1992 CMO 第 6 题

Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions:

1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ;

2) $2a_1=a_0+a_2-2$ ;

3) for arbitrary natural number $m$ , there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares.

Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.

设整数项的序列$\{a_1,a_2,\dots \}$满足以下条件：

1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$;

2) $2a_1=a_0+a_2-2$;

3) 对于任意自然数 $m$ ，序列中存在 $m$ 个连续项 $a_k, a_{k-1}, \dots ,a_{k+m-1}$ ，使得所有这些 $m$ 项都是完全平方数。

证明序列 $\{a_1,a_2,\dots \}$ 的所有项都是完全平方数。