USAMO 2024 Problem 3

Let $m$ be a positive integer. A triangulation of a polygon is $m$ -balanced if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$ -balanced triangulation of a regular $n$ -gon.

Note: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior.

Proposed by Krit Boonsiriseth