Hybrid Geo
Grab-bag of geometry problems that feature some algebra, combinatorics, or number theory. As examples, this includes geometric inequalities, combinatorial geometry, and problems involving integer distances or lattice points.
Hybrid geo is a unit about problems that feel different from usual geometry. Such a problem can be unusual in its statement (for example it might ask to prove some inequality or a combinatorial result), but it can also have an unusual solution that feels more like algebra rather than geometry (i.e. trig bash or a heavy length bash).
This is probably the most versatile geometry unit. It includes problems from all three of the following categories:
- Geometry + Algebra
- Geometry + Number theory
- Geometry + Combinatorics
What you can take from this unit
On most of the IMO shortlists, there are a few of problems that fall into this category of "unusual" problems. If you find yourself skipping those, you should consider doing this unit. In general, this unit will feel like getting out of the comfort zone, which makes it one of the most fun units to do.
Philosophy
In contrast to units such as Harmonic, Euclid alg, or Equality, Hybrid Geo does not focus on a particular technique or a philosophy. Besides that, the theoretical portion of the unit is very rich, involving stories such as Ptolemy's inequality, Fermat point, Erdos-Mordell inequality, Fagnano's problems, and the idea of convex hull.
Notable problems
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IMO shortlist 2011 G1: One of the hardest G1s on the IMO shortlist.
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IMO shortlist 2014 G2: One of the hardest G2s on the IMO shortlist.
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IMO 2001/6: This is a number theory problem in a geometry unit.
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IMO 2013/2: This is a combinatorics problem pretending to be geometry.
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(Fagnano's problem) Among triangles inscribed in an acute triangle $ABC$, which one does have have the smallest perimeter?
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(Fermat point) Among points $P$ inside a triangle $ABC$, which one minimizes the sum $PA+PB+PC$?