Maximising Can Also Be Difficult!

Algebra Level 3

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

Let a,b,ca,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 mn, \frac {m}{n} , where m,nm,n are coprime positive integers, what is m+n?m+n?

1(2a+b+c)2+1(2b+c+a)2+1(2c+b+a)2\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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