KURSCHAK 1996

Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.

Two countries ( $A$ and $B$ ) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds:

• the participants from $B$ each know an even number of people in $C$ ,
• the participants from $B$ each know an odd number of participants in $C$ .

Let $n$ and $k$ be arbitrary non-negative integers. Suppose we have drawn $2kn+1$ (different) diagonals of a convex $n$ -gon. Show that there exists a broken line formed by $2k+1$ of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only $kn$ diagonals.