2019 APMO 第 2 题

Let $m$ be a fixed positive integer. The infinite sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined in the following way: $a_{1}$ is a positive integer, and for every integer $n \geq 1$ we have

$$a_{n+1}= \begin{cases}a_{n}^{2}+2^{m} & \text { if } a_{n}<2^{m} \\ a_{n} / 2 & \text { if } a_{n} \geq 2^{m}\end{cases}$$

For each $m$, determine all possible values of $a_{1}$ such that every term in the sequence is an integer.

Answer: The only value of $m$ for which valid values of $a_{1}$ exist is $m=2$. In that case, the only solutions are $a_{1}=2^{\ell}$ for $\ell \geq 1$.

令 $m$ 为固定正整数。无限序列 $\left\{a_{n}\right\}_{n \geq 1}$ 定义如下： $a_{1}$ 是一个正整数，对于每个整数 $n \geq 1$ 我们有

$$a_{n+1}= \begin{cases}a_{n}^{2}+2^{m} & \text { if } a_{n}<2^{m} \\ a_{n} / 2 & \text { if } a_{n} \geq 2^{m}\end{cases}$$

对于每个 $m$，确定 $a_{1}$ 的所有可能值，使得序列中的每一项都是整数。

答案：$a_{1}$ 存在有效值的 $m$ 的唯一值是 $m=2$。在这种情况下，唯一的解决方案是 $a_{1}=2^{\ell}$ for $\ell \geq 1$。