KöMaL 2012 A558

Prove that there exists a constant $C$>0 for which the following statement holds: if $n$ is a positive integer and $A_1,A_2,\ldots,A_N\subset\{1,2,\ldots,n\}$ are sets such that every two of them has at least two, and every three of them has at most three elements in common, then $N$<$Cn^2$.