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-28afae13cad9c2f83696ed4d3b6799d53181d406fd475d79aab0c55ca8f96e90 | A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are amo... | [
"557"
] | [
"A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members ar... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-c1ca36e252ea059f6a994e3bb528b6fcbd192de6054b201f55e3078c6b982076 | A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\frac{P}{6^{s+2}}$. Find an explicit expression for $P$. | [
"\\( P = \\sum_{k=r}^{s} \\binom{n}{k} 6^{s-k} \\)"
] | [
"A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\\frac{P}{6^{s+2}}$. Find an explicit expression for $P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ab97e7d308ef32715af74c1ef11a4f6c7dbbfbb0dc26fa49431feb733636f36 | A set of $n^2$ checkers is arranged in an $n \times n$ square lattice. A move consists of jumping one checker over an adjacent checker onto an empty lattice point, then removing the jumped checker. Find all values of $n$ such that it is possible to perform the operations in a way that results in only one checker remain... | [
"\\( n \\) is a power of 2."
] | [
"A set of $n^2$ checkers is arranged in an $n \\times n$ square lattice. A move consists of jumping one checker over an adjacent checker onto an empty lattice point, then removing the jumped checker. Find all values of $n$ such that it is possible to perform the operations in a way that results in only one checker ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e75f8290631568af3d19c610d8a42cc4c8e22bd5978c11f08147f8bdc81d32b3 | A set of 10 positive numbers has a mean of 16. When combined with another set of 12 positive numbers, the average of the combined set is an integer. What is the third smallest possible mean of the second set? | [
"16"
] | [
"A set of 10 positive numbers has a mean of 16. When combined with another set of 12 positive numbers, the average of the combined set is an integer. What is the third smallest possible mean of the second set?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70a4923b6b5ed4c1f2c1d6260048761b3b36eadbf343c02fb63067aa7ede6172 | A set of 12 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let \( k \) be the number of points that belong to at least two of the circles. Find the smallest possible value of \( k \). | [
"12"
] | [
"A set of 12 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let \\( k \\) be the number of points that belong to at least two of the circles. Find the smallest possible value of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-480c28873f5828d9fb1581a6b3410b58818487c9610597c2975e659841d4badb | A set of 12 tokens — 3 red, 2 white, 1 blue, and 6 black — is to be distributed at random to 3 game players, 4 tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ a... | [
"389"
] | [
"A set of 12 tokens — 3 red, 2 white, 1 blue, and 6 black — is to be distributed at random to 3 game players, 4 tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\\frac{m}{n}$, where $m$ and... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 14
} | 60 | 0 | 0 |
omr-ws-5443575f460ef66acab1f27393a50ca7bfb8c4fe05a9aa1a0e976b968fd65236 | A set of 17 points is chosen on the circumference of a circle. Find the maximum number of segments which can be drawn connecting pairs of points such that each segment intersects all others inside, not on, the circle. | [
"8"
] | [
"A set of 17 points is chosen on the circumference of a circle. Find the maximum number of segments which can be drawn connecting pairs of points such that each segment intersects all others inside, not on, the circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3ad4f13596e42449dfbc0f177015bc68c952bb0b1bb79f7fe196e2b24bb4cee | A set of 1990 persons is divided into non-intersecting subsets such that: 1. No one in a subset knows all the others in the subset, 2. Among any three persons in a subset, there are always at least two who do not know each other, 3. For any two persons in a subset who do not know each other, there is exactly one person... | [
"398"
] | [
"A set of 1990 persons is divided into non-intersecting subsets such that:\n1. No one in a subset knows all the others in the subset,\n2. Among any three persons in a subset, there are always at least two who do not know each other,\n3. For any two persons in a subset who do not know each other, there is exactly on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 35,
"genselect": 8
} | 43 | 0 | 0 |
omr-ws-135bb1200e04152a01c272ded9de0bc6a98860821c5abe9c7cc0dbeaf1c900f9 | A set of 1990 persons is divided into non-intersecting subsets such that: 1. No one in a subset knows all the others in the subset, 2. Among any three persons in a subset, there are always at least two who do not know each other, 3. For any two persons in a subset who do not know each other, there is exactly one person... | [
"3"
] | [
"A set of 1990 persons is divided into non-intersecting subsets such that:\n1. No one in a subset knows all the others in the subset,\n2. Among any three persons in a subset, there are always at least two who do not know each other,\n3. For any two persons in a subset who do not know each other, there is exactly on... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df964bdc3f77dd25c412b08ca22bfa22dbc9d39a978c115358728adc61142c3e | A set of 1990 persons is divided into non-intersecting subsets such that: 1. No one in a subset knows all the others in the subset. 2. Among any three persons in a subset, there are always at least two who do not know each other. 3. For any three persons in a subset who do not know each other, there is exactly one pers... | [
"398"
] | [
"A set of 1990 persons is divided into non-intersecting subsets such that:\n1. No one in a subset knows all the others in the subset.\n2. Among any three persons in a subset, there are always at least two who do not know each other.\n3. For any three persons in a subset who do not know each other, there is exactly ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-1a1eaf94b8e5353b6008ddf7cb8c7a93dff413ae4719c6282d9b804119c352d0 | A set of 1990 persons is divided into non-intersecting subsets such that: 1. No one in a subset knows all the others in the subset. 2. Among any three persons in a subset, there are always at least two who do not know each other. 3. For any three persons in a subset who do not know each other, there is exactly one pers... | [
"3"
] | [
"A set of 1990 persons is divided into non-intersecting subsets such that:\n1. No one in a subset knows all the others in the subset.\n2. Among any three persons in a subset, there are always at least two who do not know each other.\n3. For any three persons in a subset who do not know each other, there is exactly ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-34b1b9399f777da4397a513838065e169a1f7fc825223df3c9295ff9faa31187 | A set of 1990 persons is divided into non-intersecting subsets such that: 1. No person in a subset knows all others in the subset, 2. Among any three persons in a subset, there are always at least two who do not know each other, 3. For any two persons in a subset who do not know each other, there is exactly one person ... | [
"(2, 398)"
] | [
"A set of 1990 persons is divided into non-intersecting subsets such that:\n1. No person in a subset knows all others in the subset,\n2. Among any three persons in a subset, there are always at least two who do not know each other,\n3. For any two persons in a subset who do not know each other, there is exactly one... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-66253f567581b0ea03e057de6785ed50e149ab1ae0f607ba387fed2fc40a7c3d | A set of 1998 different positive integers is given. None of these numbers can be represented as a sum of two others from the set. What is the least possible value for the largest number from such a set? | [
"3995"
] | [
"A set of 1998 different positive integers is given. None of these numbers can be represented as a sum of two others from the set. What is the least possible value for the largest number from such a set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-e4719e511336fd93ea8f2a7a6723ec7f3eb5140fb0ca0c17c60e16f11e38e035 | A set of 20 articles is prepared, and 4 of them are chosen at random for quality checking. If any one of them is found to be defective, the whole set is put under 100% screening again. If no defectives are found, the whole set is sent to the market. What is the probability that a box containing 4 defective articles wil... | [
"\\(\\frac{364}{969}\\)"
] | [
"A set of 20 articles is prepared, and 4 of them are chosen at random for quality checking. If any one of them is found to be defective, the whole set is put under 100% screening again. If no defectives are found, the whole set is sent to the market. What is the probability that a box containing 4 defective article... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-755eba23ea3d188b7512cd0b74d5ec61269c705305e2c595d1acefff7d0284a3 | A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students? | [
"8"
] | [
"A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 27,
"genselect": 6
} | 33 | 0 | 0 |
omr-ws-f8831da5b37b58538a05476db2a08075a56cf03a76e412479962c9f385e56b49 | A set of \( n \) numbers are to be added together, and before they are added, each number is rounded off to the nearest integer, thus introducing an error whose distribution is uniform. Find the mean and the variance of the round-off error of the sum. | [
"Mean = 0, Variance = \\(\\frac{n}{12}\\)"
] | [
"A set of \\( n \\) numbers are to be added together, and before they are added, each number is rounded off to the nearest integer, thus introducing an error whose distribution is uniform. Find the mean and the variance of the round-off error of the sum."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 32
} | 34 | 0 | 0 |
omr-ws-54420505739fd1301f579dba4f8e9ffb7faf3e1e56327695a93703cae73bcc38 | A set of \( n \) people participate in an online video basketball tournament. Each person may be a member of any number of 5-player teams, but no two teams may have exactly the same 5 members. The average number of complete teams among all subsets of 9 people is equal to the reciprocal of the average number of complete... | [
"238"
] | [
"A set of \\( n \\) people participate in an online video basketball tournament. Each person may be a member of any number of 5-player teams, but no two teams may have exactly the same 5 members. The average number of complete teams among all subsets of 9 people is equal to the reciprocal of the average number of c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-bef05c882129eab94bf5e9aeeaa733d106da724f7f6395325dbf941e1a2881b6 | A set of \( n \) positive integers is said to be balanced if for each integer \( k \) with \( 1 \leq k \leq n \), the average of any \( k \) numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to 2017. | [
"12859"
] | [
"A set of \\( n \\) positive integers is said to be balanced if for each integer \\( k \\) with \\( 1 \\leq k \\leq n \\), the average of any \\( k \\) numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to 2017."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df5d7bdf1ec9532e4f37a4980cf7e4486e7fd4f87de8f9440f5c732d305efbe4 | A set of balls contains $n$ balls labeled with numbers $1, 2, 3, \ldots, n$. We are given $k > 1$ such sets. We want to color the balls with two colors, black and white, such that: (a) The balls labeled with the same number are of the same color. (b) Any subset of $k+1$ balls with (not necessarily different) labels $a_... | [
"\\( k^2 + k - 2 \\)"
] | [
"A set of balls contains $n$ balls labeled with numbers $1, 2, 3, \\ldots, n$. We are given $k > 1$ such sets. We want to color the balls with two colors, black and white, such that:\n\n(a) The balls labeled with the same number are of the same color.\n\n(b) Any subset of $k+1$ balls with (not necessarily different... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5fbe4a4fda4bc9ed71a5094080c24be403900f1a653206dd6e2749eea3c0abe7 | A set of balls contains \( n \) balls labeled with numbers \( 1, 2, 3, \ldots, n \). We are given \( k > 1 \) such sets. We want to color the balls with two colors, black and white, such that: (a) The balls labeled with the same number are of the same color. (b) Any subset of \( k+1 \) balls with (not necessarily diffe... | [
"2^k - 1"
] | [
"A set of balls contains \\( n \\) balls labeled with numbers \\( 1, 2, 3, \\ldots, n \\). We are given \\( k > 1 \\) such sets. We want to color the balls with two colors, black and white, such that:\n(a) The balls labeled with the same number are of the same color.\n(b) Any subset of \\( k+1 \\) balls with (not n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9c68d1649e259c02f1bf51ff947d824a643c4627a0295886bd93487b0e314a72 | A set of cards consists of $8$ red and $7$ black cards. Three cards are dealt at random. What is the probability of obtaining three cards of the same color? | [
"\\dfrac{1}{5}"
] | [
"A set of cards consists of $8$ red and $7$ black cards. Three cards are dealt at random. What is the probability of obtaining three cards of the same color?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ff3979b668010c6f720bd1432b11cdfffdaa9ebf89cd8c61fa1c7dd4008601d | A set of cards contains 100 cards, each numbered from 1 to 100. Initially, all cards are arranged with red writing face up. Leny performs the following steps: I. Turn over all cards whose numbers are divisible by 2. II. Turn over all cards whose numbers are divisible by 3. III. Turn over all cards whose numbers are div... | [
"36"
] | [
"A set of cards contains 100 cards, each numbered from 1 to 100. Initially, all cards are arranged with red writing face up. Leny performs the following steps:\nI. Turn over all cards whose numbers are divisible by 2.\nII. Turn over all cards whose numbers are divisible by 3.\nIII. Turn over all cards whose numbers... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92acbbd094c9170d76c312f28c9d8a42c73ee1c7600dc4e226eccd89e9f8b40a | A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | [
"7"
] | [
"A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-882b6b36bfc69dbc3d331017a62ee328aa15f67c540b9750d16623648af414e0 | A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased, and the average of the remaining numbers is \( 35\frac{7}{17} \). What number was erased? | [
"7"
] | [
"A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased, and the average of the remaining numbers is \\( 35\\frac{7}{17} \\). What number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b8dc4c058e074afaf475787bd1ce1886b50e44fd8c7417e948b754a82f105c77 | A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average of the remaining numbers is $35\frac{7}{17}$. What number was erased? | [
"7"
] | [
"A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average of the remaining numbers is $35\\frac{7}{17}$. What number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e14247cef59a80f1d3e73c83094ed4689a1e16c167254e525aa3fccd192f9da4 | A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average of the remaining numbers is \(35 \frac{7}{17}\). What number was erased? | [
"7"
] | [
"A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average of the remaining numbers is \\(35 \\frac{7}{17}\\). What number was erased?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a8dc2c68251ea6e69346cc8f571116d74139a5fe810cc94c982b6ee16185294f | A set of consecutive positive integers beginning with 1 is written. One number is deleted, and the average of the remaining numbers is \(35 \frac{7}{17}\). What number was deleted? | [
"15"
] | [
"A set of consecutive positive integers beginning with 1 is written. One number is deleted, and the average of the remaining numbers is \\(35 \\frac{7}{17}\\). What number was deleted?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-800b28165abcdf939f7c8a04bd55998d3ff7c0a9b6d979a451fbd5937beb8019 | A set of discs is being placed in a stack. The thickness of each successive disc added to the stack is $1.5$ times the thickness of the disc immediately beneath it. What is the minimum number of discs that must be placed in the stack in order for the thickness of the stack to be $25$ times the thickness of the original... | [
"7"
] | [
"A set of discs is being placed in a stack. The thickness of each successive disc added to the stack is $1.5$ times the thickness of the disc immediately beneath it. What is the minimum number of discs that must be placed in the stack in order for the thickness of the stack to be $25$ times the thickness of the ori... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-81bbc457d60021108dd26f26de67e3087ad86b8bff8786a98e8cab18586f893f | A set of distinct positive integers \( S \) has the property that for every integer \( x \) in \( S \), the arithmetic mean of the set obtained by deleting \( x \) from \( S \) is an integer. Given that 1 belongs to \( S \) and that 2002 is the largest element of \( S \), what is the greatest number of elements that \(... | [
"30"
] | [
"A set of distinct positive integers \\( S \\) has the property that for every integer \\( x \\) in \\( S \\), the arithmetic mean of the set obtained by deleting \\( x \\) from \\( S \\) is an integer. Given that 1 belongs to \\( S \\) and that 2002 is the largest element of \\( S \\), what is the greatest number ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 7
} | 15 | 0 | 0 |
omr-ws-dd1828e3a58a0cea093efd1e103dbf6f1bea9371906035f42668666cdf2cc90f | A set of distinct, nonzero real numbers is placed along the circumference of a circle. Each number is equal to the product of the two numbers adjacent to it. What is the least possible number of numbers in the set? | [
"6"
] | [
"A set of distinct, nonzero real numbers is placed along the circumference of a circle. Each number is equal to the product of the two numbers adjacent to it. What is the least possible number of numbers in the set?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3df92b7fbe7be8448762266264a503ffc2e5bd8f2fd86f77fbff3017dc56cd44 | A set of eleven bank robbers decided to store their loot in a safe that can only be opened when a majority of the robbers is present. The safe is equipped with multiple locks, and each robber is given keys to some of these locks. Determine the minimum number of locks required and the number of keys each robber must hav... | [
"462 locks, 210 keys per robber."
] | [
"A set of eleven bank robbers decided to store their loot in a safe that can only be opened when a majority of the robbers is present. The safe is equipped with multiple locks, and each robber is given keys to some of these locks. Determine the minimum number of locks required and the number of keys each robber mus... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-66db80f303a473ba6acdac72e6824820a3be38d5a67346d73e023636c16084f4 | A set of integers \( A \) is called [i]fat[/i] if each of its elements is greater than or equal to the number of elements in \( A \). Let \( g_n \) denote the number of fat subsets of \( \{1, 2, \ldots, n\} \). The empty set is considered fat. Define \( g_n \) recursively. | [
"\\( g_{n+1} = g_n + F_n \\)"
] | [
"A set of integers \\( A \\) is called [i]fat[/i] if each of its elements is greater than or equal to the number of elements in \\( A \\). Let \\( g_n \\) denote the number of fat subsets of \\( \\{1, 2, \\ldots, n\\} \\). The empty set is considered fat. Define \\( g_n \\) recursively."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a81d428e1bb83767b742a13668523ac6ea2fe1760a256e127f7970c528e3d8f | A set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, \ldots, a_n)$, where each element of the set appears exactly once and $\gcd(a_i, a_{i+1}) = 1$ for $i = 1, \ldots, n$ where $a_{n+1} = a_1$. For example, $(7, 8, 9, 10, 11, 14, 13, 12)$ is a coprime cycle for $k = 7$... | [
"14"
] | [
"A set of integers has a coprime cycle if it can be listed in a cyclic list of the form $(a_1, a_2, a_3, \\ldots, a_n)$, where each element of the set appears exactly once and $\\gcd(a_i, a_{i+1}) = 1$ for $i = 1, \\ldots, n$ where $a_{n+1} = a_1$. For example, $(7, 8, 9, 10, 11, 14, 13, 12)$ is a coprime cycle for... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-531e374469d30d934bd99cd09819e02a87dfadb56b93caf8a0348ba017ac9e8d | A set of integers is called robust if it contains at least two elements and the sum of any two distinct elements is not divisible by their difference. Let $R(n)$ denote the number of robust subsets of $\{1, 2, \ldots, n\}$. Calculate $R(10)$. | [
"888"
] | [
"A set of integers is called robust if it contains at least two elements and the sum of any two distinct elements is not divisible by their difference. Let $R(n)$ denote the number of robust subsets of $\\{1, 2, \\ldots, n\\}$. Calculate $R(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-855aa4ab1672b8318707f4db56b2a76ca81cfcd4485f4db10c64eabd966ee9c3 | A set of integers is defined as *wanless* if the sum of its elements is $1$ less than a multiple of $4$. How many subsets of $\{1, 2, 3, \ldots, 2023\}$ are wanless? | [
"2^{2021}"
] | [
"A set of integers is defined as *wanless* if the sum of its elements is $1$ less than a multiple of $4$. How many subsets of $\\{1, 2, 3, \\ldots, 2023\\}$ are wanless?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6f2e3646d10eb40fa5f43fa00bfb430b46b8c71ef36e540e0371e90ace293d63 | A set of marbles can be divided in equal shares among $2$, $3$, $4$, $5$, or $6$ children with no marbles left over. What is the least number of marbles that the set could have? | [
"60"
] | [
"A set of marbles can be divided in equal shares among $2$, $3$, $4$, $5$, or $6$ children with no marbles left over. What is the least number of marbles that the set could have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa641574f22e9f7c3408182c1e97c5a29d5293796489eaaa2443a8c15cffaae4 | A set of numbers \(a_0, a_1, \ldots, a_n\) satisfies the conditions: \(a_0 = 0\) and \(a_{k+1} \ge a_k + 1\) for \(k = 0, 1, \ldots, n-1\). Given \(n = 5\) and \(a_1 = 1\), find the minimum possible value of \(\sum_{k=1}^n a_k^3 - \left(\sum_{k=1}^n a_k \right)^2\). | [
"0"
] | [
"A set of numbers \\(a_0, a_1, \\ldots, a_n\\) satisfies the conditions: \\(a_0 = 0\\) and \\(a_{k+1} \\ge a_k + 1\\) for \\(k = 0, 1, \\ldots, n-1\\). Given \\(n = 5\\) and \\(a_1 = 1\\), find the minimum possible value of \\(\\sum_{k=1}^n a_k^3 - \\left(\\sum_{k=1}^n a_k \\right)^2\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdf3762ff9b544f266d1855e2e36e4c5bced11855a6bc4f18ee8549546ef2c9d | A set of points in the plane is called full if every triple of points in the set forms the vertices of a non-obtuse triangle. What is the largest possible size of a full set? | [
"4"
] | [
"A set of points in the plane is called full if every triple of points in the set forms the vertices of a non-obtuse triangle. What is the largest possible size of a full set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-da305de4b83fc68367e6434b6c50283b17cb0dd0d4cff826ecf3baaee6c9e375 | A set of points in the plane is considered "almost symmetric" if one can remove one point so that the remaining points are rotationally symmetric of order 2 about a centre $P$. Given a set of $k$ points where $k>1$, what is the largest number of points $P$ possible? | [
"3"
] | [
"A set of points in the plane is considered \"almost symmetric\" if one can remove one point so that the remaining points are rotationally symmetric of order 2 about a centre $P$. Given a set of $k$ points where $k>1$, what is the largest number of points $P$ possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b13a117b87eb3c59916c534a765235e4acb4a9937d16551618c632262418c6f2 | A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. What is the maximum possible size of an epic set? | [
"4"
] | [
"A set of points in the plane is epic if, for every way of coloring the points red or blue, it is possible to draw two lines such that each blue point is on a line, but none of the red points are. What is the maximum possible size of an epic set?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-ea6bf47e43954b9a7e32d7d50455b6152f0b57d7d8f3a9ac12b44491943080de | A set of points is marked on the plane such that any three marked points can be covered with a disk of radius $1$. Let $S$ be the set of all marked points. Find the maximum possible value of the radius $r$ of a disk that can cover all points in $S$. | [
"1"
] | [
"A set of points is marked on the plane such that any three marked points can be covered with a disk of radius $1$. Let $S$ be the set of all marked points. Find the maximum possible value of the radius $r$ of a disk that can cover all points in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-671ce2809fc29b05b14bf7be782c76892d6c9017b69e83205eed13b028573c52 | A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Let $S$ be the set of all marked points. Define $f(S)$ as the minimum radius of a disk that can cover all points in $S$. If the maximum distance between any two points in $S$ is $d$, find the v... | [
"\\dfrac{\\sqrt{4 - d^2}}{2}"
] | [
"A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Let $S$ be the set of all marked points. Define $f(S)$ as the minimum radius of a disk that can cover all points in $S$. If the maximum distance between any two points in $S$ is $d$, find ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8dbb36a12d77763bf9fc6e1a4c541ee7302a946dbb6986d35d2f23a427f314e | A set of points on a plane is called beautiful if the following conditions are met: 1) No three points of the set lie on the same straight line. 2) Points are painted in several colors. 3) If three points of the set form a triangle with an angle of at least \(120^\circ\), then exactly two of these points are colored th... | [
"6"
] | [
"A set of points on a plane is called beautiful if the following conditions are met:\n1) No three points of the set lie on the same straight line.\n2) Points are painted in several colors.\n3) If three points of the set form a triangle with an angle of at least \\(120^\\circ\\), then exactly two of these points are... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-f39730348971a0599b08300554c4bb6c76576711507c81756e885d80df845358 | A set of poker has 52 cards with 13 cards in each suit (diamonds, clubs, hearts, spades). Each suit of cards is marked \(2, 3, \cdots, 10, J, Q, K, A\). Calculate the number of subsets with 13 cards of the poker, where \(2, 3, \cdots, 10, J, Q, K, A\) all appear and no two consecutive cards (\(A\) and \(2\), \(2\) and ... | [
"1594320"
] | [
"A set of poker has 52 cards with 13 cards in each suit (diamonds, clubs, hearts, spades). Each suit of cards is marked \\(2, 3, \\cdots, 10, J, Q, K, A\\). Calculate the number of subsets with 13 cards of the poker, where \\(2, 3, \\cdots, 10, J, Q, K, A\\) all appear and no two consecutive cards (\\(A\\) and \\(2... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e10446c6737ff53ef4f70a61624febb41e88177c574e94bbe06dd80b7b807a7c | A set of positive integers $S$, all greater than $1$, is said to cover an integer $x$ if for every pair of integers $k$ and $l$ such that $2 \geq k < l \geq x$, there exists at least one integer $y$ in $S$ for which $\left \lceil \frac{k}{y} \right \rceil \neq \left\lceil \frac{l}{y}\right \rceil$. Determine the size o... | [
"10"
] | [
"A set of positive integers $S$, all greater than $1$, is said to cover an integer $x$ if for every pair of integers $k$ and $l$ such that $2 \\geq k < l \\geq x$, there exists at least one integer $y$ in $S$ for which $\\left \\lceil \\frac{k}{y} \\right \\rceil \\neq \\left\\lceil \\frac{l}{y}\\right \\rceil$. De... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-e10b92a156bd895d3ad2f6ddfa5cf3e4eebc019628999c987ef66980222493e1 | A set of positive integers $\{a_1, \ldots, a_n\}$ is called canalero if the sum of any non-empty subset of $\{a_1, \ldots, a_n\}$ is not divisible by $n+1$. Given a set of positive integers $\{c_1, \ldots, c_n\}$, find the smallest positive integer $k$ such that $\{kc_1, \ldots, kc_n\}$ is canalero. For $n=4$ and $\{c_... | [
"1"
] | [
"A set of positive integers $\\{a_1, \\ldots, a_n\\}$ is called canalero if the sum of any non-empty subset of $\\{a_1, \\ldots, a_n\\}$ is not divisible by $n+1$. Given a set of positive integers $\\{c_1, \\ldots, c_n\\}$, find the smallest positive integer $k$ such that $\\{kc_1, \\ldots, kc_n\\}$ is canalero. Fo... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bee3ecc5b8296f6d2cafe1a015785dd8bec2d136f316c9634b7b26ef1a7e6bf0 | A set of positive integers has the properties that: - Every member in the set, apart from $1$, is divisible by at least one of $2$, $3$, or $5$. - If the set contains $2n$, $3n$, or $5n$ for some integer $n$, then it contains all three and $n$ as well. The set contains between $300$ and $400$ numbers. Exactly how many ... | [
"364"
] | [
"A set of positive integers has the properties that:\n- Every member in the set, apart from $1$, is divisible by at least one of $2$, $3$, or $5$.\n- If the set contains $2n$, $3n$, or $5n$ for some integer $n$, then it contains all three and $n$ as well.\n\nThe set contains between $300$ and $400$ numbers. Exactly... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-a7f04c49e5b330b24b1af1fcba29a0a8848f07cd1ada9fe6bc3deebc9e417844 | A set of positive integers has the properties that: - Every member in the set, apart from 1, is divisible by at least one of 2, 3, or 5. - If the set contains 2n, 3n, or 5n for some integer n, then it contains all three and n as well. The set contains between 300 and 400 numbers. Exactly how many does it contain? | [
"364"
] | [
"A set of positive integers has the properties that:\n- Every member in the set, apart from 1, is divisible by at least one of 2, 3, or 5.\n- If the set contains 2n, 3n, or 5n for some integer n, then it contains all three and n as well.\nThe set contains between 300 and 400 numbers. Exactly how many does it contai... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-46cc670104472ee8d285b3c5a8f6f2a72ab4e75e50e8825553c9bf032f33f128 | A set of positive integers has the properties that: - Every number in the set, apart from 1, is divisible by at least one of 2, 3, or 5. - If the set contains \(2n\), \(3n\), or \(5n\) for some integer \(n\), then it contains all three and \(n\) as well. - The set contains between 300 and 400 integers. How many integer... | [
"363"
] | [
"A set of positive integers has the properties that:\n- Every number in the set, apart from 1, is divisible by at least one of 2, 3, or 5.\n- If the set contains \\(2n\\), \\(3n\\), or \\(5n\\) for some integer \\(n\\), then it contains all three and \\(n\\) as well.\n- The set contains between 300 and 400 integers... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5c0e6291ea9684c67081cb7d5190e173ad1d70eccfae01e90e4862954d4c81a | A set of positive integers is called *lucky* if it contains distinct elements which sum to $7$. Compute the number of lucky subsets of $\{1, 2, 3, \ldots, 9\}$. | [
"408"
] | [
"A set of positive integers is called *lucky* if it contains distinct elements which sum to $7$. Compute the number of lucky subsets of $\\{1, 2, 3, \\ldots, 9\\}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-e234200857f0fb1f89aa8870fc2fef1247766f0796a5d5776417ab3fa6038dac | A set of positive integers is defined to be wicked if it contains no three consecutive integers. We count the empty set as a wicked set. Find the number of wicked subsets of the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). | [
"504"
] | [
"A set of positive integers is defined to be wicked if it contains no three consecutive integers. We count the empty set as a wicked set. Find the number of wicked subsets of the set \\(\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-17e870e703bdc0c43cb6e1e8909d187751ed50bc9d89db3bf10e5b86bd343976 | A set of positive integers is partitioned into 2013 infinite arithmetic sequences with differences \( r_1, r_2, \ldots, r_{2013} \). If the sum of the reciprocals of these differences is an integer, find the smallest possible value of this integer. | [
"1"
] | [
"A set of positive integers is partitioned into 2013 infinite arithmetic sequences with differences \\( r_1, r_2, \\ldots, r_{2013} \\). If the sum of the reciprocals of these differences is an integer, find the smallest possible value of this integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00f76874c4a925829321d485a22b750a50329119ad7a83fce9e69380bd83c751 | A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least 2 elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the set... | [
"30"
] | [
"A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least 2 elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-0f1e1712fbbcb92e43fba50a5a31131517ae18ebaf8f272b2d5db5c8275291ea | A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible... | [
"253"
] | [
"A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the large... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 2
} | 41 | 0 | 0 |
omr-ws-c11d69ae17cc4511bb92e30b23790c7eec8cbca0399b2e69ebbe53af1caae13e | A set of positive reals has a sum of 2014. Determine the number of numbers in the set that maximizes the product of the numbers and approximate the maximum product. | [
"Number of numbers = 741, Product = \\( \\left(\\frac{2014}{741}\\right)^{741} \\)"
] | [
"A set of positive reals has a sum of 2014. Determine the number of numbers in the set that maximizes the product of the numbers and approximate the maximum product."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 19
} | 25 | 0 | 0 |
omr-ws-1f8281f7e8f7f80cd58d5187ef1cb091380f5aee3badef410a9e33a327847c84 | A set of questions consists of 3 true/false questions and 3 multiple-choice questions with answers \( A, B, C, \) or \( D \). Someone answers all questions randomly. What is the probability that he is correct in exactly 2 questions? | [
"\\(\\frac{171}{512}\\)"
] | [
"A set of questions consists of 3 true/false questions and 3 multiple-choice questions with answers \\( A, B, C, \\) or \\( D \\). Someone answers all questions randomly. What is the probability that he is correct in exactly 2 questions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8da319da2c14e5a6cef82a85b326e24574ad38e69ef55f8968165ed863599421 | A set of real numbers is called good if it contains at least one element that is equal to the average of some nonempty subset of the set, excluding the element itself. For a finite nonempty set $S$ of real numbers, let $f(S)$ be the number of elements in $S$ that satisfy this condition. Given a set $S = \{1, 2, 3, \ldo... | [
"8"
] | [
"A set of real numbers is called good if it contains at least one element that is equal to the average of some nonempty subset of the set, excluding the element itself. For a finite nonempty set $S$ of real numbers, let $f(S)$ be the number of elements in $S$ that satisfy this condition. Given a set $S = \\{1, 2, 3... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79e4a2c7a26ced43ca2c83339e90fd08298ee4daeb2df2ea597075ad6aa65ca8 | A set of regular polygons of side length 1 is chosen such that the polygons can fill the \(360^\circ\) angle about a point \(P\). For example, a hexagon, two squares, and a triangle can fill the angle. In how many ways can the polygons be chosen? The order of placement is not important. | [
"17"
] | [
"A set of regular polygons of side length 1 is chosen such that the polygons can fill the \\(360^\\circ\\) angle about a point \\(P\\). For example, a hexagon, two squares, and a triangle can fill the angle. In how many ways can the polygons be chosen? The order of placement is not important."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-d556ab70f972f49f6500e2cf5521055ce62a806d05f97be3c1023c36f66a0089 | A set of seven integers has a median of 73, a mode of 79, and a mean of 75. What is the least possible difference between the maximum and minimum values in this set? | [
"8"
] | [
"A set of seven integers has a median of 73, a mode of 79, and a mean of 75. What is the least possible difference between the maximum and minimum values in this set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b2534f6378e942e04e6fcefae42ba492930ffeabf2f21b27f6b176a787c4fb83 | A set of squares with integer sides has an average area of $2013$. This set consists of as few squares as possible such that the above condition is satisfied, and the average perimeter of these squares is as small as possible. What are the squares in this set? | [
"89"
] | [
"A set of squares with integer sides has an average area of $2013$. This set consists of as few squares as possible such that the above condition is satisfied, and the average perimeter of these squares is as small as possible. What are the squares in this set?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-9558911f5982a3c9ebfe52eeaf64a00aab5cb14dd4ce39afbffd44c0c8437ac8 | A set of three natural numbers is called an arithmetic set if one of its elements is the average of the other two elements. Find the number of arithmetic subsets of $\{1, 2, 3, \ldots, 8\}$. | [
"12"
] | [
"A set of three natural numbers is called an arithmetic set if one of its elements is the average of the other two elements. Find the number of arithmetic subsets of $\\{1, 2, 3, \\ldots, 8\\}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-58201bd600e795a5e0afdc8ab538fc798419fa34af751a42318b36a34ab1ec9e | A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers? | [
"6"
] | [
"A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c4c4ea1740702e10c59ca80ef65a3987dd8cec0ca1152296a1e99a48f038865 | A set of three numbers is randomly drawn from the ten numbers $1, 2, 3, \ldots, 10$. Find the probability that the set has one number divisible by $2$, another number divisible by $3$, and the other number divisible by $4$. | [
"\\dfrac{13}{40}"
] | [
"A set of three numbers is randomly drawn from the ten numbers $1, 2, 3, \\ldots, 10$. Find the probability that the set has one number divisible by $2$, another number divisible by $3$, and the other number divisible by $4$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-b38808ff27d8cb00bd71276718846f79a79734e55ff5ddc2df029fda8382b986 | A set of three points is randomly chosen from a 3x3 grid. What is the probability that the points lie on the same straight line? | [
"$\\frac{2}{21}$"
] | [
"A set of three points is randomly chosen from a 3x3 grid. What is the probability that the points lie on the same straight line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-7cd207d12311509b752429f4a6b1b86dc8d0b6f77f78d6a3f85fa3ad4305f3a6 | A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? | [
"18"
] | [
"A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c7be603871a9bf42e1eb84db363dd4fb5d4fcc70a26dcf3811b4229217056564 | A set of tiles numbered from 1 to 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? | [
"18"
] | [
"A set of tiles numbered from 1 to 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-104bdb4b4384b66358e753af1cfdae32006c547e9ec8df9927f91bf57eb045b9 | A seven-digit number is called "bad" if exactly four of its digits are 0 and the rest are odd. How many seven-digit numbers are bad? | [
"1875"
] | [
"A seven-digit number is called \"bad\" if exactly four of its digits are 0 and the rest are odd. How many seven-digit numbers are bad?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-1e95c9d5ac75d1dd6379f9d2aaf7ae4aba734aa6fdbff68fcb23fb4ba5da3f8a | A seven-digit number is made up of all distinct digits 8, 7, 6, 4, 2, \( x \), \( y \) and is divisible by 3. Find all possible ordered pairs \((x, y)\). | [
"8"
] | [
"A seven-digit number is made up of all distinct digits 8, 7, 6, 4, 2, \\( x \\), \\( y \\) and is divisible by 3. Find all possible ordered pairs \\((x, y)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-756f45f82348609c145359491885448f1251ac582284cc438913f12d035b1bdf | A seven-digit telephone number has the property that if you add the square of the number formed by the first 3 digits to the square of the number formed by the last 4 digits, you get the original seven-digit phone number. What is this phone number? | [
"5882353"
] | [
"A seven-digit telephone number has the property that if you add the square of the number formed by the first 3 digits to the square of the number formed by the last 4 digits, you get the original seven-digit phone number. What is this phone number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2811eedc072515443c027dae2fcd63f114b2245e8ce7be566d5ea5cdca4c3600 | A shape is formed by two 90-degree circular arcs of radius 1. This shape is on the bottom of a unit cube and rises to the top while rotating 180 degrees. What is the surface area of the space it has passed through? | [
"\\(\\frac{\\pi^2\\sqrt{2} + 2\\pi - 4}{2}\\)"
] | [
"A shape is formed by two 90-degree circular arcs of radius 1. This shape is on the bottom of a unit cube and rises to the top while rotating 180 degrees. What is the surface area of the space it has passed through?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cb8b53c32bd4891ea1f4c5e8400fd4a804967a7abd0998561e07bd0ec6265b3 | A shark starts at the center of the ocean and swims in a pattern: east for 1 yard, north for 2 yards, up for 3 yards, west for 4 yards, south for 5 yards, down for 6 yards, and continues this pattern. By the time the shark has swum a total of 100 yards, what is the farthest distance from the center of the ocean it has ... | [
"11"
] | [
"A shark starts at the center of the ocean and swims in a pattern: east for 1 yard, north for 2 yards, up for 3 yards, west for 4 yards, south for 5 yards, down for 6 yards, and continues this pattern. By the time the shark has swum a total of 100 yards, what is the farthest distance from the center of the ocean it... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-3d3d3577bcd1f73c02f113c5e75acb7685906afe92cea58464e00696e815d270 | A shed consists of a 10-ft cube with a pyramid roof. When viewed from a great distance, directly opposite the center of a side, an observer sees an equilateral triangle atop a square. What is the surface area of the roof? | [
"200"
] | [
"A shed consists of a 10-ft cube with a pyramid roof. When viewed from a great distance, directly opposite the center of a side, an observer sees an equilateral triangle atop a square. What is the surface area of the roof?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-6b533a6e6dc3b6c4403011f86440f8a029ee94aee8f5d8d125638fb50296e1af | A sheep is tied to an arbitrary point on the circumference of a circular stone with a radius of 10 feet by a rope of length 6 feet. How much area of grass can the sheep eat? | [
"64.06"
] | [
"A sheep is tied to an arbitrary point on the circumference of a circular stone with a radius of 10 feet by a rope of length 6 feet. How much area of grass can the sheep eat?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-c8ba2bd3e44fa7bab07a788ae283b67afd6d5b86674fbe001d26a65fe5379a0e | A sheep is tied to the edge of a circular patch of grass with radius \( r \). The length of the rope is \( l \). Determine the length of the rope \( l \) if the sheep eats exactly half of the grass. | [
"l \\approx 1.1587 \\, r"
] | [
"A sheep is tied to the edge of a circular patch of grass with radius \\( r \\). The length of the rope is \\( l \\). Determine the length of the rope \\( l \\) if the sheep eats exactly half of the grass."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-700a2548c69ce687a9813547fa56324486312272f1c29458a83f8ca0c2524d86 | A sheep is tied to the edge of a circular patch of grass with radius \(r\). The length of the rope used to tie the sheep is \(l\). Find the length \(l\) of the rope if the sheep eats exactly half of the grass. | [
"\\( l = \\frac{r}{\\sqrt{2}} \\)"
] | [
"A sheep is tied to the edge of a circular patch of grass with radius \\(r\\). The length of the rope used to tie the sheep is \\(l\\). Find the length \\(l\\) of the rope if the sheep eats exactly half of the grass."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85dd1ccf5a3653df219ca446bcfeec03dd94c3d555bc172144add9bba1587fdf | A sheet of cardboard measures 20 cm by 30 cm. A square measuring \( x \) cm by \( x \) cm is cut from each corner so that an open box can be formed. a) Express the length and width of the base of the box in terms of \( x \). b) If the area of the base of the box is 264 cm², form an equation (factorized) involving the a... | [
"22 \\text{ cm} \\times 12 \\text{ cm} \\times 4 \\text{ cm}"
] | [
"A sheet of cardboard measures 20 cm by 30 cm. A square measuring \\( x \\) cm by \\( x \\) cm is cut from each corner so that an open box can be formed.\na) Express the length and width of the base of the box in terms of \\( x \\).\nb) If the area of the base of the box is 264 cm², form an equation (factorized) in... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-2904a7cbbc8bf6f52cc6ffa21dda0e6d427fc3824a7d3d87d10fceb95eead445 | A sheet of cardboard measures 20 cm × 30 cm. A square measuring \( x \) cm × \( x \) cm is cut from each corner so that an open box can be formed. a) Express the length and the width of the base of the box in terms of \( x \). b) If the area of the base of the box is 264 cm², form an equation (factorized) involving the... | [
"x = 4, dimensions: 12 cm × 22 cm × 4 cm"
] | [
"A sheet of cardboard measures 20 cm × 30 cm. A square measuring \\( x \\) cm × \\( x \\) cm is cut from each corner so that an open box can be formed.\na) Express the length and the width of the base of the box in terms of \\( x \\).\nb) If the area of the base of the box is 264 cm², form an equation (factorized) ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 29
} | 38 | 0 | 0 |
omr-ws-0d094eb63fb79caaaead206c588537821098b4004289998f56005d03a3c53b1d | A sheet of paper $ABCD$ is rectangular with width $|AB|=4$ and length $|BC|=6$. The lower right corner $B$ is folded to a point $S$ on the left edge $[AD]$, and the fold follows the segment $[RT]$, with $R$ on the bottom edge $[AB]$ and $T$ on the right edge $[BC]$. Let $x = [RB]$ and $y = [BT]$. a) What are the minimu... | [
"\\( x = 9 - 3\\sqrt{5} \\)"
] | [
"A sheet of paper $ABCD$ is rectangular with width $|AB|=4$ and length $|BC|=6$. The lower right corner $B$ is folded to a point $S$ on the left edge $[AD]$, and the fold follows the segment $[RT]$, with $R$ on the bottom edge $[AB]$ and $T$ on the right edge $[BC]$. Let $x = [RB]$ and $y = [BT]$.\na) What are the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b35a98057262967217a6e65c29602db8cdd3f4c090f9096aae0a268a86cc7c40 | A sheet of paper ABCD (labelled anticlockwise with A in the bottom left-hand corner) has AB = 1 and BC = sqrt(2). E is the point on BC such that BE = 1. F is the point on DC such that FC = CE. A fold is made along AE and then a fold is made along EF. What is the perimeter of the new shape? | [
"4"
] | [
"A sheet of paper ABCD (labelled anticlockwise with A in the bottom left-hand corner) has AB = 1 and BC = sqrt(2). E is the point on BC such that BE = 1. F is the point on DC such that FC = CE. A fold is made along AE and then a fold is made along EF. What is the perimeter of the new shape?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-784b355aa19a5eab2cfe2a20fa047992c9b83b07fea72de06e9cb3462f5635e9 | A sheet of paper has $20$ cm length by $15$ cm wide. We fold this sheet in half, parallel to its width. Then we fold the sheet double rectangular, so that two vertices opposites coincide. When unfolding the sheet, the second fold marks divide the sheet into two parts. What is the area of the dark part, in cm$^2$? | [
"150"
] | [
"A sheet of paper has $20$ cm length by $15$ cm wide. We fold this sheet in half, parallel to its width. Then we fold the sheet double rectangular, so that two vertices opposites coincide. When unfolding the sheet, the second fold marks divide the sheet into two parts. What is the area of the dark part, in cm$^2$?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-6bd2207f0e67809e1883bbfb25e9298071c78c9247be0652ae84f01f14c49ccc | A sheet of paper has a width of 21 cm. The lower corner of the sheet is folded over to reach the opposite side. Determine the width of the folded part when the length of the folded portion is minimized. | [
"10.5"
] | [
"A sheet of paper has a width of 21 cm. The lower corner of the sheet is folded over to reach the opposite side. Determine the width of the folded part when the length of the folded portion is minimized."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-136ecedcda7f9d12b08213aeaf3bb251012db8421b0b63945bf1555528c49d98 | A sheet of paper with a drawn rectangular coordinate system is folded along a straight line so that the point $(0,2)$ coincides with the point $(4,0)$. What point does the point $(7,3)$ coincide with? | [
"\\(\\left( \\frac{3}{5}, \\frac{31}{5} \\right)\\)"
] | [
"A sheet of paper with a drawn rectangular coordinate system is folded along a straight line so that the point $(0,2)$ coincides with the point $(4,0)$. What point does the point $(7,3)$ coincide with?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-84ddc3c409f4888e83c29925b35ec391db4904f82abe9accbb383cd1d5756120 | A sheet of toilet paper 100 feet long is wrapped around a cardboard roll with a radius of 1 inch. The toilet paper is 0.04 inches thick. The toilet paper wraps around the roll \( N \) times and has some leftover, but not enough to complete the next wrap around. How many more inches of toilet paper are required to compl... | [
"3.356"
] | [
"A sheet of toilet paper 100 feet long is wrapped around a cardboard roll with a radius of 1 inch. The toilet paper is 0.04 inches thick. The toilet paper wraps around the roll \\( N \\) times and has some leftover, but not enough to complete the next wrap around. How many more inches of toilet paper are required t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-821691bc9c615ec408338bf76efd22c98f0bcae1f1786870578580942d9f4867 | A shepherd has some sheep. He buys 3 more sheep. Now he has 7 times as many sheep as he would have had if he had sold 3 sheep instead of buying 3 sheep. How many sheep does he have now? | [
"7"
] | [
"A shepherd has some sheep. He buys 3 more sheep. Now he has 7 times as many sheep as he would have had if he had sold 3 sheep instead of buying 3 sheep. How many sheep does he have now?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce9b6e73f9ec22b03e88aacd66dcb1c8a30f1e021e8c97c4656012da986767ed | A shepherd left a flock of $k$ sheep to his children, distributed as follows: the oldest received $\left\lfloor\frac{k}{2}\right\rfloor$ sheep, the middle one $\left\lfloor\frac{k}{3}\right\rfloor$ sheep, and the youngest $\left\lfloor\frac{k}{5}\right\rfloor$ sheep. Knowing that there are no sheep left, determine all ... | [
"\\( k = 0, 1, 2, 3, 4, 10, 15, 16, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59 \\)"
] | [
"A shepherd left a flock of $k$ sheep to his children, distributed as follows: the oldest received $\\left\\lfloor\\frac{k}{2}\\right\\rfloor$ sheep, the middle one $\\left\\lfloor\\frac{k}{3}\\right\\rfloor$ sheep, and the youngest $\\left\\lfloor\\frac{k}{5}\\right\\rfloor$ sheep. Knowing that there are no sheep ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cda6faf1f68382839d6a9d2557b43d75e659cba22748c2547b329e680e94d8ee | A ship and a raft start simultaneously from pier $A$ and travel down the river. Having reached pier $B$, located 324 km from $A$, the ship stopped there for 18 h and then started back to $A$. At the moment when it was 180 km from $A$, a second ship, which left $A$ 40 h later than the first, overtook the raft which by t... | [
"The speed of the ships is 15 km/h and that of the river flow is 3 km/h"
] | [
"A ship and a raft start simultaneously from pier $A$ and travel down the river. Having reached pier $B$, located 324 km from $A$, the ship stopped there for 18 h and then started back to $A$. At the moment when it was 180 km from $A$, a second ship, which left $A$ 40 h later than the first, overtook the raft whic... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0723478b63101fe27661f49fe93f608b456b6cbc3f2c13e1c53ba5175e1ce294 | A ship at sea spots two lighthouses that are 3 miles apart along a straight seashore. The angles formed between the line-of-sight observations of the lighthouses and the line from the ship directly to shore are 25 degrees and 35 degrees. How far is the ship from each lighthouse? | [
"3.14"
] | [
"A ship at sea spots two lighthouses that are 3 miles apart along a straight seashore. The angles formed between the line-of-sight observations of the lighthouses and the line from the ship directly to shore are 25 degrees and 35 degrees. How far is the ship from each lighthouse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-db12d01bdbf70aba8455f325ed9350ccf2a8107a90b4baece1a505c55bb87528 | A ship at sea spots two lighthouses that are 3 miles apart along a straight seashore. The angles formed between the line-of-sight observations of the lighthouses and the line from the ship directly to shore are 25 degrees and 35 degrees. How far is the ship from shore? | [
"3 miles"
] | [
"A ship at sea spots two lighthouses that are 3 miles apart along a straight seashore. The angles formed between the line-of-sight observations of the lighthouses and the line from the ship directly to shore are 25 degrees and 35 degrees. How far is the ship from shore?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b49c40704d4c5580f3b32bdc93df01668f6fb0bc39acf16b606eae6f83e63a77 | A ship is 5 km north of an oil flag and travels on a bearing of 150 degrees. Calculate the closest distance that the ship passes by the oil flag. | [
"\\(\\frac{5}{2}\\) kilometers"
] | [
"A ship is 5 km north of an oil flag and travels on a bearing of 150 degrees. Calculate the closest distance that the ship passes by the oil flag."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-7a8c8d389d97f8dbeff564d3f9f92c6a92f2a18798a7f11c6b2229ddd8cf09e8 | A ship is 6.7 km from a lighthouse at a bearing of 308 degrees. Another ship is 8.3 km from the lighthouse at a bearing of 74 degrees. Calculate the distance between the two ships and the bearing from the first ship to the second ship. | [
"098^\\circ"
] | [
"A ship is 6.7 km from a lighthouse at a bearing of 308 degrees. Another ship is 8.3 km from the lighthouse at a bearing of 74 degrees. Calculate the distance between the two ships and the bearing from the first ship to the second ship."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75117e14670bce3a9799d48a5b9f52309b292c1e523d23886e1c3cf8a9002c89 | A ship is being loaded by cranes. Four similar cranes worked for the first two hours, then another two smaller cranes joined them, and in three hours the loading was over. If all six cranes had begun operating simultaneously, the loading would have been completed in 4.5 hours. How much time would it take one of the mor... | [
"14.4 hours"
] | [
"A ship is being loaded by cranes. Four similar cranes worked for the first two hours, then another two smaller cranes joined them, and in three hours the loading was over. If all six cranes had begun operating simultaneously, the loading would have been completed in 4.5 hours. How much time would it take one of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-843970fe462cbc91dbb0267abc79314d5402cd2cc8242e630ce7e6a9e4d74b9f | A ship is moving due west at 10 knots. You are in a speed boat 4 nautical miles from the ship, with a bearing of 142 degrees as seen from the ship. You need to catch up with the ship, and you can move at a speed of 15 knots. Determine the bearing (in degrees) you should take off at and the time (in minutes) it will tak... | [
"36.3"
] | [
"A ship is moving due west at 10 knots. You are in a speed boat 4 nautical miles from the ship, with a bearing of 142 degrees as seen from the ship. You need to catch up with the ship, and you can move at a speed of 15 knots. Determine the bearing (in degrees) you should take off at and the time (in minutes) it wil... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-293f215ef0a1ed60008aad37a1fcc28ab217e8e4f5cfa54b6e1a39d36e1ec931 | A ship is moving due west at 8 knots. You are in a speed boat \(\sqrt{2}\) nautical miles directly southeast of the ship. You need to catch up with the ship, and you can move at a speed of 16 knots. Determine the bearing (in degrees) you should take off at and the time (in minutes) it will take to reach the ship. | [
"9.1"
] | [
"A ship is moving due west at 8 knots. You are in a speed boat \\(\\sqrt{2}\\) nautical miles directly southeast of the ship. You need to catch up with the ship, and you can move at a speed of 16 knots. Determine the bearing (in degrees) you should take off at and the time (in minutes) it will take to reach the shi... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-056bab21cb51f76bdbf904844698fdf0d953bdb77aba1b9ccf3eeaa13822b39c | A ship is sailing in a straight line at 20 knots. At the port, there is a small boat whose highest speed is 12 knots. The line from the port perpendicular to the ship's route has length \(a\). The small boat's men want to get out of the port as late as possible but still in time to catch the ship. (a) When the men shou... | [
"\\arctan\\left(\\dfrac{3}{4}\\right)"
] | [
"A ship is sailing in a straight line at 20 knots. At the port, there is a small boat whose highest speed is 12 knots. The line from the port perpendicular to the ship's route has length \\(a\\). The small boat's men want to get out of the port as late as possible but still in time to catch the ship. \n(a) When the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-85c878a90d2469177696dc4cc0b96609e98b2a40da2e34881ef5c471e7de8caf | A ship is sailing on a bearing of 200º T. At 1200 hours, the captain of the ship observes a lighthouse on a bearing of 225º T, and after traveling 30 km, the captain observes the same lighthouse on a bearing of 250º T. a) Draw a diagram using A for the position of the ship at 1200 hours, L for the position of the light... | [
"1.5"
] | [
"A ship is sailing on a bearing of 200º T. At 1200 hours, the captain of the ship observes a lighthouse on a bearing of 225º T, and after traveling 30 km, the captain observes the same lighthouse on a bearing of 250º T.\na) Draw a diagram using A for the position of the ship at 1200 hours, L for the position of the... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fcfe1dabde01595e06b571dee9e0d299ae9a412a4385d4637024b6d24a6117b2 | A ship is within 10 km of a lighthouse surrounded by reefs and needs to reach the shore, which is 10 km from the lighthouse. The ship's trajectory must be predetermined. Can the ship land having sailed a distance not greater than 75 km? | [
"Yes"
] | [
"A ship is within 10 km of a lighthouse surrounded by reefs and needs to reach the shore, which is 10 km from the lighthouse. The ship's trajectory must be predetermined. Can the ship land having sailed a distance not greater than 75 km?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-45154e8cef06c3db3fe6beef2077cfb9a168940cc092683452e9847bed306c42 | A ship leaves port and sails on a bearing of S 34° W for 2.5 hours. It then turns and sails on a bearing of N 56° W for 3 hours. If the ship’s speed is 18 knots (nautical miles per hour), find the distance that the ship is from the port. | [
"50"
] | [
"A ship leaves port and sails on a bearing of S 34° W for 2.5 hours. It then turns and sails on a bearing of N 56° W for 3 hours. If the ship’s speed is 18 knots (nautical miles per hour), find the distance that the ship is from the port."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b56bcb438e789ca1b221d017f21b94ac8e82d5d3aecb68d81440163974e5cc3 | A ship leaves port at 20 miles per hour with a bearing of S 46° E. There is a warning buoy located 5 miles directly north of the port. What is the bearing of the warning buoy as seen from the ship after 4.5 hours? | [
"311.9°"
] | [
"A ship leaves port at 20 miles per hour with a bearing of S 46° E. There is a warning buoy located 5 miles directly north of the port. What is the bearing of the warning buoy as seen from the ship after 4.5 hours?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b263df2d820a8483d7b38914f74a7efd36276979bb1da144919c12a579860789 | A ship left point $A$ for point $B$. At 8 o'clock the ship overtook a boat moving at 3 km/h in the same direction. Having stayed for 10 minutes at $B$, the ship returned to $A$, meeting the boat at 8:20. The ship reached $A$ at the same time the boat reached $B$. Determine the time the boat arrived at point $B$ if it h... | [
"At 8.30 p.m"
] | [
"A ship left point $A$ for point $B$. At 8 o'clock the ship overtook a boat moving at 3 km/h in the same direction. Having stayed for 10 minutes at $B$, the ship returned to $A$, meeting the boat at 8:20. The ship reached $A$ at the same time the boat reached $B$. Determine the time the boat arrived at point $B$ if... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d84a2237b4ca937174143fc4e39f034f67182a7dc846184417a9dec0b1a1ddbb | A ship left port $A$ for port $B$, and 7.5 hours later a motor-boat followed the ship in the same direction. Halfway between $A$ and $B$ the boat caught up with the ship. When the boat reached $B$, the ship had still to cover $\frac{3}{10}$ of the entire distance. How much time will it have taken the ship to cover the ... | [
"25 hours"
] | [
"A ship left port $A$ for port $B$, and 7.5 hours later a motor-boat followed the ship in the same direction. Halfway between $A$ and $B$ the boat caught up with the ship. When the boat reached $B$, the ship had still to cover $\\frac{3}{10}$ of the entire distance. How much time will it have taken the ship to cove... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.