HMMT 十一月 2022 · 团队赛 · 第 2 题

HMMT November 2022 — Team Round — Problem 2

[25] What is the smallest r such that three disks of radius r can completely cover up a unit disk?

解析

[25] What is the smallest r such that three disks of radius r can completely cover up a unit disk? Proposed by: Gabriel Wu √ 3 Answer: 2 Solution: Look at the circumference of the unit disk. Each of the disks must be capable of covering √ 1 up at least of the circumference, which means it must be able to cover a chord of length 3. Thus, 3 √ 3 is a lower bound for r . This bound is achievable: place the three centers of the disks symmetrically 2 1 at a distance of from the center of the unit disk. 2