What is the maximum number of diagonals that can be drawn in a convex $n$-gon such that any two of them have a common interior or endpoint?
After drawing some diagonals of a convex $n$-gon in the desired manner, classify the vertices according to the number of diagonals originating from them. Let's also color, say, yellow those vertices from which only one diagonal originates, and brown those from which more than one originates. If there is a brown vertex, the diagonals originating from it are radially distributed. Among them, we call two of them extreme, and the rest, if any, internal. We will show that the other endpoint of any internal diagonal originating from a brown vertex can only be yellow. Such a diagonal divides the plane in such a way that the two extreme diagonals originating from the brown vertex end up in different half-planes. Therefore, no single segment originating from the other endpoint of the internal diagonal can intersect both of them simultaneously.

Now, let's send two soldiers to each drawn diagonal with the following command. If the diagonal has only one yellow endpoint, both soldiers should go there; otherwise, one should go to one endpoint and the other to the other endpoint. After executing the command, no more than two soldiers can arrive at any vertex, since

- a yellow vertex can only be reached by the soldiers running to the diagonal leading to it;
- a brown vertex can only be reached by the soldiers running to the two extreme diagonals.

Thus, we have at most $2n$ soldiers, or at most $n$ drawn diagonals.

![](https://cdn.mathpix.com/cropped/2024_05_02_b3e3ed6a2a066ac17bb4g-1.jpg?height=518&width=647&top_left_y=736&top_left_x=728)

Finally, we will show that if $n \geq 5$, this upper bound can be achieved. (If $n=5$, draw all additional diagonals from $C$. Among the drawn diagonals,

- those originating from $C$ have $C$ as their common endpoint,
- any diagonal originating from $C$ intersects the diagonals $E B, B D, D A$, since any line through $C$ separates the pairs $E-B, B-D, D-A$, and since the polygon is convex, the intersection point can only occur on the diagonal;
- the diagonals $E B$ and $B D$, as well as $B D$ and $D A$, have a common endpoint, and $E B$ and $D A$ intersect each other.

Thus, the desired number is $n$ if $n \geq 5$, and $n-2$ otherwise.

Note. If sides are also allowed in addition to diagonals, the maximum is always $n$.