1984 IMO Problems/Problem 3
Given points and in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point in the plane, the circle has center and radius , where is measured in radians in the range . Prove that we can find a point , not on , such that its color appears on the circumference of the circle .
Let be a sequence of positive reals such that . For each , let be the circle centered at with radius .
Because of , we can find points such that for all we have . We now forget about all the other points, and work only with the matrix .
Suppose we use colors. There must be one, , which appears infinitely many times on the first row of , in, say, points . Then cannot appear on the lines , . Next, there is a color which appears infinitely often among the points , . But then cannot appear on the lines for such . Repeating this procedure, we reach a stage where we have a row of (infinitely many actually) on which none of our colors can appear. This is a contradiction.
This solution was posted and copyrighted by grobber. The original thread for this problem can be found here: 
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