KURSCHAK 1984 Problem 3

Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$ . If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.