ACM ICPC World Finals 2008
This problem had two challenging parts, computing the line segments connecting the shafts, and then finding the actual shortest path.
The first part is “just” a pen and paper exercise in geometry (it can also be done by binary search for those who are so inclined).
The second part can be done by brute force, because of the small number of shafts (it may look as though there are 20! possible paths, but because of the non-crossing condition of the path, the actual number of possibilities is a lot smaller). I do not believe that there is a polynomial time solution for this part, but I would love to be proven wrong.