# Is it always possible to choose two subsets with the same sum?

Given two positive integers $n, m$, let $A$ be a multiset of $n$ integers taken from { $1,2,\cdots, m$ }, and $B$ be a multiset of $m$ integers taken from { $1,2,\cdots,n$ }.

Is it always possible to choose two nonempty subsets from $A$ and $B$ respectively, such that their sums are equal?

I cannot either prove or disprove it. Trying some examples made me conjecture that it is true.

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This is the problem A4 in the 54th Putnam exam. I guess you can find a solution here mat.itu.edu.tr/gungor/IMO/www.kalva.demon.co.uk/putnam/… I also suggest artofproblemsolving.com as a better forum to discuss questions at this level. –  Gjergji Zaimi Feb 22 '10 at 11:50
I see. Thanks for your reply. –  Zeyu Feb 22 '10 at 13:53