HMMT 二月 2006 · GEN1 赛 · 第 9 题

英文解析

9. Take a unit sphere S , i.e., a sphere with radius 1. Circumscribe a cube C about S ,
   and inscribe a cube D in S , so that every edge of cube C is parallel to some edge ofcube D . What is the shortest possible distance from a point on a face of C to a pointon a face of D ?
   √ √
   3 − 3 3
   Answer: = 1 − , or equivalent
   3 3
   Solution: Using the Pythagorean theorem, we know that the length of a diagonal of
   √
   a cube of edge length s is s 3. Since D is inscribed in a sphere that has diameter 2,
   √
   this means that its side length is 2 / 3.
   The distance from a face of D to a face of C will be the distance between them alongany line perpendicular to both of them; take such a line passing through the center of
   S . The distance from the center to any face of D along this line will be half the side
   √
   length of D , or 1 / 3. The distance from the center to the edge of C is the radius of
   √
   √
   S , which is 1. Therefore the desired distance is 1 − 1 / 3 = 1 − .3 3