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Nationalist Combo

Source: USEMO 2019 Problem 5

May 24, 2020
USEMO

Problem Statement

Let P\mathcal{P} be a regular polygon, and let V\mathcal{V} be its set of vertices. Each point in V\mathcal{V} is colored red, white, or blue. A subset of V\mathcal{V} is patriotic if it contains an equal number of points of each color, and a side of P\mathcal{P} is dazzling if its endpoints are of different colors.
Suppose that V\mathcal{V} is patriotic and the number of dazzling edges of P\mathcal{P} is even. Prove that there exists a line, not passing through any point in V\mathcal{V}, dividing V\mathcal{V} into two nonempty patriotic subsets.
Ankan Bhattacharya
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