BALTIC-WAY 2023 Problem 18

Let $p>7$ be a prime number and let $A$ be a subset of $\{0,1, \ldots, p-1\}$ consisting of at least $\frac{p-1}{2}$ elements. Show that for each integer $r$, there exist (not necessarily distinct) numbers $a, b, c, d \in A$ such that

$$a b-c d \equiv r (\bmod p)$$