HMMT 二月 2004 · 几何 · 第 9 题

HMMT February 2004 — Geometry — Problem 9

Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

解析

Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? Solution: 1 / 2 Imagine placing the tetrahedron ABCD flat on a table with vertex A at the top. By vectors or otherwise, we see that the center is 3 / 4 of the way from A to the bottom face, so the reflection of this face lies in a horizontal plane halfway between A and BCD . In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of 1 / 2 about A . Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling ABCD by 1 / 2 about B , C , and D . On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of ABCD with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is (volume of ABCD ) − 4 · (volume of ABCD scaled by a factor of 1 / 2) 3 = 1 − 4(1 / 2) = 1 / 2 . 3