ELMO 2013 Problem 3

Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$ . Prove that there is a positive integer $N$ such that $$N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right)$$ and for each $i = 1, 2, ..., 2013$ , there does not exist $a \in A_i$ such that $m_i$ divides $N-a$ .

Proposed by Victor Wang