2022 IMO Shortlist A5

Find all positive integers $n \geq 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leq i<j \leq n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$. (Czech Republic)

找到存在 $n$ 实数 $a_{1}<\cdots<a_{n}$ 和实数 $r>0$ 的所有正整数 $n \geq 2$，使得 $1 \leq i<j \leq n$ 的 $\frac{1}{2} n(n-1)$ 差异 $a_{j}-a_{i}$ 按某种顺序等于数字 $r^{1}， r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$。 (捷克共和国)