Problem

Source: ELMO 2013/3, by Victor Wang; also Shortlist N5

Tags: blogs, modular arithmetic, number theory, relatively prime, number theory unsolved, Chinese Remainder Theorem, Elmo



Let m1,m2,...,m2013>1 be 2013 pairwise relatively prime positive integers and A1,A2,...,A2013 be 2013 (possibly empty) sets with Ai{1,2,...,mi1} for i=1,2,...,2013. Prove that there is a positive integer N such that N(2|A1|+1)(2|A2|+1)(2|A2013|+1) and for each i=1,2,...,2013, there does not exist aAi such that mi divides Na. Proposed by Victor Wang