2025 APMO 第 5 题

Consider an infinite sequence $a_{1},a_{2},\ldots$ of positive integers such that

$$100!(a_{m} + a_{m + 1} + \cdot \cdot \cdot +a_{n})\quad \mathrm{is~a~multiple~of}\quad a_{n - m + 1}a_{n + m}$$

for all positive integers $m,n$ such that $m\leq n$

Prove that the sequence is either bounded or linear.

Observation: A sequence of positive integers is bounded if there exists a constant $N$ such that $a_{n}< N$ for all $n\in \mathbb{Z}_{>0}$ . A sequence is linear if $a_{n} = n\cdot a_{1}$ for all $n\in \mathbb{Z}_{>0}$

考虑正整数的无限序列 $a_{1},a_{2},\ldots$ ，使得

$$100!(a_{m} + a_{m + 1} + \cdot \cdot \cdot +a_{n})\quad \mathrm{是~a~多个~的}\quad a_{n - m + 1}a_{n + m}$$

对于所有正整数 $m,n$ 使得 $m\leq n$

证明该序列是有界的或线性的。

观察：如果存在常数 $N$ 使得对于所有 $n\in \mathbb{Z}_{>0}$ ，$a_{n}< N$ ，则正整数序列是有界的。如果对于所有 $n\in \mathbb{Z}_{>0}$，$a_{n} = n\cdot a_{1}$，则序列是线性的