HMMT 二月 2021 · 冲刺赛 · 第 22 题

HMMT February 2021 — Guts Round — Problem 22

[14] Let E be a three-dimensional ellipsoid. For a plane p , let E ( p ) be the projection of E onto the plane p . The minimum and maximum areas of E ( p ) are 9 π and 25 π , and there exists a p where E ( p ) is a circle of area 16 π . If V is the volume of E , compute V /π . 2

解析

[14] Let E be a three-dimensional ellipsoid. For a plane p , let E ( p ) be the projection of E onto the plane p . The minimum and maximum areas of E ( p ) are 9 π and 25 π , and there exists a p where E ( p ) is a circle of area 16 π . If V is the volume of E , compute V /π . Proposed by: Daniel Zhu Answer: 75 Solution: Let the three radii of E be a < b < c . We know that ab = 9 and bc = 25. Consider the plane p where projection E ( p ) has area 9 π . Fixing p , rotate E on the axis passing through the radius with length b until E ( p ) has area 25 π . The projection onto p will be an ellipse with radii b and r , where r increases monotonically from a to c . By Intermediate Value Theorem, there must exist a circular projection with radius b . As the area of this projection is 16 π , b = 4. Thus, 4 4 225 V = π · abc = · π = 75 π. 3 3 4