Competition problem — Number theory

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Number theory Difficulty 7.6 National olympiad, round 2 Find the answer

Let $p$ be an odd prime. An integer $x$ is called a quadratic non-residue if $p$ does not divide $x - t^2$ for any integer $t$ .
Denote by $A$ the set of all integers $a$ such that $1 \le a < p$ , and both $a$ and $4 - a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

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