Question 1
Let $a_1,a_2,\ldots,a_n$ be nonnegative integers, where $n$ is a positive integer, and let
$$A_n=\frac{a_1+\cdots+a_n}{n}.$$
Prove that
$$a_1!a_2!\cdots a_n!\ge(\lfloor A_n\rfloor!)^n,$$
where $\lfloor A_n\rfloor$ is the greatest integer not exceeding $A_n$, and $a!=1\cdot2\cdots a$ for $a\ge1$, with $0!=1$. When does equality hold?
$$A_n=\frac{a_1+\cdots+a_n}{n}.$$
Prove that
$$a_1!a_2!\cdots a_n!\ge(\lfloor A_n\rfloor!)^n,$$
where $\lfloor A_n\rfloor$ is the greatest integer not exceeding $A_n$, and $a!=1\cdot2\cdots a$ for $a\ge1$, with $0!=1$. When does equality hold?