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N5 Indonesian 2014 Shortlist

Source: Shortlist OSN Indonesia 2014, N5

May 13, 2017
number theorypigeonhole principlenational olympiad

Problem Statement

Prove that we can give a color to each of the numbers 1,2,3,...,20131,2,3,...,2013 with seven distinct colors (all colors are necessarily used) such that for any distinct numbers a,b,ca,b,c of the same color, then 2014∤abc2014\nmid abc and the remainder when abcabc is divided by 20142014 is of the same color as a,b,ca,b,c.
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