Let $n$ and $k$ be positive integers. Let $L$ be any set of $kn$ lines in the plane, no two of which are parallel. Each line in $L$ is colored one of $k$ colors, and there are $n$ lines of each color. Let $T_{i}$ be the set of all points in the plane that lie on at least one line of color $i$. Is there a circle $\mathcal{C}$ that intersects each $T_{i}$ in exactly $kn - 1$ points?
This is a generalization of problem 4 from https://cms.math.ca/Competitions/APMO/exam/apmo2015.pdf where $k = 2$, which I have found a solution to. I couldn't help but wonder if it holds for more than two colors, though. Apparently my solution to the original problem, which consists of choosing a circle tangent to two lines of different color in one of the four regions that intersects all other lines, isn't general enough to say anything about the general $k$-color case.