HMMT 二月 2005 · GEN2 赛 · 第 6 题

英文解析

6. A cube with side length 2 is inscribed in a sphere. A second cube, with faces parallelto 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 x 2 222
   of the radius is 3 = (1 + x ) + , or 3 x + 4 x − 4 = (3 x − 2)( x + 2) = 0, so x = .
   2 3