2009 USAMO 第 4 题

For $n \ge 2$ let $a_1$, $a_2$, ..., $a_n$ be positive real numbers such that

$$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$$

Prove that $\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)$.

对于 $n \ge 2$ 令 $a_1$, $a_2$, ..., $a_n$ 为正实数，使得

$$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$$

证明 $\text{max}(a_1, a_2, ... ,a_n) \le 4 \text{min}(a_1, a_2, ... , a_n)$。