Question 145
A sequence of real numbers $\{a_n\}$ satisfies the condition that $a_1 = \frac{1}{2}$, $a_{k+1} = -a_k + \frac{1}{2-a_k}$, $k = 1, 2, \dots$.
Prove the following inequality:
$$
\left( \frac{n}{2(a_1 + a_2 + \cdots + a_n)} - 1 \right)^n \le \left[ \frac{a_1 + a_2 + \cdots + a_n}{n} \right]^n \left( \frac{1}{a_1} - 1 \right) \left( \frac{1}{a_2} - 1 \right) \cdots \left( \frac{1}{a_n} - 1 \right).
$$
Prove the following inequality:
$$
\left( \frac{n}{2(a_1 + a_2 + \cdots + a_n)} - 1 \right)^n \le \left[ \frac{a_1 + a_2 + \cdots + a_n}{n} \right]^n \left( \frac{1}{a_1} - 1 \right) \left( \frac{1}{a_2} - 1 \right) \cdots \left( \frac{1}{a_n} - 1 \right).
$$