USAMO 2012

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.

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

Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (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 $m, n$ .

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.

For integer $n \geq 2$, let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $$x_{1}+x_{2}+\cdots+x_{n}=0 \text { and } x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$$ For each subset $A \subseteq\{1,2, \ldots, n\}$, 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 $S_{A} \geq \lambda$ is at most $2^{n-3} / \lambda^{2}$. For which choices of $x_{1}, x_{2}, \ldots, x_{n}, \lambda$ does equality hold?