Question 115
Let $p \gt 7$ be a prime number and let $A \subseteq \{0, 1, \dots, p-1\}$ consist of at least $\frac{p-1}{2}$ elements. Show that for each integer $r$, there are elements $a, b, c, d \in A$ such that
$$
ab - cd \equiv r \pmod{p}.
$$
$$
ab - cd \equiv r \pmod{p}.
$$