Question 49
Given $n$ black points in the plane such that no three are collinear and the distances between any two of them are pairwise distinct. Start with an arbitrary point and make it red. At each step, we draw the smallest segment such that one of its endpoints is red and the other is black, that has not yet been drawn, and that doesn't cut any other already drawn segment. Then we make its black endpoint red. Is it true that for any $n$ and for any formation of these $n$ points, after finitely many steps, we can make all the points red?