DE-BUNDESWETTBEWERB 2015

Twelve 1-Euro-coins are laid flat on a table, such that their midpoints form a regular $12$ -gon. Adjacent coins are tangent to each other. Prove that it is possible to put another seven such coins into the interior of the ring of the twelve coins.

A sum of $335$ pairwise distinct positive integers equals $100000$ . a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?

Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$ . Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$ . Both points, $X$ and $Y$ , are on the same side as $C$ with respect to line $AB$ . Show that the rays $[AX$ and $[BY$ intersect on line $CM$ .