Solution
= Solution
No example exists for $n=2$, because the unique point in $A_1\cap A_2$ would also lie in the total intersection; the one-member case is excluded by nontriviality. For every $n\geq3$, take ground set $[n]$ and define the <near-pencil>
$$
A_0=\{1,\ldots,n-1\},
\qquad
A_i=\{i,n\}\quad(1\leq i\leq n-1).
$$
Two small sets meet in $n$, each small set meets $A_0$ in its other point, and the total intersection is empty. Thus the required values are exactly $n\geq3$.