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Difference between revisions of "2005 USAMO Problems/Problem 6"

(New page: ==Problem== For <math>m</math> a positive integer, let <math>s(m)</math> be the sum of the digits of <math>m</math>. For <math>n\ge 2</math>, let <math>f(n)</math> be the minimal <math>k</...)
 
(→‎Solution)
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C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.
 
C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.
 
</cmath>
 
</cmath>
−
 
−
=Solution=
 

Revision as of 15:56, 30 March 2009

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

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