HMMT 二月 2006 · GEN1 赛 · 第 9 题
HMMT February 2006 — GEN1 Round — Problem 9
题目详情
- 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 of cube D . What is the shortest possible distance from a point on a face of C to a point on a face of D ?
解析
- 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 of cube D . What is the shortest possible distance from a point on a face of C to a point on 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 along any 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 √ √ 3 S , which is 1. Therefore the desired distance is 1 − 1 / 3 = 1 − . 3