International Mathematical Olympiad 1984 Problem 3

In the plane two different points $O$ and $A$ are given. For each point $X$ of the plane, other than $O$, denote by $a(X)$ the measure of the angle between $OA$ and $OX$ in radians, counterclockwise from $OA$ $(0 \leq a(X) < 2\pi)$. Let $C(X)$ be the circle with center $O$ and radius of length $OX + a(X)/OX$. Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ for which $a(Y) > 0$ such that its color appears on the circumference of the circle $C(Y)$.