2009 AMC 12B 第 20 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
20.
一个凸多面体 有顶点 ，，，，并有 条边。用平面 ，，， 切割该多面体，使得平面 只切过与顶点 相接的那些边。此外，没有两个平面在 的内部或表面相交。这些切割产生了 个棱锥和一个新的多面体 。 有多少条边？
A convex polyhedron has vertices and edges. The polyhedron is cut by planes in such a way that plane cuts only those edges that meet at vertex In addition, no two planes intersect inside or on The cuts produce pyramids and a new polyhedron How many edges does have?
小提示：
每条原来的边都在靠近两个端点处各被切一次，产生新的顶点。
Each original edge is cut once near each of its two endpoints, creating new vertices
大提示：
计算新产生的边（每个边端点对应一条）以及每条原边保留下来的中间部分。
Count the new edges (one per edge-endpoint) plus the surviving middle portion of each original edge
解答：
条边中的每一条都在靠近两个端点处各被切一次，所以 有 个顶点。
在顶点 处的切割会形成一个小多边形，其边数等于 的度数。对所有顶点求和得到 ，即所有边端点的总数。每条原边的中间部分也保留下来，又增加 条边。因此 有 条边。
所以正确答案是 C。
Each of the edges is cut once near each endpoint, so has vertices.
The cut at vertex creates a small polygon whose number of edges equals the degree of ; summed over all vertices this is the total number of edge-endpoints. The middle portion of each original edge also survives, adding edges. So has edges.
Thus, the correct answer is C.
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