IMO Shortlist 2002 problem N5


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2. travnja 2012.
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Let m,n\geq2 be positive integers, and let a_1,a_2,\ldots ,a_n be integers, none of which is a multiple of m^{n-1}. Show that there exist integers e_1,e_2,\ldots,e_n, not all zero, with \left|{\,e}_i\,\right|<m for all i, such that e_1a_1+e_2a_2+\,\ldots\,+e_na_n is a multiple of m^n.
Izvor: Međunarodna matematička olimpijada, shortlist 2002