Question

一个凸多面体 $P$ 有 $26$ 个顶点、 $60$ 条边和 $36$ 个面，其中 $24$ 个是三角形， $12$ 个是四边形。空间对角线是连接两个不相邻且不属于同一个面的顶点的线段。 $P$ 有多少条空间对角线？

A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular, and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $P$ have?

答案

答案：241

知识点：多面体对角线补集计数

难度评级：2070

解答：

任意一对顶点恰好决定三类对象之一：一条边、一个面的对角线，或一条空间对角线。顶点对总数为 $\binom{26}{2} = 325$ 。

其中 $60$ 对是边。 $24$ 个三角形面没有对角线，而 $12$ 个四边形面各有 $2$ 条对角线，共 $24$ 条面对角线（由于多面体是凸的，没有两个面会共享同一条对角线）。

空间对角线的条数为 $325 - 60 - 24 = 241$ 。

Every pair of vertices determines exactly one of three things: an edge, a diagonal of a face, or a space diagonal. There are $\binom{26}{2} = 325$ pairs of vertices in all.

Of these, $60$ are edges. The $24$ triangular faces have no diagonals, while each of the $12$ quadrilateral faces has $2,$ for $24$ face diagonals (no two faces share a diagonal, since the polyhedron is convex).

The number of space diagonals is $325 - 60 - 24 = 241.$

2004 AIME I 第 3 题

答案：241

难度评级：2070

解答：

其他年份的第 3 题