<p>i might be taking it in febuary, although im a senior so it wont really benefit me in any way. it would be my first time taking the series.</p>
<p>DISCLAIMER - i've never actually taken an AMC of any kid, this is based solely on my experience w/ practice problems.</p>
<p>Ok here is how id generally advise preparing for american mathematics competition series:
The most important thing to understand is that mathematics is not STUDIED. it is DONE. </p>
<p>if youre looking just to qualify for AIME, aops 1 is sufficient.</p>
<p>going through 1 and 2 should get you to USAMO easily, itll cover most tricks youd need and/or topics not taught in school. (combinatorics, number theory, more geometry.) </p>
<p>AOPS subject books are great for this also, up to maybe basic USAMO (like #1/4), and just for learning material in general.</p>
<p>on both AMC/AIME, it is important to WATCH YOUR COMPUTATIONS, especially if youre computationally challenged like me. </p>
<p>USAMO preparing is slightly more complicated. you need to know how to write proofs.</p>
<p>advice for this:
honestly the difficulty of proving things is vastly overstated.
learn a proof by contradiction, induction etc. for an example of induction see any discrete math book. for proof by contradiction just look up euclid's proof that there are infinitely many primes. </p>
<p>topics you'll need to know:
algebra: most of this is pretty basic stuff like properties of polynomials, complex #s, sequences, etc. nothing really new.
geometry: this is where it gets interesting. you can usually solve most geo problems with the tools developed in aops 2, but if you like theory you may look into stuff like projective geometry, transformations, inversions, geometry of complex numbers etc. (really not necessary though.) cut-the-knot is a cool resource here.
number theory: nothing special/advanced required here for the most part, its mostly just practice.
combinatorics: i always find this to be hard for whatever reason. honestly id recommend picking up an introductory college combinatorics text, itll be comprehensive.
graph theory: same as above, although seeing as its a college subject i honestly don't get why its on usamo.
inequalities: many pdfs available for download, its basically just algebraic manipulation. lately hasnt really appeared on USAMO, i guess.
invariants/colorings/logic/etc..: this is a new type of problems which dosent really appear on AIME/AMC. basically you look for something which dosent change. its harder than it sounds because the invariants are usually hard to find (to me they just seem random, but thats because im bad at these kind of problems.) </p>
<p>note that altough i might be making this sound easy USAMO is not really easy in any way. preparation is straightforward, problems not so much.</p>
<p>practicing problems is important here. usually after aops 2 people proceed to zeitz (art and craft of problem solving) and engel's problem solving strategies. also for individual areas of math, the books by andreescu/feng are good and have problems from AMC - olympiad level. coxeter's geometry revisited is recommended.</p>
<p>in general, the most imporant thing, to reiterate, is PRACTICE. this is a LEARNED skill. on aops i remember reading a post where an IMO gold medalist reported getting 80/150 on AMCs when they began doing those type of problems. (genius is after all mostly based on how much work you do.)</p>
<p>for example the problem posted by HiPeople above can be solved by a standard trick/process using remainder theorem, probably taught in aops 1 or 2. by doing problems you will learn the necessary tricks/processes for AMC/AIME. for usamo the same is true, but problems dont really fall to tricks, which imo is what makes it challenging.</p>