CMO, 2003

Solution:
Let $f(x, y, z)=\left(x^{3}-x\right)+\left(y^{3}-y\right)+\left(z^{3}-z\right)$. The first equation above is equivalent to $f(x, y, z)=0$. If $x, y, z \geq 1$, then $f(x, y, z) \geq 0$ with equality only if $x=y=z=1$. But if $x=y=z=1$, then the second equation is not satisfied. So in any solution to the system of equations, at least one of the variables is less than 1. Without loss of generality, suppose that $x<1$. Then
$$x^{2}+y^{2}+z^{2}>y^{2}+z^{2} \geq 2 y z>y z>x y z .$$
Therefore the system has no real positive solutions.

Solution:
We will show that the system has no real positive solution. Assume otherwise.
The second equation can be written $x^{2}-(y z) x+\left(y^{2}+z^{2}\right)$. Since this quadratic in $x$ has a real solution by hypothesis, its discriminant is nonnegative. Hence
$$y^{2} z^{2}-4 y^{2}-4 z^{2} \geq 0$$
Dividing through by $4 y^{2} z^{2}$ yields
$$\frac{1}{4} \geq \frac{1}{y^{2}}+\frac{1}{z^{2}} \geq \frac{1}{y^{2}}$$
Hence $y^{2} \geq 4$ and so $y \geq 2$, $y$ being positive. A similar argument yields $x, y, z \geq 2$. But the first equation can be written as
$$x\left(x^{2}-1\right)+y\left(y^{2}-1\right)+z\left(z^{2}-1\right)=0$$
contradicting $x, y, z \geq 2$. Hence, a real positive solution cannot exist.

Solution:
Applying the arithmetic-geometric mean inequality and the Power Mean Inequalities to $x, y, z$ we have
$$\sqrt[3]{x y z} \leq \frac{x+y+z}{3} \leq \sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}} \leq \sqrt[3]{\frac{x^{3}+y^{3}+z^{3}}{3}}$$
Letting $S=x+y+z=x^{3}+y^{3}+z^{3}$ and $P=x y z=x^{2}+y^{2}+z^{2}$, this inequality can be written
$$\sqrt[3]{P} \leq \frac{S}{3} \leq \sqrt{\frac{P}{3}} \leq \sqrt[3]{\frac{S}{3}}$$
Now $\sqrt[3]{P} \leq \sqrt{\frac{P}{3}}$ implies $P^{2} \leq P^{3} / 27$, so $P \geq 27$. Also $\frac{S}{3} \leq \sqrt[3]{\frac{S}{3}}$ implies $S^{3} / 27 \leq S / 3$,
so $S \leq 3$. But then $\sqrt[3]{P} \geq 3$ and $\sqrt[3]{\frac{S}{3}} \leq 1$ which is inconsistent with $\sqrt[3]{P} \leq \sqrt[3]{\frac{S}{3}}$. Therefore the system cannot have a real positive solution.