HMMT 二月 2006 · 冲刺赛 · 第 9 题
HMMT February 2006 — Guts Round — Problem 9
题目详情
- [6] Four unit circles are centered at the vertices of a unit square, one circle at each vertex.
What is the area of the region common to all four circles?
IX HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUNDth 2
英文原题
[6] Four unit circles are centered at the vertices of a unit square, one circle at each vertex.
What is the area of the region common to all four circles?
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IX th HARVARD-MIT MATHEMATICS TOURNAMENT, 25 FEBRUARY 2006 — GUTS ROUND
解析
英文解析
- Four unit circles are centered at the vertices of a unit square, one circle at each vertex.
What is the area of the region common to all four circles?
√
Answer: + 1 − 3π
Solution: The desired region consists of a small square and four “circle segments,”3
i.e. regions of a circle bounded by a chord and an arc. The side of this small square
°
is just the chord of a unit circle that cuts off an angle of 30 , and the circle segmentsare bounded by that chord and the circle. Using the law of cosines (in an isosceles
°
triangle with unit leg length and vertex angle 30 ), we find that the square of the
√
length of the chord is equal to 2 − 3. We can also compute the area of each circleπ 1 π 1
°
segment, namely − (1)(1) sin 30 = − . Hence, the desired region has area
12 2 12 4
( )
√
√
π 1 π
2 − 3 + 4 − = + 1 − 3.
12 4 3 2