ÖMO 2017

Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$ , respectively. Let the common point of $DF$ and the altitude through $B$ be $P$ . The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$ . Furthermore, $EQ$ intersects the altitude through $A$ in $N$ . Prove that $N$ is the midpoint of $AH$ .

Proposed by Karl Czakler

Let $S = \{1,2,..., 2017\}$ . Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$ .

Proposed by Gerhard Woeginger