IOM 2018 Problem 3

Let $k$ be a positive integer such that $p = 8k + 5$ is a prime number. The integers $r_1, r_2, \dots, r_{2k+1}$ are chosen so that the numbers $0, r_1^4, r_2^4, \dots, r_{2k+1}^4$ give pairwise different remainders modulo $p$ . Prove that the product $$\prod_{1 \leqslant i < j \leqslant 2k+1} \big(r_i^4 + r_j^4\big)$$ is congruent to $(-1)^{k(k+1)/2}$ modulo $p$ .

(Two integers are congruent modulo $p$ if $p$ divides their difference.)

Fedor Petrov