HMMT 十一月 2012 · 冲刺赛 · 第 12 题
HMMT November 2012 — Guts Round — Problem 12
题目详情
- [ 8 ] Three circles of radius 1 are drawn, whose centers form the vertices of an equilateral triangle ofside length 1. Find the area of the region common to at least two of the circles.
HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND
°
英文原题
[ 8 ] Three circles of radius 1 are drawn, whose centers form the vertices of an equilateral triangle of
side length 1. Find the area of the region common to at least two of the circles.
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HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND
解析
英文解析
- [ 8 ]
√
2 π − 3
Answer: Let A be the equilateral triangle. Let A be the area of the region outside of
△ 1
the equilateral triangle but inside the second and third circles. Define A , A analogously. We have 2
2 3
A = A = A = A =
1 2 3 k
√
( )
2 2
1 · π 1 · sin 120 4 π − 3 3 − = ,
3 2 12
√and
1 · sin 60 32
A = = .
△
2 4
Thus, the total area is
√ √ √
4 π − 3 3 3 2 π − 3
A + A + A + A = 3 · + = .
1 2 3 △
12 4 2