2017 IMO 第 1 题

For each integer $a_{0} > 1$ , define the sequence $a_{0}$ , $a_{1}$ , $a_{2}$ , ..., by

$$a_{n + 1} = \left\{ \begin{array}{ll}\sqrt{a_{n}} & \mathrm{if~}\sqrt{a_{n}} \mathrm{is~an~integer,}\\ a_{n} + 3 & \mathrm{otherwise} \end{array} \right.$$

for each $n \geq 0$ . Determine all values of $a_{0}$ for which there is a number $A$ such that $a_{n} = A$ for infinitely many values of $n$ .

对于每个整数 $a_{0} > 1$ ，定义序列 $a_{0}$ , $a_{1}$ , $a_{2}$ , ...,

a_{n + 1} = \left\{ \begin{array}{ll}\sqrt{a_{n}} & \mathrm{if~}\sqrt{a_{n}} \mathrm{is~an~integer,}\\ a_{n} + 3 & \mathrm{否则} \end{array} \right。

对于每个 $n \geq 0$ 。确定 $a_{0}$ 中存在数字 $A$ 的所有值，使得 $a_{n} = A$ 对于无限多个 $n$ 值。