HMMT 二月 2004 · GEN2 赛 · 第 9 题

英文解析

9. Given is a regular tetrahedron of volume 1. We obtain a second regular tetrahedron byreflecting 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 thebottom face, so the reflection of this face lies in a horizontal plane halfway between Aand BCD . In particular, it cuts off the smaller tetrahedron obtained by scaling theoriginal tetrahedron by a factor of 1 / 2 about A . Similarly, the reflections of the otherthree 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 thesefour smaller tetrahedra is in the intersection of ABCD with its reflection, since thereflection sends this piece to itself. So the answer we seek is just the volume of thispiece, which is
   (volume of ABCD ) − 4 · (volume of ABCD scaled by a factor of 1 / 2) = 1 − 4(1 / 2) = 1 / 2 .3