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a_1x_1 + a_2x2 + ... + a_nx_n not zero when sum a_1=0,

Source: Austrian - Polish 1990 APMC

May 7, 2020
Sumalgebracombinatoricspartition

Problem Statement

a1,a2,...,ana_1, a_2, ... , a_n is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if xix_i all belong to the same set, then a1x1+a2x2+...+anxna_1x_1 + a_2x_2 + ... + a_nx_n is non-zero.
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