Complex Nums

Complex Numbers is a unit on using complex numbers in geometry. If you were looking for complex numbers more towards algebra, you are looking for Cyclotomic, which is not really about cyclotomic polynomials. One can think of complex numbers in geometry as vectors, except their multiplicative structure allows for much more versatility.

If you've done EGMO chapter 6 before, most of the formulas should be review. One important thing to note is this blog post by Evan about arc midpoints, which has some new information and intuition on arc midpoints. Expect the problems in this unit to feel different than EGMO. This unit is also pretty middle of the road in terms of difficulty, perhaps on the easier side for D units if you hate synthetic.

Philosophy

Like bary bashing, complex is usually regarded as a bash method. However, Evan doesn't regard this as a "bash" unit. With the right setup, one could decrease the amount of work needed by a factor of 2 or even 5. To allow this right setup to appear, of course, requires thinking, so the main takeaway to get from this unit is to not turn off your brain. Evan wants you to believe that complex numbers isn't actually bash (I still think it is). In addition, there are also a few tips on actually performing the calculations, which are very helpful.

Advice

• If you don't know if you should request this unit because you think you already know complex numbers, just do it anyways. There are some lessons to be learned here not found in EGMO.

• While Evan tries to get you to think that complex nums isn't actually bash, some problems may still end up with expressions with over 10 terms in them. So don't second guess yourself when things start to get a little hairy. But of course, if you find yourself approaching the page limit for some problems, you're probably doing something wrong.

• One technique that happens to appear in the setup of a few problems is to rewrite the problem more in terms of whatever unit circle you choose. For example, if you are intersecting a line with another line, instead of using the big and annoying general intersection formula, sometimes you can intersect one of the lines with the unit circle, and now calculations get a bit easier.

• While in real life complex numbers could just be a step in a solution to a problem, in this unit all the problems can be solved outright with complex numbers. Of course, you might need one or two synthetic statements to glue your setup together, but usually nothing nontrivial.

Notable Problems

Taken from DGW:

• RMM 2019/2: Despite being problem 1 (and required) on DGW, this problem is actually quite tricky so don't be discouraged if you get stumped by the first question in the problem set.
• EGMO 2017/6: Looks pretty wtf at first sight. You might even have no idea how to approach synthetically, even less so with complex. As it turns out, it's not too bad.
• Z98CCCD6: A really weird looking problem that would be difficult to synthetic, but surprisingly clean in complex. Would recommend.
• ISL 2000/G6: One of the more famous late geo ISLs. This one is 5 clubs for a reason; a good setup goes a long way, but this time the setup is the entire problem deconstructed.
• IMO 2011/6: A notoriously hard P6. You might recognize this from EGMO. A bit more than just complex numbers is required here.