$\textbf{Mountain Hiker}$
John starts to hike up a mountain at $7:00$ am and reaches the top at $7:00$ pm. He stays at the top overnight. On the next day, he starts to hike down at $7:00$ am along the same route and reaches his starting point at $7:00$ pm. His speed during the two-day hiking varies from time to time. What is the probability that there exists one spot he passes at the exactly same time during the two days?
$\textbf{Answer}$
The probability is $100\%$.
$\textbf{Analysis}$
Imaging there is another John. Let's called him John II. John II will mirror John's first day movement exactly, but will travel on the second day. That is, John II starts to hike up at $7:00$ am while John starts to hike down. Then obviously these two will meet somewhere along the route. This is the exact time and location satisfying the given requirement of the problem. Because they will for sure meet, therefore the answer to the original question is $100\%$.
$\textbf{Note}$
This can be a mathematical problem in calculus. But as a brain teaser, it will unlikely require advanced thoerems. This solution essentially utilizes the principle of symmetry.