HMMT 二月 2005 · 冲刺赛 · 第 16 题

英文解析

16. A regular octahedron has a side length of 1. What is the distance between two oppositefaces?
    √
    Solution: 6 / 3
    Imagine orienting the octahedron so that the two opposite faces are horizontal. Projectonto a horizontal plane; these two faces are congruent equilateral triangles which (whenprojected) have the same center and opposite orientations. Hence, the vertices of theoctahedron project to the vertices of a regular hexagon ABCDEF .
    BOAF
    DCME
    Let O be the center of the hexagon and M the midpoint of AC . Now ABM is a
    √ √ √
    30-60-90 triangle, so AB = AM/ ( 3 / 2) = (1 / 2) / ( 3 / 2) = 3 / 3. If we let d denotethe desired vertical distance between the opposite faces (which project to ACE and
    √ √
    2 2 2
    BDF ), then by the Pythagorean Theorem, AB + d = 1 , so d = 1 − AB = 6 / 3.2