PUMaC 2009 · 几何（B 组） · 第 5 题

PUMaC 2009 — Geometry (Division B) — Problem 5

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? (A symmetry of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.) (Please Turn Over) 1

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.