Problem 1: One man started climbing a mountain at 5:00 AM. He spent the night up on the peak. Then, on Day 2, he set off down the mountain at 5:00 AM. Prove that at some point of the day, he was at the exact same altitude as he was at that same time the previous day.
Problem 2: Two men are planning to climb a mountain. One is climbing from the North side, the other from the South. Prove that the two men can climb the mountain in such a way that at all times, their altitudes are the same.
These problems are harder than some of you may claim. You can't simply say he can do it; you have to rigorously prove he can do it. Also, you have to prove that the conditions must be satisfied no matter how he takes his hike, not just that there exists a way for him to satisfy the conditions.