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Some sum is divisible by 2015 - 2015 PAMO Problem 3

Source: 2015 Pan-African Mathematics Olympiad Problem 3

August 26, 2015
Combinatorial Number Theorynumber theory

Problem Statement

Let a1,a2,...,a11a_1,a_2,...,a_{11} be integers. Prove that there are numbers b1,b2,...,b11b_1,b_2,...,b_{11}, each bib_i equal −1,0-1,0 or 11, but not all being 00, such that the number N=a1b1+a2b2+...+a11b11N=a_1b_1+a_2b_2+...+a_{11}b_{11} is divisible by 20152015.
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