# Lopsided ugly inequality

Algebra Level 5

The smallest value of $$x$$ which satisfies the inequality

$\frac{ (m-n)^2 } { m \left(\frac{ m+n}{2} - \sqrt{mn} \right) } < x$

for all pairs of positive integers $$(m,n)$$ such that $$m > n$$, can be expressed in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. What is the value of $$a+b$$?

Details and assumptions

The arithmetic mean of 2 numbers $$m$$ and $$n$$ is $$\frac{m+n}{2}$$.

The geometric mean of 2 numbers $$m$$ and $$n$$ is $$\sqrt{mn}$$.

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