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.
Best Answer
This isn't quite what you want, but....
On the 1971 Putnam, there was a question, show that if $n^c$ is an integer for $n=2,3,4,\dots$ then $c$ is an integer.
If you try to improve on this by proving that if $2^c$, $3^c$, and $5^c$ are integers then $c$ is an integer, you find that the proof depends on a very deep result called The Six Exponentials Theorem.
And if you try to improve further by showing that if $2^c$ and $3^c$ are integers then $c$ is an integer, well, that's generally believed to be true, but it hadn't been proved in 1971, and I think it's still unproved.