ILL 1986 Problem 48

Let $P$ be a convex $1986$ -gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$ . Starting with an arbitrary point $Q_1$ on the boundary of $P$ , define recursively a sequence of points $Q_n$ as follows: given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$ . Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$ .