Given points O and A in the plane. Every point in the plane is colored with one of a finite number of colors.
Solution Suppose the result is false.
C 1 is a spiral from O to the point of intersection of OA and C 1 . Every point of this spiral must be a different color from all points of the circle C 1 . Hence every circle center O with radius smaller than C 1 must include a point of different color to those on C 1 . Suppose there are n colors. The solutions given on this site are not complete, they are designed to be sufficient for anyone who has thought hard about the problem.