ÖMO 1987 Problem 5

Let $P$ be a point in the interior of a convex $n$ -gon $A_1 A_2 ... A_n$ $(n \ge 3)$ . Show that among the angles $\beta _{ij}=\angle A_i P A_j,1 \le i \le n$ , there are at least $n-1$ angles satisfying $90^{\circ} \le \beta_{ij} \le 180^{\circ}$ .