KURSCHAK 2014

Consider a company of $n\ge 4$ people, where everyone knows at least one other person, but everyone knows at most $n-2$ of the others. Prove that we can sit four of these people at a round table such that all four of them know exactly one of their two neighbors. (Knowledge is mutual.)

We are given an acute triangle $ABC$ , and inside it a point $P$ , which is not on any of the heights $AA_1$ , $BB_1$ , $CC_1$ . The rays $AP$ , $BP$ , $CP$ intersect the circumcircle of $ABC$ at points $A_2$ , $B_2$ , $C_2$ . Prove that the circles $AA_1A_2$ , $BB_1B_2$ and $CC_1C_2$ are concurrent.

Let $K$ be a closed convex polygonal region, and let $X$ be a point in the plane of $K$ . Show that there exists a finite sequence of reflections in the sides of $K$ , such that $K$ contains the image of $X$ after these reflections.