2006 IMO 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 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 $x$ $$\sum_{P} x^{a(P)}(1-x)^{b(P)}=1$$ 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)

令 $S$ 为平面上的有限点集，其中没有三个点在一条直线上。对于每个顶点在$S$内的凸多边形$P$，设$a(P)$为$P$的顶点数，设$b(P)$为$S$在$P$之外的点数。证明对于每个实数 $x$ $$\sum_{P} x^{a(P)}(1-x)^{b(P)}=1$$，其中顶点在 $S$ 中的所有凸多边形的总和。注意。线段、点和空集分别被视为具有 2,1 和 0 个顶点的凸多边形。 (哥伦比亚)