HMMT 二月 2012 · TEAM1 赛 · 第 7 题
HMMT February 2012 — TEAM1 Round — Problem 7
题目详情
- [ 30 ] Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?
解析
- [ 30 ] Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull? √ 3 Answer: Let the points be A, B, C, X, Y so that X and Y are on opposite sides of the plane 2 defined by triangle ABC . The volume is 1 / 3 the product of the area of ABC and sum of the distances from X and Y to the plane defined by ABC . The area of ABC is maximized when the plane containing it intersects the sphere in the largest possible cross section, and ABC is equilateral: this gives an area √ of 3 3 / 4. Then, the sum of the distances from X and Y to the plane of ABC is at most 2. This is clearly obtainable when A, B, and C form an equilateral triangle circumscribed by the equator of the √ sphere, and X and Y are at the poles, and we get a volume of 3 / 2.