So you've qualified for AIME - good job! This guide is meant to help you prepare for what will ultimately violate your confidence in your ability to do math. This first part is meant to tell you what you should generally know if you want to do decent on it. I'm going to base this on the AIME class that I took last summer:
Equations:
The type of AIME problems that are, for the most part, just plain old algebra. There's not much more to learn here. The best way to prepare for these is to go through old AIME problems about systems of equations to be familiar with the types of factorizations and substitutions that may be useful.
Complex Numbers and Polynomials:
Besides the basic a+bi notation, you should be familiar with polar representation. Also, you should know Vieta's. Newton Sums would be helpful, but if they actually come up, you can always derive them. (If you don't know what those are, go look them up. =P )
I've found that doing Olympiad problems about polynomials can actually be pretty helpful. So go bust open PSS and ACoPS for preparing for these types of problems.
Functions
Floor function, sine/cosine, logarithms, and just plain old f(x). Solving functional identities can help you be more comfortable with the function problems on the AIME. Also, going through old problems will give you more of an idea of what sort of questions to expect (do you see a pattern yet?).
Inequalities, Optimization Problems, Sequences, and Series:
AM-GM, Arithmetic sums, Geometric sums, recurrences, etc. Know them and know how to use them. Most sequence problems on the AIME will be pretty straight forward. On the other hand, most inequality / optimization problems are, as far as I've seen, pretty tricky. Again, GO THROUGH OLD PROBLEMS.
Counting:
Know your basics is the most important part here. The rest is being clever in how you apply them. Granted, it'll be helpful to know a few tricks, such as balls and urns, seating people such that nobody is next to one another, etc. The hardest counting problems on the AIME will require you to create some sort of recurrence to solve them. OLD PROBLEMS. DO THEM.
Probability:
Most of the probability problems on the AIME will actually just be a mix of counting and some other category. Know your combinatorics, conjugate counting tricks, and geometric probability, and you should do fine with these. Most of the time, the probability problems will appear to be rather tricky. The trick is seeing past this and simplifying it. This is the last time I'll tell you to do old problems. It applies to every category though.
Number Theory:
The category that most AIME qualifiers know nothing about. It's unfortunate; number theory is quite beautiful, and very common on the AIME. Things you should know are modular arithmetic and Diophantine equations. Once you do enough number theory problems, you become familiar with the tricks involved: when it's appropriate to factor (almost always), when its appropriate to expand something out (almost never), things like that.
Trigonometry and Analytic Geometry:
Almost all geometry buffs will tell you to avoid resorting to trig and coordinates to solve geometry problems. However, sometimes it will be clear that you simply need to use coordinates and trig to solve a problem. If they give you ellipses, parabolas, and trig functions in relation to a figure or angle, then its probably safe to reach for these tools. You should be familiar with the basics of coordinate geometry, and all of the trig identities. Most of the time, you'll have to use these tools to create equations, which you can then solve for your desired variable.
Euclidean Geometry:
Arguably the most important part of the AIME. Also arguably everybody's worst subject. I can't possibly list all the crap that falls into this category. The most important thing to be familiar with besides all the basic formulas are similar triangles. They come up a lot, and if you can recognize them and use them, it'll be very helpful.
In another post, I'll go more in depth on a few of these. Here's some nifty music (more MGS4 stuff).
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