2024 IMO Shortlist N3

Determine all sequences $a_1,a_2,\ldots$ of positive integers such that, for any pair of positive integers $m\le n$, the arithmetic and geometric means

$$

\frac{a_m+a_{m+1}+\cdots+a_n}{n-m+1}

$$

and

$$

(a_ma_{m+1}\cdots a_n)^{1/(n-m+1)}

$$

are both integers.

确定所有正整数数列 $a_1,a_2,\ldots$，使得对任意正整数 $m\le n$，算术平均数

$$

\frac{a_m+a_{m+1}+\cdots+a_n}{n-m+1}

$$

和几何平均数

$$

(a_ma_{m+1}\cdots a_n)^{1/(n-m+1)}

$$

都是整数。