Problem 4

Let

n

be a positive integer. Consider

2n

distinct lines on the plane, no two of which are parallel. Of the

2n

lines,

n

are colored blue, the other

n

are colored red. Let

B

be the set of all points on the plane that lie on at least one blue line, and

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

in exactly

2n-1

points, and also intersects

R

in exactly

2n-1

points.

Step 2 of 4: Enclose all pairwise intersections

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R=\{x=\pm a,\ y=\pm b\}\supset S

Detailed analysis

Choose angular bisectors of the selected pair as coordinate axes. No other line has direction between their two directions. The finite set S of all pairwise intersections can therefore be enclosed in a sufficiently large rectangle R whose sides are parallel to the axes.