Find the $(x, y)$ natural integers such that $(x+y)^{2}+3 x+y+1$ is a perfect square.
$$
\begin{gathered}
((x+y)+2)^{2}=(x+y)^{2}+4 x+4 y+4>(x+y)^{2}+3 x+y+1 \\
(x+y)^{2}<(x+y)^{2}+3 x+y+1
\end{gathered}
$$

So the square in question is $(x+y+1)^{2}$

We need to solve $(x+y+1)^{2}=(x+y)^{2}+3 x+y+1$, which simplifies to $x=y$

We also verify that if $x=y$, then $(2 x)^{2}+4 x+1=(2 x+1)^{2}$ is indeed a perfect square.

Therefore, the set of solutions is the set of pairs $(x, x)$ with $x$ being any natural number.

## Tool: the bar that divides everything

Sometimes, we encounter exercises that require verifying whether a fraction is an integer or whether a certain expression divides another. A common technique in olympiads is to gather as many expressions divisible by a certain integer $a$ using a vertical bar. It is allowed to form linear combinations of the elements divisible by $a$, and the result will always be divisible by $a$.