2023 CMO 第 1 题

Define the sequences $(a_n),(b_n)$ by

$$

\begin{aligned}

& a_n, b_n > 0, \forall n\in\mathbb{N_+}

& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}}

& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}

\end{aligned}

$$

1) If $a_{100}b_{100} = a_{101}b_{101}$ , find the value of $a_1-b_1$ ;

2) If $a_{100} = b_{99}$ , determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$ .

定义序列 $(a_n),(b_n)$

$$

\开始{对齐}

& a_n, b_n > 0, \forall n\in\mathbb{N_+}

& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}}

& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}

\结束{对齐}

$$

1) 如果 $a_{100}b_{100} = a_{101}b_{101}$ ，求 $a_1-b_1$ 的值；

2) 如果 $a_{100} = b_{99}$ ，确定 $a_{100}+b_{100}$ 和 $a_{101}+b_{101}$ 之间哪个较大。