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-e36bba856972b3f66ded22020d39f2c1073f2faf5d83e26e1594e38505967edc | Among all triangles \(ABC\) with \(BC = a\) and \(AC = b\) for which \(a + b = 2(R + r)\), find the triangles for which \(\angle C\) has the maximum value. | [
"90 degrees"
] | [
"Among all triangles \\(ABC\\) with \\(BC = a\\) and \\(AC = b\\) for which \\(a + b = 2(R + r)\\), find the triangles for which \\(\\angle C\\) has the maximum value."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-fa8a66fe806f778f0aa7555cd86f52806e7c25cb627b572219f8fb377cba3533 | Among all triangles \(ABC\) with vertex \(A\) fixed and vertices \(B\) and \(C\) on the sides of a fixed acute angle \(\angle ros\), what is the triangle that has the minimum perimeter? Note: Point \(A\) is between rays \(r\) and \(s\). | [
"ABC"
] | [
"Among all triangles \\(ABC\\) with vertex \\(A\\) fixed and vertices \\(B\\) and \\(C\\) on the sides of a fixed acute angle \\(\\angle ros\\), what is the triangle that has the minimum perimeter? Note: Point \\(A\\) is between rays \\(r\\) and \\(s\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-6d88a31133df0bc14f0010a53c244206c07c0ea2835d021baa19494c99151a4c | Among all triangles for which the median $m_c$ equals the side length $c$, find the triangle with the maximal perimeter. | [
"\\( (\\sqrt{5} + 1)c \\)"
] | [
"Among all triangles for which the median $m_c$ equals the side length $c$, find the triangle with the maximal perimeter."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-bdac55f61edd60634348a3b1d19f5198672009184ac6f4aa85edcf424624dc52 | Among all triangles formed by three vertices of a regular 12-gon inscribed in a circle with radius 1, calculate the perimeter of the triangle with the longest perimeter. Express your answer in simplest radical form. | [
"3\\sqrt{3}"
] | [
"Among all triangles formed by three vertices of a regular 12-gon inscribed in a circle with radius 1, calculate the perimeter of the triangle with the longest perimeter. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f88fad23689f3e4ee502d0578aa54ec2d1bab9f0ee87454e48b429c0d1016a2 | Among all triangles having a fixed angle \( A \) and an inscribed circle of fixed radius \( r \), determine which triangle has the least perimeter. | [
"Isosceles triangle with apex angle \\( A \\)."
] | [
"Among all triangles having a fixed angle \\( A \\) and an inscribed circle of fixed radius \\( r \\), determine which triangle has the least perimeter."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-e72201bc3a51bdcf8928c5cac4a9e051fdf17dd26140245f23273e684ba3d879 | Among all triangles that contain a square of side length 1, which triangles have the minimum area? | [
"2"
] | [
"Among all triangles that contain a square of side length 1, which triangles have the minimum area?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07e0cff3ffefb7a212466f09f472b52d1b730ce584199d7eb9e388170e439a78 | Among all triangles whose vertices are at distances 3, 5, and 7 respectively from a given point P, find the area of the triangle that has P as the orthocenter. Express your answer in simplest radical form. | [
"no_valid_answer_found"
] | [
"Among all triangles whose vertices are at distances 3, 5, and 7 respectively from a given point P, find the area of the triangle that has P as the orthocenter. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-597d47e8ead5ff12b6fc57b384e8af1bb5d398d6a4e88243fd6f3dfe318fcbe1 | Among all triangles whose vertices are at distances 3, 5, and 7 respectively from a given point P, find the area of the triangle with the largest possible area where P is the orthocenter. Express your answer in simplest radical form. | [
"no_valid_answer_found"
] | [
"Among all triangles whose vertices are at distances 3, 5, and 7 respectively from a given point P, find the area of the triangle with the largest possible area where P is the orthocenter. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5db1e3c452e28191e82398c2635ab631fde0e267eb0dc7d5a6e5890084e4766a | Among all triangles with a fixed perimeter, find the triangle that generates the maximum volume of the solid of revolution when rotated about one of its sides. | [
"An isosceles triangle with legs of length \\( \\frac{3P}{8} \\) and base of length \\( \\frac{P}{4} \\)."
] | [
"Among all triangles with a fixed perimeter, find the triangle that generates the maximum volume of the solid of revolution when rotated about one of its sides."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-f8cf0d167fb212e414f5f491652d9254d358a93d989a2cf84c9d61799ea25558 | Among all triangles with a fixed side \( l \) and a fixed area \( S \), determine for which triangles the product of the altitudes is maximum. | [
"\\text{Isosceles triangle}"
] | [
"Among all triangles with a fixed side \\( l \\) and a fixed area \\( S \\), determine for which triangles the product of the altitudes is maximum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-e3386f1ea64b4b31a0f6414f981cf710b9404945643b176bc865e1b4f45e7ab0 | Among all triangles with a fixed side length $l$ and a fixed area $S$, determine the triangles for which the product of the altitudes is maximized. | [
"Right triangles with the right angle opposite the fixed side \\(l\\)."
] | [
"Among all triangles with a fixed side length $l$ and a fixed area $S$, determine the triangles for which the product of the altitudes is maximized."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-8d7c12f0d07a10b532bd09632dd19874f055aadf7f402921357eb41d476585bb | Among all triangles with a given perimeter, find the one with the maximal radius of its incircle. | [
"equilateral triangle"
] | [
"Among all triangles with a given perimeter, find the one with the maximal radius of its incircle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-58ce15d9124ccb635d9a14d3e319de10489d4c8467e026db2965a76b9797fab8 | Among all triangles with a perimeter of 20 and a base of 8, find the area of the isosceles triangle that has the greatest area. | [
"8\\sqrt{5}"
] | [
"Among all triangles with a perimeter of 20 and a base of 8, find the area of the isosceles triangle that has the greatest area."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5cede370c314e2f4a414adad6f67b33c62680d84790db3e11bb90b7f1cc3773 | Among all triangles with a perimeter of 30 units, find the area of the triangle with the maximum possible area. Express your answer in simplest radical form. | [
"25\\sqrt{3}"
] | [
"Among all triangles with a perimeter of 30 units, find the area of the triangle with the maximum possible area. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a465d24f0b31190a8b13601707e082073915cc30426165eba8de60337c55c8f4 | Among all triangles with a perimeter of 30, find the maximum possible area. Express your answer in simplest radical form. | [
"25\\sqrt{3}"
] | [
"Among all triangles with a perimeter of 30, find the maximum possible area. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1caeb87f8957e4a3498daef5ea8b2093591a2dfec6848fd022d45a98f5194459 | Among all triangles with an inradius of $1$, find the perimeter of the triangle with the smallest possible perimeter. Express your answer in simplest radical form. | [
"6\\sqrt{3}"
] | [
"Among all triangles with an inradius of $1$, find the perimeter of the triangle with the smallest possible perimeter. Express your answer in simplest radical form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-436474779571df18661871b8456f5965291653fcaafd37220b24568b30041fd6 | Among all triangles with base $b = 10$ and area $A = 25$, calculate the perimeter of the isosceles triangle and compare it to the perimeter of a scalene triangle with the same base and area. Find the difference between these two perimeters, rounding to the nearest integer. | [
"2"
] | [
"Among all triangles with base $b = 10$ and area $A = 25$, calculate the perimeter of the isosceles triangle and compare it to the perimeter of a scalene triangle with the same base and area. Find the difference between these two perimeters, rounding to the nearest integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-163081fb68ef3e2844dedf644c9cd01f4d0aad8950c5ba9a0d67e8f0836d63e0 | Among all triangles with base $b$ and area $A$, find the perimeter of the isosceles triangle and express it in terms of $b$ and $A$. Simplify your answer as much as possible. | [
"\\dfrac{b^2 + \\sqrt{b^4 + 16A^2}}{b}"
] | [
"Among all triangles with base $b$ and area $A$, find the perimeter of the isosceles triangle and express it in terms of $b$ and $A$. Simplify your answer as much as possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d87f9a0ce97995f8650a8ea0bc69d834f56fef917393c9bb46d072f13b523edf | Among all triangles with integer side lengths and a perimeter of 20, what is the area of the triangle with the largest area? | [
"\\(6\\sqrt{10}\\)"
] | [
"Among all triangles with integer side lengths and a perimeter of 20, what is the area of the triangle with the largest area?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-38dfcd86253d8d5642643cf6d1492b89eb0eac9a00310c0672ba9c3dbfeeb46b | Among all triangles with side lengths \(a^k\), \(a^k\), and \(b^k\) (where \(k \in \mathbb{R}\) and \(0 < a < b < 2a\)), determine the value of \(k\) that maximizes the area of the triangle and find this maximum area in terms of \(a\) and \(b\). | [
" \\dfrac{ \\sqrt{ \\ln\\left( \\dfrac{b}{a} \\right) \\cdot \\ln(ab) } }{ 2 \\ln b } \\cdot a^{2k} "
] | [
"Among all triangles with side lengths \\(a^k\\), \\(a^k\\), and \\(b^k\\) (where \\(k \\in \\mathbb{R}\\) and \\(0 < a < b < 2a\\)), determine the value of \\(k\\) that maximizes the area of the triangle and find this maximum area in terms of \\(a\\) and \\(b\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-267675430d4877dc990eaa2113d389f7931b44c6bf8be3582923e7e3d129dbac | Among all triples $(a,b,c)$ of natural numbers satisfying \[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\] determine the one with the maximal value of $a$. | [
"(40, 80, 2)"
] | [
"Among all triples $(a,b,c)$ of natural numbers satisfying\n\\[(a+14\\sqrt{3})(b-14c\\sqrt{3})=2024,\\]\ndetermine the one with the maximal value of $a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc57bd7a5dd618fef69fae197fba3e5225deb2749bd55a206a6594b584665143 | Among any $(n+1)$ numbers picked at random from the $2n$ integers $1, 2, 3, \ldots, 2n$, let $f(n)$ be the minimum number of pairs $(a, b)$ such that $a$ is divisible by $b$ or $b$ is divisible by $a$. Find $f(10)$. | [
"1"
] | [
"Among any $(n+1)$ numbers picked at random from the $2n$ integers $1, 2, 3, \\ldots, 2n$, let $f(n)$ be the minimum number of pairs $(a, b)$ such that $a$ is divisible by $b$ or $b$ is divisible by $a$. Find $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d1c1956d894caf2cf900908486a4fa3165b66cb3fcd2dbc1126d8818c0f9700 | Among any $79$ consecutive natural numbers, there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which this statement fails. | [
"10^{13} - 78"
] | [
"Among any $79$ consecutive natural numbers, there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which this statement fails."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-cb50982081ea3cbe3a090f03e75911a441b10189f0167eea5e16b7ed5aaa6f5c | Among any $k+m-1$ arbitrary integers, where $m \geq k \geq 2$ and $m$ is divisible by $k$, find the minimum number of subsets of size $m$ such that the sum of the integers in each subset is divisible by $k$. Let this minimum number be $N(k, m)$. Calculate $N(3, 6)$. | [
"1"
] | [
"Among any $k+m-1$ arbitrary integers, where $m \\geq k \\geq 2$ and $m$ is divisible by $k$, find the minimum number of subsets of size $m$ such that the sum of the integers in each subset is divisible by $k$. Let this minimum number be $N(k, m)$. Calculate $N(3, 6)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-68497b3e72ed409f79fb837585f47e5ac8f1cabf7fb2fc59a69b7ff9befc98dd | Among any $n \ge 3$ segments, where the longest segment is no more than $n$ times longer than the shortest one, let $f(n)$ be the minimum number of segments that can be chosen such that they can form an acute triangle. Find the value of $f(5)$. | [
""
] | [
"Among any $n \\ge 3$ segments, where the longest segment is no more than $n$ times longer than the shortest one, let $f(n)$ be the minimum number of segments that can be chosen such that they can form an acute triangle. Find the value of $f(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-1014205dc152dc580bc0839b6eab5dfaf0ecb548fec065922741887211295823 | Among any $n$ points in the plane, there exist $1000$ points such that the pairwise distances between these $1000$ points are all distinct. Find the smallest value of $n$ for which this statement is guaranteed to be true. | [
"499501"
] | [
"Among any $n$ points in the plane, there exist $1000$ points such that the pairwise distances between these $1000$ points are all distinct. Find the smallest value of $n$ for which this statement is guaranteed to be true."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7f57aa1942c50196ee7179b59860ec93580e45eadf4663059d367732c1387889 | Among any $n+1$ whole numbers, let $f(n)$ be the minimum number of pairs whose differences are divisible by $n$. Calculate $f(10)$. | [
"1"
] | [
"Among any $n+1$ whole numbers, let $f(n)$ be the minimum number of pairs whose differences are divisible by $n$. Calculate $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94a9409e1494c70d0900d1e719715ecba848ab30afad5e242ca882245044a860 | Among any 10 distinct positive integers, let $S$ be the set of all possible values of $w$ such that there exist $x < y < z < w$ where all of the numbers below are integers or none of the numbers below is an integer. $$\frac{y}{x}, \frac{z}{x}, \frac{z}{y}, \frac{w}{x}, \frac{w}{y}, \frac{w}{z}$$ Find the sum of all ele... | [
""
] | [
"Among any 10 distinct positive integers, let $S$ be the set of all possible values of $w$ such that there exist $x < y < z < w$ where all of the numbers below are integers or none of the numbers below is an integer.\n$$\\frac{y}{x}, \\frac{z}{x}, \\frac{z}{y}, \\frac{w}{x}, \\frac{w}{y}, \\frac{w}{z}$$\nFind the s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-94f746060a68bbff501303a41ed5402824bb83dd55b272fc33a561dab6dba973 | Among any 10 real numbers, let $d$ be the smallest possible value of the difference between two different real numbers such that this difference is almost an integer. Calculate the maximum possible value of $d$. | [
"\\dfrac{1}{10}"
] | [
"Among any 10 real numbers, let $d$ be the smallest possible value of the difference between two different real numbers such that this difference is almost an integer. Calculate the maximum possible value of $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-49a15b05c83908dea9a31595f888714c9dd983874df2ab5fbb7e7e9fc388171e | Among any 12 consecutive positive integers, let $S$ be the smallest integer that is smaller than the sum of its proper divisors. For the sequence starting at $1000$, calculate the value of $S$. | [
"1000"
] | [
"Among any 12 consecutive positive integers, let $S$ be the smallest integer that is smaller than the sum of its proper divisors. For the sequence starting at $1000$, calculate the value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5c52bf0a7991691b027a3913b95d57d9c5a497dab7a109a1c20aee7f114f16c | Among any 12 consecutive positive integers, let \( S \) be the smallest integer that is smaller than the sum of its proper divisors. If the sequence starts from \( k \), find the value of \( S \) when \( k = 2023 \). | [
"2024"
] | [
"Among any 12 consecutive positive integers, let \\( S \\) be the smallest integer that is smaller than the sum of its proper divisors. If the sequence starts from \\( k \\), find the value of \\( S \\) when \\( k = 2023 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1df42582b24386f985b54cf3eb43dbc5122b668ffedcf5ef4e35cd0673b5d730 | Among any 12 consecutive positive integers, let \( S \) be the sum of the two integers which are the smallest and are also smaller than or equal to the sum of their proper divisors. Find the smallest possible value of \( S \) for the sequence starting from 1000. | [
"2002"
] | [
"Among any 12 consecutive positive integers, let \\( S \\) be the sum of the two integers which are the smallest and are also smaller than or equal to the sum of their proper divisors. Find the smallest possible value of \\( S \\) for the sequence starting from 1000."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68e6bad0250d3643bdb81ec4a200d7648ef53711a05f73176db97bd6bb52aa32 | Among any 13 points in the plane with integer coordinates, such that no three points are collinear, let $T$ be the set of all triangles formed by choosing three points from these 13 points. Determine the minimum number of triangles in $T$ whose centroid also has integer coordinates. | [
"12"
] | [
"Among any 13 points in the plane with integer coordinates, such that no three points are collinear, let $T$ be the set of all triangles formed by choosing three points from these 13 points. Determine the minimum number of triangles in $T$ whose centroid also has integer coordinates."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e38b64323e6f99a1b79a28c8c2a747c641c9d4c11b560af93086575152f8cae7 | Among any 14 consecutive positive integers, let $S$ be the set of all possible groups of 6 integers that are pairwise relatively prime. Calculate the sum of the smallest integers in each group within $S$. | [
"28"
] | [
"Among any 14 consecutive positive integers, let $S$ be the set of all possible groups of 6 integers that are pairwise relatively prime. Calculate the sum of the smallest integers in each group within $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44a7feb19a1a346eb751cd10a4b76cc160eaa0e72c96da0dd2c2d4b55055401e | Among any 16 consecutive natural numbers, let $N$ be the smallest number that is relatively prime to the other 15 numbers. If the sequence starts with 1000, find the value of $N$. | [
"1003"
] | [
"Among any 16 consecutive natural numbers, let $N$ be the smallest number that is relatively prime to the other 15 numbers. If the sequence starts with 1000, find the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f300fbf8ad04b8ada2fbf8762d67537858c20c97e87361bdd19862e53d945c9 | Among any 16 consecutive natural numbers, let $a$ and $b$ be two coprime numbers with the smallest possible sum. Calculate the maximum possible value of $a + b$. | [
"25"
] | [
"Among any 16 consecutive natural numbers, let $a$ and $b$ be two coprime numbers with the smallest possible sum. Calculate the maximum possible value of $a + b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a8327491067603c50734f82c84116a32bda682681e854228fcdacc6b10cd9d2f | Among any 16 distinct positive integers not exceeding 100, find the minimum possible value of \(a + b - c - d\) where \(a\), \(b\), \(c\), and \(d\) are four different integers such that \(a + b = c + d\). | [
"0"
] | [
"Among any 16 distinct positive integers not exceeding 100, find the minimum possible value of \\(a + b - c - d\\) where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are four different integers such that \\(a + b = c + d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e88b9331329e15024f1571e9156da31b19b6c43bffc4c503e549d559adbbc19 | Among any 17 integers, let $S$ be the sum of some 9 of them that is divisible by 9. If multiple such sums exist, choose the smallest possible positive value of $S$. Find the maximum possible value of the remainder when $S$ is divided by 81. | [
"72"
] | [
"Among any 17 integers, let $S$ be the sum of some 9 of them that is divisible by 9. If multiple such sums exist, choose the smallest possible positive value of $S$. Find the maximum possible value of the remainder when $S$ is divided by 81."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-333eac6dc91ce8055492e758ffd9be3d8246d1a40e02583938ff62ac5823e333 | Among any 18 consecutive three-digit numbers, find the maximum possible value of the smallest number that is divisible by the sum of its digits. | [
"990"
] | [
"Among any 18 consecutive three-digit numbers, find the maximum possible value of the smallest number that is divisible by the sum of its digits."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0dce12ee3bdd71fa7778165dee4abcb2fdaad05294022abeb1ef347df1afcf31 | Among any 18 consecutive three-digit numbers, let $S$ be the sum of the digits of the smallest number in the sequence that is divisible by the sum of its digits. Find the maximum possible value of $S$. | [
"18",
"24"
] | [
"Among any 18 consecutive three-digit numbers, let $S$ be the sum of the digits of the smallest number in the sequence that is divisible by the sum of its digits. Find the maximum possible value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-036ea19676b8f35b5de83e02ebd95debf299e92234683c866f34e108ee58de44 | Among any 21 distinct numbers chosen from the set $\{1, 2, 3, \ldots, 2046\}$, let $S$ be the set of all possible values of $a$ such that there exist distinct $b, c$ from the chosen numbers satisfying $bc < 2a^2 < 4bc$. Find the sum of all elements in $S$. | [
"2094080"
] | [
"Among any 21 distinct numbers chosen from the set $\\{1, 2, 3, \\ldots, 2046\\}$, let $S$ be the set of all possible values of $a$ such that there exist distinct $b, c$ from the chosen numbers satisfying $bc < 2a^2 < 4bc$. Find the sum of all elements in $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-371adef99fc54faab3948328d9d1194fca2497fb68094dbe59942b3be786fe73 | Among any 21 distinct numbers chosen from the set $\{1,2,3,\ldots,2046\}$, find the minimum possible value of $a$ such that there exist three distinct numbers $a, b, c$ satisfying \[bc < 2a^2 < 4bc.\] | [
"32"
] | [
"Among any 21 distinct numbers chosen from the set $\\{1,2,3,\\ldots,2046\\}$, find the minimum possible value of $a$ such that there exist three distinct numbers $a, b, c$ satisfying\n\\[bc < 2a^2 < 4bc.\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-56f62b0648596c135183a5b1b7b02f9d8ac4998c49e4fe33d13a89568c4dd1b1 | Among any 2n-1 integers, where n is a natural number, find the minimum value of n such that there exist n integers whose sum is divisible by n, given that the integers are consecutive and start from 1. | [
"1"
] | [
"Among any 2n-1 integers, where n is a natural number, find the minimum value of n such that there exist n integers whose sum is divisible by n, given that the integers are consecutive and start from 1."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3bc8df401175b5e89daee82644cece713c091e5e8384c662e09cd41691f5a093 | Among any 2p-1 integers, where p is a prime number, find the minimum number of distinct subsets, each containing exactly p integers, such that the sum of the integers in each subset is divisible by p. | [
"1"
] | [
"Among any 2p-1 integers, where p is a prime number, find the minimum number of distinct subsets, each containing exactly p integers, such that the sum of the integers in each subset is divisible by p."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4705e58548313cd7fc152e7df204670993abf42837154ae2f07e5eca91f898f | Among any 31 positive integers, let $S$ be the sum of a subset of 16 numbers such that $S$ is divisible by 16. Find the minimum possible value of the largest integer in this subset of 16 numbers, assuming the 31 positive integers are distinct and the smallest integer is 1. | [
"17"
] | [
"Among any 31 positive integers, let $S$ be the sum of a subset of 16 numbers such that $S$ is divisible by 16. Find the minimum possible value of the largest integer in this subset of 16 numbers, assuming the 31 positive integers are distinct and the smallest integer is 1."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-91582807d176e3d6e07c3bf85aa20cb164165d8f78a6eb4bd3c0afddb0c59c7a | Among any 6 integers, find the maximum number of pairs whose sum or difference is not divisible by 9. | [
"14"
] | [
"Among any 6 integers, find the maximum number of pairs whose sum or difference is not divisible by 9."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-f5bcafad1e149e304f369229344607b72ef44b1f5abd281f54e77a15c8b9b5e4 | Among any 6 points in the plane, let \(d\) be the maximum distance between any two points. If these points are placed within a circle of radius 1, calculate the minimum possible value of \(d\). | [
""
] | [
"Among any 6 points in the plane, let \\(d\\) be the maximum distance between any two points. If these points are placed within a circle of radius 1, calculate the minimum possible value of \\(d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-685ecf0c3a93b44da8a97e6eb510bbf61cdc057b61c41662ee52aa91e0e1a7df | Among any 7 arbitrary integers, let $S$ be the sum of the 4 integers that can be chosen such that their sum is divisible by 4. Find the remainder when the smallest possible value of $S$ is divided by 4. | [
"0"
] | [
"Among any 7 arbitrary integers, let $S$ be the sum of the 4 integers that can be chosen such that their sum is divisible by 4. Find the remainder when the smallest possible value of $S$ is divided by 4."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae02cd0a459bbdec88d84b58d06aea0c3dd95205a464b8e28ac2e55b56d14436 | Among any 7 irrational numbers, find the maximum number of irrational numbers such that the sum of any 2 of these numbers is still irrational. If there are multiple such subsets with the same maximum size, calculate the sum of the sizes of all such subsets. | [
"4"
] | [
"Among any 7 irrational numbers, find the maximum number of irrational numbers such that the sum of any 2 of these numbers is still irrational. If there are multiple such subsets with the same maximum size, calculate the sum of the sizes of all such subsets."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-29a95b999bcd43036aad3db9318aef9628040840b5d52ac6af680e26a6d83cab | Among any \( 2n-1 \) integers, determine the minimum number of subsets of size \( n \) whose sum is divisible by \( n \). Let this minimum number be \( S(n) \). Calculate \( S(5) \). | [
"1"
] | [
"Among any \\( 2n-1 \\) integers, determine the minimum number of subsets of size \\( n \\) whose sum is divisible by \\( n \\). Let this minimum number be \\( S(n) \\). Calculate \\( S(5) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e96673a958211553d03a61453c75508000e6d29b86ec91b97addf620b88bf14 | Among any \( k \) consecutive positive integers, let \( f(k) \) be the number of integers that are smaller than or equal to the sum of their proper divisors. Find the minimum possible value of \( f(36) \). | [
"8"
] | [
"Among any \\( k \\) consecutive positive integers, let \\( f(k) \\) be the number of integers that are smaller than or equal to the sum of their proper divisors. Find the minimum possible value of \\( f(36) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-cabff8cb822f4edcd626696c350e7a6702b0d9cad7a14d521263bd00366de53e | Among any finite number of regular polygons with the same perimeter of 100 units, let $A_n$ represent the area of the regular polygon with $n$ sides. Find the value of $A_{12} - A_6$. | [
"\\dfrac{625}{3} \\left(2 - \\sqrt{3}\\right)"
] | [
"Among any finite number of regular polygons with the same perimeter of 100 units, let $A_n$ represent the area of the regular polygon with $n$ sides. Find the value of $A_{12} - A_6$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0bde8349a235f7b50d88adae144aa46a6657a4081bc44c979179a573bc9b507 | Among any five lattice points chosen in the plane lattice, let $P_1, P_2, P_3, P_4, P_5$ be these points. Define $S$ as the set of midpoints of all possible segments joining these points. Calculate the minimum number of lattice points that $S$ can contain. | [
"1"
] | [
"Among any five lattice points chosen in the plane lattice, let $P_1, P_2, P_3, P_4, P_5$ be these points. Define $S$ as the set of midpoints of all possible segments joining these points. Calculate the minimum number of lattice points that $S$ can contain."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f4dd556fa9e72f55fee637a04fee0ff176a676a11814d246f57d8637a416805 | Among any seven positive integers that do not exceed \( n \), we must guarantee three integers \( a, b, c \) such that \( a + 1 \le b + c \le 4a - 1 \). What is the maximum value of \( n \)? | [
"14"
] | [
"Among any seven positive integers that do not exceed \\( n \\), we must guarantee three integers \\( a, b, c \\) such that \\( a + 1 \\le b + c \\le 4a - 1 \\). What is the maximum value of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-d01bea8b6e3668b8e6d8df0c6ab7d335d34e2ba906de5cf6159d41c23754ebb0 | Among any six people, let $x$ be the number of groups of three people where each person knows the other two, and let $y$ be the number of groups of three people where each person does not know the other two. Calculate the minimum possible value of $x + y$. | [
"2"
] | [
"Among any six people, let $x$ be the number of groups of three people where each person knows the other two, and let $y$ be the number of groups of three people where each person does not know the other two. Calculate the minimum possible value of $x + y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72a8db118f79892f85fe38d1c9a0b3e27ac10b3603d21ade0b76eede00ee62bb | Among any six persons, let $A$, $B$, and $C$ represent three who know each other, and $X$, $Y$, and $Z$ represent three who are pairwise strangers. If $N$ is the number of ways to choose such a group of six persons such that either $A$, $B$, and $C$ exist or $X$, $Y$, and $Z$ exist, calculate the remainder when $N$ is ... | [
"768"
] | [
"Among any six persons, let $A$, $B$, and $C$ represent three who know each other, and $X$, $Y$, and $Z$ represent three who are pairwise strangers. If $N$ is the number of ways to choose such a group of six persons such that either $A$, $B$, and $C$ exist or $X$, $Y$, and $Z$ exist, calculate the remainder when $N... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be31191cce9ee8a249892cac6a876e5f2b56a36c3364b2411b0970b35dabdb75 | Among any three real numbers $a$, $b$, and $c$, no two of which add up to $1$, let $x$ and $y$ be two numbers such that $\frac{xy}{x+y-1}$ is not in the interval $(0,1)$. If $a = 2$, $b = 3$, and $c = 5$, calculate the value of $\frac{xy}{x+y-1}$ for the pair $(x,y)$ that satisfies the given condition. | [
"\\dfrac{3}{2}"
] | [
"Among any three real numbers $a$, $b$, and $c$, no two of which add up to $1$, let $x$ and $y$ be two numbers such that $\\frac{xy}{x+y-1}$ is not in the interval $(0,1)$. If $a = 2$, $b = 3$, and $c = 5$, calculate the value of $\\frac{xy}{x+y-1}$ for the pair $(x,y)$ that satisfies the given condition."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-83cc4cb91d1c37b4f2cc32a62e30006d5173d6bf639d81bb5786cbac456589aa | Among eight integers with three digits each, let $\overline{a_1a_2a_3}$ and $\overline{b_1b_2b_3}$ be two such integers that form a concatenated number $a_1a_2a_3b_1b_2b_3$ divisible by 7. Find the smallest possible value of $a_1a_2a_3 + b_1b_2b_3$ if $a_1a_2a_3 < b_1b_2b_3$. | [
"207"
] | [
"Among eight integers with three digits each, let $\\overline{a_1a_2a_3}$ and $\\overline{b_1b_2b_3}$ be two such integers that form a concatenated number $a_1a_2a_3b_1b_2b_3$ divisible by 7. Find the smallest possible value of $a_1a_2a_3 + b_1b_2b_3$ if $a_1a_2a_3 < b_1b_2b_3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9d72d7fdc098cacff9f20cf9118ed22e50cf24e11399d3a13b494ba31b4d9910 | Among every subset of five dwarves, some four must lie on a circle. Find the minimum value of the maximum number of dwarves who lie on a single circle for an arrangement of $N$ dwarves. | [
"N - 1"
] | [
"Among every subset of five dwarves, some four must lie on a circle. Find the minimum value of the maximum number of dwarves who lie on a single circle for an arrangement of $N$ dwarves."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-17ad2682c2d9121853ed2a28d1cf62793bd8956dc33a05bfb45a7623dc6adb30 | Among every subset of five dwarves, some four must lie on a circle. Find the minimum value of the maximum number of dwarves who lie on a single circle for an arrangement of ten dwarves. | [
"9"
] | [
"Among every subset of five dwarves, some four must lie on a circle. Find the minimum value of the maximum number of dwarves who lie on a single circle for an arrangement of ten dwarves."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-f7e02e01419cf84699782d6b14eed041f7fdd2d19f42beea55a54eb0b28bed4b | Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils, and $216$ erasers were distributed so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. How many notebooks, pencils, and eraser... | [
"6 notebooks, 8 pencils, and 9 erasers."
] | [
"Among families in a neighborhood in the city of Chimoio, a total of $144$ notebooks, $192$ pencils, and $216$ erasers were distributed so that the largest possible number of families was covered and everyone received the same number of each material, without having any leftovers. How many notebooks, pencils, and e... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 26
} | 35 | 0 | 0 |
omr-ws-c28f30f47025bf92e4c2c64b746af95a54f22e9a233c1f579ebaa1167cc02171 | Among families with two children, one family is chosen at random and it is reported that one of the children is a boy born on a Sunday. What is the probability that the other child is a girl? | [
"\\(\\frac{13}{27}\\)"
] | [
"Among families with two children, one family is chosen at random and it is reported that one of the children is a boy born on a Sunday. What is the probability that the other child is a girl?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d2d07e58833bf490a8d11e89e42533ca8b8147dcb2df4f745f0b23a54b91dcbe | Among fifteen children who together gathered 100 nuts, determine the minimum number of pairs of children who must have gathered the same number of nuts. | [
"1"
] | [
"Among fifteen children who together gathered 100 nuts, determine the minimum number of pairs of children who must have gathered the same number of nuts."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-286105904b0ece8ac06cd10498a02d5bc7c39ef885db277dd2ebe1e0efdb1a8a | Among integers \(\{1, 2, \ldots, 10^{2012}\}\), let \( n \) be the number of numbers for which the sum of the digits is divisible by 5. What are the first three digits (from the left) of \( n \)? | [
"199"
] | [
"Among integers \\(\\{1, 2, \\ldots, 10^{2012}\\}\\), let \\( n \\) be the number of numbers for which the sum of the digits is divisible by 5. What are the first three digits (from the left) of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9abc223baf811a21d861185fc98ceab6ace123bb86bd5b09084266892529aa0e | Among nine journalists from different countries attending a press conference, each journalist speaks at most three languages, and each pair of journalists shares a common language. Determine the minimum number of journalists who must share a common language. | [
"5"
] | [
"Among nine journalists from different countries attending a press conference, each journalist speaks at most three languages, and each pair of journalists shares a common language. Determine the minimum number of journalists who must share a common language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-859f19930521d516d4ec0ed154ad1ddeaa0a68efcc77e5bc68cd004a9884a95c | Among nine mathematicians at an international conference, it is known that among any three of them, at least two speak a common language, and each mathematician can speak at most three languages. Determine the minimum number of mathematicians who must speak the same language. | [
"4"
] | [
"Among nine mathematicians at an international conference, it is known that among any three of them, at least two speak a common language, and each mathematician can speak at most three languages. Determine the minimum number of mathematicians who must speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-a25a7d6b3dd137b79a4af7b47c48036d65dbb55d6d0c9122eeeb85540e3db8f3 | Among nine mathematicians at an international conference, it is known that among any three of them, at least two speak a common language, and each mathematician speaks at most three languages. Determine the minimum number of mathematicians who must speak the same language. | [
"4"
] | [
"Among nine mathematicians at an international conference, it is known that among any three of them, at least two speak a common language, and each mathematician speaks at most three languages. Determine the minimum number of mathematicians who must speak the same language."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-4c8e3fd8010ec769efee22f0baa0bdd17a7bcd6cfafd9c9e156a2366401db148 | Among nine similar coins, one is counterfeit and lighter than the others. Using a balance only twice, how can the counterfeit coin be identified? | [
"The method described above identifies the counterfeit coin in two weighings."
] | [
"Among nine similar coins, one is counterfeit and lighter than the others. Using a balance only twice, how can the counterfeit coin be identified?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-30247e41485d78ad65ce8ae73c2cba77ed1081d930e87f95e3fe60501462b0be | Among positive integers less than or equal to 2011, denote $A$ as the sum of the integers which have a remainder of 1 when divided by 3, and denote $B$ as the sum of the integers which have a remainder of 2 when divided by 3. Find the value of $A - B$. | [
"1341"
] | [
"Among positive integers less than or equal to 2011, denote $A$ as the sum of the integers which have a remainder of 1 when divided by 3, and denote $B$ as the sum of the integers which have a remainder of 2 when divided by 3. Find the value of $A - B$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-ec7fc49732b3b081a231a3d5543e6bd157249e1206b61aadf6bc47a9e24017c8 | Among several coins with pairwise different natural number denominations, there are exactly 2 fake ones. In one check, a detector determines the total amount of denominations of all real coins in a chosen set. If the set of coins includes coins with denominations 1, 2, 3, ..., 100, what is the minimum number of checks ... | [
"8"
] | [
"Among several coins with pairwise different natural number denominations, there are exactly 2 fake ones. In one check, a detector determines the total amount of denominations of all real coins in a chosen set. If the set of coins includes coins with denominations 1, 2, 3, ..., 100, what is the minimum number of ch... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-dbd3ccba401cc44e8bcb84e4bacf9d2c1a99ac20536068cbd6ec74692e0c00f3 | Among several coins with pairwise different natural number denominations, there are exactly 3 fake ones. In one check, a detector determines the total amount of denominations of all real coins in a chosen set. Design a strategy using exactly 3 checks to identify all fake coins. Let the denominations of the coins be 1, ... | [
""
] | [
"Among several coins with pairwise different natural number denominations, there are exactly 3 fake ones. In one check, a detector determines the total amount of denominations of all real coins in a chosen set. Design a strategy using exactly 3 checks to identify all fake coins. Let the denominations of the coins b... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-57bf26cdd9149025b30d2245ed7bab29ab51cdbfc1bd888e5f2bcabe61fbcd64 | Among six points placed in a 3 by 4 rectangle, let $d$ be the smallest possible maximum distance between any two points. Calculate the value of $d^2$. | [
"5"
] | [
"Among six points placed in a 3 by 4 rectangle, let $d$ be the smallest possible maximum distance between any two points. Calculate the value of $d^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-19a61430c059fea9e31bb121925bdf78c04f295d3fc04c0eab681df2c87e5681 | Among the $3^{24}$ possible colorings of the edges of twelve consecutive triangles (where each triangle shares exactly one edge with the next), how many colorings have the property that every triangle has one edge of each color (red, green, and blue)? | [
"4098"
] | [
"Among the $3^{24}$ possible colorings of the edges of twelve consecutive triangles (where each triangle shares exactly one edge with the next), how many colorings have the property that every triangle has one edge of each color (red, green, and blue)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 5,
"tir": 3
} | 39 | 0 | 0 |
omr-ws-3b58d7fa25d0c4663686b0525d2e9cb0c39e8ff72eb5125d57c1f95ca05b2d5d | Among the $n$ inhabitants of an island, every two are either friends or enemies. Each inhabitant must make a necklace consisting of marbles such that the necklaces of two friends have at least one marble of the same type and the necklaces of two enemies differ at all marbles. Determine the minimum number of different t... | [
"25"
] | [
"Among the $n$ inhabitants of an island, every two are either friends or enemies. Each inhabitant must make a necklace consisting of marbles such that the necklaces of two friends have at least one marble of the same type and the necklaces of two enemies differ at all marbles. Determine the minimum number of differ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-eded68664ae90d723fa2052ee0756c0a1dc88e550b411c73db0552acc77ff752 | Among the 10 people seated at a round table, there is a group of 3 friends. In how many ways can the 10 people be seated at a round table if no two members of the group of 3 friends are seated next to each other? | [
"1512000"
] | [
"Among the 10 people seated at a round table, there is a group of 3 friends. In how many ways can the 10 people be seated at a round table if no two members of the group of 3 friends are seated next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e08db3d96ae3b7a2d52c5a7fb8a78bb1e88b05efd96fd218249d06373cf4742c | Among the 11 terms of an arithmetic progression, the first, the fifth, and the eleventh term are three successive terms of a certain geometric progression. Find the formula for the general term of the arithmetic progression if its first term is equal to 24. | [
"$(24,27,30,\\ldots ,54)$ and $(24,24,24,\\ldots )$"
] | [
"Among the 11 terms of an arithmetic progression, the first, the fifth, and the eleventh term are three successive terms of a certain geometric progression. Find the formula for the general term of the arithmetic progression if its first term is equal to 24."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a19c70f5e8f392d127a7b2135a2e02138aa0db675a5906c309569d3e721c476c | Among the 15 pairwise distances between 6 points in the plane, calculate the maximum number of times the same distance can occur. | [
"9"
] | [
"Among the 15 pairwise distances between 6 points in the plane, calculate the maximum number of times the same distance can occur."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-19373480c50aa87dfd3a09a636a53c6596e75f28ed5ec32aa046a44d243d7f84 | Among the 21 pairwise distances between 7 points in the plane, let $N$ be the maximum number of times a single distance can occur. Find the value of $N$. | [
"12"
] | [
"Among the 21 pairwise distances between 7 points in the plane, let $N$ be the maximum number of times a single distance can occur. Find the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3586523401c7d1095124358cf80611ffb0acb35a4dfb48197853c8407718b7e5 | Among the 48 students in Class K, 20 are girls. Among the 36 students in Class Y, 24 are girls. Four students are randomly selected from these two classes, 2 from each class. Find the probability that among the 4 selected students, at least one of them is a girl. | [
"\\(\\frac{907}{940}\\)"
] | [
"Among the 48 students in Class K, 20 are girls. Among the 36 students in Class Y, 24 are girls. Four students are randomly selected from these two classes, 2 from each class. Find the probability that among the 4 selected students, at least one of them is a girl."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-b7b973fe5bf259a70370d470c60362b43d836b4d2f3b2a4ad8f480fdc2a7e2d7 | Among the 6 real numbers: i) $2015^{\frac{1}{2014}}$ ii) $2014^{\frac{1}{2015}}$ iii) $(\frac{2015}{2014})^{\frac{2014}{2015}}$ iv) $\frac{2014}{2015}$ v) $(\frac{2014}{2015})^{\frac{2015}{2014}}$ vi) $(\frac{2014}{2015})^{\frac{2014}{2015}}$ which one is the smallest? | [
"\\((\\frac{2014}{2015})^{\\frac{2015}{2014}}\\)"
] | [
"Among the 6 real numbers:\ni) $2015^{\\frac{1}{2014}}$\nii) $2014^{\\frac{1}{2015}}$\niii) $(\\frac{2015}{2014})^{\\frac{2014}{2015}}$\niv) $\\frac{2014}{2015}$\nv) $(\\frac{2014}{2015})^{\\frac{2015}{2014}}$\nvi) $(\\frac{2014}{2015})^{\\frac{2014}{2015}}$\n\nwhich one is the smallest?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-bbd18bea23a17f58a1076e2855dc896eddfb5a6af130acb6103b6bfc2df7fe45 | Among the 8! permutations of the digits 1, 2, 3, ..., 8, find the number of arrangements such that the product of any 5 consecutive digits is divisible by 5. | [
"10080"
] | [
"Among the 8! permutations of the digits 1, 2, 3, ..., 8, find the number of arrangements such that the product of any 5 consecutive digits is divisible by 5."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-204faf95bf69701bf038b0e9fc9503dc0569d766432bab549f22a414adaa5610 | Among the 81 ordered pairs of positive digits $(a,b)$, how many of the equations \[ ax + by = 2008 \] have solutions $(x,y)$ which are ordered pairs of positive integers? | [
"70"
] | [
"Among the 81 ordered pairs of positive digits $(a,b)$, how many of the equations\n\\[ ax + by = 2008 \\]\nhave solutions $(x,y)$ which are ordered pairs of positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-afcb295c6216bb587dbeaf885f3db5134d75b9019b826f65be374e13b2d9cacb | Among the complex numbers $z$ satisfying $|z-13i|\le 5$, find the one having the maximum argument. | [
"\\( z = -\\frac{60}{13} + \\frac{144}{13}i \\)"
] | [
"Among the complex numbers $z$ satisfying $|z-13i|\\le 5$, find the one having the maximum argument."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 8,
"tir": 24
} | 78 | 0 | 0 |
omr-ws-e1d275e50cfce3d0f856e39fa51836645b87a4251b15f8b62ccd719008b0f17d | Among the complex numbers $z$ satisfying the condition $$|z-25i|\le 15,$$ find the number having the least argument. Make a drawing. | [
"12 + 16i"
] | [
"Among the complex numbers $z$ satisfying the condition $$|z-25i|\\le 15,$$ find the number having the least argument. Make a drawing."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce3b6e1370d7ba12ecc39f6134fc4eb031a13d4db95c926f7367c66c6dfe4e18 | Among the complex numbers $z$ which satisfy the condition $$|z-25i|\le 15$$ find the number having the least positive argument. | [
"$z=12+16i$. The point representing this number is the point of tangency of a ray issuing from the origin and lying in the first quadrant to a circle of radius $\\sqrt {15}$ with center at the point $(0, 25)$"
] | [
"Among the complex numbers $z$ which satisfy the condition $$|z-25i|\\le 15$$ find the number having the least positive argument."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c345c918dcab26fc08b9c1dc135538cd2b3823740bfa30f039f61deb99f9b80 | Among the convergents of $\sqrt{15}$, find a rational number that approximates $\sqrt{15}$ with accuracy to four decimal places. | [
"\\(\\frac{1677}{433}\\)"
] | [
"Among the convergents of $\\sqrt{15}$, find a rational number that approximates $\\sqrt{15}$ with accuracy to four decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-a09022adcd18ff44021c044bc7823c63a55d464a5941df98d1839cb8087a741a | Among the coordinates \((x, y)\) where \(1 \leq x, y \leq 101\), choose some points such that there does not exist 4 points which form an isosceles trapezium with its base parallel to either the \(x\)-axis or \(y\)-axis (including rectangles). Find the maximum number of coordinate points that can be chosen. | [
"201"
] | [
"Among the coordinates \\((x, y)\\) where \\(1 \\leq x, y \\leq 101\\), choose some points such that there does not exist 4 points which form an isosceles trapezium with its base parallel to either the \\(x\\)-axis or \\(y\\)-axis (including rectangles). Find the maximum number of coordinate points that can be chos... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-a1690402a96b5eb07cf8f00e2c1d75511598290fea08fc8f3144067591b13081 | Among the divisors of a natural number \( n \), there are numbers that give remainders \( 1001, 1002, \ldots, 2012 \) when divided by \( 2013 \). Find the smallest possible value of \( n \) such that among the divisors of \( n^2 \), there exist numbers that give remainders \( 1, 2, 3, \ldots, 2012 \) when divided by \(... | [
"4026"
] | [
"Among the divisors of a natural number \\( n \\), there are numbers that give remainders \\( 1001, 1002, \\ldots, 2012 \\) when divided by \\( 2013 \\). Find the smallest possible value of \\( n \\) such that among the divisors of \\( n^2 \\), there exist numbers that give remainders \\( 1, 2, 3, \\ldots, 2012 \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-533e17164874f82866e179b7f14f575d1a28c2070ae3cf0f9f8e35e07604ce73 | Among the family of planes represented by the equation \[ (4\lambda^2-\lambda +3) x-(3\lambda^2 +\lambda+4)y+\lambda(\lambda-1 )z +24\lambda^2 +\lambda +25=0, \] where \( \lambda \in \mathbb{R} \) is the parameter, determine the plane that is the farthest from the origin. | [
"5"
] | [
"Among the family of planes represented by the equation\n\\[ (4\\lambda^2-\\lambda +3) x-(3\\lambda^2 +\\lambda+4)y+\\lambda(\\lambda-1 )z +24\\lambda^2 +\\lambda +25=0, \\]\nwhere \\( \\lambda \\in \\mathbb{R} \\) is the parameter, determine the plane that is the farthest from the origin."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d015bb645114cdf79688d90d21816a7cf15f6cf92707976ca4d48a8ceb5dcb7 | Among the finite abelian groups $G$ from Problem 1, which satisfy the condition "$7^3x = 0$ for arbitrary $x \in G$"? | [
"$\\mathbb{Z}_7^6, \\mathbb{Z}_7^4 \\times \\mathbb{Z}_{7^2}, \\mathbb{Z}_7^3 \\times \\mathbb{Z}_{7^3}, \\mathbb{Z}_7^2 \\times \\mathbb{Z}_{7^2}^2, \\mathbb{Z}_7 \\times \\mathbb{Z}_{7^2} \\times \\mathbb{Z}_{7^3}, \\mathbb{Z}_{7^2}^3, \\mathbb{Z}_{7^3}^2$"
] | [
"Among the finite abelian groups $G$ from Problem 1, which satisfy the condition \"$7^3x = 0$ for arbitrary $x \\in G$\"?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-f1207c9d94facf08f5e1a2d2b1fa3114e8a26e01729a92fb90f0de52d779b810 | Among the first $2005$ positive integers, which has the largest number of divisors? | [
"1680"
] | [
"Among the first $2005$ positive integers, which has the largest number of divisors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-804e4d655484e7acc64229a95569e5fc6be455a56d89a19ee480a015b3fecd33 | Among the first 1000 positive integers, there are \( n \) whose hexadecimal representation contains only numeric digits (0-9). What is the sum of the digits of \( n \)? | [
"21"
] | [
"Among the first 1000 positive integers, there are \\( n \\) whose hexadecimal representation contains only numeric digits (0-9). What is the sum of the digits of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 45
} | 77 | 0 | 0 |
omr-ws-8f96a38102ea56e5bec1a3995d65d95f2a6f597eb4c8e819345f70ef84fd969c | Among the first 1000 positive integers, there are \( n \) whose hexadecimal representation contains only numeric digits. What is the sum of the digits of \( n \)? | [
"21"
] | [
"Among the first 1000 positive integers, there are \\( n \\) whose hexadecimal representation contains only numeric digits. What is the sum of the digits of \\( n \\)?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-77916790780d6af090abf2765899b04282a8c8b421317a9e7ce3ac9468f6057b | Among the first 2009 terms of the sequence 121, 1221, 12221, ..., how many terms are divisible by 3? | [
"670"
] | [
"Among the first 2009 terms of the sequence 121, 1221, 12221, ..., how many terms are divisible by 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-d696373aed5d82e9138ff832905bd827bee68b5c381b9dc8ec4687ddf804cd3a | Among the first 2020 positive integers, how many have exactly three odd divisors? | [
"45"
] | [
"Among the first 2020 positive integers, how many have exactly three odd divisors?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-31881161b0f0621690494e611db850d24d235afece0f74606c58308a9f116cc6 | Among the first billion positive integers, arrange the following sets in order of decreasing size: a. Palindromic numbers (e.g., 22, 121, 1145411) b. Prime numbers c. Perfect cubes | [
"Prime numbers, Palindromic numbers, Perfect cubes"
] | [
"Among the first billion positive integers, arrange the following sets in order of decreasing size:\na. Palindromic numbers (e.g., 22, 121, 1145411)\nb. Prime numbers\nc. Perfect cubes"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-a1633237cd1b22458dc75f1d6047cace18024b01444347b724d54c91bf91da89 | Among the first million Fibonacci numbers, find the smallest index $n$ such that $F_n$ ends in $999$. Calculate the sum of the digits of $n$. | [
"no_valid_answer_found"
] | [
"Among the first million Fibonacci numbers, find the smallest index $n$ such that $F_n$ ends in $999$. Calculate the sum of the digits of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5e91c69b78f6ada751195f14450ba35ec97fba948dcf2887679fa43280d7591 | Among the first thousand whole numbers find the possible triples of numbers for which the sums of their squares are divisible by their products. | [
"It can be shown that if the numbers $x$, $y$ and $z$ satisfy the indicated equality and, for instance, the inequality $z > kxy/2$ holds, then the numbers can be decreased in such a way that the same equality remains valid for them. In case $x\\le kyz/2$, $y\\le kxz/2$, $z\\le kxz/2$ and $x\\le y\\le z$ there must ... | [
"Among the first thousand whole numbers find the possible triples of numbers for which the sums of their squares are divisible by their products."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d09e2133caf113bab054517626ec7be7412ab0eb70ea22584953c55a72a79e5 | Among the fractions $\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7},$ and $\frac{6}{7}$, which fraction has a repeating decimal where the third and fourth digits form a two-digit number that is one-half of the number formed by the fifth and sixth digits? Write the fraction corresponding to this decimal... | [
"$\\frac{4}{7}$"
] | [
"Among the fractions $\\frac{1}{7}, \\frac{2}{7}, \\frac{3}{7}, \\frac{4}{7}, \\frac{5}{7},$ and $\\frac{6}{7}$, which fraction has a repeating decimal where the third and fourth digits form a two-digit number that is one-half of the number formed by the fifth and sixth digits? Write the fraction corresponding to t... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-6ecde2dc194cb15f9aed1c3d9b7c5e6d885e4dfead6d0339efa751a9e5ef2786 | Among the inscribed triangles of a given circle, find the triangle whose area is the greatest. | [
"\\text{Equilateral triangle}"
] | [
"Among the inscribed triangles of a given circle, find the triangle whose area is the greatest."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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