HMMT 二月 2003 · 冲刺赛 · 第 16 题
HMMT February 2003 — Guts Round — Problem 16
题目详情
- [8] What fraction of the area of a regular hexagon of side length 1 is within distance 1 of at least one of the vertices? 2
解析
- What fraction of the area of a regular hexagon of side length 1 is within distance of 2 at least one of the vertices? √ Solution: π 3 / 9 √ √ 2 ◦ The hexagon has area 6( 3 / 4)(1) = 3 3 / 2. The region we want consists of six 120 arcs of circles of radius 1 / 2, whichcan be reassembled into two circles of radius 1 / 2. So √ its area is π/ 2, and the ratio of areas is π 3 / 9.