Question 3
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 $\ell$ separates two points $p,q$ if they lie on opposite sides of $\ell$, with neither point on $\ell$.
Editorial clarification: the two points $p,q$ are distinct. The source says “any points”; comparing a point with itself would make that literal wording false.
Schematic illustration: only a few sample points and two of the determining lines are shown, rather than all 2004 points and all lines in $L$.
Note: A line $\ell$ separates two points $p,q$ if they lie on opposite sides of $\ell$, with neither point on $\ell$.
Editorial clarification: the two points $p,q$ are distinct. The source says “any points”; comparing a point with itself would make that literal wording false.
Schematic illustration: only a few sample points and two of the determining lines are shown, rather than all 2004 points and all lines in $L$.