# Geometry, Algebra, Combinatorics, MIXED?

Algebra Level 5

Let $$C_{0}, C_{1}, C_{2},...$$ be a sequence of circles in the Cartesian plane defined as:

1. $$C_{0}$$is the circle $$x^{2} + y^{2} = 1$$
2. For $$n = 0, 1, 2, 3, ...$$, the circle $$C_{n+1}$$ lies in the upper half plane and is tangent to $$C_{n}$$ as well as both branches of the hyperbola $$x^{2} - y^{2} = 1$$

Let $$r_{n}$$ be the radius of $$C_{n}$$. Find the length of $$r_{10}$$

Please use Wolfram Alpha only in the last step. This is a modified APMO problem.

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