2020 IMO Shortlist N3

Let $n$ be an integer with $n \geq 2$. Does there exist a sequence $\left(a_{1}, \ldots, a_{n}\right)$ of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia)

设$n$ 为$n \geq 2$ 的整数。是否存在一个正整数序列 $\left(a_{1}, \ldots, a_{n}\right)$ 并且并非所有项都相等，使得每两项的算术平均值等于该序列中某些(一项或多项)项的几何平均值？ (爱沙尼亚)