MathLabs

Problem 4

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 BB be the set of all points on the plane that lie on at least one blue line, and RR the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects BB in exactly 2n−12n-1 points, and also intersects RR in exactly 2n−12n-1 points.
Step 4 of 4: Count the intersections
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2(n−1)+1=2n−12(n-1)+1=2n-1
Detailed analysis

Every one of the other 2n-2 lines intersects C twice, and no two lines meet on C because all pairwise intersections lie inside R. The two selected lines are tangent to C and have different colors, contributing one point to each color. Thus each color set contributes exactly 2n-1 points.