I've recently discovered this page which contains a list of unsolved sublinear problems.

I thought this is a good place to spread the word since most of them have a short, understandable description — You don't have to research a lot to understand the problem — see this example.

If you want to read more about this a good place to start is this MIT page which links to some surveys.

Maybe some you will manage to solve a couple. Many of you will surely enjoy reading them!