USAMO 2005 — Days 1 and 2 Question 6

For a positive integer $m$, let $s(m)$ denote the sum of the decimal digits of $m$. A set $S$ positive integers is $k$-stable if $s(\sum_{x \in X} x) = k$ for any nonempty subset $X \subseteq S$. For each integer $n \ge 2$ let $f(n)$ be the minimal $k$ for which there exists a $k$-stable set with $n$ integers. Prove that there are constants $0 < C_1 < C_2$ with $$ C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n. $$

Proposed by Titu Andreescu, Gabriel Dospinescu (credited in the supplied source).