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-9768028b7bddfda0daa790a67d61cab9087d1e9d3ce5a43733083ecf300bece4 | A \(3n\)-digit positive integer (in base \(10\)) containing no zero is said to be quad-perfect if the number is a perfect square and each of the three numbers obtained by viewing the first \(n\) digits, the middle \(n\) digits, and the last \(n\) digits as three \(n\)-digit numbers is in itself a perfect square. (For e... | [
""
] | [
"A \\(3n\\)-digit positive integer (in base \\(10\\)) containing no zero is said to be quad-perfect if the number is a perfect square and each of the three numbers obtained by viewing the first \\(n\\) digits, the middle \\(n\\) digits, and the last \\(n\\) digits as three \\(n\\)-digit numbers is in itself a perfe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 1 |
omr-ws-37fb7515ffe19d9ecf1c5691508ab7c0eda55cb2c66773e8d840d494de0545bf | A \(4 \times 4 \times 4\) cube is composed of 64 unit cubes. The faces of 16 unit cubes are colored red. An arrangement of the cubes is "intriguing" if there is exactly 1 red unit cube in every \(1 \times 1 \times 4\) rectangular box composed of 4 unit cubes. Determine the number of "intriguing" colorings. | [
"576"
] | [
"A \\(4 \\times 4 \\times 4\\) cube is composed of 64 unit cubes. The faces of 16 unit cubes are colored red. An arrangement of the cubes is \"intriguing\" if there is exactly 1 red unit cube in every \\(1 \\times 1 \\times 4\\) rectangular box composed of 4 unit cubes. Determine the number of \"intriguing\" colori... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f7b10cfe5daa697bbaba07dae63ad5e0a05124660cfc209ae4ca63bf049cfe9 | A \(4 \times 4\) block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? \[ \begin{array}{... | [
"4"
] | [
"A \\(4 \\times 4\\) block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums?\n\n\\[\n\... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c28e1ca9f9b53eb51ae263512c5f34d7dc172213a3b51109c1d9c14caabc24b | A \(4 \times 4\) square board is colored like a chessboard and numbered from 1 to 16. Andi wants to cover the board with 7 cards of \(2 \times 1\) tiles, discarding two squares. Find the number of ways to discard two squares. | [
"64"
] | [
"A \\(4 \\times 4\\) square board is colored like a chessboard and numbered from 1 to 16. Andi wants to cover the board with 7 cards of \\(2 \\times 1\\) tiles, discarding two squares. Find the number of ways to discard two squares."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-d382f4fc11dd51f28432d9dcacb74d33dea5a03a9dbd0d6a4782e287770ea383 | A \(5 \times 5 \times 5\) cube is painted on its surface. Find the ratio of the number of painted to unpainted unit cubes. | [
"\\dfrac{98}{27}"
] | [
"A \\(5 \\times 5 \\times 5\\) cube is painted on its surface. Find the ratio of the number of painted to unpainted unit cubes."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-efe0beb47241ee4cbaa8190bb859512e70fd2ec10fa1ae3b095580965d0fd472 | A \(5 \times 5\) grid is initially uncoloured except for the four corner cells, which are coloured red, yellow, blue, and green. The task is to colour some (possibly none) of the uncoloured cells with red, blue, green, and yellow such that for each colour, the cells of that colour form a rectangle. If there are \(N\) w... | [
"0"
] | [
"A \\(5 \\times 5\\) grid is initially uncoloured except for the four corner cells, which are coloured red, yellow, blue, and green. The task is to colour some (possibly none) of the uncoloured cells with red, blue, green, and yellow such that for each colour, the cells of that colour form a rectangle. If there are... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-536883066d742b2eb0365e97b8b00138188dc72b878e45a8a35ee79ef3d392a0 | A \(6 \times 6\) square is dissected into \(9\) rectangles by lines parallel to its sides such that all these rectangles have integer sides. Let \(S\) be the sum of the areas of all unique rectangles formed. Find the maximum possible value of \(S\). | [
"30"
] | [
"A \\(6 \\times 6\\) square is dissected into \\(9\\) rectangles by lines parallel to its sides such that all these rectangles have integer sides. Let \\(S\\) be the sum of the areas of all unique rectangles formed. Find the maximum possible value of \\(S\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-728d9414fd31392218542d16edc7dfe69d81562cd58e2b84ba619f8689411665 | A and B are counting numbers starting from 1, but they skip numbers containing certain digits. A skips numbers containing the digit 7, and B skips numbers containing the digit 3. Additionally, both skip numbers containing the digit 4. What is the 2012th number that both A and B count? | [
"8905"
] | [
"A and B are counting numbers starting from 1, but they skip numbers containing certain digits. A skips numbers containing the digit 7, and B skips numbers containing the digit 3. Additionally, both skip numbers containing the digit 4. What is the 2012th number that both A and B count?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-8eaf325fce04b686a62d8e46d4c94c906554c9e7120c0f8cdfeba65cc5d36726 | A and B are playing a game on a 2017 x 1 strip where each player writes either 1 or -1 in any empty box, turn by turn. The player who writes a number with an opposite sign to an adjacent number loses. If A starts, who has the winning strategy? | [
"A has the winning strategy."
] | [
"A and B are playing a game on a 2017 x 1 strip where each player writes either 1 or -1 in any empty box, turn by turn. The player who writes a number with an opposite sign to an adjacent number loses. If A starts, who has the winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-53e1b3f34d9dfe989c39854b4628c8994de626be40f087366d812983128cfd12 | A and B are playing a game on a blackboard. At the beginning, a positive real number $x$ is fixed. First, A chooses a prime number $p \geq 7$. Then, B selects non-negative real numbers $r_1$ and $r_2$. A then chooses two integers $s$ and $t$ such that $r_1 < s < r_1 + xp$ and $r_2 < t < r_2 + xp$, and writes on the bla... | [
"\\dfrac{1}{4}"
] | [
"A and B are playing a game on a blackboard. At the beginning, a positive real number $x$ is fixed. First, A chooses a prime number $p \\geq 7$. Then, B selects non-negative real numbers $r_1$ and $r_2$. A then chooses two integers $s$ and $t$ such that $r_1 < s < r_1 + xp$ and $r_2 < t < r_2 + xp$, and writes on t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-da4bfe177ec81ed1cad7230782cdcada7d04eaa8d3f627e40c882bc604168eee | A and B are playing a game where each match can result in a win for A, a win for B, or a draw. The probability that A wins a match is 20%, and the probability that B wins a match is 30%. What is the probability that A wins the game in the 5th round? | [
"1.25%"
] | [
"A and B are playing a game where each match can result in a win for A, a win for B, or a draw. The probability that A wins a match is 20%, and the probability that B wins a match is 30%. What is the probability that A wins the game in the 5th round?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-2bf74f371cb73b0604c4d9c168e9383833b688716ceb49b9c6e89ebb476ce864 | A and B are playing a game with a starting cubic polynomial. On each turn, a player can either transform the polynomial \( f(x) \) to \( f(nx) \), \( f(n+x) \), or \( f(x) + n \) where \( n \) is a natural number, or change the sign of the coefficient of \( x^2 \). If after a finite number of turns, all coefficients of... | [
"x^3"
] | [
"A and B are playing a game with a starting cubic polynomial. On each turn, a player can either transform the polynomial \\( f(x) \\) to \\( f(nx) \\), \\( f(n+x) \\), or \\( f(x) + n \\) where \\( n \\) is a natural number, or change the sign of the coefficient of \\( x^2 \\). If after a finite number of turns, al... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-d515a959d4113db1f350c1287fc49bf8f98ff4b6600bdf6a8d5eed135796bf7f | A and B are playing the final game of the final set in a tennis match with a score of 5-5. The probability of a player winning the game while serving is \( p \) and while receiving is \( q \). If A is serving, find the respective probabilities of A and B winning the game. | [
"\\( P_A = \\frac{pq}{1 - (p^2 + q^2)} \\) and \\( P_B = 1 - \\frac{pq}{1 - (p^2 + q^2)} \\)"
] | [
"A and B are playing the final game of the final set in a tennis match with a score of 5-5. The probability of a player winning the game while serving is \\( p \\) and while receiving is \\( q \\). If A is serving, find the respective probabilities of A and B winning the game."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-7ab9a3bf4a65da24f25ff0f0cd3c2773d49bd3aa588a98fb5a5dd944b8786249 | A and B are two hunters, each shooting at a deer with one bullet. The probability that A hits the target is 0.3, and the probability that B hits the target is 0.6. What is the probability that B killed the deer? | [
"\\(\\frac{7}{9}\\)"
] | [
"A and B are two hunters, each shooting at a deer with one bullet. The probability that A hits the target is 0.3, and the probability that B hits the target is 0.6. What is the probability that B killed the deer?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-034a57e5480d095566c192b51c785859ab9fcb90fd3249d9e7773d28bcbd25d5 | A and B are two points on a circular race track. One runner starts at A running counter-clockwise, and, at the same time, a second runner starts from B running clockwise. They meet first 100 yards from A, measured along the track. They meet a second time at B and a third time at A. Assuming constant speeds, how long is... | [
"\\( 200 + 100\\sqrt{5} \\)"
] | [
"A and B are two points on a circular race track. One runner starts at A running counter-clockwise, and, at the same time, a second runner starts from B running clockwise. They meet first 100 yards from A, measured along the track. They meet a second time at B and a third time at A. Assuming constant speeds, how lo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 9
} | 19 | 0 | 0 |
omr-ws-d2870dc859222027c7256409a6cae96a292246338a40ea27aa61157763f8f6d1 | A and B can build a wall in 24 hours. After A worked alone for 7 hours, B joined A, and together they finished the rest of the work in 20 hours. How long would it take each of them to build the wall individually? | [
"A takes 42 hours, B takes 56 hours."
] | [
"A and B can build a wall in 24 hours. After A worked alone for 7 hours, B joined A, and together they finished the rest of the work in 20 hours. How long would it take each of them to build the wall individually?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 32
} | 39 | 0 | 0 |
omr-ws-779c19d28d2721852d03659dd9c83a30687ddef99bb9e0eedce01917113d3985 | A and B each have a pair of dice. A wins if the sum of the dice is 7, and B wins if the sum of the dice is 4. They throw simultaneously until one of them wins, disregarding equal throws. Find the probability that A wins. | [
"\\(\\frac{11}{16}\\)"
] | [
"A and B each have a pair of dice. A wins if the sum of the dice is 7, and B wins if the sum of the dice is 4. They throw simultaneously until one of them wins, disregarding equal throws. Find the probability that A wins."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 8,
"tir": 24
} | 73 | 0 | 0 |
omr-ws-1b2ea7f6dae729516037bf113ae698e256ca18a2cd68a0094cf7cfddbb1a48fa | A and B each have a positive integer between $1$ and $1700$. There are $26$ stones on the river bank. They take turns throwing stones into the river, starting with A. Devise a strategy that allows A and B to communicate their integers to each other using the number of stones they throw. | [
"A"
] | [
"A and B each have a positive integer between $1$ and $1700$. There are $26$ stones on the river bank. They take turns throwing stones into the river, starting with A. Devise a strategy that allows A and B to communicate their integers to each other using the number of stones they throw."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-972a8c23702de9e06964a1a14017a0e3b509177fc300f2b47e6062a204cc54c5 | A and B each take 12 turns playing with 3 dice. If a sum of 11 is thrown, A cedes one turn to B. If a sum of 14 is thrown, B cedes one turn to A. The winner is the player who first obtains the specified number of turns. Determine the probability of winning for players A and B. | [
"\\(244,140,625: 282,429,536,481\\)"
] | [
"A and B each take 12 turns playing with 3 dice. If a sum of 11 is thrown, A cedes one turn to B. If a sum of 14 is thrown, B cedes one turn to A. The winner is the player who first obtains the specified number of turns. Determine the probability of winning for players A and B."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 7
} | 9 | 0 | 0 |
omr-ws-8b5ca14b753649aca8c0e53fbf77de647b5d3a79534a668ddcf5ba59429aa645 | A and B fill a rectangular array of 16 columns and 10 rows with the numbers 1 to 160. A fills the array row-wise, while B fills it column-wise. Find the sum of the numbers that occupy the same position in both arrays. | [
"322"
] | [
"A and B fill a rectangular array of 16 columns and 10 rows with the numbers 1 to 160. A fills the array row-wise, while B fills it column-wise. Find the sum of the numbers that occupy the same position in both arrays."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-cac1b0f29ea5106c8abebe696e002349adbfbe089ed8da690b27d67dbc4c0f5f | A and B have salaries that together amount to 10,000 P.M. They spend the same amount monthly, and the ratio of their savings is 6:1. What are the minimum and maximum possible salaries that A can have? | [
"8571.43"
] | [
"A and B have salaries that together amount to 10,000 P.M. They spend the same amount monthly, and the ratio of their savings is 6:1. What are the minimum and maximum possible salaries that A can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eaa24b9ec229c485140a23973537707b53564958c2a6a742042e687f59cf0dcb | A and B independently do a multiple choice quiz with 10 questions, each having 5 options. A knows the answers to 5 questions and B knows the answers to 6 questions. Both answer the remaining questions by pure guesswork. It is already known that A answers at least one of the remaining questions correctly. What are the c... | [
"\\dfrac{507201}{1313125}"
] | [
"A and B independently do a multiple choice quiz with 10 questions, each having 5 options. A knows the answers to 5 questions and B knows the answers to 6 questions. Both answer the remaining questions by pure guesswork. It is already known that A answers at least one of the remaining questions correctly. What are ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-0173ce841f1b42c5a807bc5638bce2b7dd57f7e20677820c577484aa9c4a1f36 | A and B leave towns M and N, respectively, at the same time and drive at constant speeds back and forth between the towns. B drives at a constant speed of 40 km per hour. A arrives at town N 20 minutes after passing B, whereas B arrives at town M 45 minutes after passing A. How many minutes will it take for A and B to ... | [
"2010"
] | [
"A and B leave towns M and N, respectively, at the same time and drive at constant speeds back and forth between the towns. B drives at a constant speed of 40 km per hour. A arrives at town N 20 minutes after passing B, whereas B arrives at town M 45 minutes after passing A. How many minutes will it take for A and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-7075900340acea8f130baf388dcfb359f10712bec8934eec88222f53606b5d08 | A and B lie on a circle. P lies on the minor arc AB. Q and R (distinct from P) also lie on the circle, so that P and Q are equidistant from A, and P and R are equidistant from B. Let X be the intersection of AR and BQ. If the coordinates of A, B, and P are (0, 0), (4, 0), and (2, 2√3) respectively, find the coordinates... | [
"\\left(2, \\dfrac{2\\sqrt{3}}{3}\\right)"
] | [
"A and B lie on a circle. P lies on the minor arc AB. Q and R (distinct from P) also lie on the circle, so that P and Q are equidistant from A, and P and R are equidistant from B. Let X be the intersection of AR and BQ. If the coordinates of A, B, and P are (0, 0), (4, 0), and (2, 2√3) respectively, find the coordi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-e6269e3b9a2b7cb4d63781d183065c49f4b9873a1f87ed6c81d0e0863ab0df72 | A and B play a game on a $101 \times 101$ square grid. They take turns coloring a unit square, with A going first. A unit square cannot be colored if there are at least 2 colored squares in its row or column. The player who cannot make a move loses. Assuming both players play optimally, calculate the number of squares ... | [
"101"
] | [
"A and B play a game on a $101 \\times 101$ square grid. They take turns coloring a unit square, with A going first. A unit square cannot be colored if there are at least 2 colored squares in its row or column. The player who cannot make a move loses. Assuming both players play optimally, calculate the number of sq... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-13d34bf19669cfaae6afa073b275d047e7ee168ba96c1774b405a32afb7e6d93 | A and B play a game on a 100x100 chess board. A thinks of a position (xA, yA). B can ask how many moves at least are required for a king to get from any chosen position (x, y) to (xA, yA). The king can move one step to any of the 8 neighboring squares. What is the minimal number of questions B needs to determine (xA, y... | [
"3"
] | [
"A and B play a game on a 100x100 chess board. A thinks of a position (xA, yA). B can ask how many moves at least are required for a king to get from any chosen position (x, y) to (xA, yA). The king can move one step to any of the 8 neighboring squares. What is the minimal number of questions B needs to determine (... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-35e499d77e0489ad66a0ab2a9edf1075f9bb64c0a623a4ee0e53cec09ea1cf48 | A and B play a game on a board with integers $2, 3, \dots, n$ for some $n \ge 3$. They take turns clearing an integer, with A going first. A wins if the remaining two numbers are relatively prime. Determine all values of $n$ such that A has a winning strategy. | [
"\\( n = 3, 5, 7, 2m \\)"
] | [
"A and B play a game on a board with integers $2, 3, \\dots, n$ for some $n \\ge 3$. They take turns clearing an integer, with A going first. A wins if the remaining two numbers are relatively prime. Determine all values of $n$ such that A has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-3c98b1222c124696b100588b2c248e083ff3f378a970d54926e23671d3bc6b73 | A and B play a game on an $n \times n$ board ($n > 1$). A starts by placing a checker on the lower left corner. In each turn, the corresponding player places a checker on an empty square that is adjacent to the square where the previous checker was placed. The player who cannot make a move loses. Determine for all valu... | [
"For even \\( n \\), the second player wins; for odd \\( n \\), the first player wins."
] | [
"A and B play a game on an $n \\times n$ board ($n > 1$). A starts by placing a checker on the lower left corner. In each turn, the corresponding player places a checker on an empty square that is adjacent to the square where the previous checker was placed. The player who cannot make a move loses. Determine for al... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 16
} | 28 | 0 | 0 |
omr-ws-c4dd96248d8f38ee929128eeb80d91e1f162bd76232dc91e958b1a833d9489cc | A and B play a game on an $n \times n$ board. They take turns placing either 0 or 1 into any of the blank squares. When all the squares have been filled, B wins if the sum of all numbers in each row is even. Otherwise, A wins. Which player has a winning strategy? | [
"A"
] | [
"A and B play a game on an $n \\times n$ board. They take turns placing either 0 or 1 into any of the blank squares. When all the squares have been filled, B wins if the sum of all numbers in each row is even. Otherwise, A wins. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-9f2bcb475ea9ac4ae994f1dbacb66f940c7c778c8ecdfc0d6c7874acc0c78f40 | A and B play a game where they alternately choose an integer between 2 and 1756. A starts by saying 2. In each subsequent turn, a player must choose a number that has not been chosen before, is not greater than 1756, and is either the sum or the product of two (not necessarily distinct) earlier numbers. The player who ... | [
"B"
] | [
"A and B play a game where they alternately choose an integer between 2 and 1756. A starts by saying 2. In each subsequent turn, a player must choose a number that has not been chosen before, is not greater than 1756, and is either the sum or the product of two (not necessarily distinct) earlier numbers. The player... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-893a5936cb821df04aa6af2888a2a94bede06c45d2cc424b82d240796e697785 | A and B play a game where they choose randomly $k$ integers from $\{1,2,\dots,100\}$. If the sum of these integers is even, A wins; otherwise, B wins. For what values of $k$ do A and B have the same chance of winning? | [
"\\( k \\) is an odd number with \\( 1 \\leq k \\leq 99 \\)."
] | [
"A and B play a game where they choose randomly $k$ integers from $\\{1,2,\\dots,100\\}$. If the sum of these integers is even, A wins; otherwise, B wins. For what values of $k$ do A and B have the same chance of winning?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-6033663cb189e966405669e1e78024d6ff8430712b9517f8140742defb9b5b53 | A and B play a game which begins with a pile of 100 toothpicks. They alternate turns with A going first. On each player's turn, they must remove 1, 3, or 4 toothpicks from the pile. The player who removes the last toothpick wins the game. Who has the winning strategy? | [
"B has the winning strategy."
] | [
"A and B play a game which begins with a pile of 100 toothpicks. They alternate turns with A going first. On each player's turn, they must remove 1, 3, or 4 toothpicks from the pile. The player who removes the last toothpick wins the game. Who has the winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-17cfedfaa8e26cff5bc21f4aa55bacb03d5caaafa4f9ce6336171a1558f618ab | A and B play a game with 2018 sheets of paper numbered \(1, 2, \ldots, 2018\). A chooses some sheets to paint blue and the rest red. B writes a positive integer on each sheet such that each number (except the first sheet) is the product of the positive divisors of the previous sheet's number. B wins if the numbers on t... | [
"A"
] | [
"A and B play a game with 2018 sheets of paper numbered \\(1, 2, \\ldots, 2018\\). A chooses some sheets to paint blue and the rest red. B writes a positive integer on each sheet such that each number (except the first sheet) is the product of the positive divisors of the previous sheet's number. B wins if the numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-698a697512603ee9ef8bdd7b61702fcd14fac1f8199706353783c400668022d0 | A and B play a game with numbers $2, 3, \ldots, 50$ written on a blackboard. They fix a non-empty subset $S$ of $\{2, 3, \ldots, 50\}$. A erases all numbers in $S$. Then, starting with B, they take turns erasing one or more numbers that are not coprime with any number erased by the opponent in the previous turn. The pl... | [
"32767"
] | [
"A and B play a game with numbers $2, 3, \\ldots, 50$ written on a blackboard. They fix a non-empty subset $S$ of $\\{2, 3, \\ldots, 50\\}$. A erases all numbers in $S$. Then, starting with B, they take turns erasing one or more numbers that are not coprime with any number erased by the opponent in the previous tur... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-77b28b5929d7effa2a33130ffd37ed2dcbde469643d55b1fe21099305d29364b | A and B play a game. Each has 10 tokens numbered from 1 to 10. The board is two rows of squares. The first row is numbered 1 to 1492 and the second row is numbered 1 to 1989. On the nth turn, A places his token number n on any empty square in either row and B places his token on any empty square in the other row. B win... | [
"B has a winning strategy."
] | [
"A and B play a game. Each has 10 tokens numbered from 1 to 10. The board is two rows of squares. The first row is numbered 1 to 1492 and the second row is numbered 1 to 1989. On the nth turn, A places his token number n on any empty square in either row and B places his token on any empty square in the other row. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 4,
"tir": 6
} | 50 | 0 | 0 |
omr-ws-6c81c7ffc996379a4c3c4d2100646f45f0f6d2278cd26b66c3aae56c35cd9f08 | A and B play a match where the probability of either winning a game is $\frac{1}{4}$ and the probability of a draw is $\frac{1}{2}$. The match ends when either player wins 2 games. Find 10 times the probability that the match will be finished in 4 or fewer games. | [
"5"
] | [
"A and B play a match where the probability of either winning a game is $\\frac{1}{4}$ and the probability of a draw is $\\frac{1}{2}$. The match ends when either player wins 2 games. Find 10 times the probability that the match will be finished in 4 or fewer games."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 2
} | 22 | 0 | 0 |
omr-ws-787ddcb824664e970158348d107d4630551fb7b6006653442000e37914ce512d | A and B play a series of games where the probability of A winning a single game is \( p \) (with \( p < \frac{1}{2} \)). A gets to choose the total number of games, which must be even. To win the series, one must win more than half of the games. How many games should A choose to maximize their chances of winning the se... | [
"\\( 2k \\) games are best for \\( \\frac{k-1}{2k-1} < p < \\frac{k}{2k+1} \\)"
] | [
"A and B play a series of games where the probability of A winning a single game is \\( p \\) (with \\( p < \\frac{1}{2} \\)). A gets to choose the total number of games, which must be even. To win the series, one must win more than half of the games. How many games should A choose to maximize their chances of winn... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-d498749ec08571eaae46cf89d6998045613770baa8e7f635814a5b103da77058 | A and B play a series of games which cannot be drawn, with respective probabilities of winning a single game being \( p \) and \( q \). What is the probability that A wins \( m \) games before B wins \( n \) games? | [
"\\sum_{k=0}^{n-1} \\binom{m + k - 1}{k} p^m q^k"
] | [
"A and B play a series of games which cannot be drawn, with respective probabilities of winning a single game being \\( p \\) and \\( q \\). What is the probability that A wins \\( m \\) games before B wins \\( n \\) games?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5771456bedc0f5526ccc3dbac80b1889802443f3a180e88d93462a0a193d2c34 | A and B play a series of games. Each game is independently won by A with probability \( p \) and by B with probability \( 1-p \). They stop when the total number of wins of one of the players is two greater than that of the other player. Find the probability that A is the match winner. | [
"\\dfrac{p^2}{p^2 + (1 - p)^2}"
] | [
"A and B play a series of games. Each game is independently won by A with probability \\( p \\) and by B with probability \\( 1-p \\). They stop when the total number of wins of one of the players is two greater than that of the other player. Find the probability that A is the match winner."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d20a0277e46a7cef852c088abb5a6e0ced0b019f625f728451ecc990d372a94c | A and B play a tournament where A has a probability \( p \) of winning each game. The first to win \( n \) games wins the tournament. What is the probability that A wins the tournament? | [
"\\( P = p^n\\sum_{k=0}^{n-1}\\dbinom{n+k-1}{n-1}(1-p)^k \\)"
] | [
"A and B play a tournament where A has a probability \\( p \\) of winning each game. The first to win \\( n \\) games wins the tournament. What is the probability that A wins the tournament?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 20
} | 42 | 0 | 0 |
omr-ws-23db94166668821aa569ae22179fe6e47b24cd29ca8c07f786542612677696b2 | A and B shoot independently and stop as soon as they get their first hit. The probability of each shot for A and B are $\frac{3}{4}$ and $\frac{5}{7}$ respectively. What is the probability that B requires more shots than A to hit the target? | [
"\\( \\frac{3}{13} \\)"
] | [
"A and B shoot independently and stop as soon as they get their first hit. The probability of each shot for A and B are $\\frac{3}{4}$ and $\\frac{5}{7}$ respectively. What is the probability that B requires more shots than A to hit the target?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-dd69d9cb6ca6ba7addfd5e556d100b38011db3a19e4950b5c9744abb36eabd95 | A and B start from one vertex of a 2024x2024 grid to go to the opposite vertex using a shortest path. If they randomly choose their paths, what is the probability that their paths have only the initial and final vertices in common? | [
"\\dfrac{1}{\\dbinom{4048}{2024}}"
] | [
"A and B start from one vertex of a 2024x2024 grid to go to the opposite vertex using a shortest path. If they randomly choose their paths, what is the probability that their paths have only the initial and final vertices in common?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-a781d0954d0dd8ce790575758aa167658cee8fb464a656760c6c5c4b5d9e38e3 | A and B throw fair six-sided dice. A wins if he is the first to roll a sum of 7. B wins if he is the first to roll two throws (not necessarily consecutive) with a sum of 6 or 8. What is the probability that A wins? | [
"\\(\\frac{18}{43}\\)"
] | [
"A and B throw fair six-sided dice. A wins if he is the first to roll a sum of 7. B wins if he is the first to roll two throws (not necessarily consecutive) with a sum of 6 or 8. What is the probability that A wins?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1ecbf8d48640140dfb596d6f0cc8ac8e5c2a194c1e47c796a95fa33c85487375 | A and B toss coins where A tosses (n+1) coins and B tosses n coins. Let \( P(n) \) be the probability that A gets more heads than B. Find \( P(9) \). | [
"\\dfrac{1}{2}"
] | [
"A and B toss coins where A tosses (n+1) coins and B tosses n coins. Let \\( P(n) \\) be the probability that A gets more heads than B. Find \\( P(9) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad8db4e3921f762ebab3637b26323ad3d56f2b8d98aec05ae63c01a91235d9ca | A and C lie on a circle with center O and radius $\sqrt{50}$. The point B inside the circle is such that $\angle ABC = 90^\circ$, $AB = 6$, and $BC = 2$. Find the length of $OB$. | [
"\\(\\sqrt{26}\\)"
] | [
"A and C lie on a circle with center O and radius $\\sqrt{50}$. The point B inside the circle is such that $\\angle ABC = 90^\\circ$, $AB = 6$, and $BC = 2$. Find the length of $OB$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-069040f8e913ed9fb913ffa37f69179df4422d8f8b5934fed3154354f38443dc | A and C lie on a circle with radius $\sqrt{50}$. The point B is inside the circle such that $\angle ABC = 90^\circ$, AB = 6, BC = 2. Find OB. | [
"\\(\\sqrt{26}\\)"
] | [
"A and C lie on a circle with radius $\\sqrt{50}$. The point B is inside the circle such that $\\angle ABC = 90^\\circ$, AB = 6, BC = 2. Find OB."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 24
} | 72 | 0 | 0 |
omr-ws-038a3dfcf89f77079ae9b08b6a600c9810c04c589565e4b85cfac04684067408 | A arranged the numbers 1 through 49 into the cells of a 7*7 board. B can choose a square covering some cells of the board and ask A which numbers are found inside that square. What is the minimum number of questions B needs to ask to determine the arrangement of the numbers? | [
"12"
] | [
"A arranged the numbers 1 through 49 into the cells of a 7*7 board. B can choose a square covering some cells of the board and ask A which numbers are found inside that square. What is the minimum number of questions B needs to ask to determine the arrangement of the numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-57946d4a160c44cc133c9a3d584261d683521b48a7256d65738df25e1487e192 | A bacteria culture doubles every hour. After 5 hours, the culture contains 64,000 bacteria. Determine the initial count of bacteria, the rate of growth, and the number of bacteria after 10 hours. | [
"Initial count = 2000, Rate of growth = \\( \\ln 2 \\) per hour, Number of bacteria after 10 hours = 2048000"
] | [
"A bacteria culture doubles every hour. After 5 hours, the culture contains 64,000 bacteria. Determine the initial count of bacteria, the rate of growth, and the number of bacteria after 10 hours."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-e29d74d85013f8b0ae7216b157d6d7a25ac7c27f6558674ac4143096e53f00f3 | A bacteria culture has an initial count of $4000$. After $20$ minutes, the count is $22,400$. How long does it take for the bacteria count to double? | [
"\\( 8.02 \\) minutes"
] | [
"A bacteria culture has an initial count of $4000$. After $20$ minutes, the count is $22,400$. How long does it take for the bacteria count to double?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-6e3470f0e19d97dda06b6197ddf8cd19f77a8fdc8778f9a39f3daf4f9f5151bb | A bacteria culture starts with 1000 bacteria and the growth rate is proportional to the number of bacteria. After 2 hours, the population is 9000. a) Find an expression for the number of bacteria after \( t \) hours. b) Find the number of bacteria after 3 hours. c) In what period of time does the number of bacteria dou... | [
"a) \\( P(t) = 1000 e^{\\left(\\frac{\\ln(9)}{2}\\right)t} \\)\nb) \\( 27000 \\)\nc) \\( \\frac{\\ln(2)}{\\ln(3)} \\)"
] | [
"A bacteria culture starts with 1000 bacteria and the growth rate is proportional to the number of bacteria. After 2 hours, the population is 9000.\na) Find an expression for the number of bacteria after \\( t \\) hours.\nb) Find the number of bacteria after 3 hours.\nc) In what period of time does the number of ba... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-f1b405ad689795013deeaf5f409e55f271bd1a43c3e1ddb1b969b1d6fa87e791 | A bacteria population starts with 10,000 individuals and grows at a rate of 30% per year. Each year, 10% of the bacteria are killed by a chemical. Additionally, 10% of the remaining bacteria die after 5 years. How many bacteria are alive after 12 years? | [
"48,843"
] | [
"A bacteria population starts with 10,000 individuals and grows at a rate of 30% per year. Each year, 10% of the bacteria are killed by a chemical. Additionally, 10% of the remaining bacteria die after 5 years. How many bacteria are alive after 12 years?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8018ff3215f9ee39f869e29df7312ee0253465b5bea313f68abcc972526e12e | A bacterium has a 3/8 probability of dying and a 5/8 probability of splitting into two other bacteria. Given that the probability of the bacterium's descent never ending is in the form \( \frac{m}{n} \), with \( \gcd(m, n) = 1 \), find \( m + n \). | [
"7"
] | [
"A bacterium has a 3/8 probability of dying and a 5/8 probability of splitting into two other bacteria. Given that the probability of the bacterium's descent never ending is in the form \\( \\frac{m}{n} \\), with \\( \\gcd(m, n) = 1 \\), find \\( m + n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-28da31b2a7d6da4b34760b27ab8ee2e3733e81b41675c591a9036356c12282f3 | A badminton club with not more than 20 people contains $n$ left-handed players and $2n$ right-handed players. In a competition, every pair of players was matched with each other once, and no match ended in a tie. If the ratio of matches won by right-handed players to left-handed players is 3:4, then what is the value o... | [
"\\( n = 5 \\)"
] | [
"A badminton club with not more than 20 people contains $n$ left-handed players and $2n$ right-handed players. In a competition, every pair of players was matched with each other once, and no match ended in a tie. If the ratio of matches won by right-handed players to left-handed players is 3:4, then what is the va... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-89db1babacf307512ec211f2d001a509533c7a9d3478fbb0caeeabde906e876e | A bag consists of 10 balls, each with a weight greater than 0 grams and less than or equal to 10 grams. The total weight of all 10 balls is 20 grams. Suppose the balls are divided into two sections such that each section has a total weight of 10 grams. If the weights of the balls are represented by $w_1, w_2, \ldots, w... | [
"no_valid_answer_found"
] | [
"A bag consists of 10 balls, each with a weight greater than 0 grams and less than or equal to 10 grams. The total weight of all 10 balls is 20 grams. Suppose the balls are divided into two sections such that each section has a total weight of 10 grams. If the weights of the balls are represented by $w_1, w_2, \\ld... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d62d730dd844c93975ac69d6b2c2917e10694e2160777981efe1f3c7b77a992a | A bag consists of 30 green and 30 non-green marbles. 5 marbles are drawn randomly. What is the probability that all are green? | [
"\\(\\frac{\\binom{30}{5}}{\\binom{60}{5}}\\)"
] | [
"A bag consists of 30 green and 30 non-green marbles. 5 marbles are drawn randomly. What is the probability that all are green?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-85199886cc1056e282c15c16eef430e7992121b44be16ff6e05f25233e48d4aa | A bag consists of 30 green and 30 non-green marbles. 5 marbles are drawn randomly. What is the probability that at least one marble is green? | [
"\\( 1 - \\frac{\\binom{30}{5}}{\\binom{60}{5}} \\)"
] | [
"A bag consists of 30 green and 30 non-green marbles. 5 marbles are drawn randomly. What is the probability that at least one marble is green?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 20
} | 24 | 0 | 0 |
omr-ws-e9066508789ce44a12589af011e10e3339342c86a3bc2ebd18763b51291fdcd7 | A bag contains $1993$ red balls and $1993$ black balls. We remove two balls at a time repeatedly and: (i) discard them if they are of the same color, (ii) discard the black ball and return the red ball to the bag if they are different colors. What is the probability that this process will terminate with one red ball in... | [
"1"
] | [
"A bag contains $1993$ red balls and $1993$ black balls. We remove two balls at a time repeatedly and:\n(i) discard them if they are of the same color,\n(ii) discard the black ball and return the red ball to the bag if they are different colors.\nWhat is the probability that this process will terminate with one red... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1a8f73e7cf7a9bbb7b2f7102acf584a59f78423bbc486f4102b41be3d1bcf866 | A bag contains $2012$ balls of each of four colors. Each student in a math team class takes four balls without replacement from the bag. Compute the smallest number of students which ensures there exist two students that take the same number of balls of the same color. | [
"36"
] | [
"A bag contains $2012$ balls of each of four colors. Each student in a math team class takes four balls without replacement from the bag. Compute the smallest number of students which ensures there exist two students that take the same number of balls of the same color."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-3ee4ce560dc1c05f3549641f428dae3f7adcbd8f511ff80074cc78a3e45143b1 | A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$. Fin... | [
"\\( N = 196 \\), number of white balls is 105 or 91, and the number of black balls is 91 or 105 respectively."
] | [
"A bag contains $N$ balls, some of which are white, the others are black. Two balls are drawn at random from the bag, without replacement. It is found that the probability that the two balls are of the same color is the same as the probability that they are of different colors. It is also found that $180 < N < 220$... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 21
} | 26 | 0 | 0 |
omr-ws-b0c347d5094e88b8c8e3bb3b78fac4974d6e72ae8a4c6ec1968e1ac1db11a3b1 | A bag contains $a$ blue balls, $b$ red balls, and $c$ white balls. You will take out a ball at random, note its color, and replace it into the bag. This operation is repeated $n$ times. Find the expected value of the number of different colors that are taken out. | [
"\\( 3 - \\left(\\frac{a+b}{a+b+c}\\right)^n - \\left(\\frac{b+c}{a+b+c}\\right)^n - \\left(\\frac{c+a}{a+b+c}\\right)^n \\)"
] | [
"A bag contains $a$ blue balls, $b$ red balls, and $c$ white balls. You will take out a ball at random, note its color, and replace it into the bag. This operation is repeated $n$ times. Find the expected value of the number of different colors that are taken out."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-8ab5d40b3dc9ee4b14924f572ced3c0ffa896088a3bff479f5fb85820a2ebaa2 | A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside. If instead the black balls drawn are put back into the bag, calculate the expected number of times a black ball has been drawn when the last white ball is removed. Express your answ... | [
"b \\sum_{k=1}^{w} \\frac{1}{k}"
] | [
"A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside. If instead the black balls drawn are put back into the bag, calculate the expected number of times a black ball has been drawn when the last white ball is removed. Express your... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0618ee22c89442e81af72670ded760f6238e03ad4053a86675281c1f4119a09b | A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside. If the black balls drawn are also put aside, find an expression for the expected number of black balls that have been drawn when the last white ball is removed. | [
"\\(\\frac{wb}{w+1}\\)"
] | [
"A bag contains $b$ black balls and $w$ white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside. If the black balls drawn are also put aside, find an expression for the expected number of black balls that have been drawn when the last white ball is removed."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-75e1143ea692c0d211144b862adbfaa12f261fb380427e3ae66b19d787f1429e | A bag contains $m$ red marbles and $n$ blue marbles. At each step, Vicky takes out two marbles from the bag (not replaced). If they are of the same color, she puts a red marble in the bag; otherwise, if they are of different colors, she puts a blue marble in the bag. Vicky repeats this until only one marble remains. De... | [
"(a) blue, (b) red, (c) red, (d) blue."
] | [
"A bag contains $m$ red marbles and $n$ blue marbles. At each step, Vicky takes out two marbles from the bag (not replaced). If they are of the same color, she puts a red marble in the bag; otherwise, if they are of different colors, she puts a blue marble in the bag. Vicky repeats this until only one marble remain... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-67754deee4dc0f585ef538f8f7d8ea729a8e45e628a5d52c790bb87920f50b06 | A bag contains 10 balls, of which 3 are indistinguishable and the other 7 are distinct. In how many ways can 3 balls be chosen from the bag? | [
"64"
] | [
"A bag contains 10 balls, of which 3 are indistinguishable and the other 7 are distinct. In how many ways can 3 balls be chosen from the bag?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-d19eb617b102efa55f2854c4f410ed662c4fb2a6cf330bb61012488d9a0245e1 | A bag contains 10 black and 16 white balls. Twenty-two balls are drawn one by one until 4 white balls remain in the bag. What is the probability that the 22nd ball drawn is black? All balls are distinct. | [
"\\(\\frac{5}{11}\\)"
] | [
"A bag contains 10 black and 16 white balls. Twenty-two balls are drawn one by one until 4 white balls remain in the bag. What is the probability that the 22nd ball drawn is black? All balls are distinct."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ec56e767e4ebd64d6d71c10633a28a84e721701705239a46710dc34cee6863a9 | A bag contains 10 red balls, 10 green balls, and 10 white balls. Five balls are drawn at random from the bag without replacement. What is the probability that there are balls of all three different colors among the five? | [
"0.6789"
] | [
"A bag contains 10 red balls, 10 green balls, and 10 white balls. Five balls are drawn at random from the bag without replacement. What is the probability that there are balls of all three different colors among the five?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4cee0d0d80b2631ed265852eb7d3c3755fe03f143bd07fe0e027532f41a417fc | A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction. | [
"\\frac{15}{56}"
] | [
"A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3602de839cba5f9dac88dbdb5fc1cb151e8b8918608671a0852a31b040db2817 | A bag contains 10 red marbles, 4 green marbles, and 8 blue marbles. Given that 4 marbles are chosen at random from the bag, determine the following probabilities: a) P(2 red and 2 blue). b) P(at least 3 red). c) P(3 red and 1 green) or P(1 red and 3 green). | [
"\\dfrac{104}{1463}"
] | [
"A bag contains 10 red marbles, 4 green marbles, and 8 blue marbles. Given that 4 marbles are chosen at random from the bag, determine the following probabilities:\na) P(2 red and 2 blue).\nb) P(at least 3 red).\nc) P(3 red and 1 green) or P(1 red and 3 green)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7bab9195dda0ea23a16357d5d527a66c30796a1513acb54319f55d7183c2c8ad | A bag contains 101 marbles of two colors, green and black. The probability of choosing two black marbles without replacement is 29/50. Find the number of green marbles in the bag to the nearest whole number. | [
"24"
] | [
"A bag contains 101 marbles of two colors, green and black. The probability of choosing two black marbles without replacement is 29/50. Find the number of green marbles in the bag to the nearest whole number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-16e01ffed7cfe426877a6f7f9a42ce8abc81f57afbe992b6794786400fd3dc22 | A bag contains 12 pairs of socks. If 6 socks are picked at random, find the probability that there are exactly 2 pairs among the selected socks. | [
"\\(\\frac{{12 \\choose 2}\\cdot\\left({20 \\choose 2}-10\\right)}{{24 \\choose 6}}\\)"
] | [
"A bag contains 12 pairs of socks. If 6 socks are picked at random, find the probability that there are exactly 2 pairs among the selected socks."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8faa2ae79b97b4e3f7d1607dd6007bf06b277a8de2ad054f5e3c00824be6ef1 | A bag contains 12 tiles marked 1, 1, 2, 2, ..., 6, 6. A player draws tiles one at a time at random and holds them. If he draws a tile matching a tile he already holds, then he discards both. The game ends if he holds three unmatched tiles or if the bag is emptied. Find the probability that the bag is emptied. | [
"\\(\\frac{9}{385}\\)"
] | [
"A bag contains 12 tiles marked 1, 1, 2, 2, ..., 6, 6. A player draws tiles one at a time at random and holds them. If he draws a tile matching a tile he already holds, then he discards both. The game ends if he holds three unmatched tiles or if the bag is emptied. Find the probability that the bag is emptied."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 19,
"tir": 17
} | 36 | 0 | 0 |
omr-ws-829dfea41d1847855518d92a40a675d97a3e5b73e2940271c3a04b1ec1a32713 | A bag contains 12 tiles marked \(1, 1, 2, 2, \ldots, 6, 6\). A player draws tiles one at a time at random and holds them. If he draws a tile matching a tile he already holds, then he discards both. The game ends if he holds three unmatched tiles or if the bag is emptied. Find the probability that the bag is emptied. | [
"\\dfrac{9}{385}"
] | [
"A bag contains 12 tiles marked \\(1, 1, 2, 2, \\ldots, 6, 6\\). A player draws tiles one at a time at random and holds them. If he draws a tile matching a tile he already holds, then he discards both. The game ends if he holds three unmatched tiles or if the bag is emptied. Find the probability that the bag is emp... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-44e5908e016d5a9885d54333fc82e1cccc5535a758bf601725114259739ebd48 | A bag contains 2 red candies, 2 white candies, and 2 blue candies. If you select 3 candies with replacement, how many sequences of 3 candies could have been selected where one color appears exactly twice? | [
"144"
] | [
"A bag contains 2 red candies, 2 white candies, and 2 blue candies. If you select 3 candies with replacement, how many sequences of 3 candies could have been selected where one color appears exactly twice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-f25e76933d4f4ae1738c68c7262ac39766b082f0a5af5a94d4baa9dd378528ac | A bag contains 2 red candies, 2 white candies, and 2 blue candies. If you select 3 candies without replacement, how many sequences of 3 candies could have been selected where one color appears exactly twice? | [
"72"
] | [
"A bag contains 2 red candies, 2 white candies, and 2 blue candies. If you select 3 candies without replacement, how many sequences of 3 candies could have been selected where one color appears exactly twice?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eba99fcf8d11d1ead231d3f8eec214762c79e5ac9c71bdf4d379d8bba9439862 | A bag contains 2 white balls, 3 black balls, and 4 red balls. In how many ways can 3 balls be drawn from the bag if at least one black ball is to be included in the draw? | [
"6"
] | [
"A bag contains 2 white balls, 3 black balls, and 4 red balls. In how many ways can 3 balls be drawn from the bag if at least one black ball is to be included in the draw?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c80c0ff903c6bed4d630d38f5342b8f222efc03a918c380169688d4ea3b4b29 | A bag contains 20 balls of red, green, and blue colors in the ratio 2:3:5. After drawing one ball, 4 red balls are added to the bag. What is the probability of drawing a blue ball? | [
"\\dfrac{19}{46}"
] | [
"A bag contains 20 balls of red, green, and blue colors in the ratio 2:3:5. After drawing one ball, 4 red balls are added to the bag. What is the probability of drawing a blue ball?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9a9f6603e712c1162e19cebac343e103db614f4b78add526dfc4149b2f288a3 | A bag contains 20 blue marbles, 30 yellow marbles, and 40 red marbles. What is the minimum number of marbles that must be pulled from the bag to guarantee that at least three marbles of each color have been chosen? | [
"73"
] | [
"A bag contains 20 blue marbles, 30 yellow marbles, and 40 red marbles. What is the minimum number of marbles that must be pulled from the bag to guarantee that at least three marbles of each color have been chosen?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64b2221955d768c67573f7664d2b83070b2e5d96bdb58933e51a9d8275a52551 | A bag contains 20 books on math and physics. If 10 books are chosen and it is found that 6 of them are math books, what is the probability that there are 2 math books among the 10 books left in the bag? | [
"\\dfrac{165}{4199}"
] | [
"A bag contains 20 books on math and physics. If 10 books are chosen and it is found that 6 of them are math books, what is the probability that there are 2 math books among the 10 books left in the bag?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e3ac928991a7890ac0c9416e3da7bee28da75bd7c42394aba5c62a52cb7c6b0 | A bag contains 20 red balls and 16 blue balls. Balls are taken out uniformly at random without replacement until all balls of one color have been removed. What is the probability that the last ball taken out is red? | [
"\\(\\frac{4}{9}\\)"
] | [
"A bag contains 20 red balls and 16 blue balls. Balls are taken out uniformly at random without replacement until all balls of one color have been removed. What is the probability that the last ball taken out is red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 17
} | 26 | 0 | 0 |
omr-ws-ce30f4450ff9b5cfbf26b10eb08c67b67e897b5fc2aa81da9fdbb13598d312c1 | A bag contains 20 white balls, 30 red balls, and 40 black balls. Balls are extracted at random without replacement until the bag is empty. What is the probability that there are still black and red balls in the bag when the last white ball is drawn? | [
"\\dfrac{22}{45}"
] | [
"A bag contains 20 white balls, 30 red balls, and 40 black balls. Balls are extracted at random without replacement until the bag is empty. What is the probability that there are still black and red balls in the bag when the last white ball is drawn?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-e97ecbad272be5b9136fd44da5644edf7856e0837bb6cadc0b7f6330174e334c | A bag contains 2012 balls of each of four colors. Each student in a math team class takes four balls without replacement from the bag. Compute the smallest number of students which ensures there exist two students that take the same number of balls of each color. | [
"36"
] | [
"A bag contains 2012 balls of each of four colors. Each student in a math team class takes four balls without replacement from the bag. Compute the smallest number of students which ensures there exist two students that take the same number of balls of each color."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-e577b10e173ed2d0411f65d87a9f41a71452a94e5953eacfab8ff9b0ccfef876 | A bag contains 21 marbles: 3 red, 5 blue, 6 yellow, and 7 green. Marbles are chosen at random and removed one at a time until all marbles are removed. What is the probability that red marbles are the first to have 0 remaining in the bag? | [
"\\dfrac{757}{1680}"
] | [
"A bag contains 21 marbles: 3 red, 5 blue, 6 yellow, and 7 green. Marbles are chosen at random and removed one at a time until all marbles are removed. What is the probability that red marbles are the first to have 0 remaining in the bag?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-bdbe26cf8acb592f687d21f998973d126a8c270da87b1899552f3714495b1a7f | A bag contains 24 marbles: 10 blue and 12 green. If Jack randomly chooses 4 marbles, what is the probability that at least 3 of them will be green? | [
"\\(\\frac{2795}{10626}\\)"
] | [
"A bag contains 24 marbles: 10 blue and 12 green. If Jack randomly chooses 4 marbles, what is the probability that at least 3 of them will be green?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-82fabb969b26b26e467fb3aeb709652944d9dac8f010767bc58fddeb6d41cc36 | A bag contains 26 tiles, each labeled with a letter A through Z. Julie chooses 5 tiles at random from the bag. What is the probability that she chooses the letters of her name in any order? | [
"\\(\\frac{1}{54600}\\)"
] | [
"A bag contains 26 tiles, each labeled with a letter A through Z. Julie chooses 5 tiles at random from the bag. What is the probability that she chooses the letters of her name in any order?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c11aca99fe1da33bb29f7f60be39650390446b83d326abe471c8b7d5b10b89bb | A bag contains 26 tiles, each labeled with a letter A through Z. Julie chooses 5 tiles at random from the bag. What is the probability that she matches 3 of the letters in her name? | [
"\\(\\frac{210}{3289}\\)"
] | [
"A bag contains 26 tiles, each labeled with a letter A through Z. Julie chooses 5 tiles at random from the bag. What is the probability that she matches 3 of the letters in her name?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f0c30c5d206fdd04050427c853458a965206b40bdb9821b66ac782bb8242702 | A bag contains 3 black, 4 white, and 2 red balls, all of which are different. How many ways can at most 6 balls be selected such that the selection contains balls of all three colors? | [
"270"
] | [
"A bag contains 3 black, 4 white, and 2 red balls, all of which are different. How many ways can at most 6 balls be selected such that the selection contains balls of all three colors?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-5f8ecdd7ad8dc7a5736a6f0840ca4e315c1c9df3a9fecc7a8d41b47def05a4fb | A bag contains 3 blue balls, 5 red balls, and 4 yellow balls. One ball is taken at random, its color noted, and then put back into the bag. A second ball is taken at random, and again its color is noted. What is the probability that both balls are the same color? | [
"\\(\\frac{25}{72}\\)"
] | [
"A bag contains 3 blue balls, 5 red balls, and 4 yellow balls. One ball is taken at random, its color noted, and then put back into the bag. A second ball is taken at random, and again its color is noted. What is the probability that both balls are the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-71d9f660d07b0bba05d448599371554fdfdfa5678101c3190f738617c0af56ca | A bag contains 3 red and 3 green balls. Three balls are drawn at random from the bag and replaced with 3 white balls. Afterwards, another 3 balls are drawn at random from the bag. Find the probability that the color of the 3 balls in the second draw are all different. | [
"0.27",
"\\(\\frac{9}{50}\\)"
] | [
"A bag contains 3 red and 3 green balls. Three balls are drawn at random from the bag and replaced with 3 white balls. Afterwards, another 3 balls are drawn at random from the bag. Find the probability that the color of the 3 balls in the second draw are all different."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 5,
"tir": 47
} | 68 | 0 | 0 |
omr-ws-16e754b4d5f06fa52f56888ae5423d87917935934012d807d2875f0c728cfe2c | A bag contains 3 red marbles and 2 black marbles. Two persons, A and B, take turns drawing a marble from the bag without replacement. The first person to draw a black marble wins. What is the probability that person A, who draws first, wins? | [
"\\(\\frac{3}{5}\\)"
] | [
"A bag contains 3 red marbles and 2 black marbles. Two persons, A and B, take turns drawing a marble from the bag without replacement. The first person to draw a black marble wins. What is the probability that person A, who draws first, wins?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c3e22ba9ae39d54eb191cba10d4a7ae8ef553ba8ff11776acf6b6ddbc025da14 | A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consecutively, and the 4 violet chips are drawn ... | [
"\\frac{1}{210}"
] | [
"A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consecutively, and the 4 violet chips are d... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4cc83e6e8a4c6978c3bd061eea0c5a6e8fcc4be78c7b7dd8e141066f657566ca | A bag contains 4 balls labeled $1, 2, 3, 4$. A ball is drawn repeatedly and returned to the bag unless an even-numbered ball is drawn, in which case the process stops. Find the probability distribution of the maximum number of the ball taken out. | [
"\\[\nP(X = 1) = \\frac{1}{4}, \\quad P(X = 2) = \\frac{1}{4}, \\quad P(X = 3) = \\frac{1}{8}, \\quad P(X = 4) = \\frac{1}{8}\n\\]"
] | [
"A bag contains 4 balls labeled $1, 2, 3, 4$. A ball is drawn repeatedly and returned to the bag unless an even-numbered ball is drawn, in which case the process stops. Find the probability distribution of the maximum number of the ball taken out."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-769c9540d4c09999f4ce774afbdf1ddc020df0e52cbf76ed87d325744828f664 | A bag contains 4 balls labeled 1, 2, 3, and 4. A ball is drawn from the bag and then returned. This process is repeated until a ball labeled with an even number is drawn, at which point the process stops. Find the probability distribution of the maximum number of the ball taken out. | [
"\\dfrac{1}{2}"
] | [
"A bag contains 4 balls labeled 1, 2, 3, and 4. A ball is drawn from the bag and then returned. This process is repeated until a ball labeled with an even number is drawn, at which point the process stops. Find the probability distribution of the maximum number of the ball taken out."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-1bd9ba91193f2feae081f6b0abc585b3aeea6b0d6031d71fe2c4f5d29d576d99 | A bag contains 4 balls. If 2 balls drawn randomly are found to be white, what is the probability that all the balls in the bag are white? | [
"\\dfrac{3}{5}"
] | [
"A bag contains 4 balls. If 2 balls drawn randomly are found to be white, what is the probability that all the balls in the bag are white?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce4b73b62a91f3b3d2fb9a8dbe5f04bf63f75505caca7ef2b880b62204b6b887 | A bag contains 4 marbles of each color: red, orange, green, blue, and purple. If 3 marbles are drawn from the bag without replacement, what is the probability that the drawn marbles are all of different colors? Express the probability as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers, an... | [
"89"
] | [
"A bag contains 4 marbles of each color: red, orange, green, blue, and purple. If 3 marbles are drawn from the bag without replacement, what is the probability that the drawn marbles are all of different colors? Express the probability as a fraction \\(\\frac{a}{b}\\) where \\(a\\) and \\(b\\) are coprime positive ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-4453aa6504a75d7131704e622bf071863897dfae1a0905541307818ab4c02b30 | A bag contains 4 red marbles and 6 black marbles. 3 marbles are drawn from the bag, each marble is replaced after it is drawn, and the bag is shaken. a) Find the probability of selecting: i) 3 black marbles. ii) Exactly 2 red marbles. iii) At least 1 red marble. b) From the same bag of marbles as in part A, 2 marbles a... | [
"a) i) \\(\\frac{27}{125}\\), ii) \\(\\frac{36}{125}\\), iii) \\(\\frac{98}{125}\\), b) i) \\(\\frac{2}{15}\\), ii) \\(\\frac{8}{15}\\)"
] | [
"A bag contains 4 red marbles and 6 black marbles. 3 marbles are drawn from the bag, each marble is replaced after it is drawn, and the bag is shaken.\na) Find the probability of selecting:\ni) 3 black marbles.\nii) Exactly 2 red marbles.\niii) At least 1 red marble.\nb) From the same bag of marbles as in part A, 2... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-5bd23da5c935fd20cad6b001c47a11611163a24b85b20796c65f27ee2ed94ab2 | A bag contains 4 white and 3 black balls, and another bag contains 5 white and 2 black balls. If we take two balls from the first bag and put them into the second bag, what is the probability that the first ball drawn from the second bag is white? | [
"\\dfrac{43}{63}"
] | [
"A bag contains 4 white and 3 black balls, and another bag contains 5 white and 2 black balls. If we take two balls from the first bag and put them into the second bag, what is the probability that the first ball drawn from the second bag is white?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e853fc7a397c67a9df7a3e63c2c1c391158dd7236f904b0c404282ba4d67872a | A bag contains 5 balls of unknown color. A ball is drawn and replaced twice, and each time it is found to be red. Two balls are then drawn at once. What is the probability that both balls are red? | [
"\\dfrac{377}{550}"
] | [
"A bag contains 5 balls of unknown color. A ball is drawn and replaced twice, and each time it is found to be red. Two balls are then drawn at once. What is the probability that both balls are red?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6bded5d582eefc0619ca2e466bc8da5b4d582e32a28d22057d948edf388fb2f | A bag contains 5 dice, one of which is biased and always shows six; the others are fair. Two dice are picked at random and rolled, resulting in a double six. What is the probability of getting another double six? | [
"\\(\\frac{29}{36}\\)"
] | [
"A bag contains 5 dice, one of which is biased and always shows six; the others are fair. Two dice are picked at random and rolled, resulting in a double six. What is the probability of getting another double six?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04204a0ffdb70726527f26e3485b81d2c6fb33176b3ca08540aa3071a19f3c32 | A bag contains 5 red, 4 white, and 3 blue balls. Three balls are chosen at a time from the bag. What is the probability that the three balls are all of different colors? | [
"\\(\\frac{3}{11}\\)"
] | [
"A bag contains 5 red, 4 white, and 3 blue balls. Three balls are chosen at a time from the bag. What is the probability that the three balls are all of different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-9d6a362286957e51342d324f8090688e31e048997f3067beec0f80851e96851f | A bag contains 5 red, 4 white, and 3 blue balls. Three balls are chosen at a time from the bag. What is the probability that the three balls are all red? | [
"\\(\\frac{1}{22}\\)"
] | [
"A bag contains 5 red, 4 white, and 3 blue balls. Three balls are chosen at a time from the bag. What is the probability that the three balls are all red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
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