Question 184
Let $n$ be an integer greater than or equal to $2$. Suppose $n$ white points and $n$ black points are distributed on the circumference of a circle, and suppose we try to draw $2n$ line segments using these points so as to satisfy the following conditions:
(1) Each line segment has a white point and a black point as its end points.
(2) By tracing these line segments in order, it is possible to complete a cycle going through each of the given $2n$ points once and only once.
Prove that regardless of the way how the given $2n$ points are distributed on the circumference, it is possible to draw the line segments satisfying the conditions above in such a way that there are at most $n-1$ intersections among the line segments drawn. Here, we do not consider an end point of a line segment as an intersection point.
(1) Each line segment has a white point and a black point as its end points.
(2) By tracing these line segments in order, it is possible to complete a cycle going through each of the given $2n$ points once and only once.
Prove that regardless of the way how the given $2n$ points are distributed on the circumference, it is possible to draw the line segments satisfying the conditions above in such a way that there are at most $n-1$ intersections among the line segments drawn. Here, we do not consider an end point of a line segment as an intersection point.