CMO 2014 Problem 6

For non-empty number sets $S, T$ , define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$ . Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$ . Show that there exists a subset $D$ of $A+B$ such that

1. $D+D\subseteq 2(A+B)$ ,
2. $|D|\geq\frac{|A|\cdot|B|}{2n}$ , where $|X|$ is the number of elements of the finite set $X$ .