Question 18
Let $S(k)$ denote the sum of all digits of $k$ in base 10. Find all integers $n \ge 2$ and rational numbers $\beta \in (0,1)$, such that there exist $n$ distinct positive integers $a_1, a_2, \dots, a_n$ that satisfy: for any subset $I \subseteq \{1, 2, \dots, n\}$ with two or more elements,
$$ S\left(\sum_{i \in I} a_i\right) = \beta \cdot \sum_{i \in I} S(a_i). $$
$$ S\left(\sum_{i \in I} a_i\right) = \beta \cdot \sum_{i \in I} S(a_i). $$