If $m$ and $n$ are positive real numbers satisfying the equation $$m+4\sqrt{mn}-2\sqrt{m}-4\sqrt{n}+4n=3$$
find the value of $$\frac{\sqrt{m}+2\sqrt{n}+2014}{4-\sqrt{m}-2\sqrt{n}}$$
I came across this question in a Math Olympiad Competition and had no idea how to solve it. Can anyone help? Thanks.
Best Answer
It can be helpful to get rid of the square root symbols. If you let $m=x^2$ and $n=y^2$ with $x$ and $y$ understood to be positive, the equation becomes $x^2+4xy-2x-4y+4y^2=3$, which can be rewritten as
$$(x+2y)^2-2(x+2y)=3$$ or $$u^2-2u-3=0\qquad\text{with}\quad u=x+2y$$
The quadratic factors as $(u-3)(u+1)=0$. Recalling that $x=\sqrt m$ and $y=\sqrt n$ are supposed to be positive, this gives $u=x+2y=3$ as the only meaningful solution. We also see that the value we're after is
$${x+2y+2014\over4-(x+2y)}={3+2014\over4-3}=2017$$