2024 USAMO 第 3 题

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 $n \geq 3$ 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.

令 $m$ 为正整数。如果多边形的三角形可以用 $m$ 颜色着色，并且对于每种 $m$ 颜色，所有相同颜色的三角形的面积总和都相同，则该多边形的三角剖分是 $m$ 平衡的。查找存在规则 $n$ 边形的 $m$ 平衡三角剖分的所有正整数 $n$。注意：具有 $n \geq 3$ 条边的凸多边形 $\mathcal{P}$ 的三角剖分是通过 $\mathcal{P}$ 的 $n-3$ 对角线将 $\mathcal{P}$ 划分为 $n-2$ 个三角形，且这些对角线在多边形内部不相交。