NZ-NZMO 2023 Problem 1

For any positive integer $n$ let $n! = 1 \times 2 \times 3 \times \dots \times n$ . Do there exist infinitely many triples $(p, q, r)$ , of positive integers with $p > q > r > 1$ such that the product

$$p!\cdot q!\cdot r!$$

is a perfect square?