CONO-SUR 2018 Problem 3

Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$ a) Find all positive integers $m$ , such that $\frac {P_{2020}}{m!}$ is a perfect square. b) Prove that there are infinite many value(s) of $n$ , such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$ .