The following extremum problem in a triangle has not been solved. It looks simple, and intuitively "P = mid point of BC" seems to be the answer, but noone was able to prove it yet. The closest solution brings a set of 4 equations which was not able to be solved neither by WolframAlpha nor by Maple.
Creative Unusual Geometry Problems: Fox 322
I am walking 8 foxes at the top of my head and trying to keep their tails apart !