3. Let $C$ be the unit circle. Four distinct, smaller congruent circles $C_{1}, C_{2}, C_{3}, C_{4}$ are internally tangent to $C$ such that $C_{i}$ is externally tangent to $C_{i-1}$ and $C_{i+1}$ for $i=$ $1, \ldots, 4$ where $C_{5}$ denotes $C_{1}$ and $C_{0}$ represents $C_{4}$. Compute the radius of $C_{1}$.
Answer: $\sqrt{2}-1$
Solution: Let $O$ and $O^{\prime}$ be the centers of $C$ and $C_{1}$ respectively, and let $C_{1}$ be tangent to $C, C_{2}, C_{4}$ at $P, Q$, and $R$ respectively. Observe that $Q O R O^{\prime}$ is a square and that $P, O^{\prime}$, and $O$ are collinear. Thus, if $r$ is the desired radius, $1=r+O O^{\prime}=r+r \sqrt{2}$, so that $r=\frac{1}{\sqrt{2}+1}=\sqrt{2}-1$.