2009 CMO 第 3 题

Given two integers $m,n$ satisfying $4 < m < n.$ Let $A_{1}A_{2}\cdots A_{2n + 1}$ be a regular $2n+1$ polygon. Denote by $P$ the set of its vertices. Find the number of convex $m$ polygon whose vertices belongs to $P$ and exactly has two acute angles.

给定两个整数 $m,n$ 满足 $4 < m < n.$ 令 $A_{1}A_{2}\cdots A_{2n + 1}$ 为正则 $2n+1$ 多边形。用 $P$ 表示其顶点集。求顶点属于$P$并且恰好有两个锐角的凸$m$多边形的数量。