AT-OEMO-REG 2017

Let $x_1, x_2, \dots, x_n$ be non-negative real numbers such that $$x_1^2+x_2^2 + \dots x_9^2 \ge 25.$$ Prove that one can choose three of these numbers such that their sum is at least $5$.

Proposed by Karl Czakler

Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$.

Proposed by Theresia Eisenkölbl

The nonnegative integers $2000$, $17$ and $n$ are written on the blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed, where all three numbers remain unchanged. The player who is in turn and cannot make an allowed move loses the game.

• Prove that the game will end for every number $n$.
• Who wins the game in the case $n = 2017$?

Proposed by Richard Henner

Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. Proposed by Walther Janous