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 1 of 4: Find a differently colored adjacent pair
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ℓ1,…,ℓ2n,ℓk and ℓk+1 have different colors\ell_1,\ldots,\ell_{2n},\quad \ell_k\text{ and }\ell_{k+1}\text{ have different colors}
Detailed analysis

Rotate a generic line through a point until it becomes parallel to each given line. This orders the lines by direction. Since the colors are equally represented, two consecutive lines in this order have different colors; call them ℓk\ell_k and ℓk+1\ell_{k+1}.