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-7c056696a25ba4fd53994b4051b6b531732a21be3906526f3401df0e17cda462 | Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of the two preceding ones. It is known that each sequence contains at least one number not present in the other sequence. What is the maximum quantity of numbers that can be common to bot... | [
"2"
] | [
"Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of the two preceding ones. It is known that each sequence contains at least one number not present in the other sequence. What is the maximum quantity of numbers that can be common t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 12
} | 33 | 0 | 0 |
omr-ws-bcb980695b71779feb114fbebcd94527ef21ed0ed0d503d65df67b4031cb05cf | Two strips of width $1$ intersect in a parallelogram whose area is $2$. What is the angle between the strips? | [
"\\(30^\\circ\\)"
] | [
"Two strips of width $1$ intersect in a parallelogram whose area is $2$. What is the angle between the strips?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 24
} | 41 | 0 | 0 |
omr-ws-6c1d1b0e3f4d98a4fb6443aec3049ece128ab6256d4ce1aa2d00ebae6de184f7 | Two strips of width $1$ overlap at an angle of $\alpha$. Express the shaded area in terms of $\alpha$. | [
"$\\csc \\alpha$"
] | [
"Two strips of width $1$ overlap at an angle of $\\alpha$. Express the shaded area in terms of $\\alpha$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-28228574078006dd88f1f8322330688c2beed33f94debda90574a31e0243b77e | Two students $A$ and $B$ play a game on a $20 \times 20$ chessboard. A chess piece starts on a certain square. Players alternately move the piece to an adjacent square, and each common side between adjacent squares can only be used once. The player who cannot make a move loses. If $A$ starts the game, calculate the num... | [
"200"
] | [
"Two students $A$ and $B$ play a game on a $20 \\times 20$ chessboard. A chess piece starts on a certain square. Players alternately move the piece to an adjacent square, and each common side between adjacent squares can only be used once. The player who cannot make a move loses. If $A$ starts the game, calculate t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-135fe62af8c04b093b9d9d787e253d5274b1c901de6fc7b2babb27e91f135cf0 | Two students A and B play a game starting with a pile of 1998 matches. Players take turns, with A starting first. On each turn, a player must remove a square number (greater than or equal to 1) of matches. The player who makes the last move wins. Determine which player can force a victory and describe the winning strat... | [
"A can force a victory by removing 1936 matches on the first turn."
] | [
"Two students A and B play a game starting with a pile of 1998 matches. Players take turns, with A starting first. On each turn, a player must remove a square number (greater than or equal to 1) of matches. The player who makes the last move wins. Determine which player can force a victory and describe the winning ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 41,
"tir": 13
} | 54 | 0 | 0 |
omr-ws-379e3e67afd7db712727981829fe6cc7c29c83d0b34477397e9dd4cc03f1b9d2 | Two students alternate taking candies from one of two piles containing 72 and 30 candies, respectively. Each time, the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles? | [
"First player"
] | [
"Two students alternate taking candies from one of two piles containing 72 and 30 candies, respectively. Each time, the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-683e174983349ef56947fea5c8215b8b732bff8e5436e04b392feaf598849c61 | Two students are having a pie eating contest. The first student eats $\frac{6}{7}$ of one pie. The second student eats $\frac{3}{4}$ of one pie. How much more pie did the first student finish than the second student? Express your answer as a fraction of one pie, reduced to simplest form. | [
"\\frac{3}{28}"
] | [
"Two students are having a pie eating contest. The first student eats $\\frac{6}{7}$ of one pie. The second student eats $\\frac{3}{4}$ of one pie. How much more pie did the first student finish than the second student? Express your answer as a fraction of one pie, reduced to simplest form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fca796ea4e625fb8e529c70981b025185ac55c09861afe434743f0aeb5fbb2b1 | Two students are playing a game with a deck of five cards numbered 1 through 5. They shuffle the deck, place the cards in a stack facedown, and turn over the top card next to the stack. They take turns either drawing the top card into their hand, showing it to the other player, or drawing the faceup card and replacing ... | [
"\\dfrac{1}{2}"
] | [
"Two students are playing a game with a deck of five cards numbered 1 through 5. They shuffle the deck, place the cards in a stack facedown, and turn over the top card next to the stack. They take turns either drawing the top card into their hand, showing it to the other player, or drawing the faceup card and repla... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-4818990bbd0d3936b89a8f3d9cf439446c62bc6248135b2dce6b5484cdc8d1f2 | Two students are playing a game. The first player chooses three non-zero numbers \(a\), \(b\), and \(c\). The second player substitutes these numbers for the coefficients in the equation \( *x^2 + *x + * = 0 \) in any order. The first player wins if the resulting quadratic equation has two distinct rational roots. Find... | [
"0"
] | [
"Two students are playing a game. The first player chooses three non-zero numbers \\(a\\), \\(b\\), and \\(c\\). The second player substitutes these numbers for the coefficients in the equation \\( *x^2 + *x + * = 0 \\) in any order. The first player wins if the resulting quadratic equation has two distinct rationa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ab99f2c31cad47c9150db8f37a7b0d140004e2483b6f9c0bc342f960cfb55eb | Two students are playing a game. The first player chooses three nonzero integers \(a\), \(b\), and \(c\). The second player substitutes these numbers for the coefficients in the quadratic equation \(Ax^2 + Bx + C = 0\) in any order. The first player wins if the resulting quadratic equation has two distinct rational roo... | [
"0"
] | [
"Two students are playing a game. The first player chooses three nonzero integers \\(a\\), \\(b\\), and \\(c\\). The second player substitutes these numbers for the coefficients in the quadratic equation \\(Ax^2 + Bx + C = 0\\) in any order. The first player wins if the resulting quadratic equation has two distinct... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a7fd549d519ff5f2b770a7c0ff31fef1e8d3d3cdae025f772314e320d5961e2 | Two students attempted to solve the quadratic equation \(x^2 + bx + c = 0\). One student miscopied the middle term and obtained the solution set \{2, 3\}, while the other student miscopied the constant term and obtained the solution set \{2, 5\}. What is the correct solution set for the quadratic equation? | [
"{1, 6}"
] | [
"Two students attempted to solve the quadratic equation \\(x^2 + bx + c = 0\\). One student miscopied the middle term and obtained the solution set \\{2, 3\\}, while the other student miscopied the constant term and obtained the solution set \\{2, 5\\}. What is the correct solution set for the quadratic equation?"
... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d16e931ab5a8f20d7f9cec6cefaa94d466db5f291e58cb77b8661d03956484df | Two students began preparing for an examination at the same time and had to pass it on the same day. The first student had to read 240 pages and the second 420 pages. Each of them read the same number of complete pages every day, the first of them reading 12 pages less than the second per day. After they both read thei... | [
"The first has to read 20 pages a day and the second, 35 pages a day"
] | [
"Two students began preparing for an examination at the same time and had to pass it on the same day. The first student had to read 240 pages and the second 420 pages. Each of them read the same number of complete pages every day, the first of them reading 12 pages less than the second per day. After they both read... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc1b6cf7d2f0a251a9213c6e123b747870629644854e4180868f76fca3ea1b43 | Two students from each of four different schools will participate in a chess tournament. In the first round, each student must be paired with a student from a different school. How many different ways can the matchmaking in the first round be made? | [
"9"
] | [
"Two students from each of four different schools will participate in a chess tournament. In the first round, each student must be paired with a student from a different school. How many different ways can the matchmaking in the first round be made?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd367efaf7004f041ad5b790d58daa71f19f2c48a356c5aee9f3ea9afe85f7f2 | Two students play a game based on the total roll of two standard dice. Student A wins if a "12" is rolled first, and Student B wins if two consecutive 7s are rolled first. What is the probability that Student A will win? | [
"\\(\\frac{7}{13}\\)"
] | [
"Two students play a game based on the total roll of two standard dice. Student A wins if a \"12\" is rolled first, and Student B wins if two consecutive 7s are rolled first. What is the probability that Student A will win?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-ddedf7f182cc41de99c3c70dd6d7e3e9c0981aee9f9a5bebbfdd730f9829b96c | Two students play a game with 2019 coins on the table, taking alternating moves. The first player can take an odd number of coins from 1 to 99 in one move, and the second player can take an even number of coins from 2 to 100 in one move. The player who cannot make a move loses. Who has the winning strategy in this game... | [
"The first player has the winning strategy."
] | [
"Two students play a game with 2019 coins on the table, taking alternating moves. The first player can take an odd number of coins from 1 to 99 in one move, and the second player can take an even number of coins from 2 to 100 in one move. The player who cannot make a move loses. Who has the winning strategy in this... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-606be1e5bfd4544dd202176d3c93ce69a8d39775f581ea8a74df5ded7e4dd880 | Two students solved a quadratic equation in the variable $x$. One student copied the constant term incorrectly and got the roots $3$ and $2$. The other student copied the constant term and the coefficient of $x^2$ correctly and got the roots $-6$ and $1$. What are the correct roots of the equation? | [
"6 and -1"
] | [
"Two students solved a quadratic equation in the variable $x$. One student copied the constant term incorrectly and got the roots $3$ and $2$. The other student copied the constant term and the coefficient of $x^2$ correctly and got the roots $-6$ and $1$. What are the correct roots of the equation?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 26
} | 33 | 0 | 0 |
omr-ws-b39323e3d3d0a44843d99493cff4d74d3c5ad0ad50dd01742a7bee52f4999125 | Two students working together can complete all the housekeeping in six hours. Working alone, one student can finish the work two hours faster than the other. How many hours would it take each student to complete the housekeeping alone? | [
"10 and 8"
] | [
"Two students working together can complete all the housekeeping in six hours. Working alone, one student can finish the work two hours faster than the other. How many hours would it take each student to complete the housekeeping alone?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3867366a9c96688aa193572cb7080039d8e4351d2935b6b80999d22981fe12a0 | Two students, \( A \) and \( B \), each write down a positive integer and give it to a referee. The referee writes two integers on a blackboard, one of which is the sum of the integers written by the students. The referee then asks \( A \) if they can determine the integer written by \( B \). If \( A \) says "no," the ... | [
"3"
] | [
"Two students, \\( A \\) and \\( B \\), each write down a positive integer and give it to a referee. The referee writes two integers on a blackboard, one of which is the sum of the integers written by the students. The referee then asks \\( A \\) if they can determine the integer written by \\( B \\). If \\( A \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-8cf15493c0a89193635fc142c3fbf71101f1b4be8ce4f8f81cd78f2e79767473 | Two subsets \( A \) and \( B \) of a set \( S \) consisting of \( n \) elements are constructed randomly. Find the probability that \( A \) and \( B \) are mutually exclusive and exhaustive. | [
"\\dfrac{1}{2^n}"
] | [
"Two subsets \\( A \\) and \\( B \\) of a set \\( S \\) consisting of \\( n \\) elements are constructed randomly. Find the probability that \\( A \\) and \\( B \\) are mutually exclusive and exhaustive."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ad31a0646c68c285a01f9c6c770002eb853577b8b02f563d4387b9385b67c5f2 | Two subsets \( A \) and \( B \) of a set \( S \) consisting of \( n \) elements are constructed randomly. Find the probability that \( A \) is a subset of \( B \). | [
"\\left(\\dfrac{3}{4}\\right)^n"
] | [
"Two subsets \\( A \\) and \\( B \\) of a set \\( S \\) consisting of \\( n \\) elements are constructed randomly. Find the probability that \\( A \\) is a subset of \\( B \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ba4dee9c4bbef3a6da0f901b991e6571dfc26cc90df59ba3a2cea9641410ec19 | Two subsets of the set \( S = \{a, b, c, d, e\} \) are to be chosen so that their union is \( S \) and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? | [
"40"
] | [
"Two subsets of the set \\( S = \\{a, b, c, d, e\\} \\) are to be chosen so that their union is \\( S \\) and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-de7a3cd31dc5bed479ab101248eeac8da784ab7583fdd172197752f911e725e1 | Two substances, S and T, each contain two types of ingredients, I and G. One pound of S contains 2 ounces of I and 4 ounces of G. One pound of T contains 2 ounces of I and 6 ounces of G. A manufacturer plans to combine quantities of the two substances to obtain a mixture that contains at least 9 ounces of I and 20 ounc... | [
"\\(\\left(\\frac{7}{2} \\text{ pounds of S}, 1 \\text{ pound of T}\\right)\\)"
] | [
"Two substances, S and T, each contain two types of ingredients, I and G. One pound of S contains 2 ounces of I and 4 ounces of G. One pound of T contains 2 ounces of I and 6 ounces of G. A manufacturer plans to combine quantities of the two substances to obtain a mixture that contains at least 9 ounces of I and 20... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-d9304ccbd8acbf23e05ae4f36ace1742887e5c27b5154c03177ee866cc34923f | Two surface ships, located at coordinates (4,0,0) and (0,5,0) (in thousands of feet), detect a submarine in the directions of the vectors \(2\mathbf{i} + 3\mathbf{j} - \frac{1}{3}\mathbf{k}\) and \(18\mathbf{i} - 6\mathbf{j} - \mathbf{k}\), respectively. Four minutes ago, the submarine was at (2, -1, -1/3). Assuming th... | [
"\\((26, 23, -\\frac{1}{3})\\)"
] | [
"Two surface ships, located at coordinates (4,0,0) and (0,5,0) (in thousands of feet), detect a submarine in the directions of the vectors \\(2\\mathbf{i} + 3\\mathbf{j} - \\frac{1}{3}\\mathbf{k}\\) and \\(18\\mathbf{i} - 6\\mathbf{j} - \\mathbf{k}\\), respectively. Four minutes ago, the submarine was at (2, -1, -1... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-444f8db6c3d1f2c8e55cff8ca1722e9464400e357b37b304c5ab73aeb34d6ee6 | Two swimmers positioned at opposite sides of a rectangular swimming pool and in adjacent lanes begin to swim at the same time with constant speeds. In the first two times they are side by side, they are swimming in opposite directions: the first time they are 15 meters from one edge, and the second time they are 12 met... | [
"33"
] | [
"Two swimmers positioned at opposite sides of a rectangular swimming pool and in adjacent lanes begin to swim at the same time with constant speeds. In the first two times they are side by side, they are swimming in opposite directions: the first time they are 15 meters from one edge, and the second time they are 1... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-bf26363f52ad8ddff2caa64f95868034056e1e666d391aa5e4e789290635efaf | Two swimmers started one after the other in a 50-meter pool to cover a distance of 100 m. The speed of the second swimmer was 1.5 m/s. Having covered 21 m he caught up with the first swimmer, reached the opposite wall of the pool, returned back, and met the first swimmer $\frac{2}{3}$ second after the turn. Find the ti... | [
"1 s"
] | [
"Two swimmers started one after the other in a 50-meter pool to cover a distance of 100 m. The speed of the second swimmer was 1.5 m/s. Having covered 21 m he caught up with the first swimmer, reached the opposite wall of the pool, returned back, and met the first swimmer $\\frac{2}{3}$ second after the turn. Find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1dc318ee622cb64b9410759aa0eee0540548cf964c8f699a3be4ce1d2be561d1 | Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of time at the turns, find the number of times they pass each other. | [
"20"
] | [
"Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of time at the turns, find the number of times they pass each other."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 18
} | 18 | 0 | 0 |
omr-ws-618dbdaa7cbc076ae049663b854a459fada6a6166b9c3a5a541bc0df413fb391 | Two symmetrical axes of figure F form an angle of $46^\circ$. Find the minimum number of symmetrical axes that figure F can have. | [
"180"
] | [
"Two symmetrical axes of figure F form an angle of $46^\\circ$. Find the minimum number of symmetrical axes that figure F can have."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-05c4424521dda4d798381dca7a38c6459010ce7e31d586458c401f6a8637d152 | Two symmetrical axes of figure F intersect at an angle of $46^\circ$. Find the minimum number of symmetrical axes that figure F can have. | [
"10"
] | [
"Two symmetrical axes of figure F intersect at an angle of $46^\\circ$. Find the minimum number of symmetrical axes that figure F can have."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8271b795c000320f56d198ea0a5daaa818a1d629452c00ba7226e6255e096bbd | Two tangent circles \( c_1 \) and \( c_2 \) are given. A common tangent to the circles touches \( c_1 \) and \( c_2 \) at points \( A \) and \( B \), respectively. Segment \( AP \) is the diameter of circle \( c_1 \), and the line passing through point \( P \) is tangent to \( c_2 \) at point \( Q \). If the radius of ... | [
"2r"
] | [
"Two tangent circles \\( c_1 \\) and \\( c_2 \\) are given. A common tangent to the circles touches \\( c_1 \\) and \\( c_2 \\) at points \\( A \\) and \\( B \\), respectively. Segment \\( AP \\) is the diameter of circle \\( c_1 \\), and the line passing through point \\( P \\) is tangent to \\( c_2 \\) at point \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-2634304b5c3129d5b2dc29a77427f155ebae9914731320d77d94d6c9b9ade807 | Two tangent circles \((K, R)\) and \((\Lambda, \rho)\) are external to one another and their point of tangency is \(Z\). A straight line \((\epsilon)\) is tangent to the circles with centers \(K\) and \(\Lambda\) at points \(A\) and \(B\) respectively. A straight line \((\delta)\) is parallel to \((\epsilon)\) and is t... | [
"2"
] | [
"Two tangent circles \\((K, R)\\) and \\((\\Lambda, \\rho)\\) are external to one another and their point of tangency is \\(Z\\). A straight line \\((\\epsilon)\\) is tangent to the circles with centers \\(K\\) and \\(\\Lambda\\) at points \\(A\\) and \\(B\\) respectively. A straight line \\((\\delta)\\) is paralle... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ec0a332fd7716303e420be85199eb77ae1a8aa71e38723f89935829583aaee2 | Two tangent circles have radii 4 and 9. A common tangent is drawn. A circle is mutually tangent to the common tangent and both circles. Find the radius of this circle. | [
"\\( \\frac{36}{25} \\)"
] | [
"Two tangent circles have radii 4 and 9. A common tangent is drawn. A circle is mutually tangent to the common tangent and both circles. Find the radius of this circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-4765c4fc52cd5d587d9a251e5c3a1cd85a523d5e1d6de4ca232884871dcac6e4 | Two tangent circles of equal radii are located in triangle \(ABC\) with sides \(AB = 13\), \(BC = 12\), and \(CA = 5\), such that one circle touches sides \(BC\) and \(CA\), and the other touches sides \(BC\) and \(AB\). Find the radii of these circles. | [
"\\( \\frac{3}{2} \\)"
] | [
"Two tangent circles of equal radii are located in triangle \\(ABC\\) with sides \\(AB = 13\\), \\(BC = 12\\), and \\(CA = 5\\), such that one circle touches sides \\(BC\\) and \\(CA\\), and the other touches sides \\(BC\\) and \\(AB\\). Find the radii of these circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 10
} | 58 | 0 | 0 |
omr-ws-2cc6b0e3e54ee84451ff2f6137e0f84d70bdc653c424e70c5ac90aa333b6137e | Two tangent circles with centers \(A\) and \(B\) have radii 5 and 7, respectively. A line is drawn tangent to both circles at points \(C\) and \(D\), respectively. A line from \(A\) passes through \(BD\) at \(E\) such that \(\angle ABE = 90^\circ\). Determine the length of \(AE\). | [
"2\\sqrt{35}"
] | [
"Two tangent circles with centers \\(A\\) and \\(B\\) have radii 5 and 7, respectively. A line is drawn tangent to both circles at points \\(C\\) and \\(D\\), respectively. A line from \\(A\\) passes through \\(BD\\) at \\(E\\) such that \\(\\angle ABE = 90^\\circ\\). Determine the length of \\(AE\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-c666c48da53aa5282962cb942dbf99a5e7c9b71c4ffa3e1821dfd3da3fd12540 | Two tangent lines $PQ$ and $PR$ are drawn from an external point $P$ to a circle with center $O$. Points $Q$ and $R$ are the points of tangency, and $O$ is the midpoint of the line segment $QR$. Points $X$ and $Y$ are situated on $PQ$ and $PR$ respectively such that $XY$ is a tangent to the circle. Given that $QR = 10$... | [
"25"
] | [
"Two tangent lines $PQ$ and $PR$ are drawn from an external point $P$ to a circle with center $O$. Points $Q$ and $R$ are the points of tangency, and $O$ is the midpoint of the line segment $QR$. Points $X$ and $Y$ are situated on $PQ$ and $PR$ respectively such that $XY$ is a tangent to the circle. Given that $QR ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7d02304de9767cc880dc597567a6e4f91c0c9e33089b4d0cc26b4f208ec27a3 | Two tangent lines of a circle at points A and B intersect at point C. The midpoint of segment AB is M. Let P be a point on arc AB. Line CP intersects the circle again at point Q. Line PM intersects the circle again at point R. If the length of AQ is 12 units, calculate the length of BR. | [
"12"
] | [
"Two tangent lines of a circle at points A and B intersect at point C. The midpoint of segment AB is M. Let P be a point on arc AB. Line CP intersects the circle again at point Q. Line PM intersects the circle again at point R. If the length of AQ is 12 units, calculate the length of BR."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fef4bd555d8fc068217225d2f8f766f24bdb89b3d01842d31c33fb1c161cfbd | Two tangent lines to a circle from a point make an angle of 45 degrees. If the diameter of the circle is 8, find the area of the region bounded by the two tangent lines and the circle. | [
"\\( 16(\\sqrt{2}) + 16 - 6\\pi \\)"
] | [
"Two tangent lines to a circle from a point make an angle of 45 degrees. If the diameter of the circle is 8, find the area of the region bounded by the two tangent lines and the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-a101f738d70c67fb5b47f0c7a68e29c92e0c07edad6fb035d08ac027cdc4a9a4 | Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. Points $Q$ and $S$ are on $AT$ and $BT$, respectively, and $R$ is on the circle such that $QRST$ forms a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle. | [
"29"
] | [
"Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. Points $Q$ and $S$ are on $AT$ and $BT$, respectively, and $R$ is on the circle such that $QRST$ forms a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 23
} | 70 | 0 | 0 |
omr-ws-46cca3321ded1e7f3800fc894347725f1344e40ef5ded519edbd117b124c709f | Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. Points $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle such that $QRST$ forms a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle. | [
"29"
] | [
"Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. Points $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle such that $QRST$ forms a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-3b8a57aebbb8c6b94666abfaee4406748e691c386159d58489eac498dc554a3f | Two tangents $PA$ and $PB$ are drawn from an external point $P$ to a circle $\Gamma$ centered at $O$. $AC$ is a diameter of $\Gamma$. $BD \perp AC$ is drawn with $D$ on $AC$. If the length of $BD$ is $12$ units, calculate the length of the segment $DE$ where $E$ is the point where $PC$ intersects $BD$. | [
"6"
] | [
"Two tangents $PA$ and $PB$ are drawn from an external point $P$ to a circle $\\Gamma$ centered at $O$. $AC$ is a diameter of $\\Gamma$. $BD \\perp AC$ is drawn with $D$ on $AC$. If the length of $BD$ is $12$ units, calculate the length of the segment $DE$ where $E$ is the point where $PC$ intersects $BD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b03448a23d168f9b1780f3ec4bc1e8588f73ab49e465772a97d6662dec7d022d | Two tangents are drawn to a circle from an exterior point A, touching the circle at points B and C. A third tangent to the circle intersects segment AB at P and segment AC at R and touches the circle at Q. If AB=20, then what is the perimeter of triangle APR? | [
"40"
] | [
"Two tangents are drawn to a circle from an exterior point A, touching the circle at points B and C. A third tangent to the circle intersects segment AB at P and segment AC at R and touches the circle at Q. If AB=20, then what is the perimeter of triangle APR?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a2a49559ca69483d1c5895a6cd07247c1457af29602f044dbec971f3ee1abf0d | Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C, respectively. A third tangent intersects segment AB at P and AC at R, and touches the circle at Q. If AB = 20, find the perimeter of triangle APR. | [
"40"
] | [
"Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C, respectively. A third tangent intersects segment AB at P and AC at R, and touches the circle at Q. If AB = 20, find the perimeter of triangle APR."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-943ed5e76510ca3208bc8949e991ee3a1084f250cdb12d61411c569127124d12 | Two tangents from an external point $P$ are drawn to a circle and intersect it at $A$ and $B$. A third tangent meets the circle at $T$, and the tangents $\overrightarrow{PA}$ and $\overrightarrow{PB}$ at points $Q$ and $R$, respectively. Find the perimeter of $\triangle PQR$. | [
"2PA"
] | [
"Two tangents from an external point $P$ are drawn to a circle and intersect it at $A$ and $B$. A third tangent meets the circle at $T$, and the tangents $\\overrightarrow{PA}$ and $\\overrightarrow{PB}$ at points $Q$ and $R$, respectively. Find the perimeter of $\\triangle PQR$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c33d80f8dd0f3930263af301fd119b023454ab913a41e7cfbd94389ab62e78f | Two tangents from an external point P are drawn to a circle and intersect it at A and B. A third tangent meets the circle at T, and the tangents PA and PB at points Q and R, respectively. Find the perimeter of triangle PQR. | [
"\\( 2PA \\)"
] | [
"Two tangents from an external point P are drawn to a circle and intersect it at A and B. A third tangent meets the circle at T, and the tangents PA and PB at points Q and R, respectively. Find the perimeter of triangle PQR."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-ccf3b7fcb50f3d1ed2170d950993e163cc48dc955d7e55b07d7f289a3944ff99 | Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\angle BAC$? | [
"36"
] | [
"Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. What is the degree measure of $\\angle BAC$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-36c2b963acd4f2c37d75d0bb4b0069c58ac4d7666b6d891cf91503f85e9092c1 | Two tangents to the graph of the function $y=\sqrt {17(x^2+1)}$ intersect at right angles to a certain point of the $Oy$ axis. Write the equations of the tangents. | [
"$y=x+4,\\ y=-x+4$"
] | [
"Two tangents to the graph of the function $y=\\sqrt {17(x^2+1)}$ intersect at right angles to a certain point of the $Oy$ axis. Write the equations of the tangents."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bacf0787b7738a8e4741abc992b71d49cb46203e8cfd87273d6b06d73f81f894 | Two taxi companies, $A$ and $B$, operate in a city, and $25\%$ of the taxis belong to $A$ while $75\%$ belong to $B$. One day, a taxi caused an accident. One witness said that the taxi belonged to $A$. Two types of people live in the city: the honest who do not tell a lie at all and the liar who always tell a lie. $70\... | [
"\\dfrac{31}{88}"
] | [
"Two taxi companies, $A$ and $B$, operate in a city, and $25\\%$ of the taxis belong to $A$ while $75\\%$ belong to $B$. One day, a taxi caused an accident. One witness said that the taxi belonged to $A$. Two types of people live in the city: the honest who do not tell a lie at all and the liar who always tell a li... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8867b0b2b23dfd871f2b287f9f164184c981884703e5db70ff5c191a7cdb3759 | Two teams $A$ and $B$ play football 15 times. Each game, the team that scores the first 4 goals is the winner, and no match ends in a draw. Team $A$ wins more times than Team $B$, while Team $B$ has more goals than Team $A$. Find the biggest difference in total goals between the two teams. | [
"20"
] | [
"Two teams $A$ and $B$ play football 15 times. Each game, the team that scores the first 4 goals is the winner, and no match ends in a draw. Team $A$ wins more times than Team $B$, while Team $B$ has more goals than Team $A$. Find the biggest difference in total goals between the two teams."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5cbc6850f0b0c40d48c36459510e5ca6fd9bd9aebbe0cad42ffbb81a6b82b117 | Two teams A and B, each with 7 players, participate in a table tennis tournament. A losing player is eliminated while the winner stays to face the next player from the opposing team. The process continues until all members of a team are eliminated. What is the number of distinct ways in which the tournament can be play... | [
"6006"
] | [
"Two teams A and B, each with 7 players, participate in a table tennis tournament. A losing player is eliminated while the winner stays to face the next player from the opposing team. The process continues until all members of a team are eliminated. What is the number of distinct ways in which the tournament can be... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32daca169d98a68f270cdd61c338ebd5d328a27253a6a98ea1ebf4bc5862350c | Two teams are in a best-two-out-of-three playoff. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home and a probability $p$ of winning away. The outcomes of the games are independent. The probability that Team A... | [
"14"
] | [
"Two teams are in a best-two-out-of-three playoff. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\\frac{2}{3}$ chance of winning at home and a probability $p$ of winning away. The outcomes of the games are independent. The probability that ... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33a657855080bf282fcf865887741c723c979f3fec88cf9b8b9e4abb159808e2 | Two teams began working at 8 a.m. Having made 72 parts together, they continued working separately. At 3 p.m. it turned out that whilst the teams worked separately the first team manufactured 8 parts more than the second team. Next day the first team manufactured one part more per hour, and the second team one part les... | [
"The first team manufactured 13 workpieces, the second team 11"
] | [
"Two teams began working at 8 a.m. Having made 72 parts together, they continued working separately. At 3 p.m. it turned out that whilst the teams worked separately the first team manufactured 8 parts more than the second team. Next day the first team manufactured one part more per hour, and the second team one par... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0b9006fcbcbc1290279830516f7cf3d6702b26360e93a6dd5923b987e652a94 | Two teams of 3 swimmers each are in a pool with seven lanes. No two swimmers from the same team are allowed to be in adjacent lanes. Compute the number of possible arrangements if the swimmers themselves are distinguishable. | [
"864"
] | [
"Two teams of 3 swimmers each are in a pool with seven lanes. No two swimmers from the same team are allowed to be in adjacent lanes. Compute the number of possible arrangements if the swimmers themselves are distinguishable."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-bc094cb5cb384f1c414a2f56d6a98c7fc79a1910202c0802eafa8b0f5120f4ab | Two teams of dock workers are given a job of unloading a ship. The sum of the times it takes the first team and the second team to unload the ship, if they work separately, is equal to 12 h. Find those times if the difference constitutes 45\% of the time it would take the teams to unload the ship if they work together. | [
"$6\\frac{2}{3}$ h and $5\\frac{1}{3}$ h"
] | [
"Two teams of dock workers are given a job of unloading a ship. The sum of the times it takes the first team and the second team to unload the ship, if they work separately, is equal to 12 h. Find those times if the difference constitutes 45\\% of the time it would take the teams to unload the ship if they work tog... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dd93f7fc430dc7197b1afb1898e7e30fa3cbfba5bd3f7dc78d9613f874c5d2c | Two teams of plasterers, working jointly, plastered a house in six days. Then they plastered a club, doing three times as much work as when they plastered the house. One team began plastering the club and was then relieved by the second team, which completed the job, the first team doing twice as much work as the secon... | [
"10 days"
] | [
"Two teams of plasterers, working jointly, plastered a house in six days. Then they plastered a club, doing three times as much work as when they plastered the house. One team began plastering the club and was then relieved by the second team, which completed the job, the first team doing twice as much work as the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-135ecc2e065cc835c5c60ed412d7c0c3fc2407f25726974216bd81ff05b42486 | Two teams of plasterers, working together, plastered an apartment house in 6 days. Some other time, they worked on a club and did three times as much work as they did when plastering the house. In the club they took turns working: first the first team worked and then the second team completed the job. The first team ac... | [
"In 10 days"
] | [
"Two teams of plasterers, working together, plastered an apartment house in 6 days. Some other time, they worked on a club and did three times as much work as they did when plastering the house. In the club they took turns working: first the first team worked and then the second team completed the job. The first te... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c44bfa970f847f5a83c498231991c50cd4dda0efcc1700329f190a5a013dc534 | Two teams of stevedores were to unload a barge. The sum of the times it would take each team working individually to unload the barge is 12 hours. How much time would it take each team to unload the barge if the difference between these times is 45\% of the time it would take both teams working together to unload the b... | [
"$\\frac{20}{3}$ hours and $\\frac{16}{3}$ hours"
] | [
"Two teams of stevedores were to unload a barge. The sum of the times it would take each team working individually to unload the barge is 12 hours. How much time would it take each team to unload the barge if the difference between these times is 45\\% of the time it would take both teams working together to unload... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-703609133b037dcfa787968cfc911a76ba2a1a4007744b91df72962a69994989 | Two teams of three swimmers are in a pool with seven lanes. No two swimmers from the same team are allowed to be in adjacent lanes. Compute the number of possible arrangements if the swimmers themselves are distinguishable. | [
"648",
"\\(18 \\cdot 3! \\cdot 3!\\)"
] | [
"Two teams of three swimmers are in a pool with seven lanes. No two swimmers from the same team are allowed to be in adjacent lanes. Compute the number of possible arrangements if the swimmers themselves are distinguishable."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 38
} | 47 | 0 | 0 |
omr-ws-038bbe13d490cdf7788609f61d00d113c121cb704b653251b79ab405848936ae | Two teams of workers together dug a trench in two days. Then they began digging another trench of the same depth and width, but five times longer than the first. The first team began digging the trench alone and was then relieved by the second team. The first team dug 1.5 times more than the second team. The second tre... | [
"6 days"
] | [
"Two teams of workers together dug a trench in two days. Then they began digging another trench of the same depth and width, but five times longer than the first. The first team began digging the trench alone and was then relieved by the second team. The first team dug 1.5 times more than the second team. The secon... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b36fcb955e74e522a1fd097e20ac87ef0181952ed4e4ee42489612d2aa5940f | Two teams play a game of soccer, which team A wins 10-8. Assuming any order of the 18 goals is equally likely, what is the probability that the score was never tied after the first goal? | [
"\\(\\frac{1}{9}\\)"
] | [
"Two teams play a game of soccer, which team A wins 10-8. Assuming any order of the 18 goals is equally likely, what is the probability that the score was never tied after the first goal?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-e5f7f928fa26c5a3d7dc2849b3666e360397db8d1102024d10d94be60eac8e94 | Two teams play a series of games, stopping as soon as one of the teams has three wins. Assume the games are independent and that the chance the first team wins is an unknown parameter \( \theta \in (0,1) \). Let \( X \) denote the number of games the first team wins, and \( Y \) the number of games the other team wins.... | [
"\\( P_\\theta(X=x, Y=y) = \\theta^x (1-\\theta)^y \\) with \\( x, y \\in \\{0, 1, 2, 3\\} \\) and \\( 2 < x + y < 6 \\); minimal sufficient statistic is \\( X + Y \\); completeness not found."
] | [
"Two teams play a series of games, stopping as soon as one of the teams has three wins. Assume the games are independent and that the chance the first team wins is an unknown parameter \\( \\theta \\in (0,1) \\). Let \\( X \\) denote the number of games the first team wins, and \\( Y \\) the number of games the oth... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dace01bc75561be677a6bbae0f5989414c5a2d4a141e72b62351f2c98b370cf4 | Two teams play a tournament where 4 wins are needed to end the tournament. If both teams are equally likely to win each game, what is the probability that the tournament lasts exactly 6 games? | [
"\\(\\frac{5}{32}\\)"
] | [
"Two teams play a tournament where 4 wins are needed to end the tournament. If both teams are equally likely to win each game, what is the probability that the tournament lasts exactly 6 games?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b4971cc7e5da2cbdda474cb3401f7fb604b5d4fadcd7bfb4d6f503f9eaf391d | Two teams with a total membership of 18 were to keep a twenty-four hour watch for three days, one person at a time. For the first two days the first team was on duty, distributing their time equally among themselves. There were three girls in the second team, the rest being boys. For the time the second team was on dut... | [
"9 persons in each team"
] | [
"Two teams with a total membership of 18 were to keep a twenty-four hour watch for three days, one person at a time. For the first two days the first team was on duty, distributing their time equally among themselves. There were three girls in the second team, the rest being boys. For the time the second team was o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5e6db334b3c939df064dab095e4f5bae3d08ea0eed3eaa67361cca738a141f9 | Two teams worked together for 15 days. A third team then joined them as a result of which in 5 days the whole job was completed. The daily output of the second team is 20\% higher than that of the first team. The second and third teams together could have fulfilled the whole job in $\frac{9}{10}$ of the time it would h... | [
"16 days"
] | [
"Two teams worked together for 15 days. A third team then joined them as a result of which in 5 days the whole job was completed. The daily output of the second team is 20\\% higher than that of the first team. The second and third teams together could have fulfilled the whole job in $\\frac{9}{10}$ of the time it ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b17365a3b5c5039f14bb41b19fd706cbad3710d3f3f88fcf5504e28ddc03e289 | Two teams, A and B, each have 1000 players. Each player from team A played against each player from team B exactly once, and there were no draws. Let $S$ be the smallest subset of players from team A such that each player in team B was defeated by at least one player in $S$. Find the maximum possible value of $|S|$. | [
"1000"
] | [
"Two teams, A and B, each have 1000 players. Each player from team A played against each player from team B exactly once, and there were no draws. Let $S$ be the smallest subset of players from team A such that each player in team B was defeated by at least one player in $S$. Find the maximum possible value of $|S|... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e72c268482283af8196ba0d539b3abe783fae7c5ee392f3bd029cfec91b1b353 | Two teams, Challenger and Champion, each have $n$ people. The Challenger team chooses a positive integer $k$ (not greater than the number of members remaining in either team), and each team provides $k$ people to play in a game. The losing team has those $k$ members eliminated. This continues until one of the teams los... | [
"\\dfrac{1}{2}"
] | [
"Two teams, Challenger and Champion, each have $n$ people. The Challenger team chooses a positive integer $k$ (not greater than the number of members remaining in either team), and each team provides $k$ people to play in a game. The losing team has those $k$ members eliminated. This continues until one of the team... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-63b18d28b5d8a4f286521e3f84ab3911044fd1a1cc2de0d023b8c3c68180b36e | Two teams, Challenger and Champion, have 3 and 2 members respectively. The Challenger team chooses a positive integer $k$ (not greater than the number of members remaining in either team), and each team provides $k$ people to play in a game. The losing team has those $k$ members eliminated. This continues until one of ... | [
"\\dfrac{11}{16}"
] | [
"Two teams, Challenger and Champion, have 3 and 2 members respectively. The Challenger team chooses a positive integer $k$ (not greater than the number of members remaining in either team), and each team provides $k$ people to play in a game. The losing team has those $k$ members eliminated. This continues until on... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2fe3e3deafec2277ba9d292e9523cfa022d6d819ee3de4e77cb036644dc5b05 | Two teams, \( A \) and \( B \), play a game on a circumference. Team \( A \) has \( n \) blue flags and team \( B \) has \( n \) white flags (\( n \geq 2 \)). They take turns placing one flag each on the circumference, starting with \( A \). Once all \( 2n \) flags are placed, the territory is divided such that a point... | [
"3"
] | [
"Two teams, \\( A \\) and \\( B \\), play a game on a circumference. Team \\( A \\) has \\( n \\) blue flags and team \\( B \\) has \\( n \\) white flags (\\( n \\geq 2 \\)). They take turns placing one flag each on the circumference, starting with \\( A \\). Once all \\( 2n \\) flags are placed, the territory is d... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f838e5dab2145b7fe674c1f4c68853562664c600a176367e5143350a1197b4cc | Two teams, team A and team B, play a game where each team sends one player to fight in each round. If the player from team A wins, a player from team B joins team A. If the player from team B wins, a player from team A is eliminated. Team A starts with 100 players. How many players must team B have to have a 50% probab... | [
"10942"
] | [
"Two teams, team A and team B, play a game where each team sends one player to fight in each round. If the player from team A wins, a player from team B joins team A. If the player from team B wins, a player from team A is eliminated. Team A starts with 100 players. How many players must team B have to have a 50% p... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb78855c6879be37a35f5c2d8c926377a1d7414dd28df71dac465021495ee249 | Two tennis robots, Amy and Bob, are playing tennis. Each time Amy hits the ball, there is a 10% chance it will hit out or into the net, a 30% chance it will hit in but out of Bob's reach (winning the point), and a 60% chance Bob will hit the ball in return. Each time Bob hits the ball, there is a 20% chance it will hit... | [
"\\(\\frac{42}{73}\\)"
] | [
"Two tennis robots, Amy and Bob, are playing tennis. Each time Amy hits the ball, there is a 10% chance it will hit out or into the net, a 30% chance it will hit in but out of Bob's reach (winning the point), and a 60% chance Bob will hit the ball in return. Each time Bob hits the ball, there is a 20% chance it wil... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-50e1b64c8f7875fbedfa47f5c5ece50e98f779a11bf94e83847cbf25614e8c63 | Two thin plates \( D_{1} \) and \( D_{2} \) have masses \( m_{1} = 4 \) and \( m_{2} = 6 \), and centers of mass \( c_{1} = (1, 2) \) and \( c_{2} = (5, 3) \) respectively. Calculate the coordinates of the center of mass \( c \) of the union \( D_{1} \cup D_{2} \). | [
"\\left( \\dfrac{17}{5}, \\dfrac{13}{5} \\right)"
] | [
"Two thin plates \\( D_{1} \\) and \\( D_{2} \\) have masses \\( m_{1} = 4 \\) and \\( m_{2} = 6 \\), and centers of mass \\( c_{1} = (1, 2) \\) and \\( c_{2} = (5, 3) \\) respectively. Calculate the coordinates of the center of mass \\( c \\) of the union \\( D_{1} \\cup D_{2} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42c92ca6304e284678bf331a3e19ed0cfc61d87122a6dca4fc73a048740537dc | Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\dot... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 25
} | 42 | 0 | 0 |
omr-ws-6de53935def33612080a32758edc8210b36e12eff6eb34e68dee832516c4f352 | Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. Continue this process until the labels $1, 2, 3, \dots, 1993$ a... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. Continue this process until the labels $1, 2, 3, \\dots, 1... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 18
} | 37 | 0 | 0 |
omr-ws-49dd1ef133cb2866e919f9c77e1fdfeb86b46e7531d210fb61dd33f8e38cab99 | Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. Continue this process until the labels $1,2,3,\dots,1993$ are a... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. Continue this process until the labels $1,2,3,\\dots,1993$... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-c4123f0fb0f3831a6522c0e70981d1cb5a011b99fb98941bb85a2a20c23d4f2c | Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used.... | [
"45"
] | [
"Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \\dots, 1993$ are all... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc4e98ba9ae2cd1cc10bb5e952d5ce915f47cd5402d6bdd07d67ce9674c23a3f | Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels 1, 2, 3, ..., 1993 are all used. Som... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels 1, 2, 3, ..., 1993 are all used... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 14,
"tir": 18
} | 61 | 0 | 0 |
omr-ws-7a8157271a04569b28cbba6468439699a322a0620afadc6d311c75ade1c9ec4f | Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels 1, 2, 3, ..., 1993 are all used. Som... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels 1, 2, 3, ..., 1993 are all used... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9c23a12720bd40ae18634513ba803877c1f6b49883bc4e16e37c4972f772d1e | Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels \(1, 2, 3, \dots, 1993\) are all use... | [
"118"
] | [
"Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels \\(1, 2, 3, \\dots, 1993\\) are... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 20
} | 38 | 0 | 0 |
omr-ws-ccd1634a45f29724eac75cd9d22251b935c18f772cebbb4ab2c253d729705abc | Two three-digit numbers form a six-digit number (e.g., 263 and 458 form 263458). The six-digit number is an integer multiple of the product of the two three-digit numbers. Find the six-digit number. (Example: 167334 = 3 * (167 * 334)) | [
"167334"
] | [
"Two three-digit numbers form a six-digit number (e.g., 263 and 458 form 263458). The six-digit number is an integer multiple of the product of the two three-digit numbers. Find the six-digit number. (Example: 167334 = 3 * (167 * 334))"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 6,
"tir": 2
} | 30 | 0 | 0 |
omr-ws-e9c3e56cb4ba5415f416d4d4d084f5e06a3870a9dc2f6c6d466af02359d0313a | Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each letter has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-letter... | [
"532"
] | [
"Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each letter has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-l... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-b824e71338a4629d3e95c25e6857db9bb33dba54892b8c9e59e35014e7ca1478 | Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$, or as a $b^{}_{}$ when it should be an $a^{}... | [
"532"
] | [
"Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$, or as a $b^{}_{}$ when it should be an ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-762a4be3c279394ecfe3188ef8b53ae2425e98aba934fd627d04f609cc65adbe | Two three-letter strings, \( A \) and \( B \), are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a \( \frac{1}{3} \) chance of being received incorrectly, as an \( X \) when it should have been a \( Y \), or as a \( Y \) when it should be an \( X ... | [
"532"
] | [
"Two three-letter strings, \\( A \\) and \\( B \\), are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a \\( \\frac{1}{3} \\) chance of being received incorrectly, as an \\( X \\) when it should have been a \\( Y \\), or as a \\( Y \\) when it ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-1fc7b1a1c70d012c0c265404de97809e5709e8efc5821f6013a263e29ebf5ae4 | Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in the positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where bot... | [
"19152"
] | [
"Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in the positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane wher... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 16
} | 64 | 0 | 0 |
omr-ws-cff3eb99bec78b694d3a27ef9cb25d2f34ca01556a5dfbb1af5cbbf757a000df | Two touching circles $Y$ and $Z$ share a common tangent which meets $Y$ at $A$ and $Z$ at $B$. Let $AP$ be a diameter of $Y$ and let the tangent from $P$ to $Z$ touch $Z$ at $Q$. If the radius of circle $Y$ is $r$ and the radius of circle $Z$ is $2r$, find the length of $PQ$ in terms of $r$. | [
"2r"
] | [
"Two touching circles $Y$ and $Z$ share a common tangent which meets $Y$ at $A$ and $Z$ at $B$. Let $AP$ be a diameter of $Y$ and let the tangent from $P$ to $Z$ touch $Z$ at $Q$. If the radius of circle $Y$ is $r$ and the radius of circle $Z$ is $2r$, find the length of $PQ$ in terms of $r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef818521b1f8849c1ccaa2648dcf5a4756b7ec030190c0c70e348f9222ec6868 | Two touching circles \( S \) and \( T \) share a common tangent which meets \( S \) at \( A \) and \( T \) at \( B \). Let \( AP \) be a diameter of \( S \) and let the tangent from \( P \) to \( T \) touch it at \( Q \). If the radius of circle \( S \) is 5 units and the radius of circle \( T \) is 3 units, calculate ... | [
"10"
] | [
"Two touching circles \\( S \\) and \\( T \\) share a common tangent which meets \\( S \\) at \\( A \\) and \\( T \\) at \\( B \\). Let \\( AP \\) be a diameter of \\( S \\) and let the tangent from \\( P \\) to \\( T \\) touch it at \\( Q \\). If the radius of circle \\( S \\) is 5 units and the radius of circle \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01019d5fcc72db23da2c488a1ea4835846068e0fd7e2cb9c13770833eb5a7767 | Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? | [
"62"
] | [
"Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73e115b0728ea837097a66b1ed89bf7c4c8a8df9f355097335b1bd72160d9fb4 | Two tourists cycled towards each other, one from point A and the other from point B. The first tourist left point A 6 hours later than the second left point B. On their meeting, the first tourist had traveled 12 km less than the second. After meeting, they kept cycling with the same speed, and the first tourist arrived... | [
"\\( v_{A} = 6 \\) km/h, \\( v_{B} = 4 \\) km/h"
] | [
"Two tourists cycled towards each other, one from point A and the other from point B. The first tourist left point A 6 hours later than the second left point B. On their meeting, the first tourist had traveled 12 km less than the second. After meeting, they kept cycling with the same speed, and the first tourist ar... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-0b070a86b35db33bbcdc91536bdf4e7b6f21963d7a149bde07724945e3a7526b | Two towers, 90m and 150m tall, stand opposite each other 100m apart. A bridge is constructed 70m above the ground connecting the two towers. A rope is tied from the top of the smaller tower to the foot of the taller tower. Find the distance from the top of the smaller tower to the point where the rope crosses the bridg... | [
"\\(\\frac{20\\sqrt{181}}{9}\\)"
] | [
"Two towers, 90m and 150m tall, stand opposite each other 100m apart. A bridge is constructed 70m above the ground connecting the two towers. A rope is tied from the top of the smaller tower to the foot of the taller tower. Find the distance from the top of the smaller tower to the point where the rope crosses the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1edfbb1a1168e96d902cc1f4117a3ff415e89f2fc5aa02cc55bf3f243e9dbcad | Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, starting at midnight, a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes, starting at midnight, a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noo... | [
"5"
] | [
"Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, starting at midnight, a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes, starting at midnight, a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-0f96ec9b9d2c2220e7c505ab214807f23555b74d5502b4248846b8a255a7fc03 | Two towns, A and B, are 120 km apart. A car starts from A to travel to B, and a second car, whose speed is 20 km/h faster than the first, starts from B to travel to A at the same time. The slower car reaches B 1 hour and 48 minutes after it passes the other car. Find the speeds of both cars. | [
"40 km/h and 60 km/h"
] | [
"Two towns, A and B, are 120 km apart. A car starts from A to travel to B, and a second car, whose speed is 20 km/h faster than the first, starts from B to travel to A at the same time. The slower car reaches B 1 hour and 48 minutes after it passes the other car. Find the speeds of both cars."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 31
} | 33 | 0 | 0 |
omr-ws-33051570386fc0e10f17b4acf3e3b3129a373c44388ddd356b1a38ef0e1d9d6d | Two towns, A and B, are connected by a road. Pat begins biking from A to B and Dana begins biking from B to A at sunrise. They cross paths at noon. Pat reaches B at 5 pm, while Dana reaches A at 11:15 pm. When was sunrise? | [
"4:30 AM"
] | [
"Two towns, A and B, are connected by a road. Pat begins biking from A to B and Dana begins biking from B to A at sunrise. They cross paths at noon. Pat reaches B at 5 pm, while Dana reaches A at 11:15 pm. When was sunrise?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-ce335b5309216fa52defe950aab888cd0c6fee2fc4087f79e414511f1286e5af | Two towns, \( A \) and \( B \), are connected by a road. At sunrise, Pat begins biking from \( A \) to \( B \), and Dana begins biking from \( B \) to \( A \). They cross paths at noon. Pat reaches \( B \) at 5 pm, while Dana reaches \( A \) at 11:15 pm. When was sunrise? | [
"4:30\\,\\text{AM}"
] | [
"Two towns, \\( A \\) and \\( B \\), are connected by a road. At sunrise, Pat begins biking from \\( A \\) to \\( B \\), and Dana begins biking from \\( B \\) to \\( A \\). They cross paths at noon. Pat reaches \\( B \\) at 5 pm, while Dana reaches \\( A \\) at 11:15 pm. When was sunrise?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42e6173bf686b57e3dd1f3e313046f3bd90e4d008366b27306dc0d23e9a54b1d | Two tractors of different performance characteristics, working together, ploughed a field in $t$ days. If at first one tractor had ploughed half the field, and then the other one had completed the other half, the ploughing operation would have been completed in $k$ days. How many days would it take each tractor to plou... | [
"It would take one of the tractors $\\left(k+\\sqrt {k^2-2kt}\\right)$ days and the other, $\\left(k-\\sqrt {k^2-2kt}\\right)$ days. The problem is solvable for $\\frac{k}{2}>t$"
] | [
"Two tractors of different performance characteristics, working together, ploughed a field in $t$ days. If at first one tractor had ploughed half the field, and then the other one had completed the other half, the ploughing operation would have been completed in $k$ days. How many days would it take each tractor to... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dafc86b6a778032d0d3d26bdb9f2d4b44325fef663842d421441e53b83e37b93 | Two tractors plough a field divided into two equal parts. The tractors begin working at the same time, each on its part. Five hours after they ploughed half the field working together, it was found that it remained for the first tractor to plough 1/10 of its part, and the second tractor had to plough 4/10 more of its p... | [
"50 h"
] | [
"Two tractors plough a field divided into two equal parts. The tractors begin working at the same time, each on its part. Five hours after they ploughed half the field working together, it was found that it remained for the first tractor to plough 1/10 of its part, and the second tractor had to plough 4/10 more of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07d6f946788ca5d9ff10b939076d651f714c6f368b8dfd633ae6841c7494900e | Two tractors plough a field separated into two equal areas. Both tractors began working simultaneously, each ploughing its half. Five hours later they had ploughed half of the whole field, leaving $\frac{1}{10}$ of the area for the first tractor to complete, and $\frac{2}{5}$ for the second tractor to finish. How much ... | [
"50 hours"
] | [
"Two tractors plough a field separated into two equal areas. Both tractors began working simultaneously, each ploughing its half. Five hours later they had ploughed half of the whole field, leaving $\\frac{1}{10}$ of the area for the first tractor to complete, and $\\frac{2}{5}$ for the second tractor to finish. Ho... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0acae1b7a17adc657f6a6944fff06dd3104a7f4cedc7d8d4440458435963a360 | Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (< \alpha)$ at the same station when its rear is ... | [
"\\dfrac{3 + \\sqrt{3}}{3}"
] | [
"Two train lines intersect each other at a junction at an acute angle $\\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\\alpha$ at a station on the other line. It subtends an angle $\\beta (< \\alpha)$ at the same station when its... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d8fd351ddfeaa7538559805788c6541ae03e1911db7a453c0291bf517b2a7c72 | Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (<\alpha)$ at the same station when its rear is a... | [
"1+\\sqrt{3}"
] | [
"Two train lines intersect each other at a junction at an acute angle $\\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\\alpha$ at a station on the other line. It subtends an angle $\\beta (<\\alpha)$ at the same station when its ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e9a89f3966dbf6bf8d83c15918e52de5b6a014c82ad6388cd7be548c2f85339 | Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (<\alpha)$ at the same station, when its rear is ... | [
"\\sqrt{3}"
] | [
"Two train lines intersect each other at a junction at an acute angle $\\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\\alpha$ at a station on the other line. It subtends an angle $\\beta (<\\alpha)$ at the same station, when its... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-428d8f66e59848b9674da24b5bd4d4a80c9e0560f49afc30213df7341d1240b1 | Two train lines intersect each other at a junction at an acute angle $\theta$. When the front of a train is at the junction, it subtends an angle $\alpha$ at a station on the other line. When the rear of the train is at the junction, it subtends an angle $\beta (< \alpha)$ at the same station. Calculate the value of $\... | [
"1 + \\sqrt{3}"
] | [
"Two train lines intersect each other at a junction at an acute angle $\\theta$. When the front of a train is at the junction, it subtends an angle $\\alpha$ at a station on the other line. When the rear of the train is at the junction, it subtends an angle $\\beta (< \\alpha)$ at the same station. Calculate the va... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5c156d900cbaecb544312e6e39f010de2410b9bf6117d0a8d7f3dc7ef44956bf | Two train lines intersect each other at a junction at an acute angle $\theta$. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. When its rear is at the junction, it subtends an angle $\beta (< \alpha)$ at the same station. If $\alpha = 75^\circ$ and $\... | [
"\\dfrac{3 + \\sqrt{3}}{3}"
] | [
"Two train lines intersect each other at a junction at an acute angle $\\theta$. When the front of the train is at the junction, the train subtends an angle $\\alpha$ at a station on the other line. When its rear is at the junction, it subtends an angle $\\beta (< \\alpha)$ at the same station. If $\\alpha = 75^\\c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3d3193ea79b755fb6bf7ae1b3ad6c6ee1673a0d143b0eb7e208517abbce21c9 | Two trains -- a freight train 490 meters long and a passenger train 210 meters long -- were travelling along parallel tracks towards each other. The driver of the passenger train noticed the freight train when it was 700 meters away; 28 seconds later they passed each other. Determine the speed of each train, if we know... | [
"The speed of the freight train is equal to 10 m/s, i.e. 36 km/h and that of the passenger train, to 15 m/s. i.e. 54 km/h"
] | [
"Two trains -- a freight train 490 meters long and a passenger train 210 meters long -- were travelling along parallel tracks towards each other. The driver of the passenger train noticed the freight train when it was 700 meters away; 28 seconds later they passed each other. Determine the speed of each train, if we... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86ac34f3595786946414b6b77ae1c9814d4c4b6904289c93e07eed46d33f586e | Two trains 15km apart are traveling towards each other along the same track. The first train travels at 45 meters per second, the second at 15 meters per second. A fly is hovering just around the nose of the first train, flying at a constant rate of 75 meters per second from the first train to the second train, turns a... | [
"8"
] | [
"Two trains 15km apart are traveling towards each other along the same track. The first train travels at 45 meters per second, the second at 15 meters per second. A fly is hovering just around the nose of the first train, flying at a constant rate of 75 meters per second from the first train to the second train, tu... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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