HMMT 二月 2025 · 冲刺赛 · 第 14 题

HMMT February 2025 — Guts Round — Problem 14

[9] A parallelogram P can be folded over a straight line so that the resulting shape is a regular pentagon with side length 1. Compute the perimeter of P . ◦ ◦

解析

[9] A parallelogram P can be folded over a straight line so that the resulting shape is a regular pentagon with side length 1. Compute the perimeter of P . Proposed by: Arul Kolla √ Answer: 5 + 5 Solution: ′ ′ E D A B E C D ′ ′ In regular pentagon ABCDE (labeled clockwise), reflect ABDE across AB to obtain ABD E . Then, ′ ′ ′ CDE D is one such parallelogram P . The length of CD is √ √ 5+1 5+3 CB + BD = 1 + 2 cos ∠ CBD = 1 + 2 cos( π/ 5) = 1 + = . 2 2 √ √ 5+3 Hence, the perimeter of the desired parallelogram is 2 1 + = 5 + 5 . 2 ◦ ◦