OBM 1997 Problem 2

Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$ . Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$ . Show that $n^2 \leq 4 f(n) < 4 n^3$ .