HMMT 二月 2005 · 冲刺赛 · 第 34 题

HMMT February 2005 — Guts Round — Problem 34

[12] A regular octahedron ABCDEF is given such that AD , BE , and CF are per- pendicular. Let G , H , and I lie on edges AB , BC , and CA respectively such that AG BH CI = = = ρ . For some choice of ρ > 1, GH , HI , and IG are three edges of a GB HC IA regular icosahedron, eight of whose faces are inscribed in the faces of ABCDEF . Find ρ . 24036583

解析

A regular octahedron ABCDEF is given such that AD , BE , and CF are perpen- dicular. Let G , H , and I lie on edges AB , BC , and CA respectively such that AG BH CI = = = ρ . For some choice of ρ > 1, GH , HI , and IG are three edges GB HC IA of a regular icosahedron, eight of whose faces are inscribed in the faces of ABCDEF . Find ρ . √ Solution: (1 + 5) / 2 EJ Let J lie on edge CE such that = ρ . Then we must have that HIJ is another face of JC the icosahedron, so in particular, HI = HJ . But since BC and CE are perpendicular, √ 2 2 2 ◦ HJ = HC 2. By the Law of Cosines, HI = HC + CI − 2 HC · CI cos 60 = √ 1+ 5 2 2 2 2 HC (1 + ρ − ρ ). Therefore, 2 = 1 + ρ − ρ , or ρ − ρ − 1 = 0, giving ρ = . 2 A I G E B J H C D 24036583