Weird Geo

Are hexagons your favorite shape? Then this is the unit for you! This is a unit about those strange geometry problems that might seem impossible (weird angle conditions, hexagons, pentagons, etc.) This is about medium for both a D unit and a Z unit.

Philosophy

Something strange about many of the problems in this unit is that they can be complexed 100% of the time. In fact, when doing these problems, Evan's advice is to bash more (sorry geo mains!) A lot of the synthetic solutions to "weird geo" problems are near-impossible to find, so bashing is your best bet.

However, even when doing your bash, the algebra can be quite interesting. The reason why can be explained by degrees of freedom, which is fundamental to this unit. In OTIS units, we use the convention that degrees of freedom are up to rotation and translation but not to similarity (so, for example, a triangle has 3 degrees of freedom, and a cyclic quadrilateral has 4). Many of these problems involve counting degrees of freedom in order to set up a proper attack (so for example if the number of degrees of freedom don't actually match up with the problem statement, then we might have an inequality on our hands.)

Notable Problems

USAMO 2002/2: An excellent example of a strange equality that is actually an inequality (which can be motivated by degree counting!). A walkthrough.

RMM 2017/6: One of the most involved and intimidating walkthroughs in all of OTIS, but extremely instructive.

IMO 2005/1: Required on both versions. Has a solution that seems like magic, but if you're not a magician then there is another solution that is messier but easier to get.

Shortlist 2013 G5: An example of a strange condition that begins to unfold as you look more into the problem. One of the most famous "weird geo" problems.

USEMO 2021/3: 9 pointer on ZGX. An incredible problem, but also absurdly difficult.