HMMT 二月 2023 · 冲刺赛 · 第 15 题

HMMT February 2023 — Guts Round — Problem 15

[14] Let A and B be points in space for which AB = 1. Let R be the region of points P for which AP ≤ 1 and BP ≤ 1. Compute the largest possible side length of a cube contained within R . 2 2

解析

[14] Let A and B be points in space for which AB = 1. Let R be the region of points P for which AP ≤ 1 and BP ≤ 1. Compute the largest possible side length of a cube contained within R . Proposed by: Henry Stennes √ 10 − 1 Answer: 3 Solution: Let h be the distance between the center of one sphere and the center of the opposite face of the cube. Let x be the side length of the cube. Then we can draw a right triangle by connecting the center of the sphere, the center of the opposite face of the cube, and one of the vertices that make √ 2 x 2 2 up that face. This gives us h + ( ) = 1. Because the centers of the spheres are 1 unit apart, 2 √ √ 1 1 1 1 2 x 10 − 1 2 2 h = x + , giving us the quadratic ( x + ) + ( ) = 1. Solving yields x = . 2 2 2 2 2 3 2 2