I am reminded of this question that appeared in a regional Physics Olympiad I had appeared. I was wondering if there is a "mathematical" way of doing it.
If you start from a point $A$ at midnight along a specified path down a mountain and reach a specified point $B$ exactly 24 hours later. At this point you reverse your direction of travel and travel along the same path to point $A$ and reach $A$ exactly 24 hours later. At any point of time during these two days, your velocity can be positive, negative or zero (and of course less than $c$). Prove that there exists at least one point along the path where you were at the same time on day 1 and day 2.
It "seems" like an application of intermediate value theorem for some appropriately defined function, but I am not sure (though I will tag it as calculus for the moment). Any ideas?