[ Convex Polygons ]

Among all such numbers $n$, that any convex 100-gon can be represented as the intersection (i.e., common part) of $n$ triangles, find the smallest.
First, note that 50 triangles are sufficient. Indeed, let $\Delta_{k}$ be the triangle whose sides lie on the rays $A_{\mathrm{k}} A_{\mathrm{k}-1}$ and $A_{\mathrm{k}} A_{\mathrm{k}+1}$ and which contains the convex polygon $A_{1} \ldots A_{100}$. Then this polygon is the intersection of the triangles $\Delta_{2}, \Delta_{4}, \ldots, \Delta_{100}$. On the other hand, the 100-gon shown in the figure cannot be represented as the intersection of fewer than 50 triangles. Indeed, if three of its sides lie on the sides of one triangle, then one of these sides is the side $A_{1} A_{2}$. All sides of this polygon lie on the sides of $n$ triangles, so $2 n+1 \geq 100$, i.e., $n \geq 50$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7a1b7dea474101c41e1dg-12.jpg?height=409&width=1288&top_left_y=807&top_left_x=385)

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