^{-x}}{\sin{(5x)}+1}\). | [
"\\text{The limit diverges}"
] | [
"Analyze the convergence (or divergence) of the limit \\(\\lim_{x \\to \\infty} \\frac{e^{-x}}{\\sin{(5x)}+1}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ce81edba54285a4939ae77b75b8f50f22b27db31d1239859cec76853542f4e7e | Analyze the convergence and absolute convergence of the integral \(\int\limits_{0}^{\infty} \frac{(1+x)\ln(1+x)\cos x}{x^{\frac{3}{2}}} \, dx\). | [
"Converges conditionally but not absolutely."
] | [
"Analyze the convergence and absolute convergence of the integral \\(\\int\\limits_{0}^{\\infty} \\frac{(1+x)\\ln(1+x)\\cos x}{x^{\\frac{3}{2}}} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-0ba88b51b8ebb2badb0e8293760e0f18d26148c28a37445fea14d7d47f791f89 | Analyze the convergence of the integral ${\int_{0}^{1}\frac{dx}{\sqrt{\ln{(1/x)}}}}$. | [
"Converges"
] | [
"Analyze the convergence of the integral ${\\int_{0}^{1}\\frac{dx}{\\sqrt{\\ln{(1/x)}}}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-db2494c82460f0d92f845d3fb0a0c6e2794979adb6ae4782f70dfc2178e85c11 | Analyze the convergence of the sequence \(a_n = \frac{1 \cdot 4 \cdot \ldots \cdot (3n-2)}{2 \cdot 5 \cdot \ldots \cdot (3n-1)}\). | [
"0"
] | [
"Analyze the convergence of the sequence \\(a_n = \\frac{1 \\cdot 4 \\cdot \\ldots \\cdot (3n-2)}{2 \\cdot 5 \\cdot \\ldots \\cdot (3n-1)}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-523b814230ff27df36e525ab88f92fd170b63141df6d0b05b00c7f6f3dbb771b | Analyze the convergence of the series $\sum_{n=1}^{\infty}\frac{(n!)^2x^n}{(2n)!}$, particularly at the endpoints of the interval of convergence. | [
"(-4, 4)"
] | [
"Analyze the convergence of the series $\\sum_{n=1}^{\\infty}\\frac{(n!)^2x^n}{(2n)!}$, particularly at the endpoints of the interval of convergence."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3dbcb0a69d93a5df481b387b2edb59873fa4354841ac2d759684a67bedb51042 | Analyze the convergence of the series $\sum_{n=1}^{\infty}\frac{n!x^n}{n^n}$, particularly at the endpoints of the interval of convergence. | [
"The series converges on the interval \\((-e, e)\\) and diverges at the endpoints \\(x = e\\) and \\(x = -e\\)."
] | [
"Analyze the convergence of the series $\\sum_{n=1}^{\\infty}\\frac{n!x^n}{n^n}$, particularly at the endpoints of the interval of convergence."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8af47354d545dcdfeb3e790a330bcbf7d56dec1b44004abea953f6572ef585f7 | Analyze the convergence of the series \( \sum_{n = 2}^{\infty}\frac{\cos\left(\frac{3\pi n}{2}\right)}{\log(n)} \) and \( \sum_{n = 2}^{\infty}\frac{\sin\left(\frac{3\pi n}{2}\right)}{\log(n)} \). | [
"The series converges."
] | [
"Analyze the convergence of the series \\( \\sum_{n = 2}^{\\infty}\\frac{\\cos\\left(\\frac{3\\pi n}{2}\\right)}{\\log(n)} \\) and \\( \\sum_{n = 2}^{\\infty}\\frac{\\sin\\left(\\frac{3\\pi n}{2}\\right)}{\\log(n)} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 10
} | 23 | 0 | 0 |
omr-ws-1903a8ec5df82fc198fc94893ab14690e97bc951238b1694a74ab5677bfd9959 | Analyze the convergence of the series \( \sum_{n=1}^\infty a_{n} \), where \( a_{n}= 1/n^{2} \) if \( n \) is odd, and \( -1/n \) if \( n \) is even. | [
"\\text{The series diverges.}"
] | [
"Analyze the convergence of the series \\( \\sum_{n=1}^\\infty a_{n} \\), where \\( a_{n}= 1/n^{2} \\) if \\( n \\) is odd, and \\( -1/n \\) if \\( n \\) is even."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bd9a271c69385884379b462604ebcbe63ffdfa307ff1c7b05608366a05dd65ba | Analyze the convergence of the series \( \sum_{n=2}^{\infty}\frac{(-1)^{n}}{(n+(-1)^{n})^{s}} \). | [
"0 < s \\leq 1"
] | [
"Analyze the convergence of the series \\( \\sum_{n=2}^{\\infty}\\frac{(-1)^{n}}{(n+(-1)^{n})^{s}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-1fdba59dd4ec716c49bfbcb4a782cfb50076fd52a3d3d4228f14aa392924acd5 | Analyze the expression $(a+b+c)^5 - a^5 - b^5 - c^5$ into a factorial form. | [
"\\(5(a+b)(b+c)(c+a)(a^2 + b^2 + c^2 + ab + bc + ca)\\)"
] | [
"Analyze the expression $(a+b+c)^5 - a^5 - b^5 - c^5$ into a factorial form."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-6d44789a8b09ef54827c23fcbe71656fdb834d206823b820e9c389f76f54ea52 | Analyze the expression \(a^{10} + a^5 + 1\). | [
"(a^2 + a + 1)(a^8 - a^7 + a^5 - a^4 + a^3 - a + 1)"
] | [
"Analyze the expression \\(a^{10} + a^5 + 1\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd8393e1319454c4b23030f26dd2ae999d15fc2d8d1f9c3c2e947ef0d7b1510f | Analyze the function \( f'(x) \). Let \( M_1 \) and \( M_2 \) be points on the graph of \( f(x) \) with different \( x \)-coordinates. Determine the position of the arc \( M_1M_2 \) of the graph with respect to the segment \( M_1M_2 \). | [
"The arc \\( M_1M_2 \\) lies below the segment \\( M_1M_2 \\)."
] | [
"Analyze the function \\( f'(x) \\). Let \\( M_1 \\) and \\( M_2 \\) be points on the graph of \\( f(x) \\) with different \\( x \\)-coordinates. Determine the position of the arc \\( M_1M_2 \\) of the graph with respect to the segment \\( M_1M_2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-e55ebca52b6db1f47e5364ddbcef352d4a6b777cdc4c27d184dd43fdbc38e276 | Analyze the function \( f(x) = \arcsin\left(\frac{2x}{1+x^2}\right) \) and determine its maximum and minimum values, noting that these are not stationary points. | [
"-\\frac{\\pi}{2}"
] | [
"Analyze the function \\( f(x) = \\arcsin\\left(\\frac{2x}{1+x^2}\\right) \\) and determine its maximum and minimum values, noting that these are not stationary points."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-5178efcbae01eb0e81d9fc2fef7248095d820b3cbe0454d4ebbbb43d9c3be764 | Analyze the function \( f(x) = \frac{\sqrt{1+|x+2|}}{1-|x|} \) to determine if it is possible to sketch its graph. | [
"\\text{Yes}"
] | [
"Analyze the function \\( f(x) = \\frac{\\sqrt{1+|x+2|}}{1-|x|} \\) to determine if it is possible to sketch its graph."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cce1f434e9383813fa5a1862ec071139ab7f40e4f1210fce93bc08847664dc1 | Analyze the function \( f(x) = \frac{x}{x^2 - 9} \) to find its critical points, intervals of increase/decrease, and concavity. Specifically, determine: - The first derivative \( f'(x) \) and its critical points. - The intervals where \( f(x) \) is increasing or decreasing. - The second derivative \( f''(x) \) and its ... | [
"(0, 0)"
] | [
"Analyze the function \\( f(x) = \\frac{x}{x^2 - 9} \\) to find its critical points, intervals of increase/decrease, and concavity. Specifically, determine:\n- The first derivative \\( f'(x) \\) and its critical points.\n- The intervals where \\( f(x) \\) is increasing or decreasing.\n- The second derivative \\( f'... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-83d205e3de7e4027d15be6e0cbc106b964f472701b1665bb60e56711f2eacd09 | Analyze the function \( f(x) = \int \sin(x) \, dx \) and identify all of its features, including any unusual properties related to limits and continuity. | [
"f(x) = -\\cos(x) + C"
] | [
"Analyze the function \\( f(x) = \\int \\sin(x) \\, dx \\) and identify all of its features, including any unusual properties related to limits and continuity."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff11d84b3ab750822286f6f2719652dc28ced777c8b49bbff34d3998d945ca58 | Analyze the function \( f(x) = |x^2 - 4| + |9 - x^2| \). | [
"5"
] | [
"Analyze the function \\( f(x) = |x^2 - 4| + |9 - x^2| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-e8add572a49ecb3ff9fcbc842d8d466e20b5552602107dc258ca6658e4d7cdea | Analyze the function \( f(z) = \sin(e^{1/z}) \). | [
"z = 0 \\text{ is an essential singularity}"
] | [
"Analyze the function \\( f(z) = \\sin(e^{1/z}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-409faf62da32b020c510d92b8230e88337af1eb6afa7d5887243904f3ae1bcd2 | Analyze the function \( g(x) = f(f(x)) \) and draw its graph. Determine the position of the graph of \( g(x) \) with respect to the line \( y = x \). Find the tangents to the graph at points with \( x \)-coordinates \( 0 \) and \( 1 \). | [
"\\( y = 0 \\) and \\( y = 1 \\)"
] | [
"Analyze the function \\( g(x) = f(f(x)) \\) and draw its graph. Determine the position of the graph of \\( g(x) \\) with respect to the line \\( y = x \\). Find the tangents to the graph at points with \\( x \\)-coordinates \\( 0 \\) and \\( 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-1964517021e8fab0bcad1d3f9b28ff4701b76cf91e0b280a3019a3d6baca9a82 | Analyze the function \( h(z) = \frac{e^z - 1}{e^{2z} - 1} \) to determine its singularities and classify them. Specifically, determine if the singularity at \( z = 0 \) is removable. | [
"The singularity at \\( z = 0 \\) is removable."
] | [
"Analyze the function \\( h(z) = \\frac{e^z - 1}{e^{2z} - 1} \\) to determine its singularities and classify them. Specifically, determine if the singularity at \\( z = 0 \\) is removable."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-ba3856cb6024c358505ee0b2948a0aef9579d4395606c115bd845c2269196681 | Analyze the function \( |y| = \{ \sin x \} \), where \( \{ k \} \) denotes the fractional part function and \( |y| \) denotes the absolute value function. | [
"|y| = \\{ \\sin x \\}"
] | [
"Analyze the function \\( |y| = \\{ \\sin x \\} \\), where \\( \\{ k \\} \\) denotes the fractional part function and \\( |y| \\) denotes the absolute value function."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-3881a506629a4a9b70a49d7cc1195795b0974dc43387c2abc2d4620e0b01bed5 | Analyze the function \(f(x)\) to determine where it is increasing and decreasing, and find the value of \(x\) that maximizes the angle \(\theta\). | [
"\\( x = 4.5 \\)"
] | [
"Analyze the function \\(f(x)\\) to determine where it is increasing and decreasing, and find the value of \\(x\\) that maximizes the angle \\(\\theta\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bc9b150c8adaa48194c73b5c1173faa4736fd015bb8321c6c677b7b2cc355c4 | Analyze the functions given by the equations: \[ \rho_1 = \frac{1}{2} \left( 3 \cos \theta - 1 + \sqrt{(1 - 3 \cos \theta)^2 + 12 \cos \theta} \right) \] \[ \rho_2 = \frac{1}{2} \left( 3 \cos \theta - 1 - \sqrt{(1 - 3 \cos \theta)^2 + 12 \cos \theta} \right) \] | [
"\\rho = 3 \\cos \\theta"
] | [
"Analyze the functions given by the equations:\n\\[ \\rho_1 = \\frac{1}{2} \\left( 3 \\cos \\theta - 1 + \\sqrt{(1 - 3 \\cos \\theta)^2 + 12 \\cos \\theta} \\right) \\]\n\\[ \\rho_2 = \\frac{1}{2} \\left( 3 \\cos \\theta - 1 - \\sqrt{(1 - 3 \\cos \\theta)^2 + 12 \\cos \\theta} \\right) \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-0f7b2053bab3f2b141261c230ef6ca1fae46ffc13fe674ce4f69a000e2ffecd3 | Analyze the inequality \( e^{\frac{\ln x}{x-1}} \geq \frac{e}{3} \left(1 + \frac{1}{x} + \frac{1}{\sqrt{x}}\right) \) and determine how it was derived, considering its accuracy between \( x = 1 \) and \( x = 1.5 \). | [
"e^{\\frac{\\ln x}{x-1}} = \\frac{e}{3} \\left(1 + \\frac{1}{x} + \\frac{1}{\\sqrt{x}}\\right) \\text{ for } 1 < x \\leq 1.5"
] | [
"Analyze the inequality \\( e^{\\frac{\\ln x}{x-1}} \\geq \\frac{e}{3} \\left(1 + \\frac{1}{x} + \\frac{1}{\\sqrt{x}}\\right) \\) and determine how it was derived, considering its accuracy between \\( x = 1 \\) and \\( x = 1.5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-cc0c4eefc24abe4e555a1e6b9a4d42ba88076a7f2f639f6b0cf651b220e53fa7 | Analyze the inhomogeneous equation \( f(x^2) = R(x) f(x) + c(x) \). | [
"f(x) = \\frac{c(1)}{(1 - R(1)) \\prod_{n=1}^{\\infty} R(x^{1/2^n})} - \\sum_{k=1}^{\\infty} \\frac{c(x^{1/2^k})}{\\prod_{m=1}^{k} R(x^{1/2^m})}"
] | [
"Analyze the inhomogeneous equation \\( f(x^2) = R(x) f(x) + c(x) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fd69cb0b404c9a60d5f5a30d52bfd0299bb4626ef4bcdd5c2574286b94af102 | Analyze the limit $\lim_{y \rightarrow \infty} y E\left[\frac{1}{X}; X>y\right]$ where $X \geq 0$ is a real-valued random variable. | [
"0"
] | [
"Analyze the limit $\\lim_{y \\rightarrow \\infty} y E\\left[\\frac{1}{X}; X>y\\right]$ where $X \\geq 0$ is a real-valued random variable."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ccb14ab1df64ce164858f7ee377bcbca830e68692f0f02ef35b89f584649e5a | Analyze the monotonicity and determine whether the function \( f(x) = \int_{0}^{x} \sqrt{1 - \sin^2 t - \sin^4 t} \, dt \) is even or odd. | [
"\\( f(x) \\) is non-decreasing and odd."
] | [
"Analyze the monotonicity and determine whether the function \\( f(x) = \\int_{0}^{x} \\sqrt{1 - \\sin^2 t - \\sin^4 t} \\, dt \\) is even or odd."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-7348becc222e8bf583901a467f566bd56ff2309069437bd05273ca80303c5c53 | Analyze the number of solutions for constructing an equilateral triangle inscribed in a circle $\omega$ with a point $P$ on its perimeter, depending on the location of $P$ in $\omega$. | [
"2"
] | [
"Analyze the number of solutions for constructing an equilateral triangle inscribed in a circle $\\omega$ with a point $P$ on its perimeter, depending on the location of $P$ in $\\omega$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-c0d8f613ff65145f54a7504f84b0f96c96a6f65a79c290ad48b1033b9d4ac76f | Analyze the parametric equations \( x(T) = \sin(At) \) and \( y(T) = \cos(Bt) \). Determine the effects of the parameters \( A \) and \( B \) on the shape and orientation of the curve. Identify conditions under which the curve is closed versus open. | [
"\\text{The curve is closed if } \\frac{A}{B} \\text{ is rational and open otherwise.}"
] | [
"Analyze the parametric equations \\( x(T) = \\sin(At) \\) and \\( y(T) = \\cos(Bt) \\). Determine the effects of the parameters \\( A \\) and \\( B \\) on the shape and orientation of the curve. Identify conditions under which the curve is closed versus open."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-0956bb8458b8a7cd953a2a6f720839c35abb1ed16980e2a4f03358bd69d5a6de | Analyze the quadratic form \( P = -24x^2 - 7y^2 - 3z^2 + 14xy + 16xz - 2yz \). | [
"\\text{indefinite}"
] | [
"Analyze the quadratic form \\( P = -24x^2 - 7y^2 - 3z^2 + 14xy + 16xz - 2yz \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6033ca9f78d2fd43151e1bf363a4fe7cbc7f90b75075d49f65964446d831726b | Analyze the recurrence relation given by \( a_{0} = 1 \) and \( a_{n} = \frac{a_{n-1} + 2a_{n-2} + 3a_{n-3} + \cdots + n a_{0}}{n} \). Find as much as possible about the behavior and properties of the sequence. | [
"\\( a_n \\approx \\frac{e^{2\\sqrt{n}}}{2n^{3/4}\\sqrt{2\\pi e}} \\left(1 + O(n^{-1/2})\\right) \\)"
] | [
"Analyze the recurrence relation given by \\( a_{0} = 1 \\) and \\( a_{n} = \\frac{a_{n-1} + 2a_{n-2} + 3a_{n-3} + \\cdots + n a_{0}}{n} \\). Find as much as possible about the behavior and properties of the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 6
} | 27 | 0 | 0 |
omr-ws-cda624fac53acabd036ef58fe39ce21841a93524c752d7dc0960dc268da214c6 | Analyze the resulting equation from applying integration by parts twice to $\int e^{ax}f(x)dx$ and solve it for $f(x)$. | [
"f(x) = e^{-ax} \\frac{d}{dx} \\left( \\int e^{ax} f(x) \\, dx \\right)"
] | [
"Analyze the resulting equation from applying integration by parts twice to $\\int e^{ax}f(x)dx$ and solve it for $f(x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-9d7597af77224d43019a64b1098b67c4b0f0f70012d3a048239d3b11ed96b31c | Analyze the stability of the equilibrium point of the mass-spring system described by the differential equation \[ \frac{d}{dt}\begin{bmatrix} x\\ y\\ \end{bmatrix} = \begin{bmatrix} 0 & 1\\ -\frac{k}{m} & -\frac{f}{m}\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ \end{bmatrix}. \] | [
"\\text{The equilibrium is asymptotically stable if } f > 0 \\text{ and stable but not asymptotically if } f = 0."
] | [
"Analyze the stability of the equilibrium point of the mass-spring system described by the differential equation\n\\[\n\\frac{d}{dt}\\begin{bmatrix}\n x\\\\\n y\\\\\n \\end{bmatrix} = \n \\begin{bmatrix}\n 0 & 1\\\\\n -\\frac{k}{m} & -\\frac{f}{m}\\\\\n \\end{bmatrix} \n \\begin{bmatrix}\n x\\\\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-37e067289af960263d43e8cd391a6d67419647a7e242a606185977d3d98b2e70 | Analyze the stability of the equilibrium point of the system described by the differential equation $\frac{d}{dt}\begin{bmatrix} x\\ y\\ \end{bmatrix} = \begin{bmatrix} 0 & 1\\ -\frac{k}{m} & -\frac{f}{m}\\ \end{bmatrix} \begin{bmatrix} x\\ y\\ \end{bmatrix}$. | [
"The equilibrium point of the system is at \\( (x, y) = (0, 0) \\) and it is asymptotically stable assuming \\( f > 0 \\). The nature of the stability (overdamped, critically damped, or underdamped) depends on the relationship between \\( f^2 \\) and \\( 4km \\)."
] | [
"Analyze the stability of the equilibrium point of the system described by the differential equation $\\frac{d}{dt}\\begin{bmatrix} x\\\\ y\\\\ \\end{bmatrix} = \\begin{bmatrix} 0 & 1\\\\ -\\frac{k}{m} & -\\frac{f}{m}\\\\ \\end{bmatrix} \\begin{bmatrix} x\\\\ y\\\\ \\end{bmatrix}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 7
} | 17 | 0 | 0 |
omr-ws-66b84c0eecb023b391a021e813ee5e44092a9f46989ffb1e2bd48da97bef5440 | Analyze the uniform continuity of the function \( f: \mathbb{R}^2 \to \mathbb{R} \) defined as: \[ f(x,y) = \begin{cases} \frac{a}{\sqrt{x^2 + y^2}} & \text{if } (x,y) \in \mathbb{R}^2 - B_{d_2}(0_{\mathbb{R}^2}, 2) \\ c(x^2 + y^2) & \text{if } (x,y) \in B_{d_2}(0_{\mathbb{R}^2}, 2) \end{cases} \] on the ball \( B_{d_2... | [
"a = 8c"
] | [
"Analyze the uniform continuity of the function \\( f: \\mathbb{R}^2 \\to \\mathbb{R} \\) defined as:\n\\[ f(x,y) = \\begin{cases} \n\\frac{a}{\\sqrt{x^2 + y^2}} & \\text{if } (x,y) \\in \\mathbb{R}^2 - B_{d_2}(0_{\\mathbb{R}^2}, 2) \\\\\nc(x^2 + y^2) & \\text{if } (x,y) \\in B_{d_2}(0_{\\mathbb{R}^2}, 2)\n\\end{ca... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-87c3ed28bc9e733ddd50e8d8611527991e84ba7e28091d9dc83374ebe3548e90 | Analyze what happens on the circle of convergence of the series \( \sum_{n=0}^{\infty} \left(\frac{n}{2n^2+3}\right) z^{4n} \). | [
"The series diverges when \\( z = 1, i, -1, -i \\) and converges conditionally when \\( |z| = 1 \\) and \\( z \\neq 1, i, -1, -i \\)."
] | [
"Analyze what happens on the circle of convergence of the series \\( \\sum_{n=0}^{\\infty} \\left(\\frac{n}{2n^2+3}\\right) z^{4n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3747180a89a8c025b22f16994c9d6635c60f315dd530cee6eb9eaae9dae9a210 | Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ of integers. Banana can make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, and Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to determine Anana's $n$-tuple with certainty? | [
"n + 1"
] | [
"Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ of integers. Banana can make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, and Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to determine Anana's $n$-tuple with certainty?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-06b77c2145811779378180d2ef79b94c198580ff488f9c8b2a7d523c9812793e | Anastasia and Balthazar need to go to the grocery store, which is 100 km away. Anastasia walks at 5 km/hr, Balthazar walks at 4 km/hr, and both bike at 10 km/hr. They can drop off the bike for the other to use. What is the shortest amount of time necessary for both of them to get to the grocery store? | [
"16"
] | [
"Anastasia and Balthazar need to go to the grocery store, which is 100 km away. Anastasia walks at 5 km/hr, Balthazar walks at 4 km/hr, and both bike at 10 km/hr. They can drop off the bike for the other to use. What is the shortest amount of time necessary for both of them to get to the grocery store?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0889f8894aa6820d6954c95bdb3421674af43fb074ee857a05d0d09dd2afd8e8 | Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of two suits are split between Reid and Po-Ru's hands. Let \( N \) denote the number of ways 13 cards can be dealt to each player such that this is the case. Determine the remainder obtained when \( N \) is divided by 1... | [
"0"
] | [
"Anders, Po-Ru, Reid, and Aaron are playing Bridge. After one hand, they notice that all of the cards of two suits are split between Reid and Po-Ru's hands. Let \\( N \\) denote the number of ways 13 cards can be dealt to each player such that this is the case. Determine the remainder obtained when \\( N \\) is div... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a94471da250e5abff0dd6d17ab05bdea66099dbeacb771f263c0bd88f786118 | Anderson writes an infinitely long decimal number by randomly appending one of his favorite numbers ($1$, $7$, or $33$) to the end of the decimal he has already written. If the expected value of the number Anderson wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $... | [
"392"
] | [
"Anderson writes an infinitely long decimal number by randomly appending one of his favorite numbers ($1$, $7$, or $33$) to the end of the decimal he has already written. If the expected value of the number Anderson wrote down is of the form $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-debbb2ed531a08b1411225490b1c01b3dea96aae68d0451984d6734fb2db394e | Anderson writes an infinitely long decimal number by randomly appending one of his favorite numbers ($1$, $7$, or $33$) to the end of the decimal he has already written. If the expected value of the number Anderson wrote down is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $... | [
"392"
] | [
"Anderson writes an infinitely long decimal number by randomly appending one of his favorite numbers ($1$, $7$, or $33$) to the end of the decimal he has already written. If the expected value of the number Anderson wrote down is of the form $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-9608233491bce30e1cdc985f168dd7517e567e0d249d4bd4dd09aa9fe217ea82 | Andi, Beni, Coki, Doni, and Edo are playing a game where each player is either a mousedeer (who always tells the truth) or a wolf (who always lies). Andi says that Beni is a mousedeer. Coki says Doni is a wolf, while Edo says Andi is not a wolf. Beni says Coki is not a mousedeer, and Doni says that Edo and Andi are dif... | [
"4"
] | [
"Andi, Beni, Coki, Doni, and Edo are playing a game where each player is either a mousedeer (who always tells the truth) or a wolf (who always lies). Andi says that Beni is a mousedeer. Coki says Doni is a wolf, while Edo says Andi is not a wolf. Beni says Coki is not a mousedeer, and Doni says that Edo and Andi ar... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0031f4839b40f426e550c65e4fccec37a1db3d2ae229a1e36561615a1d39a27 | Andi, Bernan, and Code can complete a job individually in 3 hours, 4 hours, and 6 hours, respectively. Andi and Code start working together and are later joined by Bernan. The entire job is completed in 1.5 hours. How long did Bernan work? | [
"1"
] | [
"Andi, Bernan, and Code can complete a job individually in 3 hours, 4 hours, and 6 hours, respectively. Andi and Code start working together and are later joined by Bernan. The entire job is completed in 1.5 hours. How long did Bernan work?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-216755625bd785430e504cb3375fa65663685f3cce78661ec48b780efa708a15 | Andre has six cards with one number written on each side of each card. The pairs of numbers on the cards are $(5,12)$, $(3,11)$, $(0,16)$, $(7,8)$, $(4,14)$, and $(9,10)$. The cards can be placed in any order in the blank spaces of the figure. $$\underline{\quad} + \underline{\quad} + \underline{\quad} - \underline{\qu... | [
"-26"
] | [
"Andre has six cards with one number written on each side of each card. The pairs of numbers on the cards are $(5,12)$, $(3,11)$, $(0,16)$, $(7,8)$, $(4,14)$, and $(9,10)$. The cards can be placed in any order in the blank spaces of the figure. $$\\underline{\\quad} + \\underline{\\quad} + \\underline{\\quad} - \\u... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-316c6bdcb7a0d325e916badaf5de58c699e3bc8bfa2ef5bb2535b7dc48efefd7 | Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. Andrea divides the table into $33$ zones, each consisting of $3$ contiguous cells aligned vertically or horizontally. Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell such that the sum of the numbers in each zone is $5$. Let $S$ be t... | [
"17"
] | [
"Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. Andrea divides the table into $33$ zones, each consisting of $3$ contiguous cells aligned vertically or horizontally. Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell such that the sum of the numbers in each zone is $5$. Let $S$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b205432c2a4071a1233616796c2f56456e97f9c57adf5a62492b779ad62d2d5 | Andrea has completed 3/7 of a 168 km canoe trip after three days. How many kilometers per day must she average for the remaining three days to finish the trip? | [
"32"
] | [
"Andrea has completed 3/7 of a 168 km canoe trip after three days. How many kilometers per day must she average for the remaining three days to finish the trip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-321833f2df6f84479c043fc4701fe298a6ffa33e7015c9be6763d9fc1a0a358f | Andrea is an algebra teacher and wants to make an algebra problem for her students. She decides the question will be of the form \(\frac{2^ax^by^c}{x^d(ye)^f}\), but she hasn’t decided what \(a\), \(b\), \(c\), \(d\), \(e\), \(f\) should be yet. If for each letter \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), she picks an ... | [
"40"
] | [
"Andrea is an algebra teacher and wants to make an algebra problem for her students. She decides the question will be of the form \\(\\frac{2^ax^by^c}{x^d(ye)^f}\\), but she hasn’t decided what \\(a\\), \\(b\\), \\(c\\), \\(d\\), \\(e\\), \\(f\\) should be yet. If for each letter \\(a\\), \\(b\\), \\(c\\), \\(d\\),... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-db17eb3fcf584816a5dd9eb959dc7f82ac219f89f9fe6a62d719b5712d608ee9 | Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time? | [
"60"
] | [
"Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b1bf6d7a329acd56b0982823dd467f9e46831651b418a76bcc46c815e6e5ad8 | Andrej and Barbara play a game with two strips of newspaper of lengths \(a\) and \(b\). They alternately cut from any end of any of the strips a piece of length \(d\). The player who cannot make such a cut loses the game. Barbara starts the game. Determine how the lengths \(a\) and \(b\) determine the winner. | [
"Barbara wins if \\( \\lfloor \\frac{a}{d} \\rfloor + \\lfloor \\frac{b}{d} \\rfloor \\) is odd, and Andrej wins if it is even."
] | [
"Andrej and Barbara play a game with two strips of newspaper of lengths \\(a\\) and \\(b\\). They alternately cut from any end of any of the strips a piece of length \\(d\\). The player who cannot make such a cut loses the game. Barbara starts the game. Determine how the lengths \\(a\\) and \\(b\\) determine the wi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-d56e1612e923660c365788c6cd33bedde1a9d279b2c32b367b6929fd2bcc91dc | Andrew and Barbara each roll a pair of 6-sided fair dice. Points are awarded based on the value of the dice they roll. If the total value of Andrew's dice is even, one of the dice he rolled is chosen at random, and its value is added to his total points. If the total value of Barbara's dice is odd, one of the dice she ... | [
"The game is fair."
] | [
"Andrew and Barbara each roll a pair of 6-sided fair dice. Points are awarded based on the value of the dice they roll. If the total value of Andrew's dice is even, one of the dice he rolled is chosen at random, and its value is added to his total points. If the total value of Barbara's dice is odd, one of the dice... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75eab1f533d7e97c687f644e8a551b2068876eea82edd13c833f9f4aba1da985 | Andrew and Barry play a game with two heaps containing $a$ and $b$ pebbles, respectively. In each round, the player whose turn it is chooses a positive integer $k$, and the other player takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, the entire heap is taken). ... | [
"Barry"
] | [
"Andrew and Barry play a game with two heaps containing $a$ and $b$ pebbles, respectively. In each round, the player whose turn it is chooses a positive integer $k$, and the other player takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, the entire heap is tak... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-ceeb5d4bfe9dcc3ad79ac472ee2f3ad5b28e307eae86ae172e5269e3e9f584a4 | Andrew and Beatrice attempted a total of 105 free throws between them, with each person taking at least one free throw. If Andrew made exactly \( \frac{1}{3} \) of his free throw attempts and Beatrice made exactly \( \frac{3}{5} \) of her free throw attempts, what is the highest number of successful free throws they co... | [
"59"
] | [
"Andrew and Beatrice attempted a total of 105 free throws between them, with each person taking at least one free throw. If Andrew made exactly \\( \\frac{1}{3} \\) of his free throw attempts and Beatrice made exactly \\( \\frac{3}{5} \\) of her free throw attempts, what is the highest number of successful free thr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-d3d29f66f6b22eae74326e1506484644f3e5c56d3cab523d760e2ee8cc02ffa2 | Andrew and Ben each independently choose a number at random from the set of the first $2016$ positive integers. Let $a$ be Andrew's number and $b$ be Ben's number. What is the probability that $a^2 - b^2$ is divisible by $7$? | [
"\\(\\frac{13}{49}\\)"
] | [
"Andrew and Ben each independently choose a number at random from the set of the first $2016$ positive integers. Let $a$ be Andrew's number and $b$ be Ben's number. What is the probability that $a^2 - b^2$ is divisible by $7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d1cbef56a84644576602d9326e273b0c80e85d7fd8bd3587665551554750168c | Andrew and Ben each independently choose a number at random from the set of the first 2016 positive integers. Let \( a \) be Andrew's number and \( b \) be Ben's number. What is the probability that \( a^2 - b^2 \) is divisible by 7? | [
"\\dfrac{13}{49}"
] | [
"Andrew and Ben each independently choose a number at random from the set of the first 2016 positive integers. Let \\( a \\) be Andrew's number and \\( b \\) be Ben's number. What is the probability that \\( a^2 - b^2 \\) is divisible by 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e14bc9397e446bd83e6d56b7ba8e793e158a3ca3136475fd2f87c7e74ecabacc | Andrew and Ben each independently choose a number at random from the set of the first 2016 positive integers. Let \( a \) be Andrew’s number and \( b \) be Ben’s number. What is the probability that \( a^2 - b^2 \) is divisible by 7? | [
"\\(\\frac{1}{7}\\)"
] | [
"Andrew and Ben each independently choose a number at random from the set of the first 2016 positive integers. Let \\( a \\) be Andrew’s number and \\( b \\) be Ben’s number. What is the probability that \\( a^2 - b^2 \\) is divisible by 7?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c95e26344095d037dd8dd5efaf91fa69743f61fff2e35f540b037ace85ab2460 | Andrew and Bogdan ride bicycles on a rectangular route. Andrew travels the route $AEFD$, Bogdan travels $ABCD$. The speed ratio of Andrew and Bogdan is $2:7$. Both start at $S$. How many full routes will each take before their next meeting at $S$? | [
"7"
] | [
"Andrew and Bogdan ride bicycles on a rectangular route. Andrew travels the route $AEFD$, Bogdan travels $ABCD$. The speed ratio of Andrew and Bogdan is $2:7$. Both start at $S$. How many full routes will each take before their next meeting at $S$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
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