2013년 2월 8일 금요일

2012 USAMO Problems

Day 1

Problem 1

Find all integers n \ge 3 such that among any n positive real numbers a_1, a_2, \dots, a_n with \max(a_1, a_2, \dots, a_n) \le n \cdot \min(a_1, a_2, \dots, a_n), there exist three that are the side lengths of an acute triangle.
Solution

Problem 2

A circle is divided into 432 congruent arcs by 432 points. The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
Solution

Problem 3

Determine which integers n > 1 have the property that there exists an infinite sequence a_1, a_2, a_3, \dots of nonzero integers such that the equality a_k + 2a_{2k} + \dots + na_{nk} = 0 holds for every positive integer k.
Solution

Day 2

Problem 4

Find all functions f : \mathbb{Z}^+ \to \mathbb{Z}^+ (where \mathbb{Z}^+ is the set of positive integers) such that f(n!) = f(n)! for all positive integers n and such that m - n divides f(m) - f(n) for all distinct positive integers m, n.
Solution

Problem 5

Let P be a point in the plane of triangle ABC, and \gamma a line passing through P. Let A', B', C' be the points where the reflections of lines PA, PB, PC with respect to \gamma intersect lines BC, AC, AB, respectively. Prove that A', B', C' are collinear.
Solution

Problem 6

For integer n \ge 2, let x_1, x_2, \dots, x_n be real numbers satisfying x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1. For each subset A \subseteq \{1, 2, \dots, n\}, define S_A = \sum_{i \in A} x_i. (If A is the empty set, then S_A = 0.)
Prove that for any positive number \lambda, the number of sets A satisfying S_A \ge \lambda is at most 2^{n - 3}/\lambda^2. For what choices of x_1, x_2, \dots, x_n, \lambda does equality hold?
Solution
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