IMO 2006 Shortlist C3
Let S be a finite set of points in the plane such that no three of them are on a line. For each convex polygon P whose v...
Category: Combinatorics
Problem
Let S be a finite set of points in the plane such that no three of them are on a line. For each convex polygon P whose vertices are in S, let a(P) be the number of vertices of P, and let b(P) be the number of points of S which are outside P. Prove that for every real number x X P xa(P) (1 − x)b(P) = 1, where the sum is taken over all convex polygons with vertices in S. NB. A line segment, a point and the empty set are considered as convex polygons of 2, 1 and 0 vertices, respectively. (Colombia)