HMMT 二月 2009 · 冲刺赛 · 第 6 题

HMMT February 2009 — Guts Round — Problem 6

[ 6 ] Let ABC be a right triangle with hypotenuse AC . Let B be the reflection of point B across AC , ′ ′ ′ ′ ′ and let C be the reflection of C across AB . Find the ratio of [ BCB ] to [ BC B ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUND 3 5 7 9

解析

[ 6 ] Let ABC be a right triangle with hypotenuse AC . Let B be the reflection of point B across AC , ′ ′ ′ ′ ′ and let C be the reflection of C across AB . Find the ratio of [ BCB ] to [ BC B ]. ′ ′ ′ 1 ′ Answer: 1 Solution: Since C , B , and C are collinear, it is evident that [ BCB ] = [ BCC ]. It 2 ′ ′ ′ immediately follows that [ BCB ] = [ BC B ]. Thus, the ratio is 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUND 3 5 7 9