KURSCHAK 1977

Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$ .

$ABC$ is a triangle with orthocenter $H$ . The median from $A$ meets the circumcircle again at $A_1$ , and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$ . The points $B_2$ and $C_2$ are defined similarly. Show that $H$ , $A_2$ , $B_2$ and $C_2$ lie on a circle.

Three schools each have $n$ students. Each student knows a total of $n + 1$ students at the other two schools. Show that there must be three students, one from each school, who know each other.