2006 CMO 第 3 题

Let $\{a_n\}$ be a sequence such that: $a_1 = \frac{1}{2}$ , $a_{k+1}=-a_k+\frac{1}{2-a_k}$ for all $k = 1, 2,\ldots$ . Prove that

$$\left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right).$$

令 $\{a_n\}$ 为一个序列，使得： $a_1 = \frac{1}{2}$ ， $a_{k+1}=-a_k+\frac{1}{2-a_k}$ 对于所有 $k = 1, 2,\ldots$ 。证明

$$\left(\frac{n}{2(a_1+a_2+\cdots+a_n)}-1\right)^n \leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n\left(\frac{1}{a_1}-1\right)\left(\frac{1}{a_2}-1\right)\cdots \left(\frac{1}{a_n}-1\right)。$$