Let a set S of 2004 points in the plane be given, no three of which are collinear. Let L denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of S with at most two colours, such that for any points p,q of S, the number of lines in L which separate p from q is odd if and only if p and q have the same colour.
Note: A line ℓ separates two points p and q if p and q lie on opposite sides of ℓ with neither point on ℓ. combinatorics unsolvedcombinatorics