RO-TST 2013 Problem 13

Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$ -element subsets of the set $\{1,2,\ldots,2n\}$ . Determine $$\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )$$ where $[x,y]$ is the least common multiple of the integers $x$ , $y$ .