### IMO Shortlist 2006 problem C3

Kvaliteta:

Avg: 0,0Težina:

Avg: 7,0 Let be a finite set of points in the plane such that no three of them are on a line. For each convex polygon whose vertices are in , let be the number of vertices of , and let be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of , , and vertices respectively. Prove that for every real number :

, where the sum is taken over all convex polygons with vertices in .

Alternative formulation:

Let be a finite point set in the plane and no three points are collinear. A subset of will be called round if its elements is the set of vertices of a convex gon For each round subset let be the number of points from which are exterior from the convex gon Subsets with and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial

Show that the sum of polynomials for all round subsets is exactly the polynomial

, where the sum is taken over all convex polygons with vertices in .

Alternative formulation:

Let be a finite point set in the plane and no three points are collinear. A subset of will be called round if its elements is the set of vertices of a convex gon For each round subset let be the number of points from which are exterior from the convex gon Subsets with and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial

Show that the sum of polynomials for all round subsets is exactly the polynomial

Izvor: Međunarodna matematička olimpijada, shortlist 2006