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HMMT 十一月 2023 · 冲刺赛 · 第 5 题

HMMT November 2023 — Guts Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] Let ABCDE be a convex pentagon such that AB + BC + CD + DE + EA = 64 and AC + CE + EB + BD + DA = 72 . Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of ABCDE .
解析
  1. [6] Let ABCDE be a convex pentagon such that AB + BC + CD + DE + EA = 64 and AC + CE + EB + BD + DA = 72 . Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of ABCDE . Proposed by: Arul Kolla Answer: 36 Solution: A B E D C By the midsegment theorem on triangles ABC , BCD , . . . , DEA , the side lengths of the said pentagons are AC/ 2, BD/ 2, CE/ 2, DA/ 2, and EB/ 2. Thus, the answer is AC + BD + CE + DA + EB 72 = = 36 . 2 2