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-809851a8dcd9cc780ec0d8f4d7a26f10457ad062f65a948f16913af02b567d36 | Twenty-five of King Arthur's knights are seated at a round table. Three of them are chosen at random and sent off to slay a dragon. What is the probability that at least two of the three chosen knights were sitting next to each other? If this probability is written as a fraction in lowest terms, what is the sum of the ... | [
"57"
] | [
"Twenty-five of King Arthur's knights are seated at a round table. Three of them are chosen at random and sent off to slay a dragon. What is the probability that at least two of the three chosen knights were sitting next to each other? If this probability is written as a fraction in lowest terms, what is the sum of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dba965faedaf1193c81a659d64373806f31a864d2cf7d857ba00aa25bee0ea97 | Twenty-five of the numbers $1, 2, \cdots, 50$ are chosen. Twenty-five of the numbers $51, 52, \cdots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers. | [
"2525"
] | [
"Twenty-five of the numbers $1, 2, \\cdots, 50$ are chosen. Twenty-five of the numbers $51, 52, \\cdots, 100$ are also chosen. No two chosen numbers differ by $0$ or $50$. Find the sum of all $50$ chosen numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0f924d22b1ed7b43f99a0ed0034d9d61b4360d30185e21baf9a8edfe43d5fe8f | Twenty-five people are arranged in a circle. Three are selected at random. What is the probability that none of those selected were next to one another? | [
"\\(\\frac{77}{115}\\)"
] | [
"Twenty-five people are arranged in a circle. Three are selected at random. What is the probability that none of those selected were next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86798fe44e0b051cb16564103a4f9402f07bdaa446b51c4486fa110ca0683a10 | Twenty-five people form several committees. Each committee has five members, and any two committees have at most one common member. Determine the maximum number of committees. | [
"30"
] | [
"Twenty-five people form several committees. Each committee has five members, and any two committees have at most one common member. Determine the maximum number of committees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e0b4f007b713144343b66b61ea647dbfd9d673a1cd68865a0b1ecc170f8d866 | Twenty-five people form some committees with each committee having 5 members and each pair of committees having at most one common member. Determine the maximum number of committees. | [
"30"
] | [
"Twenty-five people form some committees with each committee having 5 members and each pair of committees having at most one common member. Determine the maximum number of committees."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
omr-ws-625ea84e157fe3e76dfbc45cc56f83785daa9a971d19a18cc447364ebc05b996 | Twenty-five people sit around a circular table. Three of them are chosen randomly. What is the probability that at least two of the three are sitting next to one another? | [
"\\dfrac{11}{46}"
] | [
"Twenty-five people sit around a circular table. Three of them are chosen randomly. What is the probability that at least two of the three are sitting next to one another?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8edb37cbb530a6533191744b68cf01d4c6812d708abbe2d2abb2c99390d9e57c | Twenty-five people sit around a circular table. Three of them are chosen randomly. What is the probability that two of the three are sitting next to each other? | [
"\\(\\frac{11}{46}\\)",
"\\(\\frac{21}{92}\\)"
] | [
"Twenty-five people sit around a circular table. Three of them are chosen randomly. What is the probability that two of the three are sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 31
} | 44 | 0 | 0 |
omr-ws-93cd7c645939ede2a924afdb89035d6f237251f14ff0cc223c0fdeef65b19ed7 | Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie? | [
"13"
] | [
"Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5e8a012c27587f95eaeec406c2e2f0cfc142e0c3c5da381da08208a881c00bf | Twenty-five points are arranged in a five by five grid. If two distinct points are chosen at random from the grid, what is the probability that the segment connecting these two points will not pass through any of the other points? | [
"\\(\\frac{2}{3}\\)"
] | [
"Twenty-five points are arranged in a five by five grid. If two distinct points are chosen at random from the grid, what is the probability that the segment connecting these two points will not pass through any of the other points?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 23
} | 29 | 0 | 0 |
omr-ws-6f5bfaa0aa8ff78edbfa37cc95deec0de3ffcdb090805a2712455c9fbb5c5b36 | Twenty-five students are seated in a circle around a teacher. The teacher chooses 3 students to answer questions on the blackboard. What is the probability that at least two of the three students had been sitting next to each other? | [
"\\(\\frac{11}{46}\\)"
] | [
"Twenty-five students are seated in a circle around a teacher. The teacher chooses 3 students to answer questions on the blackboard. What is the probability that at least two of the three students had been sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-0cf6bcf9fb6c0b22b99c03b750cf35c0c8bb1261650b893cbc69c5493c05d8b3 | Twenty-five workmen have completed a fifth of a project in eight days. Their foreman then decides that the project must be completed in the next 20 days. What is the smallest number of additional workmen required to complete the project on time? | [
"15"
] | [
"Twenty-five workmen have completed a fifth of a project in eight days. Their foreman then decides that the project must be completed in the next 20 days. What is the smallest number of additional workmen required to complete the project on time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-a95dbc9d3acaca78f60caf720b2119cf0b3a4de2f61a1431b1456ba1929749f9 | Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long h... | [
"3"
] | [
"Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and l... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b1639dcc17dd41d7558c094661a556c6101cf777af5b3c4fe6ec058ae5ff616f | Twenty-four liters of orange juice and forty-five liters of apple juice are poured into two empty containers $A$ and $B$. If $60\%$ of the liquid in container $A$ is orange juice and $70\%$ of the liquid in container $B$ is apple juice, how many liters of liquid are in container $B$? | [
"58"
] | [
"Twenty-four liters of orange juice and forty-five liters of apple juice are poured into two empty containers $A$ and $B$. If $60\\%$ of the liquid in container $A$ is orange juice and $70\\%$ of the liquid in container $B$ is apple juice, how many liters of liquid are in container $B$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-bc20b84f72572702ab9b8e7f4f2d867725877930b954c128ac0619623dc5bbc0 | Twenty-one people form several committees. Each committee has five members, and any two committees have at most one common member. Determine the maximum number of committees. | [
"21"
] | [
"Twenty-one people form several committees. Each committee has five members, and any two committees have at most one common member. Determine the maximum number of committees."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f8714c0f9592e1b248c20595c3a48605030f2c546b2ecdf54842dcba9dcfe979 | Twenty-one rectangles of size $3 \times 1$ are placed on an $8 \times 8$ chessboard, leaving only one free unit square. What position can the free square lie at? | [
"(3, 3), (3, 6), (6, 3), (6, 6)"
] | [
"Twenty-one rectangles of size $3 \\times 1$ are placed on an $8 \\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bc938cc007dbef6fa402559e9c626e6d401a788c254aef5690f6cc3ec22f679 | Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue, and yellow bowls are $15$, $3$, and $18$, respectively? | [
"21, 16, 11"
] | [
"Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue, and yellow bowls are $15$, $3$, and $18$, respectively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 13
} | 19 | 0 | 0 |
omr-ws-f7a4d00e866f6ff3551c6a500deed4621b383a9a3a6453db4e823f3b3ed1da62 | Twenty-seven increased by twice a number is 39. What is the number? | [
"6"
] | [
"Twenty-seven increased by twice a number is 39. What is the number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed90a4311819fc2b3a50052a0d2bbbd8c2117e81e27d84a427ea62c44ee274bb | Twenty-seven students in a class will be made into six discussion groups, each consisting of four or five students. Find the number of ways to do this. | [
"\\dfrac{27!}{(4!^3 \\cdot 5!^3 \\cdot 3!^2)}"
] | [
"Twenty-seven students in a class will be made into six discussion groups, each consisting of four or five students. Find the number of ways to do this."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-f86765536aa58785802705c0d7fa5f8bb9bd16d3546ffce19d7591322bed1b92 | Twenty-seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. These cubes are randomly arranged to form a $3 \times 3 \times 3$ cube. Given the probability that the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c}$, where $p$, $q$, and $r$ are dist... | [
"74"
] | [
"Twenty-seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. These cubes are randomly arranged to form a $3 \\times 3 \\times 3$ cube. Given the probability that the entire surface area of the larger cube is orange is $\\frac{p^a}{q^br^c}$, where $p$, $q$, and $r$ ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-eabff55ad5a7be31ceb9dc140a9aea94b98f8e0241a4b0de1f541fba0d52faa2 | Twenty-three people each with integral weight decide to play football, separating into two teams of eleven people plus a referee. To keep things fair, the teams chosen must have equal total weight. If no matter who is chosen as the referee, this can be done, find the total weight of all twenty-three people given that t... | [
"1748"
] | [
"Twenty-three people each with integral weight decide to play football, separating into two teams of eleven people plus a referee. To keep things fair, the teams chosen must have equal total weight. If no matter who is chosen as the referee, this can be done, find the total weight of all twenty-three people given t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4aefbcab83567e4be31127e2958f0fba8f48ab19f80daf991d4621cf6c51c81 | Twenty-three people of positive integral weights decide to play football. They select one person as referee and then split up into two 11-person teams of equal total weights. Given that no matter who is the referee, this can always be done, find the minimum possible value of the sum of all their weights if the lightest... | [
"23"
] | [
"Twenty-three people of positive integral weights decide to play football. They select one person as referee and then split up into two 11-person teams of equal total weights. Given that no matter who is the referee, this can always be done, find the minimum possible value of the sum of all their weights if the lig... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6888f458a5991a1a12630a0c6c38a24f5ce201cdc86cc478021d1e12d443bca4 | Twenty-two football players were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training? | [
"5"
] | [
"Twenty-two football players were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 8
} | 31 | 0 | 0 |
omr-ws-c91c9c80c3112c5fed31e3af00b6d547043509b9ce01b236a414d217512cbca3 | Twice a certain positive integer is the sum of three consecutive square numbers. Eleven times that same integer is equal to the sum of the next three consecutive squares. What is the original integer? | [
"7"
] | [
"Twice a certain positive integer is the sum of three consecutive square numbers. Eleven times that same integer is equal to the sum of the next three consecutive squares. What is the original integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b81fece2c4600f336e2860ee9973b996e2e11108bfd06af6515f76c7f5cbc016 | Twice the sum of two numbers exceeds three times their difference by $8$, while half the sum is one more than the difference. What are the numbers? | [
"(7, 3)"
] | [
"Twice the sum of two numbers exceeds three times their difference by $8$, while half the sum is one more than the difference. What are the numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 31
} | 32 | 0 | 0 |
omr-ws-1f16b41ce9a5999bc2d046a40b42038cb2450c9894b83b3b998aa1869860774f | Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times$ (the positive difference of the dice). What is the expected value of $x$? | [
"\\dfrac{2793}{20}"
] | [
"Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\\times$ (the positive difference of the dice). What is the expected value of $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b911585a8f0c2536cd5599734ea5e02223e56ff9e4ce1ab953fcb94f6c98f99 | Two $5 \times 1$ rectangles have 2 vertices in common. Determine the area of overlap. | [
"$2.4$"
] | [
"Two $5 \\times 1$ rectangles have 2 vertices in common. Determine the area of overlap."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5786ceea4ebfacaf5a06654698190c2b3abea9d3f6d8f56981c41ab7224d111 | Two $5 \times 1$ rectangles have 2 vertices in common. Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. | [
"\\(\\frac{\\sqrt{26}}{5}\\)"
] | [
"Two $5 \\times 1$ rectangles have 2 vertices in common. Determine the length of the segment between the other 2 points of intersection, $A$ and $B$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 9
} | 57 | 0 | 0 |
omr-ws-3975b94ace53cab8c8684605654bab55a078bb588ef88e038679d187837f42a8 | Two (not necessarily distinct) subsets $A_{1}$ and $A_{2}$ of $X=\{1,2,\ldots,n\}$ are chosen in arbitrary way. Find the probability of $A_{1}\bigcap A_{2}=\emptyset.$ | [
"\\left( \\dfrac{3}{4} \\right)^n"
] | [
"Two (not necessarily distinct) subsets $A_{1}$ and $A_{2}$ of $X=\\{1,2,\\ldots,n\\}$ are chosen in arbitrary way. Find the probability of $A_{1}\\bigcap A_{2}=\\emptyset.$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6ee5185fddd5267e8307a29edea5b5d409b811f1bc66b0f55f7977b983f9f52 | Two 10-digit integers are called neighbors if they differ in exactly one digit (for example, integers 1234567890 and 1234067890 are neighbors). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbors. | [
"900000000"
] | [
"Two 10-digit integers are called neighbors if they differ in exactly one digit (for example, integers 1234567890 and 1234067890 are neighbors). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbors."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-3d69995a00e6abe4d4c55a8e2ba1b5389d8cdef1bbc230582e7dc9b655714a27 | Two 10-digit integers are called neighbors if they differ in exactly one digit. Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbors. | [
"900000000"
] | [
"Two 10-digit integers are called neighbors if they differ in exactly one digit. Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbors."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7dad2bdea7c26b9c1fb0ecaf1333877ac30e6cd30eb54db31a526909f4501b2f | Two 2025-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$, form the legs of a right triangle. Find the sum of the digits in the hypotenuse. | [
"18198"
] | [
"Two 2025-digit numbers, $428\\underbrace{\\hbox{99... 99}}_{\\hbox{2019 \\,\\, 9's}}571$ and $571\\underbrace{\\hbox{99... 99}}_{\\hbox{2019 \\,\\, 9's}}428$, form the legs of a right triangle. Find the sum of the digits in the hypotenuse."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a3eceaf176f682b403e8d65bca2a9763c6cc5f0e644d471f8234bc3b5893e9e | Two 3x4 rectangles overlap at the corner so that their sides are perpendicular. The new shape has an area of 22 and a perimeter of 20. What is the distance between the two corners where the rectangles meet? | [
"\\(2\\sqrt{3}\\)"
] | [
"Two 3x4 rectangles overlap at the corner so that their sides are perpendicular. The new shape has an area of 22 and a perimeter of 20. What is the distance between the two corners where the rectangles meet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-179c3ace1e3279e2f8ace378e9fafb9368fc3431bf8e8a5d920129f054c717a0 | Two 5-digit numbers are called "responsible" if they satisfy the following conditions: - They are in the form of \(abcde\) and \(fghij\) such that \(fghij = 2(abcde)\). - All ten digits \(a\) through \(j\) are distinct. - The sum of all ten digits is 45. If two "responsible" numbers are as small as possible, what is th... | [
"22"
] | [
"Two 5-digit numbers are called \"responsible\" if they satisfy the following conditions:\n- They are in the form of \\(abcde\\) and \\(fghij\\) such that \\(fghij = 2(abcde)\\).\n- All ten digits \\(a\\) through \\(j\\) are distinct.\n- The sum of all ten digits is 45.\n\nIf two \"responsible\" numbers are as smal... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-28a531f1407c0fad27d882ef9ef6d8386dedb915afba73f41285b8167991898e | Two 5-digit numbers are called "responsible" if they satisfy the following conditions: - They are in the form of \(abcde\) and \(fghij\) such that \(fghij = 2(abcde)\). - All ten digits \(a\) through \(j\) are distinct. - The sum of all ten digits is 45. If two "responsible" numbers are as small as possible, what is th... | [
"24"
] | [
"Two 5-digit numbers are called \"responsible\" if they satisfy the following conditions:\n- They are in the form of \\(abcde\\) and \\(fghij\\) such that \\(fghij = 2(abcde)\\).\n- All ten digits \\(a\\) through \\(j\\) are distinct.\n- The sum of all ten digits is 45.\n\nIf two \"responsible\" numbers are as smal... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aee2858d8eef40cfb1b124776b344348486fecfc3bc46f91d5914c3424668394 | Two 5-digit numbers are called "responsible" if they satisfy the following conditions: 1. They are in the form of $abcde$ and $fghij$ such that $fghij = 2(abcde)$. 2. All ten digits, $a$ through $j$, are distinct. 3. The sum of all ten digits is 45. If two "responsible" numbers are as small as possible, what is the sum... | [
"22"
] | [
"Two 5-digit numbers are called \"responsible\" if they satisfy the following conditions:\n1. They are in the form of $abcde$ and $fghij$ such that $fghij = 2(abcde)$.\n2. All ten digits, $a$ through $j$, are distinct.\n3. The sum of all ten digits is 45.\n\nIf two \"responsible\" numbers are as small as possible, ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-bc5a814e760c2d16b286958c77cdc6d6e1f848982299a96cc1f346ca2090ce11 | Two 500-meter long escalators are side by side, both going upwards at 1 meter per second. Jinsong places his backpack on one escalator and at the exact same time begins running up his escalator. When he reaches the top, he immediately begins running down the stairs and when he reaches the bottom of the stairs he immedi... | [
"3"
] | [
"Two 500-meter long escalators are side by side, both going upwards at 1 meter per second. Jinsong places his backpack on one escalator and at the exact same time begins running up his escalator. When he reaches the top, he immediately begins running down the stairs and when he reaches the bottom of the stairs he i... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-28e2eacea98bd2b8825bcca7e3fc63e13d0d46e7d9ff88ab241885a49d2d59c4 | Two 6-sided dice are relabeled with positive integers (not {1,2,3,4,5,6}). If the probability distribution of the sums of the numbers rolled on these dice is identical to that of standard 6-sided dice, find the sum of all possible values for the largest number on the first die. | [
"8"
] | [
"Two 6-sided dice are relabeled with positive integers (not {1,2,3,4,5,6}). If the probability distribution of the sums of the numbers rolled on these dice is identical to that of standard 6-sided dice, find the sum of all possible values for the largest number on the first die."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-d2e17ea221dd06bbebb737b37d602d8c86a80139600dc02f409c1cfdbfd87c07 | Two 6-sided dice, one red and one green, are rolled. What is the probability that the red die shows an odd number and the green die shows a number that is a perfect square? | [
"\\frac16"
] | [
"Two 6-sided dice, one red and one green, are rolled. What is the probability that the red die shows an odd number and the green die shows a number that is a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48909fed0fc4dda7734bafbc28f37f179d1cb5cf96663751b1c959ee2ed665e4 | Two A's and two B's are randomly placed inside a 3x3 grid. What is the probability that the line connecting the two B's intersects? (Note: A line is extended in both directions.) | [
"\\(\\frac{52}{63}\\)"
] | [
"Two A's and two B's are randomly placed inside a 3x3 grid. What is the probability that the line connecting the two B's intersects? (Note: A line is extended in both directions.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18380d05e7c414a71181c53a2c70b1c38d880a52efb34b0876ac0cee42a50327 | Two acute triangles \(ABC\) and \(A_1B_1C_1\) are such that \(B_1\) and \(C_1\) lie on \(BC\), and \(A_1\) lies inside the triangle \(ABC\). Let \(S\) and \(S_1\) be the areas of triangles \(ABC\) and \(A_1B_1C_1\) respectively. If \(AB = 13\), \(AC = 15\), and \(BC = 14\), calculate the maximum possible value of \(\fr... | [
"3"
] | [
"Two acute triangles \\(ABC\\) and \\(A_1B_1C_1\\) are such that \\(B_1\\) and \\(C_1\\) lie on \\(BC\\), and \\(A_1\\) lies inside the triangle \\(ABC\\). Let \\(S\\) and \\(S_1\\) be the areas of triangles \\(ABC\\) and \\(A_1B_1C_1\\) respectively. If \\(AB = 13\\), \\(AC = 15\\), and \\(BC = 14\\), calculate th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4701b40c96b4720530857da6eae41232109226161f0ee66fe97d22c0905781a | Two acute-angled triangles, $\triangle ABC$ and $\triangle DEF$, share the same circumcircle. The Euler circle (nine-point circle) of $\triangle ABC$ passes through the midpoints of two sides of $\triangle DEF$. If $M$ and $N$ are the midpoints of $DE$ and $DF$ respectively, and the radius of the Euler circle of $\tria... | [
"25"
] | [
"Two acute-angled triangles, $\\triangle ABC$ and $\\triangle DEF$, share the same circumcircle. The Euler circle (nine-point circle) of $\\triangle ABC$ passes through the midpoints of two sides of $\\triangle DEF$. If $M$ and $N$ are the midpoints of $DE$ and $DF$ respectively, and the radius of the Euler circle ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48d25002d8d55d01d9eeef23f25ad9323bde8f8e771b00fa456f2a652928d939 | Two adjacent sides of a rhombus form a $60$-degree angle. If each side of the rhombus measures $2$ cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form. | [
"2\\sqrt{3}"
] | [
"Two adjacent sides of a rhombus form a $60$-degree angle. If each side of the rhombus measures $2$ cm, what is the area of the rhombus, in square centimeters? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d25e5358aa3dcbb301db4e2ae7c78e52781d081bf6b64ad7c02e934ae7995f9e | Two adjacent vertices of a square are (-4,5) and (-3,7). What are all possible points that could be the other vertices of this square? | [
"\\((-6, 4), (-2, 2), (-2, 8), (-6, 6)\\)"
] | [
"Two adjacent vertices of a square are (-4,5) and (-3,7). What are all possible points that could be the other vertices of this square?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-e264432471276fbc089c9dbb58f8e0eafe2de482863d169b8563dc28751793b8 | Two adjacent vertices of a square are on a circle of radius \( R \) and the other two vertices lie on a tangent to the circle. What is the length of the side of the square? | [
"\\(\\frac{8R}{5}\\)"
] | [
"Two adjacent vertices of a square are on a circle of radius \\( R \\) and the other two vertices lie on a tangent to the circle. What is the length of the side of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-5921ee617636764d570dcab62e10331fbfc3f55a4f7799102d1f17314d6d0570 | Two administrators each have a 15-minute project to complete and expect to access a shared database between 4 and 6 p.m. today. Neither will call after 5:45 p.m. and neither will call more than once. At least one of them will be successful. What is the probability that both administrators will complete their project? | [
"\\(\\frac{36}{49}\\)"
] | [
"Two administrators each have a 15-minute project to complete and expect to access a shared database between 4 and 6 p.m. today. Neither will call after 5:45 p.m. and neither will call more than once. At least one of them will be successful. What is the probability that both administrators will complete their proje... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 24
} | 70 | 0 | 0 |
omr-ws-26cba2632081d813317807943ef863a7fbde51a2a5dfed186d811f96e98af979 | Two agents choose speeds $x_1$ and $x_2$ respectively, with utility $x_i^2$ for each agent. The probability of a mutual accident is $1-\frac{1}{x_1 x_2}$. If an accident occurs, it imposes a cost of 2 on agent 1 and 3 on agent 2. Assuming utility is linear in money, find the socially optimal speeds $x_1$ and $x_2$ that... | [
"\\sqrt[4]{\\dfrac{5}{2}}"
] | [
"Two agents choose speeds $x_1$ and $x_2$ respectively, with utility $x_i^2$ for each agent. The probability of a mutual accident is $1-\\frac{1}{x_1 x_2}$. If an accident occurs, it imposes a cost of 2 on agent 1 and 3 on agent 2. Assuming utility is linear in money, find the socially optimal speeds $x_1$ and $x_2... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-42cb1ccc4ec6a34378a0eb32607a966839fd185d72473df0f4376a38cabc99b1 | Two aircraft are flying at an altitude of 2000m and are 147km apart, both heading to the same airfield. The first aircraft is traveling at a bearing of 119 degrees with an average speed of 206 km/h and is expected to land in 24.5 minutes. The second aircraft, flying at a bearing of 78 degrees, is expected to land 5 min... | [
"406"
] | [
"Two aircraft are flying at an altitude of 2000m and are 147km apart, both heading to the same airfield. The first aircraft is traveling at a bearing of 119 degrees with an average speed of 206 km/h and is expected to land in 24.5 minutes. The second aircraft, flying at a bearing of 78 degrees, is expected to land ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70aef4fd97d669219da28a494135191e07f226ba85d8fad9f3dede8df6a5de12 | Two airplanes are approaching Dayton International Airport. The first airplane is a 747 jet, 400 miles to the east, traveling at 325 MPH and descending at 70 ft/min from an altitude of 10,000 ft. The second airplane is a 10-passenger prop plane, 260 miles to the south, traveling at 211.25 MPH and climbing at 24.8 ft/mi... | [
"\\dfrac{16}{13}"
] | [
"Two airplanes are approaching Dayton International Airport. The first airplane is a 747 jet, 400 miles to the east, traveling at 325 MPH and descending at 70 ft/min from an altitude of 10,000 ft. The second airplane is a 10-passenger prop plane, 260 miles to the south, traveling at 211.25 MPH and climbing at 24.8 ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-afba257694e824383df024b1c02cfe9e744151c1f5e14a8f09b9a9fc60d13aa4 | Two airplanes are approaching Dayton International Airport. The first airplane is a 747 jet, 400 miles to the east, traveling at 325 MPH and descending at 70 ft/min from an altitude of 10,000 ft. The second airplane is a 10-passenger prop plane, 260 miles to the south, traveling at 211.25 MPH and climbing at 24.8 ft/mi... | [
"\\dfrac{16}{13} \\text{ hours}"
] | [
"Two airplanes are approaching Dayton International Airport. The first airplane is a 747 jet, 400 miles to the east, traveling at 325 MPH and descending at 70 ft/min from an altitude of 10,000 ft. The second airplane is a 10-passenger prop plane, 260 miles to the south, traveling at 211.25 MPH and climbing at 24.8 ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-5056bd2bb08d77c80355e4198c6bd866d17372f4c7a891ce3542d8aea90da12f | Two airplanes are flying horizontally at the same altitude along paths that are $120^\circ$ apart and at the same speed of $v$ km/h. At a certain moment one of the airplanes reaches the point where the two paths intersect. At that moment the other airplane is $a$ km away (not having reached the intersection). How soon ... | [
"In $\\frac{a}{2v}$ h"
] | [
"Two airplanes are flying horizontally at the same altitude along paths that are $120^\\circ$ apart and at the same speed of $v$ km/h. At a certain moment one of the airplanes reaches the point where the two paths intersect. At that moment the other airplane is $a$ km away (not having reached the intersection). How... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1c2d11ef342ac5d9861929df0c3eccfba7f201984c24db84e10d5710295e5bc | Two airplanes simultaneously take off from $A$ and $B$. Flying towards each other, they meet at a distance of $a$ km from the midpoint of $AB$. If the first airplane took off $b$ hours later than the second, they would meet after passing half the distance from $A$ to $B$. If, conversely, the second airplane took off $b... | [
"s=8a, v_1=\\frac{8a}{3b}, v_2=\\frac{8a}{5b}"
] | [
"Two airplanes simultaneously take off from $A$ and $B$. Flying towards each other, they meet at a distance of $a$ km from the midpoint of $AB$. If the first airplane took off $b$ hours later than the second, they would meet after passing half the distance from $A$ to $B$. If, conversely, the second airplane took o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7bf89e5d3d1356d706d5ceb67c68d76c6d54d0964c8cd0da4116ea85fdc06f57 | Two alarm clocks are set to ring at specific intervals. The first alarm clock, which is 3 minutes fast, first rings when it reads 10:14 and then rings every 9 minutes. The second alarm clock, which is 4 minutes fast, first rings when it reads 10:19 and then rings every 7 minutes. What is the correct time when both alar... | [
"10:29"
] | [
"Two alarm clocks are set to ring at specific intervals. The first alarm clock, which is 3 minutes fast, first rings when it reads 10:14 and then rings every 9 minutes. The second alarm clock, which is 4 minutes fast, first rings when it reads 10:19 and then rings every 7 minutes. What is the correct time when both... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-b192086000bee43bc257f88639129acb2d981221675e2ca4171232477b4a5514 | Two alloys each contain two metals. The ratio of the metals contained in the first alloy is 1:2, and in the second 3:2. In what ratio must these alloys be taken to obtain a new alloy with a ratio of the metals of 8:7? | [
"$1:3$"
] | [
"Two alloys each contain two metals. The ratio of the metals contained in the first alloy is 1:2, and in the second 3:2. In what ratio must these alloys be taken to obtain a new alloy with a ratio of the metals of 8:7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca0f181ac60d7965ba5613a549f2ff4ee3f6e6223078a1cb47de54313c777239 | Two altitudes of a triangle are both tangents to its circumcircle. Find the angles of the triangle. | [
"\\(120^\\circ, 30^\\circ, 30^\\circ\\)"
] | [
"Two altitudes of a triangle are both tangents to its circumcircle. Find the angles of the triangle."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-8bae55f13b9cfebf4eb7f128395622b6ea66e5c093bfb123195baaa63f52bd6a | Two altitudes of a triangle are of length $1$ and $3$. What is the largest number $x$ so that the length of the third altitude must be larger than $x$? | [
"\\(\\frac{3}{4}\\)"
] | [
"Two altitudes of a triangle are of length $1$ and $3$. What is the largest number $x$ so that the length of the third altitude must be larger than $x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-523c7bef1942569f57be0f77a598f6e067e5261edace777820221f9dbe2e2f0d | Two altitudes of a triangle have lengths $12$ and $15$. What is the maximum possible length of the third altitude? | [
"60"
] | [
"Two altitudes of a triangle have lengths $12$ and $15$. What is the maximum possible length of the third altitude?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-7ee6b356e01e376a8bf0af321ab34c78b91bbf692c817ec7a0acd4e95129245b | Two and a half centimeters from the end of a 95 cm diameter of a circle, the diameter intersects a chord and splits the chord into two segments, one of which has length 10 cm. How long is the chord? | [
"\\(\\frac{265}{8}\\)"
] | [
"Two and a half centimeters from the end of a 95 cm diameter of a circle, the diameter intersects a chord and splits the chord into two segments, one of which has length 10 cm. How long is the chord?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 2,
"tir": 22
} | 40 | 0 | 0 |
omr-ws-1b9ad1af424ccf8aaa5aab98cec6b741fafd76168943cd5d8badb9aad15b0e1c | Two and a half liters of 96\% acid solution were poured off from the vessel and 2.5 liters of 80\% solution of the same acid were added, then again 2.5 liters were poured off and 2.5 liters of 80\% acid solution were added. As a result, the vessel contained 89\% acid solution. Find the capacity of the vessel. | [
"10 liters"
] | [
"Two and a half liters of 96\\% acid solution were poured off from the vessel and 2.5 liters of 80\\% solution of the same acid were added, then again 2.5 liters were poured off and 2.5 liters of 80\\% acid solution were added. As a result, the vessel contained 89\\% acid solution. Find the capacity of the vessel."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f6f10f07278bf3c2946180653f473491d06d7ad8537db99afb45d1dc14f954b | Two and a half work crews take two and a half days to clear two and a half hectares of forest. How many work crews are required to clear four hectares of forest in two days? | [
"5"
] | [
"Two and a half work crews take two and a half days to clear two and a half hectares of forest. How many work crews are required to clear four hectares of forest in two days?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf5e564fc419da2eebf59b6e9aa6f33b653a4cdc38dc4d3017e478426dff8c8e | Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth. | [
"28.4"
] | [
"Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b7d296a447aa5d05d6ca2a35ed8481153ae656bf26190626747edc7b7e703cb | Two angles of a triangle measure 30 degrees and 105 degrees, and the side between these two angles measures 2 cm. What is the perimeter of the triangle? | [
"\\(\\frac{8 + 3\\sqrt{6} + 2\\sqrt{2}}{4}\\)"
] | [
"Two angles of a triangle measure 30 degrees and 105 degrees, and the side between these two angles measures 2 cm. What is the perimeter of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a6829b671d00dbbc13d510d1363218cdea866b7a1e23b581b86a6ebeb3f84fa | Two ants are located at the same vertex of a regular octagon. Each minute, each ant moves to one of its adjacent vertices with equal probability. Find the expected number of minutes it will take them to be on opposite vertices (exactly 4 vertices apart). | [
"008",
"8"
] | [
"Two ants are located at the same vertex of a regular octagon. Each minute, each ant moves to one of its adjacent vertices with equal probability. Find the expected number of minutes it will take them to be on opposite vertices (exactly 4 vertices apart)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 29
} | 29 | 0 | 0 |
omr-ws-52b7df35dcc8f03761ca75407053a04a5d1ca3e097ded590bc8cd40c49faba81 | Two ants are on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. What is the probability that both ants return to their starting points after 4 moves? | [
"\\(\\frac{49}{729}\\)"
] | [
"Two ants are on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. What is the probability that both ants return to their starting points after 4 moves?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-90bc548303a42fb237951fa0cee027946f436ffcd280309c583ee4e342ea719f | Two ants are on two distinct vertices of a tetrahedron. Each minute, each ant randomly moves to one of the three adjacent vertices. If \( m \) minutes are expected to pass before the two ants occupy the same vertex, what is the greatest integer less than or equal to \( m \)? | [
"4"
] | [
"Two ants are on two distinct vertices of a tetrahedron. Each minute, each ant randomly moves to one of the three adjacent vertices. If \\( m \\) minutes are expected to pass before the two ants occupy the same vertex, what is the greatest integer less than or equal to \\( m \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-97209e8f8241ef907e0e927daa90e28595cdac7aa1e5c4f166d764d51f78d296 | Two ants are placed at opposite corners B and D of a 4x4 square grid (16 unit squares). Each ant travels to the opposite corner at a constant speed of 1 unit per second, taking exactly 8 seconds to reach their destination. The ants travel along the segments of the unit squares. Find the probability that the two ants wi... | [
"\\dfrac{1}{70}"
] | [
"Two ants are placed at opposite corners B and D of a 4x4 square grid (16 unit squares). Each ant travels to the opposite corner at a constant speed of 1 unit per second, taking exactly 8 seconds to reach their destination. The ants travel along the segments of the unit squares. Find the probability that the two an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-51069d4bea4c41ffc6e9e0908a4fb23c77420914c1015aae036290be5b182c33 | Two ants begin on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. Find the probability that both ants return to their starting position after 4 moves. | [
"\\dfrac{49}{729}"
] | [
"Two ants begin on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. Find the probability that both ants return to their starting position after 4 moves."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-22f34680b3d88b62a8367bb94f9542dbad8190ba6269bfce36b5b1480ba0e8bf | Two ants crawl along the sides of a $7 \times 7$ board, each passing through all 64 vertices exactly once and returning to their starting points. What is the smallest possible number of sides covered by both ants? | [
"16"
] | [
"Two ants crawl along the sides of a $7 \\times 7$ board, each passing through all 64 vertices exactly once and returning to their starting points. What is the smallest possible number of sides covered by both ants?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 5
} | 50 | 0 | 0 |
omr-ws-ac7eed4dcfc75ae34442f15f3f6f2f1977f64f583d973a6c5d53ed5f54095be0 | Two ants crawl out of points $B$ and $C$ on a circle at the same time, moving along the arc $BC$ towards each other. The product of the distances from the ants to point $A$ remains constant throughout their movement. Let $P(t)$ be the position of the point where the line connecting the ants at time $t$ touches a fixed ... | [
"r"
] | [
"Two ants crawl out of points $B$ and $C$ on a circle at the same time, moving along the arc $BC$ towards each other. The product of the distances from the ants to point $A$ remains constant throughout their movement. Let $P(t)$ be the position of the point where the line connecting the ants at time $t$ touches a f... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f8ef753bb0ca721f41628dbec5536f5f1fd97d1a6110b07cabd5dda56853469c | Two ants sit on the surface of a regular tetrahedron with edge length $2$. Calculate the maximum distance one ant must travel to meet the other ant, assuming both ants can only move along the surface of the tetrahedron. Express your answer in terms of $\pi$. | [
""
] | [
"Two ants sit on the surface of a regular tetrahedron with edge length $2$. Calculate the maximum distance one ant must travel to meet the other ant, assuming both ants can only move along the surface of the tetrahedron. Express your answer in terms of $\\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-aba77539aa57f25db229a1be074a96d8e33c875e99e469405a186795125d9925 | Two ants start at opposite corners of a grid. Ant X starts on the southwest corner and moves either one edge east or one edge north each unit of time. Ant Y starts on the northeast corner and moves either one edge west or one edge south each unit of time. Each ant must always move each unit of time. What is the probabi... | [
"\\(\\frac{\\binom{2n}{n}}{2^{2n}}\\)"
] | [
"Two ants start at opposite corners of a grid. Ant X starts on the southwest corner and moves either one edge east or one edge north each unit of time. Ant Y starts on the northeast corner and moves either one edge west or one edge south each unit of time. Each ant must always move each unit of time. What is the pr... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 6,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-abfc55f0396f61629e9dc21f2a494e927a599c8a1e9b08908b8762d690b3ed29 | Two ants start at points $(-1,1)$ and $(1,1)$ and walk to the right along the parabola $y=x^2$. Their midpoint moves along the line $y=1$ with constant speed $1$. What is the speed of the left ant when it first hits the line $y=1/2$? | [
"3\\sqrt{3} - 3"
] | [
"Two ants start at points $(-1,1)$ and $(1,1)$ and walk to the right along the parabola $y=x^2$. Their midpoint moves along the line $y=1$ with constant speed $1$. What is the speed of the left ant when it first hits the line $y=1/2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-519d3764ea939f66286ec6399a3502ce95815d01e231e6089bc63e49aeff578e | Two ants walk from point A to point B on a circular wire frame. There is another concentric circular wire frame outside the first one. The ants can transfer between the two frames perpendicularly to the tangents of the circles. For which range of central angles is it shorter for the ants to walk to the outer frame, the... | [
"\\theta > 2 \\arccos\\left(\\frac{r}{R}\\right)"
] | [
"Two ants walk from point A to point B on a circular wire frame. There is another concentric circular wire frame outside the first one. The ants can transfer between the two frames perpendicularly to the tangents of the circles. For which range of central angles is it shorter for the ants to walk to the outer frame... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-ec9f66268380dc0f8fdbe50c8774d4599de971b3e1aa98c10befd3b2882b41d4 | Two ants, $A$ and $B$, begin on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. Find the probability that both ants return to their starting positions after 4 moves. | [
"\\(\\frac{49}{729}\\)"
] | [
"Two ants, $A$ and $B$, begin on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. Find the probability that both ants return to their starting positions after 4 moves."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-384371e675f400ca54a755dcde1c47a597cfbcf8d21a85c13fb2dae3c5215dba | Two arbitrary circles \( W_1 \) and \( W_2 \) intersect at points \( A \) and \( B \). Circles \( C_1 \) and \( C_2 \) are tangent to \( W_1 \) and \( W_2 \) at points \( E, F \) and \( C, D \) respectively. Let \( P \) be the point of concurrency of lines \( CD \), \( EF \), and \( AB \). If the coordinates of \( A \)... | [
"(5, 6)"
] | [
"Two arbitrary circles \\( W_1 \\) and \\( W_2 \\) intersect at points \\( A \\) and \\( B \\). Circles \\( C_1 \\) and \\( C_2 \\) are tangent to \\( W_1 \\) and \\( W_2 \\) at points \\( E, F \\) and \\( C, D \\) respectively. Let \\( P \\) be the point of concurrency of lines \\( CD \\), \\( EF \\), and \\( AB \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c10bdc0ecbf289bcd7c598404b84559b60d4791eeab2035260d0213b75f284e4 | Two arbitrary lines \(L_1\) and \(L_2\) intersect at \(O\). Three circles \((O_1)\), \((O_2)\), and \((O_3)\) are tangent to lines \(L_1\) and \(L_2\). Circles \((O_1)\) and \((O_2)\) have two other common tangents different from \(L_1\) and \(L_2\), and let \(S_{12}\) be their intersection point. Similarly, we get \(S... | [
"0"
] | [
"Two arbitrary lines \\(L_1\\) and \\(L_2\\) intersect at \\(O\\). Three circles \\((O_1)\\), \\((O_2)\\), and \\((O_3)\\) are tangent to lines \\(L_1\\) and \\(L_2\\). Circles \\((O_1)\\) and \\((O_2)\\) have two other common tangents different from \\(L_1\\) and \\(L_2\\), and let \\(S_{12}\\) be their intersecti... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6160f2c956c1a11b1027b446f72b17b221174e12f92ad47b9c3b1604822a5a6e | Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$? | [
"1000"
] | [
"Two arithmetic sequences $A$ and $B$ both begin with 30 and have common differences of absolute value 10, with sequence $A$ increasing and sequence $B$ decreasing. What is the absolute value of the difference between the 51st term of sequence $A$ and the 51st term of sequence $B$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fddae9d6c4f65ef78776db5a2341904f4ea9aabc815e9fd675eec2728ccd651 | Two asteroids in space are hurtling directly toward each other. Initially, they are 2012 kilometers apart. Asteroid A travels at a rate of 1720 kilometers per hour. Asteroid B travels at a rate of 1280 kilometers per hour. How many kilometers apart are the two asteroids one minute before they crash? | [
"50\n\nExplanation:\nThe key to solving the problem is understanding the rate at which the distance between the two asteroids decreases. Each asteroid is moving towards each other, so their relative speed is the sum of their individual speeds. \n\nAsteroid A travels at 1720 kilometers per hour, and Asteroid B trave... | [
"Two asteroids in space are hurtling directly toward each other. Initially, they are 2012 kilometers apart. Asteroid A travels at a rate of 1720 kilometers per hour. Asteroid B travels at a rate of 1280 kilometers per hour. How many kilometers apart are the two asteroids one minute before they crash?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-708e6d9f38467cc34b0a23dc2cdc2c147a6a6396bcc28e5cd597792ebad85ac1 | Two athletes start a 10,000-meter race on a 400-meter track. One runs 60-second laps, and the other runs 68-second laps. On which lap will the faster runner overtake the slower runner? | [
"10"
] | [
"Two athletes start a 10,000-meter race on a 400-meter track. One runs 60-second laps, and the other runs 68-second laps. On which lap will the faster runner overtake the slower runner?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9556c586206acdb58b8181f3d4a6a652e62ed863f305fa812b23c7a6a7184b3f | Two babies A and B are playing a game with 2022 bottles of milk. Each bottle has a maximum capacity of 200ml and initially holds 30ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least 100ml of milk and drink half of it. (2) Pick two bottles with less than 100ml of mi... | [
"A has a winning strategy."
] | [
"Two babies A and B are playing a game with 2022 bottles of milk. Each bottle has a maximum capacity of 200ml and initially holds 30ml of milk. Starting from A, they take turns and do one of the following:\n(1) Pick a bottle with at least 100ml of milk and drink half of it.\n(2) Pick two bottles with less than 100m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 4
} | 26 | 0 | 0 |
omr-ws-600898bca1fee337af7b972ca99df8f297591ba3b0d221ff42943eb7a0bc062f | Two bags contain some numbers, and the total number of numbers is prime. When the number 170 is transferred from one bag to the other, the average in both bags increases by one. If the total sum of all numbers is 2004, find the number of numbers. | [
"167"
] | [
"Two bags contain some numbers, and the total number of numbers is prime. When the number 170 is transferred from one bag to the other, the average in both bags increases by one. If the total sum of all numbers is 2004, find the number of numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4718f893e5fa84590642e4bcb21cf98132c8fb4391096f146edb71d66efa8ad6 | Two bags contain some numbers. The total number of numbers is prime. When '170' is transferred from bag-1 to bag-2, the average of the numbers in both bags increases by 1. If the total sum of the numbers is 2004, find the number of numbers. | [
"167"
] | [
"Two bags contain some numbers. The total number of numbers is prime. When '170' is transferred from bag-1 to bag-2, the average of the numbers in both bags increases by 1. If the total sum of the numbers is 2004, find the number of numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-5788040c97e92cbea966cfa8e0b79d8106b09b6fcbb4938bf5aeb0de20b1bfa5 | Two bags each contain 5 red and 3 black balls. Two persons, A and B, each draw one ball at random from their respective bags with replacement until both draw a black ball. Find the probability that the maximum number of trials required is n. | [
"\\(\\frac{9}{64} \\left(\\frac{55}{64}\\right)^{n-1}\\)"
] | [
"Two bags each contain 5 red and 3 black balls. Two persons, A and B, each draw one ball at random from their respective bags with replacement until both draw a black ball. Find the probability that the maximum number of trials required is n."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-517c2f88980e32554a6c7586fd7a8d5979927ab0c27608fbde7ff20574db8d37 | Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that you win $2 for each black ball selected and lose $1 for each white ball selected. Let \( X \) denote your winnings. What are the probabilities associated with each value of \( X \)? | [
"\\[\nP(X = -2) = \\frac{4}{13}, \\quad P(X = -1) = \\frac{16}{91}, \\quad P(X = 0) = \\frac{32}{91}, \\quad P(X = 1) = \\frac{8}{91}, \\quad P(X = 2) = \\frac{6}{91}, \\quad P(X = 3) = \\frac{32}{91}, \\quad P(X = 4) = \\frac{6}{91}\n\\]"
] | [
"Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that you win $2 for each black ball selected and lose $1 for each white ball selected. Let \\( X \\) denote your winnings. What are the probabilities associated with each value of \\( X \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3d637877b0d016641c8dd5385dcccd14a6a22248813c9569b9da0734155acff | Two balls are placed in the middle of one of the sides of a parallelogram-shaped billiard table. They hit so that they bounce off different adjacent sides and then both hit the same point on the opposite side. One ball travels twice the distance before bouncing than after. Find the ratio of the lengths of the path segm... | [
"2"
] | [
"Two balls are placed in the middle of one of the sides of a parallelogram-shaped billiard table. They hit so that they bounce off different adjacent sides and then both hit the same point on the opposite side. One ball travels twice the distance before bouncing than after. Find the ratio of the lengths of the path... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-83046ac8d01974f5463528393714ae5483fd5bcc5f0a1c6f5ad9cc5dcd76fc0a | Two balls lie on the ground touching. If one ball has a radius of $8$ inches and the point of contact is $10$ inches above the ground, what is the radius of the other ball? | [
"\\(\\frac{40}{3}\\)"
] | [
"Two balls lie on the ground touching. If one ball has a radius of $8$ inches and the point of contact is $10$ inches above the ground, what is the radius of the other ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-76fb25aea76bfc31ba13b6281ffde07994a6f2100e84c423141dce3cd6984a81 | Two balls with radii 1cm, centers initially 102 cm apart, roll towards each other along a straight line. Ball A rolls at a speed of 1.5cm/s while Ball B rolls at a speed of 0.5cm/s. How many complete revolutions does Ball A go through before the two balls collide? | [
"11"
] | [
"Two balls with radii 1cm, centers initially 102 cm apart, roll towards each other along a straight line. Ball A rolls at a speed of 1.5cm/s while Ball B rolls at a speed of 0.5cm/s. How many complete revolutions does Ball A go through before the two balls collide?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-8c1e43c8f22ac0ea5731da2c21b53910c7762aada68b86f34bf40823cff9db7b | Two bands of pirates, Black Beard and the Straw Hat Pirates, play a game on the archipelago Barrantes - $n$. The archipelago is constructed by starting with a main island (Humberto) and iteratively adding and connecting islands as described. The game rules are that the first player conquers an island, and each subseque... | [
"3"
] | [
"Two bands of pirates, Black Beard and the Straw Hat Pirates, play a game on the archipelago Barrantes - $n$. The archipelago is constructed by starting with a main island (Humberto) and iteratively adding and connecting islands as described. The game rules are that the first player conquers an island, and each sub... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46bd905141faba77b4f323beb04ff369c23fd0db8c6902adc9e8cc9e7882d1ca | Two base 120 digits (not necessarily different) are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base-120 product is 1? | [
"\\(\\frac{1}{450}\\)"
] | [
"Two base 120 digits (not necessarily different) are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base-120 product is 1?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-6b3444c230e4c4492485d2db0134ec28462812e2abc17a34e38546ed8a1b8cb4 | Two base-120 units digits (not necessarily different) are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base-120 product is 1? | [
"\\(\\frac{1}{450}\\)"
] | [
"Two base-120 units digits (not necessarily different) are chosen randomly and their product is written in base 120. What is the probability that the units digit of the base-120 product is 1?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-7c9a5a327f3a6ed6b5280e72ddadb3f6e4a11e91f86742b398741dcce2089a5b | Two beams, each 3 units in length, are connected to form a sloped 'wall' and a 'roof'. The base of the structure is 4.5 units. Determine the height of the vertical line X. | [
"3.46567393556"
] | [
"Two beams, each 3 units in length, are connected to form a sloped 'wall' and a 'roof'. The base of the structure is 4.5 units. Determine the height of the vertical line X."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-502fd59e9f40b52ddd18b8d9b1363103bee49fd5317788234005672feaaff610 | Two beasts, Rosencrans and Gildenstern, play a game on a circle with $n$ points ($n \geq 5$). On their turn, each beast draws a chord between a pair of points such that any two chords have a shared point (either intersect or have a common endpoint). The game ends when a player cannot draw a chord. The last beast to dra... | [
"n \\text{ is even}"
] | [
"Two beasts, Rosencrans and Gildenstern, play a game on a circle with $n$ points ($n \\geq 5$). On their turn, each beast draws a chord between a pair of points such that any two chords have a shared point (either intersect or have a common endpoint). The game ends when a player cannot draw a chord. The last beast ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-7e5247b26393a427a43d81ccadd9a61796347bbb0f293b71a972668593c6b725 | Two bees start at the same spot and fly at the same rate. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling when th... | [
"A east, B west"
] | [
"Two bees start at the same spot and fly at the same rate. Bee $A$ travels $1$ foot north, then $1$ foot east, then $1$ foot upwards, and then continues to repeat this pattern. Bee $B$ travels $1$ foot south, then $1$ foot west, and then continues to repeat this pattern. In what directions are the bees traveling wh... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 23,
"tir": 15
} | 38 | 0 | 0 |
omr-ws-620adf8deb88921fa810ba2a23e3658573697f88773c0efea2ca551f8abc29e2 | Two beetles sit at vertices $A$ and $H$ of a cube $ABCDEFGH$ with edge length $40\sqrt{110}$ units. The beetles start moving simultaneously along $AC$ and $HF$ with the speed of the first beetle twice that of the other one. What will be the shortest distance between the two beetles? | [
"440"
] | [
"Two beetles sit at vertices $A$ and $H$ of a cube $ABCDEFGH$ with edge length $40\\sqrt{110}$ units. The beetles start moving simultaneously along $AC$ and $HF$ with the speed of the first beetle twice that of the other one. What will be the shortest distance between the two beetles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8861113bb0af31503f695fc43441ef3ab24a32014bb88f6af231ddaf70f9e373 | Two beetles sit at vertices A and H of a cube ABCDEFGH, where the length of each edge is $4\sqrt{110}$. The beetles start moving simultaneously along AC and HF, with the speed of the first beetle being twice that of the second. What will be the shortest distance between the beetles? | [
"40"
] | [
"Two beetles sit at vertices A and H of a cube ABCDEFGH, where the length of each edge is $4\\sqrt{110}$. The beetles start moving simultaneously along AC and HF, with the speed of the first beetle being twice that of the second. What will be the shortest distance between the beetles?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-693f79468730f0bf7e040ff880d02502c41306a3239acf6c33568e8c8d860b83 | Two bells ring every 6 and 10 times per hour, respectively. Both bells ring at the start of the hour. After how much time will the bells ring again at the same time? Express your answer in hours. | [
"0.5"
] | [
"Two bells ring every 6 and 10 times per hour, respectively. Both bells ring at the start of the hour. After how much time will the bells ring again at the same time? Express your answer in hours."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-abe61cce13827e916f772c9a9bd3094c22809c592332fb29d3340c8ebf35f60a | Two bicyclists are 7/8 of the way through a mile-long tunnel when a train approaches the closer end at 40 mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast do the bicyclists ride? | [
"30 mph"
] | [
"Two bicyclists are 7/8 of the way through a mile-long tunnel when a train approaches the closer end at 40 mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast do the bicyclists ride?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-9e9725c9fe8cf6eab09166be59db3c6f6892c56368c21019906aea1faf1016b6 | Two bicyclists are seven-eighths of the way through a mile-long tunnel when a train approaches the closer end at 40 mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast did they ride? | [
"30 mph"
] | [
"Two bicyclists are seven-eighths of the way through a mile-long tunnel when a train approaches the closer end at 40 mph. The riders take off at the same speed in opposite directions and each escapes the tunnel as the train passes them. How fast did they ride?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 8,
"tir": 30
} | 83 | 0 | 0 |
omr-ws-d1308adbf33c895f5cf77b2b6e5a27cbe21a086eac4e25f36ff470da8262618b | Two birds sit on top of towers that are 30 ft and 40 ft tall, respectively, and are 50 ft apart. Each bird flies at the same rate to a fountain between the towers and arrives at the same time. How far from each tower is the fountain? | [
"18 and 32",
"18 ft and 32 ft"
] | [
"Two birds sit on top of towers that are 30 ft and 40 ft tall, respectively, and are 50 ft apart. Each bird flies at the same rate to a fountain between the towers and arrives at the same time. How far from each tower is the fountain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 55
} | 83 | 0 | 0 |
omr-ws-004f4e52c9d43ae661f5d4d17bfc00a1831f8139e0e69fe2bd6eafb4692398d0 | Two blue devils and 4 angels go trick-or-treating. They randomly split up into 3 non-empty groups. Let $p$ be the probability that in at least one of these groups, the number of angels is nonzero and no more than the number of devils in that group. If $p = \frac{m}{n}$ in lowest terms, compute $m + n$. | [
"76"
] | [
"Two blue devils and 4 angels go trick-or-treating. They randomly split up into 3 non-empty groups. Let $p$ be the probability that in at least one of these groups, the number of angels is nonzero and no more than the number of devils in that group. If $p = \\frac{m}{n}$ in lowest terms, compute $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-84ba98046c59db6c49a31f096edd467a6b00df5056430d1623188eced078f5e3 | Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) Expres... | [
"16\\sqrt{3}"
] | [
"Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) E... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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