CN-TST 2021 Problem 21

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$ , such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i).$$ where $S(n)$ denotes sum of digits of decimal representation of $n$ .