Lopsided ugly inequality

Algebra Level 5

(mn)2m(m+n2mn)<x \frac{ (m-n)^2 } { m \left(\frac{ m+n}{2} - \sqrt{mn} \right) } < x

The smallest value of xx which satisfies the inequality above for all pairs of positive integers (m,n) (m,n) such that m>n m > n , can be expressed in the form ab \frac{a}{b} , where aa and bb are positive coprime integers. What is the value of a+ba+b?

Details and assumptions

The arithmetic mean of 2 numbers mm and nn is m+n2 \frac{m+n}{2} .

The geometric mean of 2 numbers mm and nn is mn \sqrt{mn} .

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