AT-OEMO-FED1 2020

Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$ . Proof that $$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$ When does equality occur?

(Walther Janous)

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$ . On the sides $AC$ and $BC$ , the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$ . Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$ , respectively. Prove that $FG$ is half as long as $AB$ .

(Walther Janous)

On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as $a, b, c$ such that $a >gcd(b, c)$ , then $a$ gets replaced by $a-gcd(b, c)$ . The game ends if there is no way to denote the numbers such that $a >gcd(b, c)$ .

Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers.

(Theresia Eisenkölbl)

Determine all positive integers $N$ such that $$2^N-2N$$ is a perfect square.

(Walther Janous)