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.
I'm collecting such problems for many years. The current collection is here, it is about 80 problems.
At the moment, I have just few problems in topology and in geometric group theory and just one in algebraic geometry.
Thank you all for nice problems --- I gave bounty to one, but would give it to 4 problems if I could :). Some of them are in the collection by now --- please post more...