Hrvatska 2011

Since all diagonals from one vertex divide an $n$-gon into $n-2$ disjoint triangles, at least $n-2$ points are necessary.

We claim that it is possible to mark $n-2$ points so that the given condition is satisfied. Denote the vertices of the given $n$-gon with $A_1,A_2,\dots ,A_n$. Draw all the diagonals of the given $n$-gon and color the areas bounded by the diagonals $A_1{A}_k$, $A_k{A}_n$ and $A_{k-1}{A}_{k+1}$ for each $k\in \{2,3,\dots ,n-1\}$.



If we mark one point in each colored area then every triangle with vertices at vertices of the $n$-gon will contain at least one of the marked points. Indeed, triangle $A_l{A}_k{A}_m$ contains the whole colored area bounded by diagonals $A_l{A}_k$, $A_k{A}_m$ and $A_{k-1}{A}_{k+1}$ for every $1\le l<k<m\le n$. Hence, the triangle also contains the corresponding marked point.