AM-GM Won't Work Here

Algebra Level 5

\[\large x^5 y + y^5 z + z^5 x \]

Let \(x, y,\) and \(z\) be non-negative reals such that \(x+y+z=1\).

The maximum value of the above expression can be represented as \(\dfrac {a^b}{c^d}\), where \(a\) and \(c\) are not perfect powers, and \(a,b,c,d\) are positive integers. Find the value of \(a+b+c+d\).

\[\] Bonus: Generalize this for the expression \(x^n y + y^n z + z^n x\).


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