Cesko-Slovacko-Poljsko, 2013

Let us examine the diophantine equation $x^2 + a x + b = y^2$ with unknown integers $x$ and $y$. It can be transformed into the form $(2x + 2y + a)(2x - 2y + a) = a^2 - 4b$. Since we assume $b$ is not a perfect square, $a^2 - 4b \neq 0$. There are only finitely many ways to write $a^2 - 4b$ as a product of two integers. Each such factorization gives two linear equations for $x, y$ which have at most one integer solution. Thus there are only finitely many such $x$.