I like the Jordan curve theorem, because apparently no one even thought of it as something that needed to be proven until the 1880s, and then it turned out to be remarkably difficult to prove.
The statement of the theorem can basically be summed up as "every closed, non-self-intersecting continuous curve in the plane has an inside and an outside." which is glaringly obvious
@Austin_Dern
Also fun, the Brouwer fixed-point theorem implies that there must always be two antipodal points on the earth with exactly the same temperature and atmospheric pressure.
And the hairy ball theorem says that there must always be at least one point with no wind.
@Austin_Dern I don't think so.
@Austin_Dern okay, important question: what's the track's grade?
@Austin_Dern With no specification of the grade it's trivial, then--at 90° grade (i.e. vertical track) there is no way to prevent it from hitting one side if the speed profile is, say, zero.
@Austin_Dern not that you'd ever encounter a grade of more than like 3° on actual rail tracks but hey you said no specification
@Austin_Dern ∈ that case my instinct is to say that there does always exist such an angle.
@Austin_Dern oh, uh, that was supposed to be in that case. i-n is the compose sequence for ∈, I guess I hit compose by mistake
@Austin_Dern I'm an engineer, I assume reasonable constraints on everything by default. I deal with physical systems!
@Austin_Dern Because if it's flat or at a constant angle the answer is obviously yes, but if there are changing slopes it gets trickier