Problem

Source: 2021ChinaTST test4 day1 P3

Tags: number theory, sum of digits, decimal representation, China TST, China, number theory proposed



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 $n$.