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Find greatest positive integer for which there exist ...

Source: Romanian IMO Team Selection Test TST 1996, problem 2

September 27, 2005
number theory proposednumber theory

Problem Statement

Find the greatest positive integer nn for which there exist nn nonnegative integers x1,x2,…,xnx_1, x_2,\ldots , x_n, not all zero, such that for any ε1,ε2,…,εn\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n from the set {−1,0,1}\{-1, 0, 1\}, not all zero, ε1x1+ε2x2+⋯+εnxn\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n is not divisible by n3n^3.
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