Question 4
Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
$$
A_{n} = \frac{a_{1} + a_{2} + \cdots + a_{n}}{n}
$$
Prove that
$$
a_{1}! a_{2}! \ldots a_{n}! \geq \left( \left\lfloor A_{n} \right\rfloor ! \right)^{n}
$$
where $\left\lfloor A_{n} \right\rfloor$ is the greatest integer less than or equal to $A_{n}$, and $a! = 1 \times 2 \times \cdots \times a$ for $a \geq 1$ (and $0! = 1$). When does equality hold?
$$
A_{n} = \frac{a_{1} + a_{2} + \cdots + a_{n}}{n}
$$
Prove that
$$
a_{1}! a_{2}! \ldots a_{n}! \geq \left( \left\lfloor A_{n} \right\rfloor ! \right)^{n}
$$
where $\left\lfloor A_{n} \right\rfloor$ is the greatest integer less than or equal to $A_{n}$, and $a! = 1 \times 2 \times \cdots \times a$ for $a \geq 1$ (and $0! = 1$). When does equality hold?