THE 68th ROMANIAN MATHEMATICAL OLYMPIAD

The sets we search contain elements $a$, $3a$, with $3a\le 2017$, so $a\le 672$. For $a=1$, we have $\{1,3\}\subset B$, and the possible sets are $\{1,2,3\}$ and $\{1,3,4\}$. For $a\in \{2,3,4,\dots ,672\}$ we have the sets $\{a-1,a,3a\}$, $\{a,a+1,3a\}$, $\{a,3a-1,3a\}$ and $\{a,3a,3a+1\}$, which are all distinct, giving a total of $4\cdot 671=2684$ sets. So the number of sets with the required properties is $2686$.