HMMT 二月 2018 · 几何 · 第 4 题

HMMT February 2018 — Geometry — Problem 4

A paper equilateral triangle of side length 2 on a table has vertices labeled A , B , C . Let M be the point on the sheet of paper halfway between A and C . Over time, point M is lifted upwards, folding the triangle along segment BM , while A , B , and C remain on the table. This continues until A and C touch. Find the maximum volume of tetrahedron ABCM at any time during this process.

解析

A paper equilateral triangle of side length 2 on a table has vertices labeled A , B , C . Let M be the point on the sheet of paper halfway between A and C . Over time, point M is lifted upwards, folding the triangle along segment BM , while A , B , and C remain on the table. This continues until A and C touch. Find the maximum volume of tetrahedron ABCM at any time during this process. Proposed by: Dhruv Rohatgi √ 3 Answer: 6 View triangle ABM as a base of this tetrahedron. Then relative to triangle ABM , triangle CBM rotates around segment BM on a hinge. Therefore the volume is maximized when C is farthest from triangle ABM , which is when triangles ABM and CBM are perpendicular. The volume in this case √ √ 1 3 can be calculated using the formula for the volume of a tetrahedron as · 1 · 1 · 3 = . 6 6