IMO Shortlist 2006 problem C3

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Dodao/la: arhiva
2. travnja 2012.
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. A line segment, a point, and the empty set are considered as convex polygons of 2, 1, and 0 vertices respectively. Prove that for every real number x:

\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1, where the sum is taken over all convex polygons with vertices in S.

Alternative formulation:

Let M be a finite point set in the plane and no three points are collinear. A subset A of M will be called round if its elements is the set of vertices of a convex A -gon V(A). For each round subset let r(A) be the number of points from M which are exterior from the convex A -gon V(A). Subsets with 0,1 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 A of M construct the polynomial
P_A(x) = x^{|A|}(1 - x)^{r(A)}.
Show that the sum of polynomials for all round subsets is exactly the polynomial P(x) = 1.
Izvor: Međunarodna matematička olimpijada, shortlist 2006