Mathematical Olympiad in Ukraine

From the first two equations of our system we obtain: $x^3 - y^3 = 2y^3 - 2z^3 + y - z = (y-z)(2y^2 + 2yz + 2z^2 + 1)$, analogously, we can easily get equalities: $y^3 - z^3 = (z-x)(2z^2 + 2zx + 2x^2 + 1)$, $z^3 - x^3 = (x-y)(2x^2 + 2xy + 2y^2 + 1)$. It is easy to see that $(2y^2 + 2yz + 2z^2 + 1) = y^2 + z^2 + (y+z)^2 + 1 > 0$. If we suppose that $y > z$, then from the first equality we have that $x > y$, then applying third equality we conclude: $z > x > y > z$ - contradiction. The case when $y < z$ is completely analogous. Hence we obtain $x = y = z$. and solving the equation we get the answer $(1,1,1)$.