1. Determine all integers $x$ such that $x^{2}+3 x+24$ is a perfect square.
Solution. In the set of integers $\mathbb{Z}$, we need to solve the equation $x^{2}+3 x+24=y^{2}$. This equation is equivalent to the equation $(2 x+3)^{2}+87=4 y^{2}$, or to the equation

$$
(2 y+2 x+3)(2 y-2 x-3)=3 \cdot 89
$$

From the last equation, we get the possibilities $2 x+2 y+3= \pm 1, \pm 3, \pm 29, \pm 87$ and the corresponding values of $2 y-2 x-3$. By solving these systems of equations, we get $x \in\{-23,-8,5,20\}$. By direct verification, we see that they indeed satisfy the conditions of the problem.