BG-MO 2004 Problem 6

Let $p$ be a prime number and let $0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $k$ be the number of distinct residues modulo $p$ that $a_{i}+b_{j}$ give when $i$ runs from 1 to $m$ , and $j$ from 1 to $n$ . Prove that

a) if $m+n > p$ then $k = p$ ;

b) if $m+n\leq p$ then $k\geq m+n-1$ .