1. For which values of the real parameter $a$ does the system of equations

$$
x+y=a^{3}-a \quad \text { and } \quad x y=a^{2}
$$

have real solutions $x$ and $y$?
II/1. Express $x=a^{3}-a-y$, substitute $x$ into the second equation and we get $\left(a^{3}-a-y\right) y=a^{2}$ or $y^{2}+y\left(a-a^{3}\right)+a^{2}=0$. The quadratic equation has real solutions if its discriminant is non-negative, that is $\left(a-a^{3}\right)^{2}-4 a^{2} \geq 0$. Simplify the inequality and we get $a^{2}\left(a^{2}+1\right)\left(a^{2}-3\right) \geq 0$. The inequality holds if $a=0$ or $a^{2} \geq 3$.