VN-TST 1998 Problem 2

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$ . The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$ . Prove that for each positive integer $n \geq 3$ , the equation: $$(p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8)$$ is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$ , and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$ .