MathNet 2016-part-06 — 184 problems (EN–VI) Question 89

There is a grid of equilateral triangles with a distance $1$ between any two neighboring grid points. An equilateral triangle with side length $n$ lies on the grid so that all of its vertices are grid points, and all of its sides match the grid. Now, let us decompose this equilateral triangle into $n^2$ smaller triangles (not necessarily equilateral triangles) so that the vertices of all these small triangles are all grid points, and all these small triangles have equal areas.
Prove that there are at least $n$ equilateral triangles among these smaller triangles.