ISL 1970 Problem 32

Let there be given an acute angle $\angle AOB = 3\alpha$ , where $\overline{OA}= \overline{OB}$ . The point $A$ is the center of a circle with radius $\overline{OA}$ . A line $s$ parallel to $OA$ passes through $B$ . Inside the given angle a variable line $t$ is drawn through $O$ . It meets the circle in $O$ and $C$ and the given line $s$ in $D$ , where $\angle AOC = x$ . Starting from an arbitrarily chosen position $t_0$ of $t$ , the series $t_0, t_1, t_2, \ldots$ is determined by defining $\overline{BD_{i+1}}=\overline{OC_i}$ for each $i$ (in which $C_i$ and $D_i$ denote the positions of $C$ and $D$ , corresponding to $t_i$ ). Making use of the graphical representations of $BD$ and $OC$ as functions of $x$ , determine the behavior of $t_i$ for $i\to \infty$ .