USAMO 2005 Problem 6

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