KöMaL 2013 B4525

Given $n$ points in the interior of a regular triangle $\mathcal{T}$ , prove that $\mathcal{T}$ can be covered with 2$n$+1 regular triangles whose sides are parallel to the sides of $\mathcal{T}$ such that they contain none of the $n$ points in their interior. Also prove that the covering is not necessarily possible with 2$n$ triangles.