Super Complex
A more difficult version of the complex numbers unit.
Unit Summary
This unit is essentially a more difficult version of the complex numbers unit, as you may have inferred from the title. It includes a good number of challenging problems that may take a lot of time, beyond the time given in any standard proof-based exams, if the setup is done without much thought. This unit focuses on actually approaching the problem and thinking about how to complex bash it instead of simply giving you problems that require a lot of calculation. In order to do these problems, you need to make synthetic geometry observations to optimize your calculations. In order to help you confirm if your setup is indeed proper, each problem has a page limit to see whether you are going in the right direction. It is recommended to do the D-level Complex Nums unit before trying this unit.
Philosophy
Only bash when you need to...
The unit introduces how to optimize bashing with synthetic observations, projective geometry, etc. Remember not to just plow through a bash without actually thinking about the problem.
Should I take this unit?
If you are a synthetic geometry lover who hates bashing, you may ignore this unit. However, if you are a geometry basher who relies on brute force instead of creativity, it is highly recommended that you take this unit since it will help you learn how to optimize your calculations and complete the problem in less time, since obviously not all geometry problems are bash-able in an hour if you do not use a creative setup. For those who are new to complex numbers, take the B-level or D-level Complex Nums unit first.
Notable Problems
- Shortlist 2018 G7: An interesting problem that uses complex numbers along with diagram intuition and other basic observations to optimize calculations.
- IMO 2019 P6: A fun problem using a straightforward, complex Bash approach; routine work for experts.
- USA TST 2006 P6: An extremely enjoyable problem that uses rotation and some angle chasing but finishes with complex numbers.
- IMO 2011 P6: A difficult yet fun problem that uses a creative complex number setup that is quite hard to see.