What is the smallest number of points that need to be marked inside a convex $n$-gon so that within any triangle with vertices at the vertices of the $n$-gon, there is at least one marked point?
Since the diagonals emanating from one vertex divide an $n$-gon into $n-2$ triangles, $n-2$ points are necessary.

From the figure, it can be understood how to manage with $n-2$ points: it is sufficient to mark one point in each shaded triangle. Indeed, inside the triangle $A_{\mathrm{p}} A_{\mathrm{q}} A_{\mathrm{r}}$, where $p<q<r$, there is always a shaded triangle adjacent to the vertex $A^{q}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_39c3940b0a0b7bafc849g-30.jpg?height=1026&width=1221&top_left_y=1351&top_left_x=424)

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