HMMT 二月 2012 · TEAM1 赛 · 第 7 题

英文解析

7. [ 30 ] Five points are chosen on a sphere of radius 1. What is the maximum possible volume of theirconvex hull?
   √
   Answer: Let the points be A, B, C, X, Y so that X and Y are on opposite sides of the plane 3
   defined by triangle ABC . The volume is 1 / 3 the product of the area of ABC and sum of the distances 2
   from X and Y to the plane defined by ABC . The area of ABC is maximized when the plane containingit 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 isclearly 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.