ÖMO 2022

Let $ABC$ be an isosceles triangle with base $AB$ . We choose a point $P$ inside the triangle on altitude through $C$ . The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$ . Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.

(Walther Janous)

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$ .

(b) Show that a corresponding decomposition into $30$ squares is also possible.

(Walther Janous)