2009 IMO 第 3 题

Suppose that $s_{1},s_{2},s_{3},\ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences $s_{s_{1}}$ , $s_{s_{2}}$ , $s_{s_{3}}$ , ...and $s_{s_{1} + 1}$ , $s_{s_{2} + 1}$ , $s_{s_{3} + 1}$ , ...are both arithmetic progressions. Prove that the sequence $s_{1}$ , $s_{2}$ , $s_{3}$ , ...is itself an arithmetic progression.

假设 $s_{1},s_{2},s_{3},\ldots$ 是一个严格递增的正整数序列，使得子序列 $s_{s_{1}}$ 、 $s_{s_{2}}$ 、 $s_{s_{3}}$ 、...和 $s_{s_{1} + 1}$ 、 $s_{s_{2} + 1}$ 、 $s_{s_{3} + 1}$ , ...都是算术级数。证明序列 $s_{1}$ , $s_{2}$ , $s_{3}$ , ...本身是一个算术级数。