Contest problems with connections to deeper mathematics

Solution 1:

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.