TUYMAADA 2009 J4

Each of the subsets $A_1$ , $A_2$ , $\dots,$ $A_n$ of a 2009-element set $X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $X$ there is a 24-element subset $B$ containing neither of the sets $A_1$ , $A_2$ , $\dots,$ $A_n$ .