KöMaL 2006 A409

For a positive integer $m$, 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 $s\left(\sum\limits_{x\in X}x\right)=k$ for any nonempty subset $X \subset S$. Prove that there are constants 0<$C_1$<$C_2$ with $C_1 log_{10} n \le f$($n$)$\le C_2 log_{10} n$.