Question 1
There are 44 distinct holes along a straight groove and 2017 ants. Each ant emerges from one hole, crawls at constant velocity to a different hole, and disappears into it. Let $T$ be the set of all emergence and disappearance times. The ants' velocities are pairwise different, and $|T|\le45$. Prove that at least two ants never meet. Two ants meet if, at some instant, both occupy the same position in the groove, including their emergence and disappearance instants.