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Blue and red lines

Source: APMO 2015 Problem 4

March 30, 2015
combinatorial geometryAPMO

Problem Statement

Let nn be a positive integer. Consider 2n2n distinct lines on the plane, no two of which are parallel. Of the 2n2n lines, nn are colored blue, the other nn are colored red. Let B\mathcal{B} be the set of all points on the plane that lie on at least one blue line, and R\mathcal{R} the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects B\mathcal{B} in exactly 2n−12n - 1 points, and also intersects R\mathcal{R} in exactly 2n−12n - 1 points.
Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand
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