1990 APMO 第 2 题

Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers, and let $S_{k}$ be the sum of products of $a_{1}, a_{2}, \ldots, a_{n}$ taken $k$ at a time.
Show that

$$

S_{k} S_{n-k} \geq\binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \quad \text { for } \quad k=1,2, \ldots, n-1

$$

令$a_{1}、a_{2}、\ldots、a_{n}$为正实数，并令$S_{k}$为$a_{1}、a_{2}、\ldots、a_{n}$每次取$k$的乘积之和。

表明

$$S_{k} S_{n-k} \geq\binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \quad \text { for } \quad k=1,2, \ldots, n-1$$