1999 USAMO 第 4 题

Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that

$$a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}.$$

Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.

设 $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) 为实数，使得

$$

a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{和} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}。

$$

证明 $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$。