HMMT 二月 2005 · 几何 · 第 5 题

HMMT February 2005 — Geometry — Problem 5

A cube with side length 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube?

解析

A cube with side length 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube? Solution: 2 / 3 √ First note that the long diagonal of the cube has length 2 3, so the radius of the √ sphere is 3. Let x be the side length of the smaller cube. Then the distance from the center of the sphere to the far face of the smaller cube is 1 + x , while the distance from √ x 2 the center of the far face to a vertex lying on the sphere is . Therefore, the square 2 2 2 x 2 2 of the radius is 3 = (1 + x ) + , or 3 x + 4 x − 4 = (3 x − 2)( x + 2) = 0, so x = . 2 3