CN-LEAGUE 2014 Problem 3

Let $S=\{1,2,3,\cdots,100\}$ . Find the maximum value of integer $k$ , such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.