HMMT 二月 2020 · 几何 · 第 1 题

HMMT February 2020 — Geometry — Problem 1

Let DIAL , F OR , and F RIEN D be regular polygons in the plane. If ID = 1, find the product of all possible areas of OLA . ◦

解析

Let DIAL , F OR , and F RIEN D be regular polygons in the plane. If ID = 1, find the product of all possible areas of OLA . Proposed by: Andrew Gu 1 Answer: 32 Solution: Focusing on F RIEN D and F OR first, observe that either DIO is an equilateral triangle or O is the midpoint of ID . Next, OLA is always an isosceles triangle with base LA = 1. The possible √ 3 distances of O from LA are 1 and 1 ± as the distance from O to ID in the equilateral triangle case 2 √ 3 is . 2 A A I 1 2 R E O 2 O 1 F N L L D 1 2 The three possibilities are shown in the diagram as shaded triangles 4 O L A , 4 O L A , and 1 1 1 2 2 2 4 O L A . 1 2 2 The product of all possible areas is thus ( ) ( ) √ √ 3 3 1 · 1 − · 1 + 2 2 1 1 = = . 3 5 2 2 32 ◦