Croatian Mathematical Competitions

We have
$$\begin{aligned} a^5 + a + 1 &= (a^5 - a^4) + (a^4 - a^3) + (a^3 - a^2) + a^2 + a + 1 \\ &= (a^2 + a + 1)(a^3 - a^2 + 1). \end{aligned}$$
If $a^2 + a + 1 = 0$, then $a^2(a - 1) = -(a + 1)(a - 1) = 1 - a^2 = a + 2$.
Since $a = \frac{-1 \pm i\sqrt{3}}{2}$, it follows that $a^2(a - 1) = \frac{3}{2} \pm \frac{\sqrt{3}}{2} i$.
If $a^3 - a^2 + 1 = 0$, then $a^2(a - 1) = a^3 - a^2 = -1$.
Therefore, the observed expression can take on values $-1, \frac{3}{2}, \frac{\sqrt{3}}{2} i$ and $\frac{3}{2} - \frac{\sqrt{3}}{2} i$.