ILL 1992 Problem 70

Let two circles $A$ and $B$ with unequal radii $r$ and $R$ , respectively, be tangent internally at the point $A_0$ . If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$ , and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$ , prove that $$\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.$$