# Find The Maximum

Algebra Level 4

As $$x,y,z$$ range over all positive real numbers satisfying $$xy+yz+zx= xyz(x+y+z),$$ the maximum value of $\dfrac{1}{(2x+y+z)^2} + \dfrac{1}{(x+2y+z)^2} + \dfrac{1}{(x+y+2z)^2}$ can be expressed as $$\dfrac{a}{b},$$where $$a,b$$ are coprime positive integers. Find $$a+b.$$

Details and assumptions

• This problem is not original.
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