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2022 Bundeswettbewerb Mathematik, 4
Tags: combinatorics , geometry , enclosing circles , colored points
Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1. Prove that there exists a circle with diameter $\sqrt{2}$ such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.