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HMMT 二月 2016 · 冲刺赛 · 第 3 题

HMMT February 2016 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Let P ROBLEM Z be a regular octagon inscribed in a circle of unit radius. Diagonals M R , OZ meet at I . Compute LI .
解析
  1. [ 5 ] Let P ROBLEM Z be a regular octagon inscribed in a circle of unit radius. Diagonals M R , OZ meet at I . Compute LI . Proposed by: Evan Chen √ Answer: 2 If W is the center of the circle then I is the incenter of 4 RW Z . Moreover, P RIZ is a rhombus. It √ √ √ follows that P I is twice the inradius of a 1-1- 2 triangle, hence the answer of 2 − 2. So LI = 2. Alternatively, one can show (note, really) that the triangle OIL is isosceles.