Maximising Can Also Be Difficult!

Algebra Level 4

\[\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} = a+b+c\]

Let \(a,b,c\) be positive real numbers such that the above equation is satisfied. If the maximum value of the expression below is in the form of \( \frac {m}{n} ,\) where \(m,n\) are coprime positive integers, what is \(m+n?\)

\[\dfrac{1}{\left(2a+b+c\right)^2}+\dfrac{1}{\left(2b+c+a\right)^2}+\dfrac{1}{\left(2c+b+a\right)^2} \]

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