Problem

Source: Bundeswettbewerb Mathematik 2022, Round 2, Problem 4

Tags: enclosing circles, colored points, geometry, combinatorics



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.