I have the following question from USAMO 2005:
"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 $C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n$"
I did find a solution here :
https://artofproblemsolving.com/wiki/index.php?title=2005_USAMO_Problems/Problem_6"
But I am not understanding what exactly is the question about. Can anyone please try explaining the problem and its solution in a simpler manner.
Note: I would also like to present what I have understood from the complicated framing of the question. What I basically understood that I need to find a set $S$ containing $n$ elements such that sum of all the subsets of the set $S$ is equal and I need to form the set in such a way that the sum of digits of all the above mentioned subsets should be a minimum and for that minimum, I need to prove that two such constants satisfying the above conditions must exist. If the above interpretation of mine is correct, then for any given $n$, I form the set S in such a way that all its elements consist of (n-1) 9's. Note that this a blind conjecture with no solid basis