2004 IMO Shortlist S03

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.

*Proposed by Horst Sewerin, Germany*

令 ${n}$ 和 $k$ 为正整数。平面上有给定的 ${n}$ 个圆。它们中的每两个在两个不同的点相交，并且它们确定的所有交点都是成对不同的(即没有三个圆有公共点)。没有三个圆圈有共同点。每个交叉点必须使用 $n$ 个不同颜色中的一种进行着色，以便每种颜色至少使用一次，并且每个圆上恰好出现 $k$ 个不同颜色。查找 $n\geq 2$ 和 $k$ 中可能进行这种着色的所有值。

*由德国 Horst Sewerin 提出*