Let
![n \geq 3](/media/m/5/4/8/54807b3bf99aa939833fe57bf8d891d3.png)
and consider a set
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
of
![2n - 1](/media/m/d/f/8/df883cf64f335e260ddf6ffbf1141afc.png)
distinct points on a circle. Suppose that exactly
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
points from
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
. Find the smallest value of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
so that every such coloring of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
points of
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
is good.
%V0
Let $n \geq 3$ and consider a set $E$ of $2n - 1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$. Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.