PUMaC 2009 · 几何（A 组） · 第 3 题
PUMaC 2009 — Geometry (Division A) — Problem 3
题目详情
- A polygon is called concave if it has at least one angle strictly greater than 180 . What is the maximum number of symmetries that an 11-sided concave polygon can have?
解析
- A polygon is called concave if it has at least one angle strictly greater than 180 . What is the maximum number of symmetries that an 11-sided concave polygon can have? Solution. 1. An 11-gon can have only axes passing through a vertex and an opposite side. If it had exactly 2 such axes, they would have to be perpendicular and the polygon would have ◦ an even number of sides. If it had exactly 3 axes, each would have to be at an angle of 60 with the next one, and the number of sides of the polygon would be a multiple of 3. Since our polygon has a concave angle, any number of axes of symmetry above 3 would imply at least 12 sides.