Do the following:

First, plot the nun's position on the trail (x axis, with her starting position at the origin and her end position somewhere to the right) against the time of day (y axis) for when she goes up. You will get a line, probably not a straight line, that goes monotonically up and to the RIGHT. Color that line red.

Next, on the same graph, plot the nun's position on the trail vs. time of day for when she comes back. You will get a line that goes monotonically up and to the LEFT. Color that line blue.

Now look at the graph. There is absolutely no doubt that SOMEWHERE on that graph, the two lines intersect. Both start at y=0, one on the left (origin) and one on the right. They both end at some positive time on the y axis, one going up and to the right and the other going up and to the left. The red line will end at the same x position that the blue line starts, and the blue line will end at the same x position that the red line starts.

In other words, there is no possible way the two lines cannot cross. They HAVE to cross. What's more, as long as the nun keeps moving and doesn't go backward on the path, there can only be one point where they cross.

The odds are 100% that the lines will cross. No way around it. And at that point, she will be at EXACTLY the same place that she was the previous day at EXACTLY the same time.