2024 USAMO Problems/Problem 3
Let be a positive integer. A triangulation of a polygon is -balanced if its triangles can be colored with 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 colors. Find all positive integers for which there exists an -balanced triangulation of a regular -gon. Note: A triangulation of a convex polygon with sides is any partitioning of into triangles by diagonals of that do not intersect in the polygon's interior.