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[
"Mathematics -> Algebra -> Other"
] | 8
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nLet \\( n (\\geq 2) \\) be a positive integer. We aim to find the minimum \\( m \\) such that the(...TRUNCATED)
|
1 + \left\lceil \frac{n}{2} \right\rceil
|
china_team_selection_test
| ["\nOkay, let's try to tackle this problem. So, we need to find the minimum m such that there exists(...TRUNCATED)
|
[
24576,
21703,
24576,
22618
] |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nIn an acute scalene triangle \\(ABC\\), points \\(D, E, F\\) lie on sides \\(BC, CA, AB\\), respe(...TRUNCATED)
|
1
|
usa_team_selection_test
| ["\n\nOkay, so I need to solve this geometry problem about an acute scalene triangle with orthocente(...TRUNCATED)
|
[
24576,
9983,
15848,
10967
] |
[
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nA tournament is a directed graph for which every (unordered) pair of vertices has a single direct(...TRUNCATED)
|
\lceil \log_2 n \rceil
|
usa_team_selection_test
| ["\nOkay, let's try to tackle this problem. So, the question is about finding the minimum directed-e(...TRUNCATED)
|
[
12274,
24576,
21094,
24576
] |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 8.5
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\n\nTo determine whether there exist positive reals \\(a_0, a_1, \\ldots, a_{19}\\) such that the p(...TRUNCATED)
|
\text{Yes}
|
china_national_olympiad
| ["\nOkay, so I need to figure out if there exist positive real numbers a₀, a₁, ..., a₁₉ such(...TRUNCATED)
|
[
14250,
12247,
22221,
19172
] |
[
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nLet \\( p \\) be a prime. We arrange the numbers in \\( \\{1, 2, \\ldots, p^2\\} \\) as a \\( p \(...TRUNCATED)
|
2(p!)^2
|
china_national_olympiad
| ["\nOkay, so I need to figure out how many good arrangements there are for a p x p matrix where p is(...TRUNCATED)
|
[
17979,
24576,
19777,
24576
] |
[
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nLet the physicist label the usamons as \\(1, 2, \\ldots, 2015\\). Define \\(x_i = 0\\) if usamon (...TRUNCATED)
|
\text{No}
|
usa_team_selection_test
| ["\nOkay, so I need to figure out if there's a way for the physicist to isolate two usamons that are(...TRUNCATED)
|
[
14095,
24576,
24576,
22788
] |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Algebra -> Other"
] | 8
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nThere are \\(2022\\) equally spaced points on a circular track \\(\\gamma\\) of circumference \\((...TRUNCATED)
|
2042222
|
usa_team_selection_test_for_imo
| ["\nOkay, so there's this problem about Bunbun the Bunny hopping around a circular track with 2022 e(...TRUNCATED)
|
[
24576,
24576,
19103,
22905
] |
[
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 9
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nTo determine the minimum number of players needed to participate in the series such that the set (...TRUNCATED)
|
\frac{1}{2} \max A + 3
|
china_national_olympiad
| ["\nOkay, so I need to figure out the minimum number of players required in a table tennis doubles c(...TRUNCATED)
|
[
24576,
16306,
13687,
16037
] |
["Mathematics -> Geometry -> Plane Geometry -> Other","Mathematics -> Discrete Mathematics -> Combin(...TRUNCATED)
| 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nGiven a set of 100 points in the plane, we want to determine the maximum number of harmonic pairs(...TRUNCATED)
|
3750
|
usa_team_selection_test
| ["\nOkay, so I need to figure out the maximum number of harmonic pairs among 100 points in the plane(...TRUNCATED)
|
[
24576,
20505,
19911,
24576
] |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons",
"Mathematics -> Number Theory -> Other"
] | 7
| "<|im_start|>system\nYou are a world-class Olympiad mathematics competitor. \nThe following problem (...TRUNCATED)
| "\nTo determine the largest integer \\( n \\) for which it is possible to draw a convex \\( n \\)-go(...TRUNCATED)
|
561
|
usa_team_selection_test
| ["\n\nOkay, so I need to figure out the largest integer n for which it's possible to draw a convex n(...TRUNCATED)
|
[
19793,
19037,
14091,
15803
] |
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