Cool Algebra!

First, rationalize the denominator. $\large\frac{1}{\sqrt{2011+\sqrt{2011^{2}-1}}}$ $\times$ $\large\frac{\sqrt{2011-\sqrt{2011^{2}-1}}}{\sqrt{2011-\sqrt{2011^{2}-1}}}$

$\hookrightarrow$ $\sqrt{2011-\sqrt{2011^{2}-1}}$

Using the identity $\sqrt{a±\sqrt{b}}$ = $\large\sqrt{\frac{a+\sqrt{a^{2}-b}}{2}}$ ± $\large\sqrt{\frac{a-\sqrt{a^{2}-b}}{2}}$ ,

$\sqrt{2011-\sqrt{2011^{2}-1}}$ = $\large\sqrt{\frac{2011+\sqrt{2011^{2}-(2011^{2}-1)}}{2}}$ - $\large\sqrt{\frac{2011-\sqrt{2011^{2}-(2011^{2}-1)}}{2}}$

$\hookrightarrow$ $\large\sqrt{\frac{2011+\sqrt{1}}{2}}$ - $\large\sqrt{\frac{2011-\sqrt{1}}{2}}$

$\hookrightarrow$ $\large\sqrt{\frac{2012}{2}}$ - $\large\sqrt{\frac{2010}{2}}$

$\hookrightarrow$ $\large\sqrt{1006}$ - $\large\sqrt{1005}$

Hence, $m = 1006$ and $n = 1005$ .

$\boxed{m + n = 2011}$

Do square of rhs, to find 2*sqrt(mn) do sqrt(m) + sqrt(n) whole square -( sqrt(m)-sqrt(n))whole square whole by 4. To find sqrt m minus sqrt n, rationalise lhs. Use obtained term to find soln

Ràtionalise L.H.S. then square both sides and compare.