2002 APMO 第 1 题

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?

令 $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ 为非负整数序列，其中 $n$ 为正整数。

让

$$A_{n}=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}$$

证明

$$a_{1}!a_{2}!\ldots a_{n}!\geq\left(\left\lfloor A_{n}\right\rfloor!\right)^{n}$$

其中 $\left\lfloor A_{n}\right\rfloor$ 是小于或等于 $A_{n}$ 的最大整数，并且 $a!=1 \times 2 \times \cdots \times a$ 对于 $a \geq 1$ (和 $0!=1$ )。平等何时成立？