HMMT 二月 2016 · 冲刺赛 · 第 3 题

英文解析

3. [ 5 ] Let P ROBLEM Z be a regular octagon inscribed in a circle of unit radius. Diagonals M R , OZmeet 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.