Mathematical and Physical Journal
for High Schools
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Problem A. 409. (October 2006)

A. 409. For a positive integer m, let s(m) be the sum of the digits of m. For n\ge2, 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\subsetS. Prove that there are constants 0<C1<C2 with C1log10n\lef(n)\leC2log10n.

U.S.A. Mathematical Olympiad, 2005

(5 pont)

Deadline expired on November 15, 2006.

Sorry, the solution is available only in Hungarian. Google translation



13 students sent a solution.
5 points:Dobribán Edgár, Fischer Richárd, Gyenizse Gergő, Hujter Bálint, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Korándi Dániel, Lovász László Miklós, Nagy 224 Csaba, Nagy 235 János, Nagy 314 Dániel, Sümegi Károly, Tomon István.

Problems in Mathematics of KöMaL, October 2006