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HMMT 二月 2003 · 冲刺赛 · 第 16 题

HMMT February 2003 — Guts Round — Problem 16

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [8] What fraction of the area of a regular hexagon of side length 1 is within distanceof at least one of the vertices?1 2
解析

英文解析

  1. What fraction of the area of a regular hexagon of side length 1 is within distance ofat least one of the vertices?2
    √
    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.