Problem

Source: Romania National Olympiad 2023

Tags: algebra, quadratic equation



We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$. a) Show that the equation has $2$ distinct real solutions. b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$