HMMT 二月 2013 · 冲刺赛 · 第 31 题
HMMT February 2013 — Guts Round — Problem 31
题目详情
英文原题
- [ 20 ] Let ABCD be a quadrilateral inscribed in a unit circle with center O . Suppose that ∠ AOB =
° ′ ′
∠ COD = 135 , BC = 1. Let B and C be the reflections of A across BO and CO respectively. Let
′ ′
H and H be the orthocenters of AB C and BCD , respectively. If M is the midpoint of OH , and
1 2 1
′ ′
O is the reflection of O about the midpoint of M H , compute OO . 2
解析
英文解析
- [ 20 ] Let ABCD be a quadrilateral inscribed in a unit circle with center O . Suppose that ∠ AOB =
° ′ ′
∠ COD = 135 , BC = 1. Let B and C be the reflections of A across BO and CO respectively. Let
′ ′
H and H be the orthocenters of AB C and BCD , respectively. If M is the midpoint of OH , and
1 2 1
′ ′
O is the reflection of O about the midpoint of M H , compute OO .
√ √2
Answer: (8 − 6 − 3 2) Put the diagram on the complex plane with O at the origin and A at 1
′ 2 ′ 24