How much overlap is there between AoPS contest prep books and their standard curriculum?

[lewelma]

Because of all the lovely help and advice from this board, my ds is excited about prepping for the qualifying exam for the Math Olympiad Summer Program. In NZ, this exam is a 1 month, take-home exam with "junior" problems similar to ones on the Mathpath qualifying exam ("senior" problems are much harder). I would like to buy him some books to study and don't want to buy material that he has already covered in AoPS's other books. He has finished or is currently working on:

 

Intro to Algebra

Intro to Geometry

Intro to Counting and Probability

Intro to Number Theory

 

I am particularly wondering about the contest prep books: "Volume 1 the Basics" and "Volume 2 and Beyond". Or is it better to just work through previous exams? Are these prep books designed for multiple choice exams or are they for the 1-month proof-based exams?

 

Also, he needs to understand how to write proofs effectively and what types of proofs are possible. What books would you recommend for a beginner?

 

 

Thanks,

 

Ruth in NZ

[wapiti]

I would ask AoPS directly for a personal recommendation with all the info you have posted here. They usually respond quickly.

[Kathy in Richmond]

Because of all the lovely help and advice from this board, my ds is excited about prepping for the qualifying exam for the Math Olympiad Summer Program. In NZ, this exam is a 1 month, take-home exam with "junior" problems similar to ones on the Mathpath qualifying exam ("senior" problems are much harder). I would like to buy him some books to study and don't want to buy material that he has already covered in AoPS's other books. He has finished or is currently working on:

 

Intro to Algebra

Intro to Geometry

Intro to Counting and Probability

Intro to Number Theory

 

I am particularly wondering about the contest prep books: "Volume 1 the Basics" and "Volume 2 and Beyond". Or is it better to just work through previous exams? Are these prep books designed for multiple choice exams or are they for the 1-month proof-based exams?

 

Also, he needs to understand how to write proofs effectively and what types of proofs are possible. What books would you recommend for a beginner?

 

The AoPS contest prep books are great for any kind of math olympiad prep, but aren't specifically targeted to the type of month-long take home exams you're talking about. The problems sets pull mostly stuff from past Mathcounts, AMC, Mandelbrot (run by Sam Vandervelde who often teaches at MathPath), Mu Alpha Theta, etc, exams, which are typically done in one sitting, not over the course of days (though I have seen the occasional IMO or MOP problem go by--way tough! there really is no upper limit in the second book. No one could solve *every* problem inside it).

 

They'd still be great to have on hand for your son; I refer to my copies frequently. I have sticky notes in my copy of Vol 1 bookmarking the geometry sections. They're really great summaries of techniques and also how to tackle and think about tough geometry problems (they're often the hardest category on math contests!). If he's finished all of those AoPS intro texts, then he's ready for Vol 2, though. There's plenty of challenging material in that one, including partitions & generating functions in combinatorics; concurrency & homothecy & inversion in geometry; quadratic congruences, Fermat's theorem, Diophantine equations in number theory; graph theory...these are all tools needed for a kid going the math olympiad route in order for him to develop those longer proof-based problems. The texts are a handy toolbox. Volume 2 kept my kids busy all the way through high school. And he'd still learn how to write up problems and tackle proofs. The answer guides, as always, are very nicely done with lots of detail and good examples of mathematical writing. There is a brief intro to formal proof techniques at the end of Vol 1 & also the beginning of Vol 2 covering three major types: proof by contradiction, mathematical induction, and the pigeonhole principle.

 

I'd recommend also getting a copy of The Art and Craft of Problem Solving by Paul Zeitz. Paul was a member of the first USA IMO math team, has been involved in training math olympians, and has taught at MathPath in the past. He's a great guy with a knack for explaining things well. His book is excellent training in how to think mathematically (the first half includes problem solving strategies and proof techniques), and also for content areas (the second half applies those techniques to algebra, number theory, combinatorics, and calculus). I think it's a good complement to the AoPS volumes: the presentation is slightly more advanced (especially the proof techniques), though the AoPS books are better for geometry and have a greater quantity of practice problems with full solutions. The Zeitz book just has hints, not solutions. According to the Preface, there is an accompanying Instructors Resource Manual with solutions, but I've never tried to purchase it.

 

For further work in proofs, I'd recommend old USAMTS exams. The USAMTS contest consists of five problems at a time, and kids are given about a month to work on them and write them up for grading by teams of AoPS graders & NSA mathematicians. They're high school level, though, so they get tough fast. I'd recommend going back to the earliest years; the problems seemed easier to me in the old days (before AoPS came online & started training so many of these kids!). Each exam has published solutions, so he can compare his solutions to those.

 

Good luck to your son, Ruth. Sounds like you have a wonderful math year planned. :001_smile:

[lewelma]

Thanks Kathy. This is very helpful!

 

Interestingly, my ds finds the geometry problems the easiest. That is the only category that he can solve in the "seniors" section!

 

Do you have a recommendation for game theory? There are definitely a few of those on past exams.

[Kathy in Richmond]

Interestingly, my ds finds the geometry problems the easiest. That is the only category that he can solve in the "seniors" section!

 

He and my daughter are two of a kind then, with those visual spatial & creative strengths that help them in geometry. :001_smile:

 

Do you have a recommendation for game theory? There are definitely a few of those on past exams.

 

 

Game theory is tougher to recommend!

 

My kids learned about it at summer camps. I spent some time yesterday and today going through their camp notebooks trying to figure out how their instructors approached it. In MathPath, John Conway (co-author of Winning Ways for Your Mathematical Plays, the bible of combinatorial game theory) gave colloquia on game theory (dots and boxes, game of life, and the like) where he introduced the subjects by playing the games with the kids. Then he walked them through a variety of problems & solutions. In Mathcamp, Alfonso Garcia-Saz taught their game theory course, again via examples like those in Conway's books. Then he turned them loose on problem sets.

 

Most resources on the market labelled game theory have an applied flavor & are often written from an economist's point of view. They're interesting (typically covering the prisoner's dilemma, Nash equilibria, competition, warfare, etc), but they're not exactly what you're looking for. This includes Coursera game theory (which I took last year & liked), Yale OCW game theory, and most textbooks that I found via google search. The types of game theory problems that come up in math olympiads and math camp admissions quizzes have a combinatorial, not economic, flavor. They are simple problems to pose, but often terribly difficult to solve. It sometimes seems that each one has a solution approach all its own.

 

So where could you find helps in combinatorial game theory? some ideas:

 

Winning Ways for Your Mathematical Plays by Berlecamp, Conway, & Guy is a pricey 4-volume set. It's accessible and overwhelming at the same time (I only own volume 1). Many, many games are described and analyzed, but there's no real unifying discussion. It feels like lots of disparate examples, albeit fun ones! And no problem sets are included to test your understanding. For a pre-college student, I'd rather invest my money in Conway's Book of Numbers if I were going to splurge.

 

A couple of handouts (here & here) my kids received in their Mathcamp game theory class. Again, these are lecture via examples followed by homework sets. No answers, though.

 

Arthur Engel's Problem-Solving Strategies has a short chapter on classic game theory (analyzes NIM, etc). It's a dense book, though, and I ususally recommend it after the AoPS and Zeitz problem solving volumes. Again, it teaches game theory through examples and problem sets. Solutions are included this time.

 

What I think I'd do in your shoes would be first to learn basic combinatorics and proof techniques (such as the material in AoPS problem solving volume 2) so that your son has the right tools in his toolbox. Then I'd have him use USAMTS problems to gain experience. I'd select game theory problems from past exams - there should be quite a few - and just dive in and try to solve. Read the problem statement for understanding, play with it for an hour or two, and then study the solution. You'd be mimicking what the summer camp teachers do. Next pull a couple more and try to have him solve them on his own (over a few days). Try hard, but if he can't solve them, have him read and study the solution again. Put them aside and try them again after some time has passed. Eventually he'll get the hang of it! They'll get easier over time.

 

Here are a couple examples to start with from the USAMTS archives:

Problem # 4/1/15 and solution

Problem # 3/4/17 and solution

 

hope that helps!

[lewelma]

Oh, Kathy, you are wonderful!

 

Now that he knows there is a camp in NZ, he is very motivated to try to get in. Unfortunately, it is THE Math Olympiad camp, and they only take 24 high school students. He has NO interest in multi-choice, timed math exams, but because the qualifying exam is the month long proof kind he is quite excited. The "seniors" problems are for rising 11th and 12th graders and the juniors for the younger students. 8th grade here is high school, but given that you have to apply 2/3rds of the way through 7th grade, I am not sure that the intermediate school teachers will have a clue about this program, so my ds might be the only rising 8th grader taking the exam. Not sure if they will take his age into account -- I am really curious about the selection criteria. I have warned him that he is not likely to get in this year, but has a good shot for it as a rising 9th grader given that there were only 120 kids that took the exam last year.

 

I've told him that the *quality* of his proofs are what is key, not just the general idea of the answer. I suggested seeing a proof like an essay -- make an outline, write a draft, edit, proof-read, and then copy it over. We will definitely need to practice this skill. He is currently very "hasty", and says "I get it, I get it, you just do xxxx." Then when I ask him to write down the proof, he is not very interested. I don't think he is clear that he needs practice not only answering the questions, but also doing the boring work of writing out a clear-cut proof. But he is happy to do it with geometry, so at least I have a place to start.

 

I am still looking for a math mentor and have a few leads.

[lewelma]

Just found Conway's Book of Numbers in our local library!!!!

[Kathy in Richmond]

Now that he knows there is a camp in NZ, he is very motivated to try to get in. Unfortunately, it is THE Math Olympiad camp, and they only take 24 high school students. He has NO interest in multi-choice, timed math exams, but because the qualifying exam is the month long proof kind he is quite excited. The "seniors" problems are for rising 11th and 12th graders and the juniors for the younger students. 8th grade here is high school, but given that you have to apply 2/3rds of the way through 7th grade, I am not sure that the intermediate school teachers will have a clue about this program, so my ds might be the only rising 8th grader taking the exam. Not sure if they will take his age into account -- I am really curious about the selection criteria. I have warned him that he is not likely to get in this year, but has a good shot for it as a rising 9th grader given that there were only 120 kids that took the exam last year.

 

It's interesting how the training camp qualification process varies in different countries!

 

The US math olympiad summer training camp (MOSP) admissions criteria have changed over the years. Right now there are easier selection criteria for ninth graders or for girls training for the China Girls Olympiad than there are for grades ten through twelve kids competing for spots on the US IMO team. But everyone has to go through the timed, multiple-choice AMC 10 or 12 along the way to qualification, followed by AIME (timed, but not multiple choice), and USAJMO or USAMO (6 questions, 9 hours over 2 days, proof-based exams). There's no way to avoid the timed math exams here, especially with tens of thousands of kids entering the competition annually. [ETA: actually there is a way to avoid the AMC 10/12 via USAMTS scores instead, but it's extremely difficult to do well enough to advance to USAMO level that way]

 

I've told him that the *quality* of his proofs are what is key, not just the general idea of the answer. I suggested seeing a proof like an essay -- make an outline, write a draft, edit, proof-read, and then copy it over. We will definitely need to practice this skill. He is currently very "hasty", and says "I get it, I get it, you just do xxxx." Then when I ask him to write down the proof, he is not very interested. I don't think he is clear that he needs practice not only answering the questions, but also doing the boring work of writing out a clear-cut proof. But he is happy to do it with geometry, so at least I have a place to start.

 

Yes, it's a lot like the essay writing process. My son wasn't very interested in the writing process at first, either (resistant might be the word!) It helped him to see examples of good solutions to past olympiad problems, to take some graded AoPS classes online, to go to camps, & to write the USAMTS, where he was evaluated by people other than me :-)

 

I am still looking for a math mentor and have a few leads.

 

good luck!!

[MBM]

There's no way to avoid the timed math exams here, especially with tens of thousands of kids entering the competition annually. [ETA: actually there is a way to avoid the AMC 10/12 via USAMTS scores instead, but it's extremely difficult to do well enough to advance to USAMO level that way]

 

Hi, Kathy :seeya:

 

My son is hoping to score high enough on the USAMTS so that he can skip the AMCs. He was elated to learn about the USAMTS because he hates *fast* math! LOL. He's doing okay so far. Guess we'll see once the 3rd round gets graded.

 

:)

 

Good luck to your son, Lewlelma!

[Kathy in Richmond]

Hi, Kathy :seeya:

 

My son is hoping to score high enough on the USAMTS so that he can skip the AMCs. He was elated to learn about the USAMTS because he hates *fast* math! LOL. He's doing okay so far. Guess we'll see once the 3rd round gets graded.

 

Hi MBM!

 

Do you know how they're assigning USAMTS scores to equivalent AMC scores this year?

 

All I remember is that in my kids' day, they assigned a 100 on the USAMTS (perfect score) to be a score of 100 on the AMC 12. Didn't seem quite right at the time, since the AMC 12 is out of 150 points. Have they changed it since?

[Belacqua]

 

Hi, Kathy :seeya:

 

My son is hoping to score high enough on the USAMTS so that he can skip the AMCs. He was elated to learn about the USAMTS because he hates *fast* math! LOL. He's doing okay so far. Guess we'll see once the 3rd round gets graded.

 

:)

 

Good luck to your son, Lewlelma!

 

Good luck to all!

 

Something I've been curious about: if a kid qualifies for AIME with a USAMTS score, how do they work the USAMO qualifying index (as that factors in the AMC score)?

[MBM]

I don't know. All I (sort of) know is that for the USAMTS they take 3 exams, each worth 25 points, and I think the minimum to get in is 68? From there, I have no idea how they assign scores with the AMCs.

 

My son just finished semester finals this morning and is hanging out with a group of friends downtown. I'll ask him when he gets home.

 

Btw, he might meet Paul Sally at Young Scholars for the first time tomorrow. Should be interesting. I will have to remind ds to turn off his cell phone and not wear a hat!

[Kathy in Richmond]

I just found this on the AMC website in their FAQ/AIME section:

 

Q. AIME #15. What score do you need on USAMTS to qualify for AIME?

A. The minimum score required to qualify for AIME from USAMTS is a score of 54 out of 60 possible points

(90% of the possible points) after 2 rounds of USAMTS. In computing the USAMO index, we use the percent on the USAMTS (after two rounds) plus 10 times the AIME score. For example, if you scored 54 on the USAMTS, (90\% of a possible 60 points after two rounds), and get a score of 11 on the AIME, then your index would be 90 + 10*11 = 200. If you took both the AMC 10 or AMC 12, and the USAMTS your index is the maximum of the two ways calculating the index.

 

So it looks like they're still computing the index in a similar way. I think that most kids who score 90% on the USAMTS would probably get over a 90 on the AMC12, but it's a safeguard against a bad AMC day (& the 12 has certainly become tougher in recent years!)

 

Good luck, everyone! (& waving hi to Belacqua, too!)

[MBM]

Thanks, Kathy. Interesting. My son might sit for the AMCs but they had to take them super early last year, around 6:15 or so (I don't remember). They have to get them done before the early bird classes that start at 7:10.

 

Good luck to everyone taking the tests!

[Kathy in Richmond]

Thanks, Kathy. Interesting. My son might sit for the AMCs but they had to take them super early last year, around 6:15 or so (I don't remember). They have to get them done before the early bird classes that start at 7:10

 

Oh my goodness, that's early! I don't know if I could make my brain function at that hour of the morning. Maybe the kids could request an after-school exam administration this year?

[Guest lotr]

Thanks for the thread and all of the wonderful info.