Algebra Level 5

Let $$x,y,z$$ be positive real numbers such that $$x^2 + y^2 + z^2 = 2(xy + yz+zx)$$. Find the maximum value of

$\frac{x^{2016}+y^{2016}+z^{2016}}{(x+y+z)\big(x^2+y^2+z^2\big) \ldots \big(x^{62}+y^{62}+z^{62}\big)\big(x^{63}+y^{63}+z^{63}\big)}.$

 Bonus: Generalize this. That is, find the conditions for $$x,y,z$$ such that $\frac{x^{N}+y^{N}+z^{N}}{\displaystyle \prod_j (x^{a_j}+y^{a_j}+z^{a_j})}$ attains its maximum given that $$\displaystyle \sum_{j}a_j = N$$.

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