Problem 3
Let S be a set of 2004 points in the plane, no three collinear, and let L be the set of all lines determined by pairs of points of S. A line separates two points when they lie on opposite sides of it and neither lies on the line. Prove that S can be colored with at most two colors so that two points have the same color exactly when an odd number of lines in L separates them.
Step 2 of 5: Reduce the claim to a three-point parity identity
Detailed analysis
The prescribed colors give q and r the same color exactly when n(p,q)+n(p,r) is even. Thus it remains to prove, for distinct p,q,r, that n(p,q)+n(p,r)+n(q,r) is odd.