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"Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number (...TRUNCATED) | "1. **Affine Transformation and Line Directions**:\n By an affine transformation, we can assume th(...TRUNCATED) | 91 | Combinatorics | math-word-problem | aops_forum | e67ecad5-1a1b-548f-bb64-0d6a2f726825 | [
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"(IMO SL 2019 C4)\n\nOn a flat plain in Camelot, King Arthur builds a labyrinth $\\mathfrak{L}$ cons(...TRUNCATED) | "We will show that the only possible value is $n+1$, regardless of the labyrinth $\\mathfrak{L}$ cho(...TRUNCATED) | n+1 | Combinatorics | math-word-problem | olympiads | fc6bb733-006a-542f-862d-e4892c797ac3 | [
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"[ Dirichlet's Principle (continued). ] [Partitions into pairs and groups; bijections]\n\nBefore a c(...TRUNCATED) | "a) The first two cards can \"encode\" the suit of the second card, the next two cards can encode th(...TRUNCATED) | 23 | Combinatorics | math-word-problem | olympiads | efa80055-745e-5557-93f4-7595cfa9756e | [
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"There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no (...TRUNCATED) | "1. **Understanding the Problem:**\n We need to determine the maximum number of planes that can be(...TRUNCATED) | \frac{n(n-1)}{2} | Combinatorics | math-word-problem | aops_forum | 659af099-5671-5bf7-81e7-7a839d4e7574 | [
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"Question 5: In an $n \\times n$ grid, 101 cells are colored blue. It is known that there is a uniqu(...TRUNCATED) | "The minimum possible value of $n$ is 101.\nFirst, prove the more general conclusion below.\nLemma G(...TRUNCATED) | 101 | Combinatorics | math-word-problem | olympiads | 211450e4-6329-5df8-b4a1-8751d992e4e3 | [
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] | ["<think>\nOkay, so I need to solve this problem from the Bulgarian Mathematical Olympiad. Let me re(...TRUNCATED) | [
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"6. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$ there are always (...TRUNCATED) | "When $n=8$, divide 8 points into two groups, each with 4 points, connect red edges between points i(...TRUNCATED) | 9 | Combinatorics | math-word-problem | olympiads | 7315319b-d4fe-51cf-be24-43a4dc1334b8 | [
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"A teacher wants to divide the $2010$ questions she asked in the exams during the school year into t(...TRUNCATED) | "1. **Define the problem and constraints:**\n - We have 2010 questions.\n - We need to divide th(...TRUNCATED) | 6 | Combinatorics | math-word-problem | aops_forum | 5ff5038f-b6bb-55ac-80cc-7602aa555c8f | [
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"22. Find the number of triangles whose sides are formed by the sides and the diagonals of a regular(...TRUNCATED) | "22. Answer: 287.\nA triangle can be formed using $3,4,5$ or 6 vertices.\n\nSo the total number is\n(...TRUNCATED) | 287 | Combinatorics | math-word-problem | olympiads | cc5c6db2-3364-57ad-9a1e-b83741fd9ce4 | [
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] | ["<think>\nOkay, so I need to find the number of triangles that can be formed using the sides and di(...TRUNCATED) | [
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"1. A class of students learned at the award ceremony that the number of students (a student can sub(...TRUNCATED) | "- 1. B.\n\nThe class won a total of\n$$\n18+12+6+3+3+2=44 \\text { (awards), }\n$$\n\nThose who won(...TRUNCATED) | B | Combinatorics | MCQ | cn_contest | cacb4756-8cff-5fe6-8d2c-cbd3e870bb9f | [
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"Exercise 3. Consider 2017 lines in the plane, which intersect pairwise at distinct points. Let $E$ (...TRUNCATED) | "Solution to Exercise 3 The minimum $m$ sought is $m=3$ and, with what follows, it will be quite evi(...TRUNCATED) | 3 | Combinatorics | math-word-problem | olympiads | cab91b9a-cc46-5564-90b0-3b6999feab82 | [
true,
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] | ["<think>\nOkay, so I have this problem here about coloring intersection points of 2017 lines in the(...TRUNCATED) | [
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