AM-GM Won't Work Here

Algebra Level 5

x5y+y5z+z5x\large x^5 y + y^5 z + z^5 x

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

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

Bonus: Generalize this for the expression xny+ynz+znxx^n y + y^n z + z^n x.


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