21.10*. 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?
21.10. Since the diagonals emanating from one vertex divide an $n$-sided polygon into $n-2$ triangles, $n-2$ points are necessary.

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-396.jpg?height=415&width=575&top_left_y=248&top_left_x=862)

Fig. 21.3

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-396.jpg?height=612&width=727&top_left_y=735&top_left_x=794)

Fig. 21.4

From Fig. 21.4, 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 $A_{p} A_{q} A_{r}$, where $p<q<r$, there is always a shaded triangle adjacent to the vertex $A_{q}$.