Solution:
$x = y = z = 3$
Step by step solution:
We have:
$x^2 + y^2 + z^2 = xy + yz + zx$
$\to 2x^2 + 2y^2 + 2z^2 - 2xy + 2yz - 2zx = 0$
$\to (x-y)^2 + (y-z)^2 + (z - x)^2 = 0$
$\to \begin{cases}x - y = 0\\y - z = 0\\z - x = 0\end{cases}$
$\to x= y = z$
So that:
$x^{2015} + y^{2015} + z^{2015} = 9^{1008}$
$\Leftrightarrow 3.x^{2015} = 3^{2016}$
$\Leftrightarrow x^{2015} = 3^{2015}$
$\Leftrightarrow x = 3$
Therefore:
$x = y = z = 3$
