HMMT 十一月 2008 · 冲刺赛 · 第 30 题
HMMT November 2008 — Guts Round — Problem 30
题目详情
- [ 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 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst
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英文原题
[ 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 ?
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1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND
解析
英文解析
- [ 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 ?
Answer: Triangles ACF and BCF share a height, so the ratio of their areas is AF/BF = 1 / 2.1
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 6
ACM and BCM is also 1 / 2. Similarly, the ratio of the areas of ABM and BCM is 1 / 2. But the sum 5
of the areas of ACM , BCM , and ABM is 1, so the area of ACM is . Then the area of EM C is 2 / 31
the area of ACM , because they share heights, so their areas are in the same ratio as their bases. The 4
2 · 1 1
area of EM C is then = .
3 · 4 6
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst
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