HMMT 十一月 2011 · 冲刺赛 · 第 9 题

英文解析

9. [ 7 ] Unit circle Ω has points X, Y, Z on its circumference so that XY Z is an equilateral triangle. Let
   W be a point other than X in the plane such that triangle W Y Z is also equilateral. Determine the area of the region inside triangle W Y Z that lies outside circle Ω.
   √
   3 3 − π
   °
   Answer: Let O be the center of the circle. Then, we note that since ∠ W Y Z = 60 = ∠ Y XZ ,
   that Y W is tangent to Ω. Similarly, W Z is tangent to Ω. Now, we note that the circular segment 3
   corresponding to Y Z is equal to the area of Ω less the area of triangle OY Z . Hence, our total area 1
   is 3
   √ √ √
   1 3 3 1 3 3 3 − π
   [ W Y Z ] − [Ω] + [ Y OZ ] = − π + = .
   3 4 3 4 3
   Ω Y
   X O W1 Z