We all know that problems from for example the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you like, and, that you would all add the connection it has.
The most interesting part would be to see solutions to these problems using both elementary methods, and also with the more abstract "deeper methods". Hopefully, the more abstract method should make a solution easier. This would be nice examples on how abstraction could make problems easier.
I hope this is not too subjective.