4. Let $n \geqslant 3$ be an integer. Let $\mathscr{C}_{1}, \mathscr{C}_{2}$, and $\mathscr{C}_{3}$ be the boundaries of three convex $n$-gons in the plane such that the intersection of any two of them is a finite set of points. Find the maximum possible number of points in the set $\mathscr{C}_{1} \cap \mathscr{C}_{2} \cap \mathscr{C}_{3}$.

(Turkish)
4. The points in $\mathscr{C}_{1} \cap \mathscr{C}_{2} \cap \mathscr{C}_{3}$ are the vertices of a convex polygon $\mathscr{P}$. Consider one of its sides $A B$. It belongs to at most one of the boundaries of the $n$-gons $\mathscr{C}_{1}, \mathscr{C}_{2}, \mathscr{C}_{3}$ (the intersection of any two does not contain the entire segment), which means that the line $A B$ cuts off at least one vertex from each of the remaining two $n$-gons. Thus, each side of $\mathscr{P}$ cuts off at least two of the total $3 n$ vertices of $\mathscr{C}_{1}, \mathscr{C}_{2}, \mathscr{C}_{3}$, so $\mathscr{P}$ cannot have more than $\frac{3 n}{2}$ sides. The number $m=\left[\frac{3 n}{2}\right]$ can be achieved. Let $A_{1} A_{2} \ldots A_{m}$ be a convex $m$-gon, $a_{k}(1 \leqslant k \leqslant m)$ the line through vertices $A_{k}, A_{k+1}$, and $b_{k}$ $(1 \leqslant k \leqslant 3 n-m)$ any line through $A_{k}$ that has no other common points with the $m$-gon $\left(A_{n+1}=A_{1}, b_{n+1} \neq b_{1}\right)$. It is sufficient to define $\mathscr{C}_{i}(i=1,2,3)$ as the polygon determined by all (defined) lines $a_{k}$ and $b_{l}$, where $k \equiv l+1 \equiv i$

![](https://cdn.mathpix.com/cropped/2024_06_04_d03ed0fb1e330b7a364dg-3.jpg?height=51&width=797&top_left_y=951&top_left_x=230)
indeed $n$-gons.

![](https://cdn.mathpix.com/cropped/2024_06_04_d03ed0fb1e330b7a364dg-3.jpg?height=546&width=780&top_left_y=498&top_left_x=1069)