World Tour (Japan)

Algebra Level 4

\[ {\displaystyle \left(\sum _{n=1}^{99}\sqrt{10+\sqrt{n}} \right)} \div { \displaystyle \left(\sum _{n=1}^{99}\sqrt{10-\sqrt{n}}\right) } \]

If the expression above is equal to \( a+b\sqrt{c}\) for integers \(a,b,c\) with \(c\) square-free, find \(a+b+c\).

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