Question bank USAMO 2012

Find all integers $n \ge 3$ such that among any $n$ positive real numbers
$a_1$, $a_2$, …, $a_n$ 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.

A circle is divided into 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 $n \gt 1$ have the property
that there exists an infinite sequence $a_1$, $a_2$, $a_3$, …
of nonzero integers such that the equality

\[ a_k + 2a_{2k} + \dots + na_{nk} = 0 \]

holds for every positive integer $k$.

Find all functions $f \colon \mathbb N \to \mathbb N$ 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 through $P$.
Let $A'$, $B'$, $C'$ be the points where the
reflections of lines $PA$, $PB$, $PC$ with respect to $\gamma$
intersect lines $BC$, $CA$, $AB$ respectively.
Prove that $A'$, $B'$, $C'$ are collinear.

For integer $n \ge 2$,
let $x_1$, $x_2$, …, $x_n$ 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 $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$.
For which choices of $x_1$, $x_2$, …, $x_n$, $\lambda$ does equality hold?