# Circus

Geometry Level 5

The Red and Blue circles share the same center.

The radius of Red circle equals $\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}}$ and the radius of Blue circle equals $3 - \frac{1}{4} + \frac{1\cdot3}{4\cdot6} - \frac{1\cdot3\cdot5}{4\cdot6\cdot8} + \cdots$ If the hatched area equals $\bigg(\frac{L}{M\sqrt{N}}-1\bigg)\pi+M$ where $$L,M,N$$ are positive co-prime integers with $$N$$ being square-free. Find $$L+M+N$$.

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