HMMT 二月 2001 · 冲刺赛 · 第 10 题

英文解析

10. [8] Two concentric circles have radii r and R > r . Three new circles are drawn sothat they are each tangent to the big two circles and tangent to the other two new circles.
    Find .Rr
    Solution: The centers of the three new circles form a triangle. The diameter of the newcircles is R − r , so the side length of the triangle is R − r . Call the center of the concentriccircle O , two vertices of the triangle A and B , and AB ’s midpoint D . OA is the average Rsin(30) sin(90)
    R + rand r , namely . Using the law of sines on triangle DAO , we get = ⇒ R = 3 r ,
    2 AD AOso = 3 .Rr