IMO 1984 Problem 3

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$ , where $\angle AOX$ is measured in radians in the range $[0,2\pi)$ . Prove that we can find a point $X$ , not on $OA$ , such that its color appears on the circumference of the circle $C(X)$ .