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Adding weights to numbers such that they sum to 0

Source: 2015 China Tst 1 Day 2 Q2

March 14, 2015
number theoryDiophantine equation

Problem Statement

FIx positive integer nn. Prove: For any positive integers a,b,ca,b,c not exceeding 3n2+4n3n^2+4n, there exist integers x,y,zx,y,z with absolute value not exceeding 2n2n and not all 00, such that ax+by+cz=0ax+by+cz=0
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