IMO 1987 SL 3

Origin: FIN

Problem

Does there exist a second-degree polynomial p(x, y) in two variables such that every nonnegative integer n equals p(k, m) for one and only one ordered pair (k, m) of nonnegative integers?

Solution

The answer: yes. Set p(k, m) = k + [1 + 2 + \cdot \cdot \cdot + (k + m)] = (k + m)2 + 3k + m . It is obviously of the desired type.