HMMT 二月 2013 · 冲刺赛 · 第 31 题

HMMT February 2013 — Guts Round — Problem 31

[ 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

[ 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 √ √ 1 Answer: (8 − 6 − 3 2) Put the diagram on the complex plane with O at the origin and A at 4 ′ 2 ′ 2