MathDB

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 \minus{} x)^{b(P)}} \equal{} 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

M

will be called round if its elements is the set of vertices of a convex A \minus{}gon

V(A).

For each round subset let

r(A)

be the number of points from

M

which are exterior from the convex A \minus{}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

M

construct the polynomial P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. Show that the sum of polynomials for all round subsets is exactly the polynomial P(x) \equal{} 1.
Proposed by Federico Ardila, Colombia

Problem Statement