HMMT 十一月 2008 · 冲刺赛 · 第 30 题

HMMT November 2008 — Guts Round — Problem 30

[ 14 ] Alice has an equilateral triangle ABC of area 1. Put D on BC , E on CA , and F on AB , with BD = DC , CE = 2 EA , and 2 AF = F B . Note that AD, BE , and CF pass through a single point M . What is the area of triangle EM C ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND 2 2

解析

[ 14 ] Alice has an equilateral triangle ABC of area 1. Put D on BC , E on CA , and F on AB , with BD = DC , CE = 2 EA , and 2 AF = F B . Note that AD, BE , and CF pass through a single point M . What is the area of triangle EM C ? 1 Answer: Triangles ACF and BCF share a height, so the ratio of their areas is AF/BF = 1 / 2. 6 By the same method, the ratio of the areas of AM F and BM F is 1 / 2. So, the ratio of the areas of 5 ACM and BCM is also 1 / 2. Similarly, the ratio of the areas of ABM and BCM is 1 / 2. But the sum 1 of the areas of ACM , BCM , and ABM is 1, so the area of ACM is . Then the area of EM C is 2 / 3 4 the area of ACM , because they share heights, so their areas are in the same ratio as their bases. The 2 · 1 1 area of EM C is then = . 3 · 4 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND 2 2