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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