1. (HUN) Solve the following system of equations:
$$
\begin{aligned}
x+y+z & =a, \\
x^{2}+y^{2}+z^{2} & =b^{2}, \\
x y & =z^{2},
\end{aligned}
$$
where $a$ and $b$ are given real numbers. What conditions must hold on $a$ and $b$ for the solutions to be positive and distinct?
1. This is a problem solvable using elementary manipulations, so we shall state only the final solutions. For $a=0$ we get $(x, y, z)=(0,0,0)$. For $a \neq 0$ we get $(x, y, z) \in\left\{\left(t_{1}, t_{2}, z_{0}\right),\left(t_{2}, t_{1}, z_{0}\right)\right\}$, where
$$
z_{0}=\frac{a^{2}-b^{2}}{2 a} \quad \text { and } \quad t_{1,2}=\frac{a^{2}+b^{2} \pm \sqrt{\left(3 a^{2}-b^{2}\right)\left(3 b^{2}-a^{2}\right)}}{4 a} .
$$

For the solutions to be positive and distinct the following conditions are necessary and sufficient: $3 b^{2}>a^{2}>b^{2}$ and $a>0$.