IMO 1984 SL 8

Origin: ROM

Problem

In a plane two different points O and A are given. For each point X ̸= O of the plane denote by \alpha(X) the angle AOX measured in radians (0 \leq\alpha(X) < 2\pi) and by C(X) the circle with center O and radius OX + \alpha(X) OX . Suppose each point of the plane is colored by one of a finite number of colors. Show that there exists a point X with \alpha(X) > 0 such that its color appears somewhere on the circle C(X).

Solution

Suppose that the statement of the problem is false. Consider two arbitrary circles R = (O, r) and S = (O, s) with 0 < r < s < 1. The point X \inR with \alpha(X) = r(s −r) < 2\pi satisfies that C(X) = S. It follows that the color of the point X does not appear on S. Consequently, the set of colors that appear on R is not the same as the set of colors that appear on S. Hence any two distinct circles with center at O and radii less than 1 have distinct sets of colors. This is a contradiction, since there are infinitely many such circles but only finitely many possible sets of colors.