Let the sets be $A_1, A_2, \dots , A_n$. Let $A_1 = \{a,b\}$ and suppose $A_2 = \{a,c\}$ where $a,b,c$ are distinct elements. Now, any other $A_i$ must contain either $a$ or $b$ (and not both) or $a$ or $c$ (and not both). Thus, $A_i = \{b,c\}$ or $\{a,x\}$ for some element $x$. Because $n > 3$, no $A_i$ can be $\{b,c\}$ or else some other $A_j = \{a,x\}$ with $x \neq b,c$ and has no common element with $A_i = \{b,c\}$. Thus, all the sets contain $a$... qed