2003 IMO Shortlist S20

Let $b$ be an integer greater than $5$ . For each positive integer $n$ , consider the number $$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5,$$ written in base $b$ .

Prove that the following condition holds if and only if $b = 10$ : *there exists a positive integer $M$ such that for any integer $n$ greater than $M$ , the number $x_n$ is a perfect square.*

*Proposed by Laurentiu Panaitopol, Romania*

令 $b$ 为大于 $5$ 的整数。对于每个正整数 $n$ ，考虑数字 $$x_n = \underbrace{11\cdots1}_{n - 1}\underbrace{22\cdots2}_{n}5，$$ 以 $b$ 为基数编写。

证明以下条件成立当且仅当 $b = 10$ 时：*存在正整数 $M$，使得对于任何大于 $M$ 的整数 $n$，数字 $x_n$ 是完全平方数。*

*由罗马尼亚 Laurentiu Panaitopol 提议*