### IMO Shortlist 2006 problem C3

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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 Show that the sum of polynomials for all round subsets is exactly the polynomial $P(x) = 1.$
Izvor: Međunarodna matematička olimpijada, shortlist 2006