Olympiad problems- how to prepare

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yugrey

2326 posts

yugrey

#1 h Aug 6, 2010, 5:40 PM • 2 Y

Y by Adventure10, Mango247

OK, well, I waste a lot of time, and so I don't think I'm time-efficient enough to do WOOT this year, sadly. However, my last 3 practice AIME scores have been 7,5, and 9, and I feel I can consistently score 7-9 (the 5 was late at night, and was very computational, and had lots of geo, one of which is good for my AIME performance. Luckily I take AIME for real in the morning/afternoon) and I feel I am getting to the point (after a few more practice AIMEs) where I feel I can start transitioning to Olympiad problems. However, I feel this will be hard without WOOT, but I may be able to take Olympiad Geo. Will going through practice tests really work? I plan to go from early USAMO's and CMO's to harder contests and problems, but olympiad problems are hard. Luckily, I have solved one (the JMO #1 in practice, does that count?) but I feel that practice tests alone won't work (well, they did for AIME, with a great MathPath breakout course, but I don't know). My ultimate goal next year is to win JMO. I want to find a book/books or other ways to study that give me a fighting chance. USAJMO problems, I feel, are at or a little above my level. I've heard ACOPS and PSS are good, but I can't stand how ACOPS doesn't have solutions. I think I'll go for these books (ACOPS and PSS) nonetheless, but does anyone have any suggestions?

EDIT: Also, I'm OK at, but not very comfortable with logs, preventing me from doing the last step of Click to reveal hidden text (spoilers) and for trig, I know sum identities, law of cosines and sines, sort of extended law of signs, but no product identities or stuff like that. How should I learn these so as not to be defeated by a trig or log question on JMO?

Thank you for reading through this long post and thanks in advance for advice,

-Yugrey

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speedcuber96

58 posts

speedcuber96

#2 h Aug 6, 2010, 6:16 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

I would suggest taking part in USAMTS. Some of the problems are similar to JMO problems and will be good practice.

As for books, if you think you are beyond the intermediate series of AoPS and AoPS Vol. 2, ACoPS and PSS, I'm quite sure that PSS and ACoPS are the best books to follow through with.

Doing basic olympiad problems from contests all around the world will really help. AoPS has an excellent collection of problems out there.

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yugrey

2326 posts

yugrey

#3 h Aug 6, 2010, 6:26 PM • 2 Y

Y by Adventure10, Mango247

I don't know if I can do everything in the Intermediate series yet, but I'm ready competition-wise to start Olympiad problems. USAMTS seems good though, Any other suggestions?

Thanks in advance again,

Yugrey

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Jason27603

629 posts

Jason27603

#4 h Aug 6, 2010, 9:23 PM • 6 Y

Y by jam10307, Adventure10, Mango247, theSpider, and 2 other users

You really don't need to know the sum-to-product or product-to-sum trig identities for Olympiads. I highly doubt that there are many people who have them memorized (I don't bother). Those can be derived relatively quickly by adding and subtracting the appropriate sum identities from each other. On an Olympiad, this shouldn't take up too much of your time. Also, even if you do forget the trig identities, most of them can be easily derived.
Also, its not knowing formulas that will help you, its knowing when and how to apply them. For example, I was defeated by problem #6 on the JMO last year because I kept on bashing away at similar triangles and looking for ratios. I never tried trig (well, trig is like advanced ratios, but whatever), and so missed the pwnage Law of Cosines solution.
Also, I am not sure that I have ever seen an Olympiad problem requiring logs, but if you want to do well on the AMCs and AIME, you'd better know logs. So take Algebra II or something.
Also, be warned that winning the JMO is likely to be quite challenging. Last year you had to have a 35+ to be a winner. That means that you can only miss ONE problem. Last year I had a good first day, got every problem right, then missed the last two problems on the second day (yes, including the really easy #5).

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kevin7

175 posts

kevin7

#5 h Aug 6, 2010, 9:59 PM • 5 Y

Y by Adventure10, Mango247, and 3 other users

The USAJMO is a lot more elementary than the USAMO; 3/6 of the problems on last year's JMO required nothing but logic, and another two needed just angle chasing and the law of cosines.

I made it to HM last year knowing only the content of AoPS 1, so I suppose if you wanted to win it, mastering AoPS 1 and 2 would be enough. PSS would be serious overkill, for sure. As that guy above me said though, the required score to win is extremely high, so what you really want is precision with a few theorems, not piles and piles of them.

Practicing on problems to 'get used' to the kinds of problems also would be helpful. USAMTS would help with writing up rigorous proofs, I suppose.

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yugrey

2326 posts

yugrey

#6 h Aug 6, 2010, 10:17 PM • 2 Y

Y by Adventure10, Mango247

Jason27603 wrote:

You really don't need to know the sum-to-product or product-to-sum trig identities for Olympiads. I highly doubt that there are many people who have them memorized (I don't bother). Those can be derived relatively quickly by adding and subtracting the appropriate sum identities from each other. On an Olympiad, this shouldn't take up too much of your time. Also, even if you do forget the trig identities, most of them can be easily derived.
Also, its not knowing formulas that will help you, its knowing when and how to apply them. For example, I was defeated by problem #6 on the JMO last year because I kept on bashing away at similar triangles and looking for ratios. I never tried trig (well, trig is like advanced ratios, but whatever), and so missed the pwnage Law of Cosines solution.
Also, I am not sure that I have ever seen an Olympiad problem requiring logs, but if you want to do well on the AMCs and AIME, you'd better know logs. So take Algebra II or something.
Also, be warned that winning the JMO is likely to be quite challenging. Last year you had to have a 35+ to be a winner. That means that you can only miss ONE problem. Last year I had a good first day, got every problem right, then missed the last two problems on the second day (yes, including the really easy #5).

OK, this and the post above me had some good points, but I would prefer to look up all these identities and know their derivations. I'm officially learning logs. Anyway, I'm complaining so much about theorems because I'm young (13, going into 8th grade) and I prefer thought and problem-solving. That's what makes me good, because I don't know Cauchy-Schwartz, Melanauss's, and a great many other theorems but I have good problem solving skills, in my opinion, to go against by bad "theorem-knowing" skills. Big problems for me are speed, computation, and knowing theorems. Of course, I could use preparation that helps insight and thought much more than anyone, but I need extra help on my theorems and identities, since i've only been doing competitions for a year. By the way, to Kevin, I would be fine with an HM too, but I'd really like a win. I'm just scared of an inequality/trig log/theorem-heavy geometry J3/J6. Anyway, thanks for the suggestions. I hope they keep coming.

-Yugrey.

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vcez

178 posts

vcez

#7 h Aug 6, 2010, 11:46 PM • 4 Y

Y by Adventure10, Mango247, and 2 other users

yugrey wrote:

had lots of geo

lol why do people hate geo? AIME geo is trivial.

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Poincare

1341 posts

Poincare

#8 h Aug 7, 2010, 12:35 AM • 6 Y

Y by Adventure10, Mango247, and 4 other users

AIME geo is not trivial for a lot of us (*cough* me *cough*). It isn't really very nice/helpful to say that. The poster said that he wasn't good at geo and he was taking the Olympiad geo class, it doesn't help by telling him that AIME geo is trivial, that just makes him feel worse. IMHO, since he's only in 8th grade (same as me, and I get 4 or 5's on AIMEs), a 9 on an AIME is very good, screwing the geometry completely.

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yugrey

2326 posts

yugrey

#9 h Aug 7, 2010, 1:35 AM • 2 Y

Y by Adventure10, Mango247

First of all, I'm not offended when people express their opinions that something is trivial (I do this all the time). I am not that bad at geo. However, I am very slow at it so I think I could do Oly geo better than AIME geo. My big geo weakness is auxiliary lines. That's why I plan to take Oly geo, since I could do 4 problems on the pretest (2 right, one gave up and looked at the answer, then without particularly thinking of the answer, realized what the answer was, one made a little stupid mistake with the whole idea correct- none of those 4 involved auxiliary lines). I'm relatively decent at similarity (relative to my other subjects, I'm not saying absolutely good or bad, because that would either insult me or someone reading this and after all, it's relative), but I think I know the Intro Books if they are AMC Geo. Yet, there's no Intermediate Geometry. I heard Challenging Problems in Geo was good. Also, I think PSS and ACOPS will probably help a lot for JMO, and both help problem solving/logic, which is what you need on JMO since there are no inequalities you have to bash with theorems and even the geo is almost theorem-free, as opposed to USAMO, which is intended for people with more problem solving experience (although I bet USAMO is better than JMO once you've seen your fair share of theorems.)

General note on theorems: OK the advice is that theorems/identities are not the thing to focus on. I agree, but they may be more important than you think. Since most algebra AIME problems involve factorizations of polynomials, I think about how screwed I would be if I thought $(a+b)^2=a^2+b^2$ . I can't do olympiad inequalities, since, as I said before, IDK Cauchy and I know AM-GM but have never really practiced using it.

Another thing to say: The harder a competition is, the more a like it and the better I am at it relatively. I missed AIME by 1.5 points this year and if I had taken the AIME, I probably would have got 3-5 (those were my scores a little later, when I started to look at AIMEs). My brain has grown since then and I have used it, but I think 3-5 is still good for a non-AIME qualifier even on this AMC. AMC 12,, maybe not. I took AMC 10, though. Also, I solved my unofficial JMO problem rigorously when I was at 5 on a Practice AIME.

Please continue to make more great suggestions- what books, practice tests, AoPS classes.

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Aryth

888 posts

Aryth

#10 h Aug 7, 2010, 2:29 AM • 4 Y

Y by Adventure10 and 3 other users

yugrey wrote:

I can't do olympiad inequalities, since, as I said before, IDK Cauchy and I know AM-GM but have never really practiced using it.

http://www.math.cmu.edu/~ploh/docs/math/b2-inequalities.pdf

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yugrey

2326 posts

yugrey

#11 h Aug 7, 2010, 2:45 AM • 1 Y

Y by Adventure10

How would you complete that ugly square on the warmup? Seems interesting though.

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AwesomeToad

4535 posts

AwesomeToad

#12 h Aug 7, 2010, 2:57 PM • 2 Y

Y by Adventure10, Mango247

kevin7 wrote:

The USAJMO is a lot more elementary than the USAMO.

What about the earlier USAMOs (before 2000 etc., or even before 1990)? I hope to do the USAMO next year, or USAJMO, but since I have almost no experience with Olympiad problems, USAJMO and the like difficulty seem like good places to start.

Unfortunately, I bombed the USAJMO day 1, so I've kind of wasted the only USAJMO there is.
I'm wondering if there are other contests of similar difficulty. I'm considering (as starting points) these, but I'd like some of your opinions about these:

- The last 5 problems on AIME, since the USAJMO problems are roughly the level of 13-15 on AIME.
- Really early USAMOs (e.g. before 2000, or even before 1990); these seem easier, but I'm not sure about its relative difficulty to the USAJMO.
- Canadian Math Olympiad; I don't know a lot, but a lot of people have talked about this.
- USAMTS (I'm sure someone already mentioned this)

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darkdieuguerre

1118 posts

darkdieuguerre

#13 h Aug 7, 2010, 3:18 PM • 2 Y

Y by Adventure10 and 1 other user

Have you seen the ratings here? The Junior Balkan Mathematical Olympiad is probably of similar difficulty, though with only one year of the USAJMO, it's hard to say.

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orl

3647 posts

orl

#14 h Aug 7, 2010, 4:46 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

I suggest guys to have a look at the contest section and NMO forum to check some contests. For those who would like to add some problems from the NMO forum to the contests section please pm admin nsato. And whenever you solve problems from a particular competition as offered on AoPS please add a some information to the ranking wiki. Cheers.

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yugrey

2326 posts

yugrey

#15 h Aug 8, 2010, 3:15 PM • 2 Y

Y by Adventure10, Mango247

National Olympiads look hard. Are there any other good books to suggest (remember, this is for 7-9 scorers on the AIME, we're not that good yet). Is there anything better than ACOPS and PSS? Early AIME problems are comparable to late AMC problems, but can I get a list of competitions comparable to late AIME problems and some goods books to use them with?

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BarbieRocks

1102 posts

BarbieRocks

#16 h Aug 8, 2010, 4:09 PM • 2 Y

Y by Adventure10, Mango247

HMMT Has some good Medium-Hard AIME Problems.

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speedcuber96

58 posts

speedcuber96

#17 h Aug 8, 2010, 4:24 PM • 2 Y

Y by Adventure10, Mango247

So does the Stanford Math Tournament (SMT). They archives of old problems:

http://sumo.stanford.edu/smt/

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huang

83 posts

huang

#18 h Aug 8, 2010, 8:28 PM • 2 Y

Y by Adventure10, Mango247

Jason27603 wrote:

Also, I am not sure that I have ever seen an Olympiad problem requiring logs

Well, a popular solution to 2009 Iberoamerican #2 was with logs http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=29&year=2009

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yugrey

2326 posts

yugrey

#19 h Aug 12, 2010, 5:34 PM • 2 Y

Y by Adventure10, Mango247

Yay! I got 4/5 problems on the 1972 USAMO.

Should I just keep working through early USAMOs? It turns out there is a small chance I get to take WOOt, not a 0 chance!

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MathWise

326 posts

MathWise

#20 h Aug 12, 2010, 6:33 PM • 2 Y

Y by Adventure10, Mango247

yugrey wrote:

How would you complete that ugly square on the warmup? Seems interesting though.

I think there's actually a typo in it: the 176 before the > should be a 179, and the > should be a $\geq$ (or just replace the 176 with 180).
Solution to corrected problem

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