USAMO 2020 — Days 1 and 2 Question 3

Let $p$ be an odd prime. An integer $x$ is a quadratic non-residue if $x\not\equiv t^2\pmod p$ for every integer $t$. Let $A$ consist of integers $1\le a\lt p$ such that both $a$ and $4-a$ are quadratic non-residues. Find the remainder of $\prod_{a\in A}a$ modulo $p$.

Proposed by Richard Stong, Toni Bluher. Source: Evan Chen’s third-party solution notes.

Source video link: https://youtu.be/r7j0oRtpErA?t=3240