Odd points even triangles

This is a problem from the British (or maybe Balkan, no idea) math olympiad.


You are given a set $S$ of $2005$ points (no three collinear) in the 2D plane. For each point $P$ in $S$, you count the number of triangles (formed by points in $S$) within which $P$ lies ($P$ must be strictly inside the triangle).

Show that this number is even, irrespective of the point $P$.


[I don't remember the solution to this one, so medium.]