DE-BUNDESWETTBEWERB 2019

An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$ . Show that there are two dominoes forming a $2 \times 2$ square.

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$ . We form the two six-digit numbers $SCHLAF$ and $FLACHS$ .

Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$ .

Remark: The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$ , respectively. Show that the lines $EF$ and $GH$ are parallel.