AT-OEMO-BEG 2022

Show that for all real numbers $x$ and $y$ with $x > -1$ and $y > -1$ and $x + y = 1$ the inequality $$\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23$$ holds. When does equality apply?

(Walther Janous)

You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$ . The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)

A semicircle is erected over the segment $AB$ with center $M$ . Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$ . The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$ . Prove that $PM = PS$ holds.

(Karl Czakler)

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$ .

(Karl Czakler)