A monk climbs a mountain one day, descends it the next; prove he repeats a spot
A monk starts at dawn and reaches the summit by dusk on day one, walking at whatever varying pace he likes, even stopping to rest. On day two, he leaves the summit at dawn and reaches the bottom by dusk, again at any pace. Prove that there must be some time of day when he was at exactly the same point on the path on both days.
Reveal the answer
Imagine both trips happening on the same day: one monk climbing up, a second (identical) monk climbing down starting from where the real monk ended up. The two imaginary monks must meet somewhere on the path, since one ends above the other's start and vice versa, and their positions change continuously. That meeting point is the proof, a simple application of the Intermediate Value Theorem to an everyday scenario.