Ukrainian National Mathematical Olympiad - Third Round (Second Tour)

From the first equation $-1 \leq a \leq 1$ and $-1 \leq b \leq 1$, therefore $0 \leq 1+a \leq 2$ and $0 \leq 1+b \leq 2$. Add both equations and get
$$a^2(1+a) + b^2(1+b) = 0.$$
As both items are non-negative, their sum equals zero if and only if every item equals $0$. So $a, b \in \{-1, 0\}$. From the first equation it follows that one variable is not zero, from the second equation it follows that this variable equals $-1$. Thus, the equations are satisfied by such pairs of numbers $(a, b): (-1, 0)$ and $(0, -1)$, hence $a^3 + b^2 = -1$ or $a^3 + b^2 = 1$.