HMMT 二月 2006 · 冲刺赛 · 第 6 题

HMMT February 2006 — Guts Round — Problem 6

[6] Suppose ABC is a triangle such that AB = 13 , BC = 15, and CA = 14. Say D is the midpoint of BC , E is the midpoint of AD , F is the midpoint of BE , and G is the midpoint of DF . Compute the area of triangle EF G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUND ⌊ ⌋ ⌊ ⌋ x x 2

解析

Suppose ABC is a triangle such that AB = 13 , BC = 15, and CA = 14. Say D is the midpoint of BC , E is the midpoint of AD , F is the midpoint of BE , and G is the midpoint of DF . Compute the area of triangle EF G . 21 Answer: 4 √ Solution: By Heron’s formula, [ ABC ] = 21(21 − 15)(21 − 14)(21 − 13) = 84. [ DEF ] [ BDE ] [ ABD ] Now, unwinding the midpoint conditions yields [ EF G ] = = = = 2 4 8 [ ABC ] 84 21 = = . 16 16 4