It'd be very easy to cheat on math though.

Agreed. With the internet, you can find out the solution to pretty much any math problem.

someone can make a multiplayer game with tourney features and we must become champ?

If someone were to do that, would it be a valid idea for a special competition? Something that's play-with-other-people-at-same-time would not make sense (networked games that aren't something like cards or puzzles are usually inherently unfair, so screw that), but something built with a score or time attack in mind and anti-cheated could work.

Earlier I had the idea of possibly doing a different type of music competition where members could record their self actually playing an instrument. This would only work if enough MFGGers knew how to play an instrument, but I figured that the idea is worth throwing out there.

There's nothing in music competition rules that say it can't be someone playing an instrument anyway. I think when Yoshiman was still around he used to play some of the instruments himself for music comps.

Sorry if this reply is late, but I have a few things to say.
1. Cooking Competitions: I know that most people aren't willing to send food overseas or pay for shipping costs to someone regionally, but an alternative would be for each member to design a recipe, including step-by-step instructions on what to do. They could post the recipe in the topic and have the person who may be judging replicate the food MFGGers make.

That's actually a really good idea.