1994 USAMO 第 4 题

Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$

$$

\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).

$$

令 $\, a_1, a_2, a_3, \ldots \,$ 为满足 $\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,$ 的正实数序列，对于所有 $\, n \geq 1$。证明，对于所有 $\, n \geq 1, \,$

$$\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right)。$$