prmo 2012 question 12

$$ \text{If } \frac{1}{\sqrt{2011} + \sqrt{2011^2 - 1}} = \sqrt{m} - \sqrt{n}, \text{ where } m \text{ and } n \text{ are positive integers, what is the value of } m + n?
$$
Solution:
Multiply up and down with $$ {\sqrt{2011} - \sqrt{2011^2 - 1}} $$ to get: $$ {\sqrt{2011} - \sqrt{2011^2 - 1}} $$ If this is equal to: $$ \sqrt{m} - \sqrt{n} $$ then squaring both sides we get: $$ 2011 - \sqrt{2011^2 - 1} = m + n - 2\sqrt{mn} $$ so $$ 2011 - \sqrt{2012.2010} = m + n - 2\sqrt{mn} $$ and $$ 2011 - 2\sqrt{1006.1005} = m + n - 2\sqrt{mn} $$ and $$ 1006 + 1005 - 2\sqrt{1006.1005} = m + n - 2\sqrt{mn} $$ so m = 1006, n = 1005, m + n = 2011.

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