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Dense Colors in a Circle

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October 31, 2024
algebraChilecombinatoricscombinatorial geometryColoring

Problem Statement

The points inside a circle Γ \Gamma are painted with n≥1 n \geq 1 colors. A color is said to be dense in a circle Ω \Omega if every circle contained within Ω \Omega has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within Γ \Gamma .
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