2016 IMO Shortlist C3

Let $n$ be a positive integer relatively prime to 6 . We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours. $\mathbf{C 4}$. Find all positive integers $n$ for which we can fill in the entries of an $n \times n$ table with the following properties: - each entry can be one of $I, M$ and $O$; - in each row and each column, the letters $I, M$ and $O$ occur the same number of times; and - in any diagonal whose number of entries is a multiple of three, the letters $I, M$ and $O$ occur the same number of times.

令 $n$ 为与 6 互质的正整数。我们用三种颜色绘制正则 $n$ 边形的顶点，以便每种颜色都有奇数个顶点。证明存在一个等腰三角形，其三个顶点颜色不同。 $\mathbf{C 4}$。查找所有正整数 $n$，我们可以将其填充到具有以下属性的 $n \times n$ 表的条目中： - 每个条目可以是 $I、M$ 和 $O$ 之一； - 在每一行和每一列中，字母$I、M$和$O$出现的次数相同； - 在任何条目数为三的倍数的对角线上，字母 $I、M$ 和 $O$ 出现的次数相同。