Just do a plot of the position of the hikers on the trail over time. x-axis is time of day. y-axis is distance along the trail.

Hiker 1 starts near the bottom left (beginning of day, ground), and moves toward the top right (end of day, summit). Hiker 2 starts near the top left (beginning of day, summit), and moves toward the bottom right (end of day, ground). Maybe due to changing sunrise/sunset times, the start/end times for one path are shifted SLIGHTLY to the left/right for one relative to the other. But that's irrelevant.

At some point during the day, these curves have to cross. At that intersection, the hikers are in the exact same spot along the path at the exact same time of day. Speed does not matter. One hiker could stop - or both could stop. They could hike in circles for a bit if they want. But time can't go backward. Eventually they will cross. It doesn't matter what time and point, but there will be one.