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IMO ShortList 2002, number theory problem 5

Source: IMO ShortList 2002, number theory problem 5

September 28, 2004

modular arithmeticnumber theoryIMO Shortlistgenerating functionsroots of unitycomplex numbersHi

Problem Statement

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