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Summed up product divisible by 1001

Source: IMO LongList 1988, Poland 3, Problem 60 of ILL

November 3, 2005
number theory unsolvednumber theory

Problem Statement

Given integers a1,…,a10,a_1, \ldots, a_{10}, prove that there exist a non-zero sequence {x1,…,x10}\{x_1, \ldots, x_{10}\} such that all xix_i belong to {−1,0,1}\{-1,0,1\} and the number ∑i=110xi⋅ai\sum^{10}_{i=1} x_i \cdot a_i is divisible by 1001.
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