1989 CMO 第 2 题

Let $x_1, x_2, \dots ,x_n$ ( $n\ge 2$ ) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$ . Prove that:$$\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.$$

设 $x_1, x_2, \dots ,x_n$ ( $n\ge 2$ ) 为满足 $\sum^{n}_{i=1}x_i=1$ 的正实数。证明：$$\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}。$$