AT-OEMO-REG 2021

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds.

(Karl Czakler)

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$ . The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$ . Let $E$ denote the intersection of $CD$ and $AB$ . Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$ .

(Karl Czakler)

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard.

(Karl Czakler)

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$ , $y | (z + 1)$ and $z | (x + 1)$ .

(Walther Janous)