AMC/AIME prep/AOPS questions

[nsguy1350]

Hey guys. First off, I didn't know whether or not to put this in the AMC/AIME section or not, but I don't think it really matters that much. Anyways, I've been looking at the Art of Problem Solving (AoPS or AOPS) forum and bookstore, and I've heard these books are excellent for preparation of those contests. I will order some this summer and try to work through them. I'll have plenty of questions, so if anyone else (I know a few people already) would like to study with me and several other people, you are welcome to join! It's nice to do something a bit more intensive than UIL.

For the books, pre-tests and post-tests are provided, to help determine if the book would be necessary for a potential buyer. Currently, I am thinking about buying the Introduction to Geometry, Introduction to Number Theory, Introduction to Probability and Counting, Intermediate Algebra, Precalculus, Intermediate Counting and Probability (which I've heard is not really necessary for AMC/AIME, since the Intro is generally sufficient. I also heard this is an INTENSIVE book, and possibly the hardest AOPS class,) and if I ever have time, the Calculus book. I've heard that these books are much more in-depth than your ordinary Algebra/Geometry/Precalculus etc. courses, so that's why I'm considering buying the introduction. Also, there are two volumes that are specifically designed for contest preparation, so I'll do those later, after I get through a good portion of the Intro/Intermediate series.

Anyone care to join me? Any advice, plans, or suggestions?

Also, one question, from the Intermediate Algebra post-test:
Simplify [unparseable or potentially dangerous latex formula].
Answer:[unparseable or potentially dangerous latex formula].
What is an efficient way of solving for that? My friend and I have tried several different methods, and none have worked out. Now I am seeking help. Thanks!

[rcoker7]

fourth roots are messy, so we should just take 2 square roots instead. square roots of this sort are usually in the form a - sqrt(b). Squaring this gives us a^2 + b - 2a(sqrt(b)). We can set this equal to the first equation by separating rational and irrational parts(this technique will work the majority of the time). This gives us 161 = a^2 + b and -72(sqrt(5)) = -2a(sqrt(b)). Solving this gives us 2 solutions, but we must take the positive one to continue, which is 9 - sqrt(80). This could be simplified but it would be a waste of time. Using the same method, we'll end up with either sqrt(5) - 2 or 2 - sqrt(5).

[007math]

Man, you are gonna be busy this summer! And that is a lot of books :O (which I see as a lot of money). I bought the AoPS Volume 1 & 2 over 4 years ago. Never have gone through one of them fully. lol, I haven't even opened one of them in high school.

[nsguy1350]

Hmm... I'll try it Ross. We also did a two square root process, but instead of finding [unparseable or potentially dangerous latex formula], we didn't factor out from the square root and got harder to deal with answers. Umm, that might be a bit vague, but I have a new idea and I'll see how it goes. Thanks!

And oh Adeesh, I'm not doing all of those books this summer. I plan on buying a couple and then going through them, and then doing some more over the school year, where my school can provide them (talked to my coach, we might have an afterschool AMC/AIME thing now, and they can provide books. Interesting.)

[SumitG]

This fourth root question actually falls very much in line with Galois Theory. You can even figure out the roots to polynomials with this value as a root which are present in other fields. You also can tell whether if you reduced the field you were working in to something like the rationals if your root would still exist, and what other roots might be guaranteed existence if it did exist. It's a pretty fascinating topic.

[nsguy1350]