Let
![{n}](/media/m/a/d/f/adffbb64b30e201a792a0cd756b4c2a3.png)
and
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be positive integers. There are given
![{n}](/media/m/a/d/f/adffbb64b30e201a792a0cd756b4c2a3.png)
circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwisely 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](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
distinct colors so that each color is used at least once and exactly
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
distinct colors occur on each circle. Find all values of
![n\geq 2](/media/m/e/d/b/edbb3c15913fef4235c90cca2333a608.png)
and
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
for which such a coloring is possible.
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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 pairwisely 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.