Beautiful But So Dangerous

Algebra Level 4

\[ \small \frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\]

Over all positive reals \(x, y \) and \(z\), the maximum of the expression above can be expressed as \(\dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).


Source: VMO 2014 problem 2 day 2.
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