A convex 2004-sided polygon P is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of P thick if all other 2001 vertices of P lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.
Problem
Source: MOP 2005 Homework - Black Group #11
Tags: geometry, circumcircle, combinatorics unsolved, combinatorics