1989 APMO 第 1 题

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive real numbers, and let

$$S=x_{1}+x_{2}+\cdots+x_{n}$$

Prove that

$$

\left(1+x_{1}\right)\left(1+x_{2}\right) \cdots\left(1+x_{n}\right) \leq 1+S+\frac{S^{2}}{2!}+\frac{S^{3}}{3!}+\cdots+\frac{S^{n}}{n!}

$$

设 $x_{1}, x_{2}, \ldots, x_{n}$ 为正实数，并令

$$S=x_{1}+x_{2}+\cdots+x_{n}$$

证明

$$\left(1+x_{1}\right)\left(1+x_{2}\right) \cdots\left(1+x_{n}\right) \leq 1+S+\frac{S^{2}}{2!}+\frac{S^{3}}{3!}+\cdots+\frac{S^{n}}{n!}$$