Topic for a math paper competiton

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I'm looking for a topic for a national Math Paper competiton for High School students. The problem is that in the paper I have to study some unsolved before problem. My main interest is mathematical analysis but I know it's difficult to study something there on my level. So far I thought about finding the maximum volume/area of an n-hedron circumscribed on a unit sphere and related problems. I'd be really grateful for any other ideas, preferably not in geometry.