AT-OEMO-BEG 2011 Problem 2

Let $p$ and $q$ be real numbers. The quadratic equation $$x^2 + px + q = 0$$ has the real solutions $x_1$ and $x_2$ . In addition, the following two conditions apply: (i) The numbers $x_1$ and $x_2$ differ from each other by exactly $1$ . (ii) The numbers $p$ and $q$ differ from each other by exactly $1$ . Show that then $p$ , $q$ , $x_1$ and $x_2$ are integers.

(G. Kirchner, University of Innsbruck)