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-266110681b8619cf467feda5d0d4ca18bfc769cae061df2c7463a96d30b88330 | An equilateral triangle with sides of length 25 is partitioned into 625 identical smaller triangles by lines parallel to its sides. Each small triangle can be labeled with either 1 or -1. The product of the number inside each small triangle and the numbers in all its neighboring triangles (sharing a side) must equal 1.... | [
"2"
] | [
"An equilateral triangle with sides of length 25 is partitioned into 625 identical smaller triangles by lines parallel to its sides. Each small triangle can be labeled with either 1 or -1. The product of the number inside each small triangle and the numbers in all its neighboring triangles (sharing a side) must equ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-28d80531e787ef85f9a686378e35de8b9add2f1d946300f0e69cafa35b78c63d | An equilateral triangle, a circle, and a square have the same perimeter. Of the three shapes, which one has the largest area? | [
"circle"
] | [
"An equilateral triangle, a circle, and a square have the same perimeter. Of the three shapes, which one has the largest area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-516e229afe637f5fa92620cf1224a6fbc47e746fa70af695f796c1c593bb1442 | An equilateral triangle, square, and regular hexagon each with side length $1$ are situated in $3D$ space such that they share a common vertex $A$. The edge shared by the square and the hexagon is $\overline{AB}$, the edge shared by the hexagon and the triangle is $\overline{AC}$, and the edge shared by the triangle an... | [
"15"
] | [
"An equilateral triangle, square, and regular hexagon each with side length $1$ are situated in $3D$ space such that they share a common vertex $A$. The edge shared by the square and the hexagon is $\\overline{AB}$, the edge shared by the hexagon and the triangle is $\\overline{AC}$, and the edge shared by the tria... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b049647dc5fd0cfd081f7f8831842d612872c15b37b62722734e547dcbf092a | An equilateral triangular pizza is to be divided by a straight cut passing through a point chosen by the mother. Determine the largest fraction of the pizza that the mother can guarantee for herself by choosing an appropriate point for the cut. | [
"\\dfrac{4}{9}"
] | [
"An equilateral triangular pizza is to be divided by a straight cut passing through a point chosen by the mother. Determine the largest fraction of the pizza that the mother can guarantee for herself by choosing an appropriate point for the cut."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-7d6aabf71a90a12318f4a313c995e65fe1f00437d566a69d3ce03c2b750ecc9b | An error occurred when two numbers were multiplied, one of which is larger than the other by 10: the tens digit in the product was decreased by 4. When the answer was verified by dividing the resulting product by the smaller factor, the quotient obtained was equal to 39 and the remainder to 22. Find the factors. | [
"31 and 41"
] | [
"An error occurred when two numbers were multiplied, one of which is larger than the other by 10: the tens digit in the product was decreased by 4. When the answer was verified by dividing the resulting product by the smaller factor, the quotient obtained was equal to 39 and the remainder to 22. Find the factors."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b7b7dd2b61db8b76898f18d18c8e7685607a6029aa33b1ebb319da5aa04c827 | An escalator is moving downwards. Sam takes 40 steps to reach the top of the escalator while going up and 10 steps to reach the bottom while coming down. What is the total number of steps on the escalator when it is not moving, given that Sam walks at a constant speed? | [
"20"
] | [
"An escalator is moving downwards. Sam takes 40 steps to reach the top of the escalator while going up and 10 steps to reach the bottom while coming down. What is the total number of steps on the escalator when it is not moving, given that Sam walks at a constant speed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94858009d115782c7d865ba214b2a9b5feb8bfe6a1503bdb6b199b01842e7591 | An escalator with \( n \) uniform steps visible at all times descends at a constant speed. Two boys, \( A \) and \( Z \), walk down the escalator steadily as it moves, with \( A \) negotiating twice as many escalator steps per minute as \( Z \). \( A \) reaches the bottom after taking 27 steps while \( Z \) reaches the... | [
"54"
] | [
"An escalator with \\( n \\) uniform steps visible at all times descends at a constant speed. Two boys, \\( A \\) and \\( Z \\), walk down the escalator steadily as it moves, with \\( A \\) negotiating twice as many escalator steps per minute as \\( Z \\). \\( A \\) reaches the bottom after taking 27 steps while \\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-173cc84c1fe537e578436aa19b88fa4add6e38ed6386b9ff01a93933a31e284b | An escalator's speed and number of visible steps are constant. Walking up the escalator at a brisk rate permits 42 steps. Walking at half that rate permits 24 steps. How many steps are visible? | [
"168"
] | [
"An escalator's speed and number of visible steps are constant. Walking up the escalator at a brisk rate permits 42 steps. Walking at half that rate permits 24 steps. How many steps are visible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-4e2d0d40822bc32e616a5bf86e4a3e318e4ee8561e633e18857156f2cf29248b | An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1$. What is the least number of circles needed to m... | [
"64"
] | [
"An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1$. What is the least number of circles needed... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-c070a8bf3b8efb5030a1aab03aeff70f1ec1493465a940102b1a9aba2882ee26 | An even number of people are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Let $n$ be the number of people. Calculate the minimum value of $n$ such that there are at least two persons with exactly 3 people between them both before and after the br... | [
"8"
] | [
"An even number of people are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Let $n$ be the number of people. Calculate the minimum value of $n$ such that there are at least two persons with exactly 3 people between them both before and after t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c023385fd0efccdb48fe516f4eaf5b2c6b0470cf7932ee6707225cbb6b01dc42 | An even number of people sit at a circular table. After a break, they sit down again in a different order. Let $n$ be the number of people. Find the minimum value of $n$ such that there must be at least two people who have the same number of people sitting between them before and after the break. | [
"4"
] | [
"An even number of people sit at a circular table. After a break, they sit down again in a different order. Let $n$ be the number of people. Find the minimum value of $n$ such that there must be at least two people who have the same number of people sitting between them before and after the break."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8d808a95a4ed325d67fb03754a24cfce07bd80488282bb0bbb65a122b69ca6fb | An even number of persons are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Let $n$ be the number of persons. Calculate the minimum value of $n$ such that there are at least two persons with the same number of persons between them as before the br... | [
"6"
] | [
"An even number of persons are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Let $n$ be the number of persons. Calculate the minimum value of $n$ such that there are at least two persons with the same number of persons between them as before t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cddefe2275976fe552f077452ff44d15a9662453d400bc63b0f0ddfc6cee38ed | An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Let $P(n)$ be the minimum number of pairs of politicians such that the number of people sitting between them is the same as before the break, for any arrangement of $n$ politicians. Find $P(... | [
"5"
] | [
"An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Let $P(n)$ be the minimum number of pairs of politicians such that the number of people sitting between them is the same as before the break, for any arrangement of $n$ politicians. Fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e88434df03df10dd034dd680a636f99fed2cf32e7414c4f2a6d0890b2436aa64 | An even positive integer \( n \) is called nice if the set \( \{1, 2, \dots, n\} \) can be partitioned into \( \frac{n}{2} \) two-element subsets such that the sum of the elements in each subset is a power of 3. Find the number of nice positive integers which are smaller than \( 3^{2022} \). | [
"\\( 2^{2022} - 1 \\)"
] | [
"An even positive integer \\( n \\) is called nice if the set \\( \\{1, 2, \\dots, n\\} \\) can be partitioned into \\( \\frac{n}{2} \\) two-element subsets such that the sum of the elements in each subset is a power of 3. Find the number of nice positive integers which are smaller than \\( 3^{2022} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 5
} | 25 | 0 | 0 |
omr-ws-f389a6a76fe8766c44ccee52341ed451452647619aca55a70316b8597bfabd0c | An event is either a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. What is the probability of exactly 50 hits in the first 100 trials? | [
"\\(\\frac{1}{99}\\)"
] | [
"An event is either a hit or a miss. The first event is a hit, the second is a miss. Thereafter, the probability of a hit equals the proportion of hits in the previous trials. What is the probability of exactly 50 hits in the first 100 trials?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-796a50041ea470638212a1bdcb23a166206365f07f63a073de0878f2ff23d653 | An event occurs periodically in $x$ consecutive years, followed by a break of $y$ consecutive years. The event occurred in $1964$, $1986$, $1996$, and $2008$, and did not occur in $1976$, $1993$, $2006$, and $2013$. Determine the first year after $2008$ in which the event will occur again. | [
"2018"
] | [
"An event occurs periodically in $x$ consecutive years, followed by a break of $y$ consecutive years. The event occurred in $1964$, $1986$, $1996$, and $2008$, and did not occur in $1976$, $1993$, $2006$, and $2013$. Determine the first year after $2008$ in which the event will occur again."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c5792084a39ca238915c5608ac4ebe511cd46ab2ae054db7b3d4dd87af46d80 | An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of 101 ships. Some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Each carrier can provide each destroyer... | [
"2601"
] | [
"An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of 101 ships. Some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Each carrier can provide each dest... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-f3afaef7592ad480a88ba202dce8f2a89a80a186901e2010de5e447a909bae1a | An evil ghost is haunting a 2017 x 2017 feet field divided into 1x1 foot cells. At the beginning, the ghost rests on a random cell. Marcus can shoot at any cell, and if the ghost is hit, it moves to an adjacent cell (sharing a side). If not hit, the ghost stays in the same cell. Marcus cannot determine if he has hit th... | [
"\\(\\frac{3 \\times 2017^2 - 1}{2}\\)"
] | [
"An evil ghost is haunting a 2017 x 2017 feet field divided into 1x1 foot cells. At the beginning, the ghost rests on a random cell. Marcus can shoot at any cell, and if the ghost is hit, it moves to an adjacent cell (sharing a side). If not hit, the ghost stays in the same cell. Marcus cannot determine if he has h... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 6,
"tir": 9
} | 17 | 0 | 0 |
omr-ws-5e28d12c81147d63cec3d08514eb7255f95b48794306dc9140e2abfa0fa94f51 | An exam center is preparing question papers for 160 students, with each paper containing 9 questions from 9 different topics (one question per topic). They allow up to 2 collisions, meaning at most 2 out of the 9 questions can be the same for any two papers. What is the minimum number of questions that should be prepar... | [
"6"
] | [
"An exam center is preparing question papers for 160 students, with each paper containing 9 questions from 9 different topics (one question per topic). They allow up to 2 collisions, meaning at most 2 out of the 9 questions can be the same for any two papers. What is the minimum number of questions that should be p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-14776772c0490114797d07548c9c2822538780b07cc1bfa3911ca27c379690d1 | An exam consisting of six questions is sat by 2006 children. Each question is marked either right or wrong. Any three children have right answers to at least five of the six questions between them. Let $N$ be the total number of right answers achieved by all the children. Find the least possible value of $N$. | [
"9900"
] | [
"An exam consisting of six questions is sat by 2006 children. Each question is marked either right or wrong. Any three children have right answers to at least five of the six questions between them. Let $N$ be the total number of right answers achieved by all the children. Find the least possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51eed7d707b85927c6881752740e511fdce2c0bb1557133c101cd05d9c2595e3 | An exam consisting of six questions is sat by 2006 children. Each question is marked either right or wrong. Any three children have right answers to at least five of the six questions between them. Let \( N \) be the total number of right answers achieved by all the children. Find the least possible value of \( N \). | [
"12024"
] | [
"An exam consisting of six questions is sat by 2006 children. Each question is marked either right or wrong. Any three children have right answers to at least five of the six questions between them. Let \\( N \\) be the total number of right answers achieved by all the children. Find the least possible value of \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-816f637b565506f3085aab7139ceaaaec92f8fe7e4b2d296fd70087d9505f7c1 | An exam consists of 5 questions, each worth 20 points. For an average student, the probability distribution for the points earned on each question is as follows: 20 points with 40%, 15 points with 30%, 10 points with 20%, and 0 points with 10%. What is the most probable total score for an average student? | [
"80"
] | [
"An exam consists of 5 questions, each worth 20 points. For an average student, the probability distribution for the points earned on each question is as follows: 20 points with 40%, 15 points with 30%, 10 points with 20%, and 0 points with 10%. What is the most probable total score for an average student?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-cc7aef5eeb5ca5659ecd4a2f15464d254f818174d49e04637225d2952eb0caa0 | An exam has 30 multiple-choice problems. A contestant who answers \( m \) questions correctly and \( n \) incorrectly (and does not answer \( 30 - m - n \) questions) gets a score of \( 30 + 4m - n \). A contestant scores \( N > 80 \). A knowledge of \( N \) is sufficient to deduce how many questions the contestant sco... | [
"119"
] | [
"An exam has 30 multiple-choice problems. A contestant who answers \\( m \\) questions correctly and \\( n \\) incorrectly (and does not answer \\( 30 - m - n \\) questions) gets a score of \\( 30 + 4m - n \\). A contestant scores \\( N > 80 \\). A knowledge of \\( N \\) is sufficient to deduce how many questions t... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-ee80fe83f588da867e6e22bc376057da5169d44dd796a4164e4ecd1b2f9dd7b3 | An exam has 4 multiple choice questions, each with 3 choices. A group of students took the exam, and for each question, each student chose one of the three choices. It is observed that for any 3 students in this group, there is at least one question where the answers from these 3 students are all different. What is the... | [
"9"
] | [
"An exam has 4 multiple choice questions, each with 3 choices. A group of students took the exam, and for each question, each student chose one of the three choices. It is observed that for any 3 students in this group, there is at least one question where the answers from these 3 students are all different. What i... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf43cb54c4ed76c5baec6b5a7dc0358f3333528a4f0ef9c35a9d38b05fd8f7f2 | An exam paper consists of 5 multiple choice questions, each with 4 different choices. 2000 students take the exam, and each student chooses exactly one answer per question. Find the smallest value of \( n \) for which it is possible that among any \( n \) of the students' answer sheets, there exist 4 of them such that ... | [
"1025"
] | [
"An exam paper consists of 5 multiple choice questions, each with 4 different choices. 2000 students take the exam, and each student chooses exactly one answer per question. Find the smallest value of \\( n \\) for which it is possible that among any \\( n \\) of the students' answer sheets, there exist 4 of them s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-411c40f6c2d39924c928c5abd1317c89b8d181b9a41c56359c4908db0e341eaf | An exam paper consists of 5 multiple-choice questions, each with 4 different choices. 2000 students take the test, and each student chooses exactly one answer per question. Find the smallest value of $n$ for which it is possible that among any $n$ of the student's answer sheets, there exist 4 of them such that any two ... | [
"1025"
] | [
"An exam paper consists of 5 multiple-choice questions, each with 4 different choices. 2000 students take the test, and each student chooses exactly one answer per question. Find the smallest value of $n$ for which it is possible that among any $n$ of the student's answer sheets, there exist 4 of them such that any... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ee73491a9902c16ac1508dba83e749002ae6590fb247c0fc7f46f3f455311bb8 | An exam paper consists of five multiple-choice questions, each with four different choices. 2000 students take the test, and each student chooses exactly one answer per question. Find the smallest value of \( n \) for which it is possible that among any \( n \) of the students' answer sheets, there exist four of them s... | [
"25"
] | [
"An exam paper consists of five multiple-choice questions, each with four different choices. 2000 students take the test, and each student chooses exactly one answer per question. Find the smallest value of \\( n \\) for which it is possible that among any \\( n \\) of the students' answer sheets, there exist four ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c7efb540b6ab2eb5369b0b48e9067d19f1bb3ad2073b7ff2f1e7cc7f5bfadcc | An examination centre has 50 rooms, with 10 candidates assigned to each room. Each candidate is absent with probability \( p \), where \( 0 < p < 1 \), independently of the other candidates. What is the probability that at least one room will have full attendance? | [
"\\( 1 - \\left(1 - (1 - p)^{10}\\right)^{50} \\)"
] | [
"An examination centre has 50 rooms, with 10 candidates assigned to each room. Each candidate is absent with probability \\( p \\), where \\( 0 < p < 1 \\), independently of the other candidates. What is the probability that at least one room will have full attendance?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-79cf984eb7929b88ef40c418c9c21e691a19253c785275808472d5f850ee01d6 | An examination comprises 2 papers each with a total of 100 marks. In order to pass the examination, a candidate must score at least 45 marks in each paper and at least 100 marks on the 2 papers combined. Only integer marks are given. Find the total number of possible ways in which a candidate scores at least 45 marks i... | [
"55"
] | [
"An examination comprises 2 papers each with a total of 100 marks. In order to pass the examination, a candidate must score at least 45 marks in each paper and at least 100 marks on the 2 papers combined. Only integer marks are given. Find the total number of possible ways in which a candidate scores at least 45 ma... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f989d332ee9b62de7315d14bbc649d6ccbeed483c6c050a0ebfe499946fface5 | An examination has 20 questions. For each question, the marks that can be obtained are either $-1$, $0$, or $4$. What is the number of possible total marks that a student can score in the examination? | [
"95"
] | [
"An examination has 20 questions. For each question, the marks that can be obtained are either $-1$, $0$, or $4$. What is the number of possible total marks that a student can score in the examination?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-0eed6c07598c2c23230c8463588910513dfa7173561eab7e2a726d9afc44ae57 | An examination has 20 questions. For each question, the marks that can be obtained are either −1, 0, or 4. Let S be the set of possible total marks that a student can score in the examination. What is the number of elements in S? | [
"95"
] | [
"An examination has 20 questions. For each question, the marks that can be obtained are either −1, 0, or 4. Let S be the set of possible total marks that a student can score in the examination. What is the number of elements in S?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-581c990f333b76efb5e93f22d44d2918b1eba54903d9c28af01e0a0edd9de5c2 | An examination paper includes questions on four topics taken from a curriculum of 45 topics. A student has not revised 15 of the topics. What is the probability that he gets a paper in which he can answer all the questions? | [
"$\\binom{30}{4}/\\binom{45}{4}$"
] | [
"An examination paper includes questions on four topics taken from a curriculum of 45 topics. A student has not revised 15 of the topics. What is the probability that he gets a paper in which he can answer all the questions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba0014871fd7f53d4dd32276d51a1dffabab2fa4831032a23209583c291a4242 | An examinee can score either 0, 1, or 10 marks on each of 9 questions, each worth 10 marks. How many possible total scores can the examinee achieve? | [
"55"
] | [
"An examinee can score either 0, 1, or 10 marks on each of 9 questions, each worth 10 marks. How many possible total scores can the examinee achieve?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-22f01d7c658068abc920c7b0e09d86107221656c1b001f84018260f04ef9c548 | An experiment has \( n \) possible outcomes, all equally likely. Each trial is independent. We repeat the trials until we have two trials with the same outcome. What is the expected number of trials \( f(n) \)? Find an asymptotic expression for \( f(n) \). | [
"\\sqrt{\\frac{\\pi n}{2}}"
] | [
"An experiment has \\( n \\) possible outcomes, all equally likely. Each trial is independent. We repeat the trials until we have two trials with the same outcome. What is the expected number of trials \\( f(n) \\)? Find an asymptotic expression for \\( f(n) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64e26949babd80536f0c0df041f36c686d7aa044de4886ff8328272b6ed58ffe | An exponential sequence has the first term \( x \) and common ratio \( x \) (where \( x \neq 1 \)). Find the sum of the first \( n \) terms. If \( x < 1 \), find the sum to infinity. | [
"\\( S_n = x \\frac{1 - x^n}{1 - x} \\) and \\( S_{\\infty} = \\frac{x}{1 - x} \\)"
] | [
"An exponential sequence has the first term \\( x \\) and common ratio \\( x \\) (where \\( x \\neq 1 \\)). Find the sum of the first \\( n \\) terms. If \\( x < 1 \\), find the sum to infinity."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 23
} | 23 | 0 | 0 |
omr-ws-c295b6b5718dddf2a1f2c0d59017fcd7630cffe5adb91958b9e98b62bcc34046 | An extensible string of length \( a \) is hung horizontally at points \( A \) and \( B \), which are a distance \( a \) apart. Given that the string has a non-negligible weight, find the maximum vertical displacement of the string from the horizontal line connecting \( A \) and \( B \). | [
"0"
] | [
"An extensible string of length \\( a \\) is hung horizontally at points \\( A \\) and \\( B \\), which are a distance \\( a \\) apart. Given that the string has a non-negligible weight, find the maximum vertical displacement of the string from the horizontal line connecting \\( A \\) and \\( B \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-b23a9f7a9fa4336eac3b4252bc08fc81a808bf2048c5852fc1b467e06523cc12 | An exterior angle of a regular polygon contains \( n \) degrees. If \( 35 < n < 45 \), what values may \( n \) have? | [
"36, 40"
] | [
"An exterior angle of a regular polygon contains \\( n \\) degrees. If \\( 35 < n < 45 \\), what values may \\( n \\) have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 30
} | 36 | 0 | 0 |
omr-ws-cac3a33301f3e6f0a189e0f763bec81fc9ec2f4a36a62ebfda1ff7a6f60a84a7 | An exterior common tangent to two non-intersecting circles with centers $O_1$ and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the point ... | [
"2\\sqrt{6}"
] | [
"An exterior common tangent to two non-intersecting circles with centers $O_1$ and $O_2$ touches them at the points $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects the circles at the points $B_1$ and $B_2$ respectively. $C$ is the point where the straight lines $A_1B_1$ and $A_2B_2$ meet. $D$ is the p... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-2e8e98d2ce2ad54e9c8ba1f3bb35bf280abfc52d9ac81642eccff3075b670b99 | An exterior tangent to two circles of radii 5 cm and 2 cm is 1.5 times longer than their interior tangent. Determine the distance between the centers of the circles. | [
"$O_1O_2=9$ cm"
] | [
"An exterior tangent to two circles of radii 5 cm and 2 cm is 1.5 times longer than their interior tangent. Determine the distance between the centers of the circles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bea02d8412c32833cd1bb9b18e007ff022409f594695793282641fb3833b5918 | An homothety of the plane sends the line \( x = 1 \) to the line \( x = 2 \) and the line \( y = 2 \) to the line \( y = 1 \). Determine the equation of the line that contains the fixed point(s) of this homothety, if any. If there are no fixed points, enter "None". | [
"x + y = 3"
] | [
"An homothety of the plane sends the line \\( x = 1 \\) to the line \\( x = 2 \\) and the line \\( y = 2 \\) to the line \\( y = 1 \\). Determine the equation of the line that contains the fixed point(s) of this homothety, if any. If there are no fixed points, enter \"None\"."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-6119f70f300c160e8495c33a8053cfa319cdcd948cf6ee519e05a543dff64216 | An hourglass is formed from two identical cones. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half the d... | [
"52.5 minutes"
] | [
"An hourglass is formed from two identical cones. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of sand in the lower cone to be half ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f19f1b1067d042381877a7bbcdcc12be84395c3ffbdcaf8a0d4dc3919645deef | An iPod automatically selects a song to play from a playlist consisting of \( m \) songs with respective probabilities \( p_1, p_2, \ldots, p_m \) such that \( \sum_{i=1}^m p_i = 1 \). Each selection is independent. Let \( X \) be the total number of songs required to be played before all \( m \) songs have been played... | [
"\\sum_{k=1}^m (-1)^{k-1} \\sum_{1 \\leq i_1 < i_2 < \\cdots < i_k \\leq m} \\left( p_{i_1} + p_{i_2} + \\cdots + p_{i_k} \\right) \\left( 1 - (p_{i_1} + p_{i_2} + \\cdots + p_{i_k}) \\right)^{n-1}"
] | [
"An iPod automatically selects a song to play from a playlist consisting of \\( m \\) songs with respective probabilities \\( p_1, p_2, \\ldots, p_m \\) such that \\( \\sum_{i=1}^m p_i = 1 \\). Each selection is independent. Let \\( X \\) be the total number of songs required to be played before all \\( m \\) songs... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04335c04dce71a97952e7ebf2cd4a29d1db104543b895ee9c64defb46c3f30da | An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is a natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e., $V_n - V_{n+1} = a L_n$, for every $n$, where $a$ is a positive cons... | [
"4"
] | [
"An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is a natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e., $V_n - V_{n+1} = a L_n$, for every $n$, where $a$ is a positive... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f71414477ac83434effe28582ffc8c0ed9cb628301b713ca7cc072d3215d5c2f | An ice cream cone has a height of 5 inches, and half of a sphere is sticking out of the cone. How deep into the cone must one eat so that the remaining volume of ice cream (both cone and hemisphere) is equal to the volume already eaten? | [
"\\( \\frac{3}{2} \\) inches."
] | [
"An ice cream cone has a height of 5 inches, and half of a sphere is sticking out of the cone. How deep into the cone must one eat so that the remaining volume of ice cream (both cone and hemisphere) is equal to the volume already eaten?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f2e2866b7cac8526b5c8b7939ecf4b6f050cdf848a850872e9e5698619b73556 | An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone? | [
"1"
] | [
"An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02fbb701eb562ed6ab76b69f828b41f2f70b015c2316cafe718bb8d1b6b42b0e | An ice cream parlor sells a sundae special consisting of two scoops of ice cream with four toppings. They have $20$ toppings from which to choose, and they will allow you to choose the same topping up to four times, as long as the total number of choices is exactly four. How many different combinations of choices of fo... | [
"6385"
] | [
"An ice cream parlor sells a sundae special consisting of two scoops of ice cream with four toppings. They have $20$ toppings from which to choose, and they will allow you to choose the same topping up to four times, as long as the total number of choices is exactly four. How many different combinations of choices ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98c8b8b37ec3178fd5a9267b5b0fb58382e92390f2c8c868bd8719143f7b61d6 | An ice cream seller has only one kind of ice cream priced at $10. There are 400 customers waiting, with 100 customers having only $20 bills and 300 customers having only $10 bills. In how many orders can the customers arrive such that the seller always has at least one $10 bill available (except at the very beginning)? | [
"\\dfrac{1}{2} \\dbinom{400}{100}"
] | [
"An ice cream seller has only one kind of ice cream priced at $10. There are 400 customers waiting, with 100 customers having only $20 bills and 300 customers having only $10 bills. In how many orders can the customers arrive such that the seller always has at least one $10 bill available (except at the very beginn... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-8d2f7eaf3d944c8203589df66fb0e3929f0f8354c15d9e36ca316579e5c632d8 | An ice cream shop offers 6 kinds of ice cream. What is the greatest number of two scoop sundaes that can be made such that each sundae contains two types of ice cream and no two sundaes are the same combination? | [
"15"
] | [
"An ice cream shop offers 6 kinds of ice cream. What is the greatest number of two scoop sundaes that can be made such that each sundae contains two types of ice cream and no two sundaes are the same combination?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12a3c3e97672c09ad08d04e9368f996effd9d3b7f7d9bc18d5078af51e38449c | An ice cube tray with 10 spaces is filled such that each space has a random chance of being filled $\frac{1}{3},\ \frac{2}{3},$ or $\frac{3}{3}$ of the way, with no space being empty. If the total amount of water must equal $\frac{20}{3}$ spaces, how many ways are there for them to be filled? | [
"6"
] | [
"An ice cube tray with 10 spaces is filled such that each space has a random chance of being filled $\\frac{1}{3},\\ \\frac{2}{3},$ or $\\frac{3}{3}$ of the way, with no space being empty. If the total amount of water must equal $\\frac{20}{3}$ spaces, how many ways are there for them to be filled?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-989197ce35b22d6e5714bfd93ec886cfa8d142514090bcac1db205fc8b06cb4d | An icosahedron is a polyhedron that has 20 faces which are all equilateral triangles. How many pairs of parallel edges does an icosahedron have? | [
"15"
] | [
"An icosahedron is a polyhedron that has 20 faces which are all equilateral triangles. How many pairs of parallel edges does an icosahedron have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8e2e9f03220bb006fade0c3ceaa30efc75e6d00d6cc4ab4cfab32a42c9f8a9c1 | An ideal fruit salad consists of 3 strawberries, 8 blueberries, 5 raspberries, 4 blackberries, and 2 kiwis. How many pieces of fruit will be wasted if you want to make the maximum number of whole fruit salads with 34 strawberries, 58 blueberries, 36 raspberries, 34 blackberries, and 20 kiwis? | [
"26"
] | [
"An ideal fruit salad consists of 3 strawberries, 8 blueberries, 5 raspberries, 4 blackberries, and 2 kiwis. How many pieces of fruit will be wasted if you want to make the maximum number of whole fruit salads with 34 strawberries, 58 blueberries, 36 raspberries, 34 blackberries, and 20 kiwis?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4e70b90da8d76239c3c4f9a2057a387e7545f5763ec14bc0c2793d1f3aa563ca | An incident ray passing through the point (-2,2) is reflected by the line \( x - 2y + 1 = 0 \) and passes through the point (0,7). Find the equation of the normal at the point of incidence. | [
"\\( y = -2x - 2 \\)"
] | [
"An incident ray passing through the point (-2,2) is reflected by the line \\( x - 2y + 1 = 0 \\) and passes through the point (0,7). Find the equation of the normal at the point of incidence."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db4f23894ba36855b9056d1e231388b365a85d52d49f5078d8486ded9d2d832c | An incircle of a triangle \(ABC\) touches its sides \(AB\), \(BC\), and \(CA\) at \(C'\), \(A'\), and \(B'\) respectively. Let \(M\), \(N\), \(K\), and \(L\) be the midpoints of \(C'A\), \(B'A\), \(A'C\), and \(B'C\) respectively. The line \(A'C'\) intersects lines \(MN\) and \(KL\) at \(E\) and \(F\) respectively; lin... | [
"1"
] | [
"An incircle of a triangle \\(ABC\\) touches its sides \\(AB\\), \\(BC\\), and \\(CA\\) at \\(C'\\), \\(A'\\), and \\(B'\\) respectively. Let \\(M\\), \\(N\\), \\(K\\), and \\(L\\) be the midpoints of \\(C'A\\), \\(B'A\\), \\(A'C\\), and \\(B'C\\) respectively. The line \\(A'C'\\) intersects lines \\(MN\\) and \\(K... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b41e8bbd5fdf9ba633bc00c5580fbac890ebeababf170c36c3f363036f6e048 | An inclined plane passing through a diameter of the base of a solid circular cylinder of diameter $D$ cm, placed vertically, cuts the curved surface of the cylinder at a maximum height of $H$ cm. What is the volume of the cutout portion in cubic centimeters? | [
"\\dfrac{1}{6} H D^2"
] | [
"An inclined plane passing through a diameter of the base of a solid circular cylinder of diameter $D$ cm, placed vertically, cuts the curved surface of the cylinder at a maximum height of $H$ cm. What is the volume of the cutout portion in cubic centimeters?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-4b44b4d1daba32f18f595d071b40adf9510757cb6d1b92454d6caacc5065a636 | An increase in the output of a factory as compared to that in the preceding year is $p\%$ for the first year and $q\%$ for the second year. What should the percent increase of the output be for the third year fur the average annual increase of the output for three years to he equal to $r\%$? | [
"$\\displaystyle\\frac{3r-p-q-\\displaystyle\\frac{pq}{100}}{\\left(1+\\displaystyle\\frac{p}{100}\\right) \\left(1+\\displaystyle\\frac{q}{100}\\right)}\\%$"
] | [
"An increase in the output of a factory as compared to that in the preceding year is $p\\%$ for the first year and $q\\%$ for the second year. What should the percent increase of the output be for the third year fur the average annual increase of the output for three years to he equal to $r\\%$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e883636281d65a1b5590d128fec8d5adb415bffb6774892e20b4f49dbd3995d6 | An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Calculate the sum of ... | [
"96"
] | [
"An increasing arithmetic sequence with infinitely many terms is determined as follows. A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Calculate the su... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d1acf15336b1afabb504372706a8b5ac2b58c2681473c2baadba797d3fc89ea6 | An increasing function $f \colon \mathbb N \to \mathbb N$ satisfies the equation $$f(f(n))=3n$$ for any $n \in \mathbb N$. Find all values of the parameter $\lambda \in \mathbb Q$ for which the equation $$f(\lambda x)=\lambda f(x)$$ has at least one solution in $\mathbb N$. | [
"3^k"
] | [
"An increasing function $f \\colon \\mathbb N \\to \\mathbb N$ satisfies the equation $$f(f(n))=3n$$ for any $n \\in \\mathbb N$. Find all values of the parameter $\\lambda \\in \\mathbb Q$ for which the equation $$f(\\lambda x)=\\lambda f(x)$$ has at least one solution in $\\mathbb N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9f36795d04054dd0c69934069273d52dbb3360944de74a14941b07c55ee01ddc | An increasing function \( f \) defined on \([0,1]\) satisfies the following conditions: 1. \( f(x) + f(1-x) = 1 \) 2. \( f\left(\frac{x}{6}\right) = \frac{1}{2} f(x) \) Find \( f\left(\frac{1}{2013}\right) \). | [
"\\(\\frac{1}{2}\\)"
] | [
"An increasing function \\( f \\) defined on \\([0,1]\\) satisfies the following conditions:\n1. \\( f(x) + f(1-x) = 1 \\)\n2. \\( f\\left(\\frac{x}{6}\\right) = \\frac{1}{2} f(x) \\)\n\nFind \\( f\\left(\\frac{1}{2013}\\right) \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-761703f6abb2d3409e3965693c1a4bcaf09f8d6ce1f7582b94ea27e8d0372a05 | An increasing sequence \(a_{1}, a_{2}, \dots, a_{k}\) of positive integers satisfies \(a_{1} = 2024\) and the condition \(a_{m} + a_{n} \geq a_{m+n} + \lvert m - n \rvert\) for all positive integers \(m\) and \(n\) such that \(m + n \leq k\). What is the maximum possible number of terms in this sequence? | [
"2025"
] | [
"An increasing sequence \\(a_{1}, a_{2}, \\dots, a_{k}\\) of positive integers satisfies \\(a_{1} = 2024\\) and the condition \\(a_{m} + a_{n} \\geq a_{m+n} + \\lvert m - n \\rvert\\) for all positive integers \\(m\\) and \\(n\\) such that \\(m + n \\leq k\\). What is the maximum possible number of terms in this se... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04ebbfa0ffd7d6ca70068da30e77784ddf6a9069b815e356b41c2836a485ff92 | An increasing sequence of integers starting with 1 has the property that if \( n \) is any member of the sequence, then so also are \( 3n \) and \( n + 7 \). Also, all the members of the sequence are solely generated from the first number 1; thus the sequence starts with \( 1, 3, 8, 9, 10, \ldots \) and \( 2, 4, 5, 6, ... | [
"7"
] | [
"An increasing sequence of integers starting with 1 has the property that if \\( n \\) is any member of the sequence, then so also are \\( 3n \\) and \\( n + 7 \\). Also, all the members of the sequence are solely generated from the first number 1; thus the sequence starts with \\( 1, 3, 8, 9, 10, \\ldots \\) and \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f22f110b29c2aab771fac8ec119fc884157ee798e982c21bd0291b56c5db5696 | An increasing sequence of positive integers has 4 terms. The first three terms form an arithmetic progression, and the last three terms form a geometric progression. The difference between the first and last terms is 30. Find the sum of all four terms. | [
"129"
] | [
"An increasing sequence of positive integers has 4 terms. The first three terms form an arithmetic progression, and the last three terms form a geometric progression. The difference between the first and last terms is 30. Find the sum of all four terms."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a34476615c5613ff0f16f0872042060a7eee2f324464b7b94f6a8da9af9081b7 | An individual bets \$1 each day on a game where a three-digit number is selected. If the number is guessed correctly, the payout is \$800 for each dollar bet. Given that there is a new winning number each day, how much money is the individual expected to win over a year? | [
"\\(-\\frac{365}{9}\\)"
] | [
"An individual bets \\$1 each day on a game where a three-digit number is selected. If the number is guessed correctly, the payout is \\$800 for each dollar bet. Given that there is a new winning number each day, how much money is the individual expected to win over a year?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-1583a30425953d30061252dd59d1ecd3d7a3dba2a3cf993121742c411233505c | An industry uses two types of machines (Machine A and Machine B) that produce pollutants (200 BO and 100 HC). Two solutions, C and D, can reduce the pollutants. Solution C reduces Machine A by 30 BO and 18 HC, and Machine B by 20 BO and 22 HC. Solution D reduces Machine A by 50 BO and 30 HC, and Machine B by 80 BO and ... | [
"4800"
] | [
"An industry uses two types of machines (Machine A and Machine B) that produce pollutants (200 BO and 100 HC). Two solutions, C and D, can reduce the pollutants. Solution C reduces Machine A by 30 BO and 18 HC, and Machine B by 20 BO and 22 HC. Solution D reduces Machine A by 50 BO and 30 HC, and Machine B by 80 BO... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-a407a33a45178177b7549e04b50a6468d3e3321a822475288f95bad5b3b81024 | An inextensible thread of length $x$ is used to form a convex pentagon (Figure B) consisting of a square and a right-angled isosceles triangle, where the base of the triangle corresponds to one side of the square. Determine the side length of the square in terms of $x$. | [
"\\(\\frac{x(3 - \\sqrt{2})}{7}\\)"
] | [
"An inextensible thread of length $x$ is used to form a convex pentagon (Figure B) consisting of a square and a right-angled isosceles triangle, where the base of the triangle corresponds to one side of the square. Determine the side length of the square in terms of $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-549b4e7ecfabf727dcbac9c88d4630ed9128a2bb56b0b08ba8688ecf66fb67ee | An inextensible thread of length $x$ is used to form a convex pentagon (Figure C) consisting of a square and an isosceles triangle with a $120^\circ$ angle at the tip, where the base of the triangle corresponds to one side of the square. Determine the side length of the square in terms of $x$. | [
"\\(\\frac{x (6 - \\sqrt{3})}{22}\\)"
] | [
"An inextensible thread of length $x$ is used to form a convex pentagon (Figure C) consisting of a square and an isosceles triangle with a $120^\\circ$ angle at the tip, where the base of the triangle corresponds to one side of the square. Determine the side length of the square in terms of $x$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d356c36f2e404f67ca591a45120f4e3b9c67c69139f6b93c566d783f6e85c152 | An infection is spreading on an \( 8 \times 8 \) chessboard. An infected cell remains infected forever, and on every step a healthy cell becomes infected if at least two of its neighbors are already infected, where two cells are called neighbors if they share a side. Can all cells become infected if initially (1) 8 cel... | [
"Yes for 8 infected cells, No for 7 infected cells."
] | [
"An infection is spreading on an \\( 8 \\times 8 \\) chessboard. An infected cell remains infected forever, and on every step a healthy cell becomes infected if at least two of its neighbors are already infected, where two cells are called neighbors if they share a side. Can all cells become infected if initially\n... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-81c97bec1940afbad0e23249b0fb3a324b744470e282112bb3e229ccf2093aa6 | An infectious rumor is spreading through the University of Pennsylvania. At the start of the first day, only three people know the rumor. However, throughout the day each person who knows the rumor tells it to two new people who do not know the rumor. Also, every evening, a third of the people who know the rumor forget... | [
"10"
] | [
"An infectious rumor is spreading through the University of Pennsylvania. At the start of the first day, only three people know the rumor. However, throughout the day each person who knows the rumor tells it to two new people who do not know the rumor. Also, every evening, a third of the people who know the rumor f... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e99cf669f327fb1506cbab984afef594249a497b597ff49e5565541cf975fb5 | An infinite castle has rooms labeled \(1, 2, 3, \ldots\). If room \(n\) is on the same hall as rooms \(2n + 1\) and \(3n + 1\) for every \(n\), what is the maximum possible number of different halls in the castle? | [
"1"
] | [
"An infinite castle has rooms labeled \\(1, 2, 3, \\ldots\\). If room \\(n\\) is on the same hall as rooms \\(2n + 1\\) and \\(3n + 1\\) for every \\(n\\), what is the maximum possible number of different halls in the castle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-8013fcb5f8ef3651bc86b6fe775a75cbee83d75e9edc966863b63e2af1c25ca6 | An infinite checkerboard is divided by a horizontal line into upper and lower halves. A number of checkers are placed on the board below the line. A "move" consists of one checker jumping horizontally or vertically over a second checker, and removing the second checker. What is the minimum value of $n$ which will allow... | [
"16"
] | [
"An infinite checkerboard is divided by a horizontal line into upper and lower halves. A number of checkers are placed on the board below the line. A \"move\" consists of one checker jumping horizontally or vertically over a second checker, and removing the second checker. What is the minimum value of $n$ which wil... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-d79fa0d5328e2034fc99afd650d19b9654e54f4821f8819bad315f5c11d74636 | An infinite chessboard consists of $1 \times 1$ squares. A flea starts on a white square and makes jumps by $a$ to the right and $b$ upwards, where $a$, $b$, and $\frac{a}{b}$ are irrational. Let $N$ be the smallest positive integer such that after $N$ jumps, the flea lands on a black square. Given $a = \sqrt{2}$ and $... | [
"2"
] | [
"An infinite chessboard consists of $1 \\times 1$ squares. A flea starts on a white square and makes jumps by $a$ to the right and $b$ upwards, where $a$, $b$, and $\\frac{a}{b}$ are irrational. Let $N$ be the smallest positive integer such that after $N$ jumps, the flea lands on a black square. Given $a = \\sqrt{2... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1ef48b3ad5081f279b77e56bb3c2d0dc19db7505daca430ed87e09400c0f5a5 | An infinite chessboard consists of 1×1 squares. A flea starts on a white square and makes jumps by $a$ to the right and $b$ upwards, where $a$, $b$, and $\frac{a}{b}$ are irrational. Let $N$ be the smallest positive integer such that after $N$ jumps, the flea lands on a black square. If $a = \sqrt{2}$ and $b = \sqrt{3}... | [
"2"
] | [
"An infinite chessboard consists of 1×1 squares. A flea starts on a white square and makes jumps by $a$ to the right and $b$ upwards, where $a$, $b$, and $\\frac{a}{b}$ are irrational. Let $N$ be the smallest positive integer such that after $N$ jumps, the flea lands on a black square. If $a = \\sqrt{2}$ and $b = \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f8dcb3822a621268fcb377e9e12ed7aa6110b03a57b0e2e7d0521da624264a1 | An infinite decreasing geometric sequence has the first term equal to $1$ and each term is $k$ times greater than the sum of all the following terms. Find the value of $k$. | [
"\\( k > 0, k < -2 \\)"
] | [
"An infinite decreasing geometric sequence has the first term equal to $1$ and each term is $k$ times greater than the sum of all the following terms. Find the value of $k$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06f4c4eb3808c7873e1bae56a5b42a959364a5c0485f7109d9cc3a1c7d6e53b0 | An infinite family of concentric circles with radii 1, 2, 3, ... is drawn in the plane. A line segment of length 1 is randomly placed in the plane such that the angle it makes with the positive x-axis and the midpoint of the segment are uniformly distributed. What is the probability that the segment intersects one of t... | [
"\\dfrac{2}{\\pi}"
] | [
"An infinite family of concentric circles with radii 1, 2, 3, ... is drawn in the plane. A line segment of length 1 is randomly placed in the plane such that the angle it makes with the positive x-axis and the midpoint of the segment are uniformly distributed. What is the probability that the segment intersects one... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-494c26c1123f9d4fc241706c998c6894dabbd03f55622c259516081aef15981d | An infinite geometric sequence $\{a_n\}$ has a sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n}$. | [
"4"
] | [
"An infinite geometric sequence $\\{a_n\\}$ has a sum $\\sum_{n=0}^{\\infty} a_n = 3$. Compute the maximum possible value of the sum $\\sum_{n=0}^{\\infty} a_{3n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-69c0a93f25779c01733ce71fc42fd75a6710693151c73dcee76467ecbf0a14fe | An infinite geometric sequence \(a_1, a_2, a_3, \dots\) satisfies \(a_1 = 1\) and \[\dfrac{1}{a_1a_2} + \dfrac{1}{a_2a_3} + \dfrac{1}{a_3a_4} + \cdots = \dfrac{1}{2}.\] The sum of all possible values of \(a_2\) can be expressed as \(m + \sqrt{n}\), where \(m\) and \(n\) are integers and \(n\) is not a positive perfect ... | [
"102"
] | [
"An infinite geometric sequence \\(a_1, a_2, a_3, \\dots\\) satisfies \\(a_1 = 1\\) and \\[\\dfrac{1}{a_1a_2} + \\dfrac{1}{a_2a_3} + \\dfrac{1}{a_3a_4} + \\cdots = \\dfrac{1}{2}.\\] The sum of all possible values of \\(a_2\\) can be expressed as \\(m + \\sqrt{n}\\), where \\(m\\) and \\(n\\) are integers and \\(n\\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f2d18e81ade436a8c0cdcedf69901b9484b42b6a0bb38f7512c1cb7e0514deb | An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ | [
"6"
] | [
"An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-223a1afb0488b5c600305d7e98546f553c64a076f2890efff83d4dbeb504e78c | An infinite geometric series has a second term of $36$ and a fourth term of $4$. What is the sum of all the possible sums of this geometric series? | [
"81"
] | [
"An infinite geometric series has a second term of $36$ and a fourth term of $4$. What is the sum of all the possible sums of this geometric series?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2652febc8ff5a0aeb04d686463f4035646d3f3da52e827f1db674b5d34065389 | An infinite geometric series has common ratio $-1/5$ and sum $16.$ What is the first term of the series? | [
"\\frac{96}{5}"
] | [
"An infinite geometric series has common ratio $-1/5$ and sum $16.$ What is the first term of the series?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-683266da242778040f6531aa0b21df04263f77bcc7e32d5e11185307f82209cc | An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series? | [
"\\frac{105}{2}"
] | [
"An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8096ac05a624b3b6fea6d3919cb0a4d168b3af689a95b6750d68921ce0c30377 | An infinite geometric series has common ratio $\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence? | [
"\\frac{-100}{9}"
] | [
"An infinite geometric series has common ratio $\\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-14ccaedef5962712d3dfd006acc675550f8d6589495a682dce91a9ff8bcab65d | An infinite geometric series has first term $328$ and a sum of $2009$. What is its common ratio? | [
"\\frac{41}{49}"
] | [
"An infinite geometric series has first term $328$ and a sum of $2009$. What is its common ratio?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee012723db6e399983eeaf1807b68d8a8c30f84b5fb8eed9e958519f02d0054e | An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\frac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n.$ | [
"802"
] | [
"An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\\frac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n.$"
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-6f2c429440b4acf0fc656e758ad67a95df0e0918007545fb97219c487f9b6b36 | An infinite geometric series with initial term \( a \) and common ratio \( b \) sums to \( b - a \). What is the maximum possible value of \( a \)? | [
"\\( 3 - 2\\sqrt{2} \\)"
] | [
"An infinite geometric series with initial term \\( a \\) and common ratio \\( b \\) sums to \\( b - a \\). What is the maximum possible value of \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-5ac0bb01cf49c0fe22d23144361ae649e752f85b4a8b2c4dceea63fc8e95fbf0 | An infinite grid of squares with coordinates $x,y \in \mathbb{Z}^2$ is filled with $X$s and $O$s such that no three consecutive squares in a row, column, or diagonal have the same symbol. Given that $s(0,0) = X$, where $s(x,y)$ denotes the symbol at coordinate $(x,y)$, find $s(4,4)$. | [
"X"
] | [
"An infinite grid of squares with coordinates $x,y \\in \\mathbb{Z}^2$ is filled with $X$s and $O$s such that no three consecutive squares in a row, column, or diagonal have the same symbol. Given that $s(0,0) = X$, where $s(x,y)$ denotes the symbol at coordinate $(x,y)$, find $s(4,4)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-f7c22d99692dad4f439e5d939275b8a74b82efa3f9e09f20aa908ecff5427280 | An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red, and all other cells are white. Initially, we are in the red cell. In one move, we can move to an adjacent cell (sharing a side). Find the number of sequences of \( n \) moves such that no cell is visited more... | [
"4! / (4 - n)!"
] | [
"An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red, and all other cells are white. Initially, we are in the red cell. In one move, we can move to an adjacent cell (sharing a side). Find the number of sequences of \\( n \\) moves such that no cell is visit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d116cbe20116e565d30e0b84419acda3102cd592e02a2b6a416669ce6ec6abd6 | An infinite line of people, $\text{James}_1, \text{James}_2, \text{James}_3, \dots$, are standing in a line. On $\text{James}_i$'s turn, he can either: - Take half of the money $x$ in the "money pool" and give the other half to $\text{James}_{i-1}$, with a genie then giving $\text{James}_{i-1}$ an additional $\frac{x}{... | [
"\\text{Take the money}"
] | [
"An infinite line of people, $\\text{James}_1, \\text{James}_2, \\text{James}_3, \\dots$, are standing in a line. On $\\text{James}_i$'s turn, he can either:\n- Take half of the money $x$ in the \"money pool\" and give the other half to $\\text{James}_{i-1}$, with a genie then giving $\\text{James}_{i-1}$ an additi... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-4ad8416de6359bd563f388eedbd9f90ba823abe59e7d3cac6e7636d09f2f2179 | An infinite number of boxes are placed in a row and numbered $1-\infty$ from left to right. A man places balls in the boxes according to the following pattern: - In step 1, he places a ball in box 1. - In every subsequent step, he places a ball in the smallest numbered box that does not already contain a ball and then ... | [
"42"
] | [
"An infinite number of boxes are placed in a row and numbered $1-\\infty$ from left to right. A man places balls in the boxes according to the following pattern: \n- In step 1, he places a ball in box 1.\n- In every subsequent step, he places a ball in the smallest numbered box that does not already contain a ball ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a55f1c266ded8edc98c56113d136de70a2a466a709fcf0ee7d706d23b682040 | An infinite number of buckets, labeled $1$, $2$, $3$, $\ldots$, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket $k$ is $2^{-k}$. Given that all three balls land in the same bucket $B$ and that $B$ is even, find the... | [
"191"
] | [
"An infinite number of buckets, labeled $1$, $2$, $3$, $\\ldots$, lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket $k$ is $2^{-k}$. Given that all three balls land in the same bucket $B$ and that $B$ is even, fi... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-f1373f4df31ba5820bd4c389953a13d3072063e74105f4f23729a6de81e5017c | An infinite number of coins with probabilities of \( \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \ldots \) of coming up heads are all flipped. What is the probability that exactly \( 1 \) of them comes up heads? | [
"\\dfrac{3}{8}"
] | [
"An infinite number of coins with probabilities of \\( \\frac{1}{4}, \\frac{1}{9}, \\frac{1}{16}, \\ldots \\) of coming up heads are all flipped. What is the probability that exactly \\( 1 \\) of them comes up heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f73e1ab8d241dc6dd54b7f9e7330b1e13dfaefcf1bc28fd8dd78b0d3e653edcc | An infinite number of congruent spheres are packed in space hexagonally. To the nearest tenth, what percent of the space is in the interior of one of these spheres? Note: Hexagonal packing in three dimensions (also known as face-centered cubic packing) involves arranging spheres such that each layer of spheres is place... | [
"74.0%"
] | [
"An infinite number of congruent spheres are packed in space hexagonally. To the nearest tenth, what percent of the space is in the interior of one of these spheres?\n\nNote: Hexagonal packing in three dimensions (also known as face-centered cubic packing) involves arranging spheres such that each layer of spheres ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 8
} | 26 | 0 | 0 |
omr-ws-8c93cc9f14c4be933bf7518c5e215dd745bc32ba491a953fc73decbcae43bc39 | An infinite number of parallel lines, spaced one metre apart, cover an infinite floor. If a rod of negligible thickness and 2 metres long is thrown up in the air, what is the probability that the rod lands and touches one of the lines? | [
"\\dfrac{2}{3} + \\dfrac{4 - 2\\sqrt{3}}{\\pi}"
] | [
"An infinite number of parallel lines, spaced one metre apart, cover an infinite floor. If a rod of negligible thickness and 2 metres long is thrown up in the air, what is the probability that the rod lands and touches one of the lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-e35645d8cac7fc8e6cfbef12592f36cb22bad59d43269dad25bd51fc01d6e24f | An infinite number of participants are registered under the numbers $1, 2, \ldots$. Each participant with number $n$ has at least $n$ friends among the remaining participants (friendship is mutual). There is a hotel with an infinite number of double rooms. Let $R(n)$ be the minimum number of rooms required to accommoda... | [
"5"
] | [
"An infinite number of participants are registered under the numbers $1, 2, \\ldots$. Each participant with number $n$ has at least $n$ friends among the remaining participants (friendship is mutual). There is a hotel with an infinite number of double rooms. Let $R(n)$ be the minimum number of rooms required to acc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2f17bf378ee025de8f3f8e5285ef2d8499858e84b8ea006ece5ba28c6f650f7 | An infinite number of people are lined up, each given either a black hat or a white hat. Each person can see all the hats in front of them but not their own or those behind them. They can discuss a strategy beforehand. Find a strategy that guarantees only finitely many wrong guesses when each person, starting from the ... | [
"1"
] | [
"An infinite number of people are lined up, each given either a black hat or a white hat. Each person can see all the hats in front of them but not their own or those behind them. They can discuss a strategy beforehand. Find a strategy that guarantees only finitely many wrong guesses when each person, starting from... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-5b6335f04a5d37822c1a741a0606a928396309a89f2a08fb39d06e338ba8d23e | An infinite number of squares on an infinitely square grid paper are painted red. Determine the minimum value of $k$ such that you can draw a number of squares on the paper, with sides along the grid lines, satisfying: (1) no square in the grid belongs to more than one square (an edge, on the other hand, may belong to ... | [
"4"
] | [
"An infinite number of squares on an infinitely square grid paper are painted red. Determine the minimum value of $k$ such that you can draw a number of squares on the paper, with sides along the grid lines, satisfying:\n(1) no square in the grid belongs to more than one square (an edge, on the other hand, may belo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-795b499efdd0468ff3982aa9b1a37821cdc2d7b116193dd40acbf501b31aca3e | An infinite number of wise men stand in line such that the first sees everyone else, the second sees everyone starting with the third, and so on. Each wise man is given a white or black cap. They must simultaneously name the color of their own cap. Only those who guess correctly will go to lunch. The wise men can agree... | [
"1"
] | [
"An infinite number of wise men stand in line such that the first sees everyone else, the second sees everyone starting with the third, and so on. Each wise man is given a white or black cap. They must simultaneously name the color of their own cap. Only those who guess correctly will go to lunch. The wise men can ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-dbac9c49aa75eebed671d09c5a05709a202916207f2afba6e09ec518ab42f3d9 | An infinite plane is completely tessellated by congruent right triangles with integer length sides and hypotenuse $c$. A coin with radius $1$ is tossed randomly onto the plane. The probability that the coin lands on at least one line segment is $\tfrac{23}{144}$. Find the remainder when the sum of all possible values o... | [
"35"
] | [
"An infinite plane is completely tessellated by congruent right triangles with integer length sides and hypotenuse $c$. A coin with radius $1$ is tossed randomly onto the plane. The probability that the coin lands on at least one line segment is $\\tfrac{23}{144}$. Find the remainder when the sum of all possible va... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e972c3e2c98e45969959dbdb10e2d78ae25cd1bfc00b83587b23dcd848e17038 | An infinite plane is tiled with hexagons of side length 2. If a coin of radius 1 is thrown such that it lands on any point in the plane with uniform probability, what is the probability that it is completely inside a hexagon? Express your answer as a common fraction in simplest form. | [
"\\dfrac{4 - 2\\sqrt{3}}{3}"
] | [
"An infinite plane is tiled with hexagons of side length 2. If a coin of radius 1 is thrown such that it lands on any point in the plane with uniform probability, what is the probability that it is completely inside a hexagon? Express your answer as a common fraction in simplest form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-ce83928f324e669c98e8f6b493c788b0803118f360de9f6863b61452398410a3 | An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping? | [
"4.5 cm²"
] | [
"An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-e6d6bdd248fcac7f5d71516702e15a213ca1ccbd22684c72c6aeacae8add28e7 | An infinite sequence \( x_1, x_2, \ldots \) of positive integers satisfies \[ x_{n+2} = \gcd(x_{n+1}, x_n) + 2006 \] for each positive integer \( n \). Does there exist such a sequence which contains exactly \( 10^{2006} \) distinct numbers? | [
"No"
] | [
"An infinite sequence \\( x_1, x_2, \\ldots \\) of positive integers satisfies \\[ x_{n+2} = \\gcd(x_{n+1}, x_n) + 2006 \\] for each positive integer \\( n \\). Does there exist such a sequence which contains exactly \\( 10^{2006} \\) distinct numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-67222728114df31644dcfb8ace3733e2739a76f926e4a9e9a3d9da413f775c68 | An infinite sequence \((x_n)_{n \geq 1}\) is given by its first term \(x_1 = \frac{5}{2}\), and the relation \(x_{n+1} = \frac{x_n + 1}{\lfloor x_n \rfloor}\) for all positive integers \(n\). Calculate the value of \(x_{10}\). | [
"\\dfrac{127}{64}"
] | [
"An infinite sequence \\((x_n)_{n \\geq 1}\\) is given by its first term \\(x_1 = \\frac{5}{2}\\), and the relation \\(x_{n+1} = \\frac{x_n + 1}{\\lfloor x_n \\rfloor}\\) for all positive integers \\(n\\). Calculate the value of \\(x_{10}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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