[lell]

the case where n=2, k=1 was an IMO problem around 2001/2002 (and the parent article says it is first from spivak '97) and it's pretty easy to see the generalization if you know about fundamental groups. I used to pose the n arbitrary, k=1 case to other math students :D (parent article claims this version is first from mathpuzzle.com, 2002)