Question 162
There are $44$ distinct holes on a line and $2017$ ants. Each ant crawled up from a hole, then moved to another hole and crawled down. Denote $T$ as the set of time points that the ants crawled up or crawled down from some hole. Suppose that the speed of the ants are pairwise distinct and they did not change their own speed. Prove that if $|T| \le 45$ then there exist two ants did not meet.
Note that two ants met when there exists a time point such that they were at the same location on the line, including the holes.
Note that two ants met when there exists a time point such that they were at the same location on the line, including the holes.