USAMO Prep

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123s

87 posts

123s

#1 h Apr 10, 2007, 9:16 PM • 6 Y

Y by Adventure10, Mango247, and 4 other users

Does anyone have any advice on how (to learn) to write clear, concise, and rigorous proofs, and/or any advice on preparing for the USAMO in general?

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jb05

1086 posts

jb05

#2 h Apr 10, 2007, 10:27 PM • 4 Y

Y by Adventure10 and 3 other users

Preparing: do old contests (be very, very patient - spend the entire 4.5 hours at least once just to develop patience). If you can't make any progress with them, just read Engel (Problem Solving Strategies). Actually read that even if you can do them (that's how I managed to go from averaging 0 problems per USAMO at the AIME to one or two when I took it last year). If that's too hard, sorry but you probably aren't going to do too well this year

. Prepare for next time.

There is a misconception that there's something tricky and mysterious about proofs. Just pretend that you're explaining it to a friend, but keep in mind that the friend can't ask you questions afterwards (so be clear and rigorous, but skip arithmetic and obvious stuff). If you are really worried, ask someone with more experience for advice. You can even post your proofs on these forums.

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dasherm

854 posts

dasherm

#3 h Apr 11, 2007, 2:13 AM • 2 Y

Y by Adventure10, Mango247

jb05 wrote:

If you can't make any progress with them, just read Engel (Problem Solving Strategies).

Is this a book? If not, is it available on the web?

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davidyko

815 posts

davidyko

#4 h Apr 11, 2007, 2:16 AM • 2 Y

Y by Adventure10, Mango247

Yup, it's a book. I believe it is in the Bookstore on this site.

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Teki-Teki

553 posts

Teki-Teki

#5 h Apr 12, 2007, 7:38 PM • 4 Y

Y by Adventure10, Mango247, and 2 other users

to clarify, it is a pale yellow book (sticky-note color) with purplish text on the front, in case you want to make sure.

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davidyko

815 posts

davidyko

#6 h Apr 12, 2007, 7:41 PM • 2 Y

Y by Adventure10, Mango247

And it's filled with lots of cool math.

Not that I'm up to that level yet.

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noneoftheabove

257 posts

noneoftheabove

#7 h Apr 12, 2007, 9:05 PM • 2 Y

Y by Adventure10, Mango247

about how long does it take to get through engel??

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davidyko

815 posts

davidyko

#8 h Apr 12, 2007, 10:47 PM • 2 Y

Y by Adventure10 and 1 other user

You definitely won't get through it in a jiffy, from skimming through its contents. Even though I haven't done any of it yet.
Okay, I'll shut up about things I don't know about now.

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Klebian

1143 posts

Klebian

#9 h Apr 13, 2007, 12:17 AM • 3 Y

Y by Adventure10 and 2 other users

Thank you.

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Arvind_sn

524 posts

Arvind_sn

#10 h Apr 13, 2007, 12:48 AM • 4 Y

Y by Adventure10, Mango247, and 2 other users

Hmm I have Engel... it's definitely NOT something you can read in a couple of months. As a matter of fact, I'm betting that even the best mathematicians in the nation could get something out of it by reading it even if they've read it before. It will take you at least a year to read, but I remember a friend who said that he sort of "skimmed" over a few chapters in the book and made red MOP. If you are as much of a genius as he is, go for it! If not, there's always next year

But Engel definitely taught me a few techniques on how to tackle Olympiad problems. However, the difficulty is quite high. If you want something a bit easier to read (you can get the first chapter of Engel on Amazon's "Search Inside!" feature), you could try Larson's "Problem Solving Through Problems"

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Boy Soprano II

7935 posts

Boy Soprano II

#11 h Apr 13, 2007, 2:05 AM • 6 Y

Y by Adventure10, Mango247, theSpider, and 3 other users

A math book is not a novel. I don't see the point of purposely trying to do every single problem in a worthwhile book on problem solving. You can read through the lessons in very little time, and it is probably worthwhile to do so. To do the more than one thousand problems would obviously take a very long time. If you can do a problem in five minutes, then it is far too easy for you to be doing that many. So I don't think you should be trying to "get through Engel". Instead, try working on some problems from it, and enjoy them, instead of seeing them as so many hurdles to jump.

Anyhow, Engel has a somewhat different flavor from the USAMO. You'll notice it most in the inequalities and the geometry, I think. But it's still good practice.

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DPopov

1398 posts

DPopov

#12 h Apr 13, 2007, 2:31 AM • 3 Y

Y by Adventure10, Mango247, theSpider

Engel is definitely not meant to be read straigth through. I would skip around and try to learn various techniques. Then attempt a few of the olympiad problems and try to apply the techniques but don't try to do every problem; it's not necessary or efficient.

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Phelpedo

2444 posts

Phelpedo

#13 h Apr 13, 2007, 3:09 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Don't think you can walk away with a mastery of the material without doing a good number of problems, though. Also, just going straight from the problems to the solutions is not a good idea.

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junggi

1813 posts

junggi

#14 h Apr 13, 2007, 3:12 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

how many problems per chapter should we do?(percentage wise or something)

@phelpedo: nice, 2007 post count(at the time of my post)

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DPopov

1398 posts

DPopov

#15 h Apr 13, 2007, 5:29 AM • 4 Y

Y by Adventure10, Mango247, and 2 other users

Don't get me wrong, you definitely need to do a good deal of problems to master the technique. However, this will take some time and it's almost impossible to do before this year's USAMO. I would say that if you work on 15 olympiad level problems from each chapter in addition to the exercises presented within the context, you should obtain a pretty decent understand of the material. Of course it's always a good idea to review with more problems. The only way to really practice for USAMO is by doing lots and lots of problems (at least this is my opinion).

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DPatrick

4398 posts

DPatrick

#37 h May 3, 2007, 3:09 PM • 6 Y

Y by Adventure10, Mango247, and 4 other users

As mentioned above, USAMTS problems are generally not nearly as difficult as USAMO problems. On the other hand, #1 and #4 on most USAMOs are often not any more difficult that hard USAMTS problems. In fact, rrusczyk and I were talking last week about how this year's USAMO #1 would have made a great USAMTS problem.

The other thing about the USAMTS is that it will give you experience writing full solutions to problems, as opposed to the AIME, where your solutions can have holes but you can still get the right answer. We see a lot of papers for the USAMTS where a student has the right answer at the end and think they deserve a 5 (a perfect score), and we only award a 1 or 2 or 3 because the solutions has major gaps. This is even more significant on the USAMO, since a solution with a major gaps or errors is likely to only receive 0 or 1 or 2 (out of 7).

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darkprince

215 posts

darkprince

#38 h May 4, 2007, 3:29 AM • 5 Y

Y by Adventure10, Mango247, Mango247, Mango247, and 1 other user

Would it be better to learn proof-writing first, or to learn to solve more difficult problems like those on the AIME if I'm preparing for USAMO? I should say that I've had limited experience with proofs, having done only USAMTS so far...But will that be bad in the long term?
darkprince

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mysmartmouth

2104 posts

mysmartmouth

#39 h May 4, 2007, 3:48 AM • 1 Y

Y by Adventure10

darkprince wrote:

Would it be better to learn proof-writing first, or to learn to solve more difficult problems like those on the AIME if I'm preparing for USAMO? I should say that I've had limited experience with proofs, having done only USAMTS so far...But will that be bad in the long term?
darkprince

I think it would be much more beneficial being able to solve AIME problems before learning to write proofs. I would focus extensively on the AIME, without completely ignoring proofs. Once you think you are capable of solving #1 and #4 type problems on a regular basis, it may be time to move on to more extensive proof practice. Books by authors such as Engel and Larson are good, since they teach problem solving techniques through proofs. Another problem with putting USAMO practice before AIME practice is that you need to do well enough on the AIME to even qualify for USAMO. This seems to be of secondary importance, but I can cite several examples of USAMO qualifiers who decided to focus extensively on USAMO and then didn't qualify for it the next year.

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Phelpedo

2444 posts

Phelpedo

#40 h May 4, 2007, 3:54 AM • 1 Y

Y by Adventure10

Engel and Larson usually only show the investigation leading up to the solution. Their solutions should not be a model for formal proofs. Look at the AMC website's USAMO solutions for formal proofs (although their #1 is sketchy, eh).

If you're having a hard time with USAMTS problems, there would be little point in tackling USAMO problems at this point. I wouldn't worry too much about not being prepared enough for USAMO - problem solving is problem solving. Although there are many techniques unique to olympiad problems, it's not like AIME problem solving ability and Olympiad problem solving ability are mutually exclusive.

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darkprince

215 posts

darkprince

#41 h May 4, 2007, 5:54 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Quote:

If you're having a hard time with USAMTS problems, there would be little point in tackling USAMO problems at this point. I wouldn't worry too much about not being prepared enough for USAMO - problem solving is problem solving.

Phelpedo, do you suggest becoming good at general problem solving (so that I'll be able to solve AIME and USAMTS relatively easily) before moving to Olympiad problem solving, and learn proof writing along the way?

That sounds reasonable. Until I can solve the Olympiad problems, I won't try to write clear, concise solutions to those problems.

Hopefully, that'll change soon, because this is my last shot at USAMO.

Also, Engel's book helps in covering topics that I might not have had prior knowledge about, right? If so, then I definitely need it, not so much for proof writing, but for concepts that I might not have had full exposure to.

mysmartmouth, thank you. I'll try for scores of 12-15 on AIME before proof work; that gives me a definite plan to work with, since that's closer to where I am right now (i can consistently get around 5 or 6).

Just as a side note, would excelling in AIME problem solving help with USAMTS?

darkprince

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Boy Soprano II

7935 posts

Boy Soprano II

#42 h May 4, 2007, 8:35 PM • 4 Y

Y by Adventure10 and 3 other users

Most of the solutions I have read in Engel were complete. I have not seen Larson. On the other hand, the best models of olympiad solutions that I have seen are in the USAMO/TST/IMO pamphlets that the MAA publishes. The solutions are better than those on the website, and there are also more alternate solutions. By the way, what is sketchy about the solutions to problem 1 on the website? I read three good solutions.

Quote:

Phelpedo, do you suggest becoming good at general problem solving (so that I'll be able to solve AIME and USAMTS relatively easily) before moving to Olympiad problem solving, and learn proof writing along the way?

I don't understand—olympiad problem solving is a particular difficulty (and perhaps topic) range of general problem solving. It is good to spend some time on AIMEs if you haven't. However, if you usually can score a five to an eight on AIMEs, I see no reason not to start easy olympiad problems. Twelve to fifteen on the AIME is significantly higher than I have yet scored, and I was at red MOP last year. Trying to pass that mark before starting olympiad problems is overkill—a parallel of trying to learn to multiply a hundred three-digit number in your head in three minutes before studying basic algebra. Some of last year's red MOPpers got eights on the AIME, but they were still capable of tackling easy olympiad problems.

I happen to disagree with mysmartmouth about proofwriting skills. If you have solved a problem and you can't communicate your solution effectively to other mathematicians, you might as well not have solved it. In addition, if you put proofwriting off until you are solving olympiad problems, you will probably impose some psychology on yourself that associates proofs with difficult problems. In any case, there is no magic trick to it, no secret formula—just write down a sequence of logical thoughts that lead you to the desired conclusion. You don't have to be pedantic. If you can't do this, you probably haven't really solved the problem. As you do this more often, your form and style will improve.

Also, I did practically no practice for the AIME this year and concentrated exclusively on USAMO, and I still made USAMO by a wide margin even after three stupid mistakes on the AIME. This does not demonstrate that in general, exclusive USAMO practice will probably qualify you for USAMO, but then, a few examples in the other direction will not demonstrate the opposite. All I know is that for me, olympiad problems make AIME practice sufficiently obsolete. Furthermore, olympiad problems are deeper, and a better use of your mind. When you have spent enough time on olympiad problems, AIME problems seem to have negligible difficulty.

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Phelpedo

2444 posts

Phelpedo

#43 h May 4, 2007, 10:36 PM • 5 Y

Y by Adventure10, Mango247, and 3 other users

Proofwriting is really a fairly simple thing. However, many people claim to be bad at proofwriting when really the issue is that they haven't actually solved the problem rigorously. For example, I've heard a lot of people (including myself) attribute a less-than-7 score on the USAMO'06 #2 to bad proofwriting (probably something of the nature that you thought the numbers should be consecutive but couldn't prove it), when really it's just not solving the problem completely.

Of course there are tricky things like saying "all steps reversible" at the end of inequality solutions (if you write it backwards).

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MellowMelon

5850 posts

MellowMelon

#44 h May 5, 2007, 11:20 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

This year's #2 and #4 showed especially well that one of the biggest problems people have with proof writing, even some veterans, is making assumptions that aren't always true. For #2, it was things like all disks equal radius, or a set of 3 disks is going to be tangent. For #4, most people's proofs of 8025 as a limit failed consider all possible 8026 cell dinosaurs.

When you prove something, especially that some number is a maximum, make sure you leave no possibility out. In the same vein, don't make any large intuitive leaps. That's how you figure the solution out, not write the proof.

I disagree that their #1 is sketchy. They were concise, but it was a complete proof(s). Just an easy problem.

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Phelpedo

2444 posts

Phelpedo

#45 h May 5, 2007, 2:21 PM • 2 Y

Y by Adventure10, Mango247

As I said, I don't really believe that the two mistakes you mentioned were due to bad proof-writing. I would be surprised if not every "veteran" who argued that there had to be 3 tangent disks of equal radius knew he/she was handwaving but just didn't know what to do about it. Similarly for #4, it's not "bad proof-writing," it's just not solving the problem.

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darkprince

215 posts

darkprince

#46 h May 5, 2007, 6:34 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Quote:

Also, I did practically no practice for the AIME this year and concentrated exclusively on USAMO, and I still made USAMO by a wide margin even after three stupid mistakes on the AIME. This does not demonstrate that in general, exclusive USAMO practice will probably qualify you for USAMO, but then, a few examples in the other direction will not demonstrate the opposite.
Boy Soprano II

I see...I can get around 6 or 7 in AIME. I should prepare a little more, then begin easy Olympiad problems? I guess that works, but are easy Olympiad problems approximately the difficulty of mid-level AIME problems? I've looked at some at Kalva.net, but they seem to be far more intricate and difficult to end up in the right place, and with all the right steps, too. I don't know, it might be too much of a leap for me. I'll try though. Thanks!

darkprince

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paladin8

3237 posts

paladin8

#47 h May 5, 2007, 7:11 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

You probably want to get a little more comfortable with AIME problems. I think once you think you "know" how to solve 10 that's good enough and will provide you with a solid buffer in case you make stupid mistakes.

For easy olympiad problems, I recommend checking out Canada's or India's NMO's since the US ones are usually too difficult for people just starting olympiad problems. In recent years, however, they have tried to accomodate for such people I think.

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MellowMelon

5850 posts

MellowMelon

#48 h May 6, 2007, 2:04 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Phelpedo wrote:

As I said, I don't really believe that the two mistakes you mentioned were due to bad proof-writing. I would be surprised if not every "veteran" who argued that there had to be 3 tangent disks of equal radius knew he/she was handwaving but just didn't know what to do about it. Similarly for #4, it's not "bad proof-writing," it's just not solving the problem.

You have a point with experienced people knowing they're doing that. However, from my own personal experience, I think a lot of Olympiad novices think they've gotten a solution to a problem when they really haven't. They write their (incorrect) solution and then waste lots of time on a problem they don't end up solving, when otherwise they might realize they need to dedicate more time to the first one. Granted, whether they could solve the problem with that time is a different matter that you mentioned, but not spending that time filling gaps is the part of "bad proofing" that I'm referring to.

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noneoftheabove

257 posts

noneoftheabove

#49 h May 6, 2007, 2:59 AM • 2 Y

Y by Adventure10 and 1 other user

What if a solver cant solve that gap wouldn't potential partial credit be better than not writing a bad proof?

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Altheman

6194 posts

Altheman

#50 h May 6, 2007, 6:39 AM • 2 Y

Y by Adventure10, Mango247

well obviously you give all the ideas you have, but it is really important to spend time trying to rigorize your solutions...i felt i really learned a lot when i actually wrote up solutions for this years usamo...it is a lot harder than i thought...lol

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JSteinhardt

947 posts

JSteinhardt

#51 h May 6, 2007, 12:58 PM • 5 Y

Y by Adventure10, Mango247, and 3 other users

Also, a proof with a small gap that is along the lines of the intended solution is probably only worth a $5$ . Anything beyond that takes it down to like a $1$ or a $2$ , at most. On this USAMO, if you assumed that the three circles were tangent, or of equal radius, you would probably get very few points, possibly none, because that is basically the entire point of the problem.

So while it is certainly worth it to write up your progress on a problem, you shouldn't expect it to get that many points unless the progress was pretty significant.

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