Combinatorics? Hmm, not bad at all. Particularly if there's a bijection involved.
Number theory? Yes, if I'm lucky and if I don't leave out a pesky 0 while dividing.
Algebra? I stand some chance. A little bit. Just hope I don't run out of variable names.
Geometry? Next problem, please.

This blog is basically me trying to teach myself geometry. Because I fail so hard at it. I'm starting this on the first day of RL school, too. Of course first days are always light and easy so let's start with the bits of geometry that go with my best subject.

It's become apparent that me writing the above is just me procrastinating the actual math, so here goes.

Helly's Theorem. If there's a finite bunch of convex sets and any three of them have a point in common then all of the sets have a point in common.

This theorem does not have anything to do with any of the below problems, but it's fun to prove, and it is also the only thing here that I actually solved recently.

1. A point is in a convex polygon. Prove that at least one perpendicular from the point to one of the polygon's sides will intersect that polygon's side (and not either of its extensions.)
(Okay, this is really easy and not really combi. Bonus point if you can do it in one sentence.)

2. Does there exist a finite set of points in the plane such that, for any point , there exist at least (distinct) points in that are distance 1 from ? Bonus if you can fit in a square with single-digit area.

3. Given a finite sets of points on the plane, prove that you can color them black and white so that on any line parallel to one of the coordinate axes, there are either an equal number of black and white points, or their numbers differ by one.

4. Construct a finite set of points so that any point in that set has exactly three points that are closest to it (that is, equally close to , and closer to than any other points.)

Okay, I'm cheating by putting up random problems I did a long time ago. I just like having this on the first day of school. Oh well.