HMMT 二月 2005 · 冲刺赛 · 第 34 题
HMMT February 2005 — Guts Round — Problem 34
题目详情
英文原题
- [12] A regular octahedron ABCDEF is given such that AD , BE , and CF are perpendicular. 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 IAregular icosahedron, eight of whose faces are inscribed in the faces of ABCDEF . Findρ .
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解析
英文解析
- A regular octahedron ABCDEF is given such that AD , BE , and CF are perpendicular. 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 IAof a regular icosahedron, eight of whose faces are inscribed in the faces of ABCDEF .
Find ρ .
√
Solution: (1 + 5) / 2
Let J lie on edge CE such that = ρ . Then we must have that HIJ is another face of e jthe icosahedron, so in particular, HI = HJ . But since BC and CE are perpendicular,JC
√
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 = .
A2
BEGI
DCHJ
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