2024 IMO Shortlist A6

Let $a_0,a_1,a_2,\ldots$ be an infinite strictly increasing sequence of positive integers such that for each $n\ge 1$ we have

$$

a_n\in\left\{\frac{a_{n-1}+a_{n+1}}2,\sqrt{a_{n-1}a_{n+1}}\right\}.

$$

Let $b_1,b_2,\ldots$ be an infinite sequence of letters defined by

$$

b_n=\begin{cases}

A, & \text{if } a_n=\frac{a_{n-1}+a_{n+1}}2,\\

G, & \text{otherwise.}

\end{cases}

$$

Prove that there exist positive integers $n_0$ and $d$ such that for all $n\ge n_0$ we have $b_{n+d}=b_n$.

设 $a_0,a_1,a_2,\ldots$ 是一个无限严格递增的正整数列，并且对每个 $n\ge 1$，

$$

a_n\in\left\{\frac{a_{n-1}+a_{n+1}}2,\sqrt{a_{n-1}a_{n+1}}\right\}.

$$

定义字母序列 $b_1,b_2,\ldots$ 为

$$

b_n=\begin{cases}

A, & \text{若 } a_n=\frac{a_{n-1}+a_{n+1}}2,\\

G, & \text{否则。}

\end{cases}

$$

证明存在正整数 $n_0$ 和 $d$，使得对所有 $n\ge n_0$ 都有 $b_{n+d}=b_n$。