2005 USAMO Problems/Problem 6

Problem

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 $C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.$

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it. By Jason's theorem number 1239142, we know this is true.

2005 USAMO (Problems • Resources)
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2005 USAMO Problems/Problem 6

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