2021 CMO 第 6 题

$P$ is a convex polyhedron such that:**(1)** every vertex belongs to exactly $3$ faces.**(1)** For every natural number $n$ , there are even number of faces with $n$ vertices.

An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$ , the number of faces with $n$ vertices on each side are the same. (assume this is possible)

Show that the number of times the ant turns left is the same as the number of times the ant turn right.

$P$ 是一个凸多面体，这样：**(1)** 每个顶点恰好属于 $3$ 个面。**(1)** 对于每个自然数 $n$ ，有偶数个具有 $n$ 个顶点的面。

一只蚂蚁沿着 $P$ 的边缘行走，形成一个非自相交循环，将这个多面体的面分为两侧，这样对于每个自然数 $n$ ，每侧具有 $n$ 顶点的面的数量是相同的。 (假设这是可能的)

证明蚂蚁向左转的次数与蚂蚁向右转的次数相同。