Question 3
$A$ is a set of $n$ non-negative integers. We say it has property $P$ if the set $\{x + y \mid x, y \in A\}$ has $\frac{n(n+1)}{2}$ 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 $P$. Show that $\frac{n^2}{4} \le f(n) \lt n^3$.
(If you still have time, try to improve these bounds. For example, try to prove that $f(p) \lt 2p^2$ for all primes $p$.)
(If you still have time, try to improve these bounds. For example, try to prove that $f(p) \lt 2p^2$ for all primes $p$.)