I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you have a problem like this please post it here.
Remarks:
I have been collecting such problems for many years. The current collection is at arXiv; the paper version is available at amazon.
At the moment, I have just a few problems in topology and in geometric group theory and only one in algebraic geometry.
Thank you all for nice problems --- I gave bounty to one, but would give itdecided to 4 problems if I could :). Some of them areadd bounty once in a while and choose the collection already ---best problem please post more..(among new or old).