Promote this post!

last point. +1

As for relaxing, I like to do frivolous math right before contests and it helps. For instance, if its right before the AMC I go play ftw and do alcumus problems purely for fun. The problems aren't that hard so its fairly relaxing (and not the mention it can be fun if you do it with friends), and they'll also help you be less error prone for the competition!

Kind of related to "right training":

I've seen people put lots of stress on contest math and not as much on undergrad, even though they plan to go into math. Here are lessons I learned the hard way:
* Being very good at contests but not good at undergraduate mathematics is markedly less satisfying than being decent at both.
* Learning undergraduate math can help with some of the harder-end problems on many competitions.
* Putting off undergrad math until the end of high school and finding out that you could have done all of it three years ago is not a pleasant feeling.

Granted, undergrad math is very different from contest or high school mathematics, but you should at least look into it before you say "Sounds hard, I'll do it next year, gotta prepare for the AMCs instead"

I spent the night before a contest reading http://www.math.uiuc.edu/K-theory/305/nowmovo.pdf, so on contest day I'm like "hey math is actually possible to do!" which is a rather pleasant mood to be in~

Note: You may not understand some of the terminology in the paper I linked. I don't understand a lot of it, but this just makes the difficulty harder, so then I'm even more pleasantly surprised by how possible contests are to do!

On a more serious note (though I was actually being half-serious with the preceding comments), I think that being able to learn in a very piecemeal fashion is a very useful thing to do. This does not mean that you read a book, say "this is way too hard", read another book, say "this is way too easy", and repeat 10 times.

What it means is that you should learn how to learn without focusing on one thing. This is because in many cases, subjects are connected in ways where the connection is actually the key to learning those subjects well.

As just one example, I think number theory and polynomial theory needs to be learned simultaneously to really start to appreciate either subject. (By this, I am NOT suggesting doing this via learning abstract algebra, although that would help. The abstract reasons for this similarity go far beyond a course in abstract algebra and are not fully understood. If the program turns out to work, then in some sense, the integers are actually a polynomial ring in disguise. But what "turns out to work" means here is unclear; supplying a proof of the Riemann hypothesis would count though (In my opinion, the approach is by far the most interesting approach to RH, although it is not known to work at all.)

Comparing number theory and polynomial theory makes a few things much clearer. It highlights how evaluating a polynomial at a point and how taking mods are in some sense, the same thing. It also highlights the only thing in olympiad number theory that really separates it from polynomial theory: The Frobenius endomorphism. So many people fail to grasp the importance of the Frobenius endomorphism, and it becomes completely clear if you have a deep understanding of the connection between number theory and polynomial theory. Good luck solving problem 10 on http://www.artofproblemsolving.com/Forum/blog.php?u=69901&b=82673& without understanding this connection.

What this means practically is to really understand math, you need to form a lot of connections between the things you're doing. Thus, knowing how to explore is a really powerful tool for learning mathematics. The only real "fundamentals" in contest math are solving problems for which you've seen a similar problem before, and while quite useful at times, I think that ultimately you will eventually need to know how to explore.

This means that:

1.) I will extend v_Enhance's point: The choice of what you are doing not really matter, but you should try, at some point, doing a lot of different things. (One nice thing about Volume 1/2 is that it, by nature, does a lot of different things, so focusing on Volume 1/2 is actually a pretty good idea if you're not already quite good at piecemeal learning.) What this does is it really forces you to learn how to make connections.

2.) On the other hand, having the ability to fill in gaps in your knowledge is very useful. So I am not suggesting spending 100% of you time doing stuff like in 1.), but you should know how to do both part 1.) and part 2.)

3.) People sometimes focus super-hard on learning and thus, by focusing so hard, learn less than they would have otherwise. The reason this happens because they are not doing 1.), and are making less connections than they would have. Similarly, people sometimes fail to solve problems because they focus too much on solving them and don't bother understanding the problem.

4.) People also sometimes jump around too much and have a ton of gaps in what they know. This happens because they are not doing 2.) enough.

5.) So how do you balance 1.) and 2.)? You balance that by doing the appropriate ratio for your sense of curiosity. So basically, do what you feel like doing.

6.) People tend to care a lot about importance. But what does it mean for some topic to be important? One answer (which I like very much) is that an important topic is one that is connected to many other topics. What this means, is that yes, if you understand an important topic, many other things will become clear to you. But what this also means is that "importance" is quite a personal notion: The thing which is most important for you to learn will be the thing which has the most connections to things you already know. So therefore, you should do the thing you are most curious about, as it will usually be pretty high up on this personal scale. Care more about that than you do about "what book is best".

7.) Reading stuff that you are not able to fully understand is often a great idea, if you are interested in the stuff you're reading for a better reason than "it has a hard-sounding name." In fact, I spent around 70% of my time reading things too hard for me.

Hmm is the starbucks link supposed to go to a starcraft website?

what you should do is stop criticizing.

He's not criticizing... cire_il just thinks it's funny that the url on the word Starbucks links to Starcraft.
(Do you know what criticizing means?)

An extension to 1=2's post: Many of the problems at the end of every section in an undergrad math book are "contest" level. A few examples of such books are Linear Algebra by Jim Hefferon, An Introduction to Probability Theory and its Application by William Feller (classic and very challenging), and Introduction to Mathematical Statistics by Robert Hogg and Allen Craig (classic and some what challenging).

Yes, but it doesn't follow that those are the sort to appear on high school contests.

I don't think you read the opening post. If you did, I don't think you bothered taking the time to understand it. The advice is golden for pretty much any level of math (though, admittedly, it may not be ideal for very young children just being introduced to math - e.g. young children preparing for something like MATHCOUNTS). Not only that, but the advice given transcends math and is useful for essentially any serious intellectual area (Physics, Chess, etc.)

References to a no longer existing post. -- mod

I agree. Many posts by sincere people regarding how to best prepare for certain competitions are ignored by a flood of links to this thread. I'm sure that the majority of those who concur with OP were at one point also "looking for the right training".

This post as it stands gives great advice. This advice is 100% applicable to the threads asking "What should I do to get better?" However, this does not make it necessarily the right answer. If someone is truly looking for what they should do to get better, I feel that this post is a good way to set the mindset of how to go about it, which can then be supplemented by actual ways to get better.

The issue is many people simply ask about how they should prepare for AIME or USAMO without any real question. They simply figure that AOPS has a lot of successful people that excel at both contests, so why not see what they did? Unfortunately, that's not how it works - that's what this post is saying. There is no "right" training.

The point brought up isn't a bad one, and I agree that many people on AOPS link to this post too much. However, it does bring up many points that may be lost in a contest prep discussion.

The Wiki has a ranking of competition difficulties: http://www.artofproblemsolving.com/Wiki/index.php/A:CR. I don't know how accurate it is but it might be worth looking at.

Keep in mind that within any particular olympiad, the difficulty of the problems is not uniform. For a standard six-problem olympiad, we often have

A particular suggestion might be the All-Russian olympiads, as these are also divided into grades, and Russian problems tend to be less technical than, say, the USAMO. (And there's lots of nice problems.)

I think recommending Russian problems to someone who finds USAJMO too hard is a bit crazy....

Is there any other sequence of olympiads you would suggest for me to try?

No, Russian problems are nice.

But whatever, here are possible contests:

http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=21&sid=8a09a53f932791469233c1c45afb2d5c

http://www.math.toronto.edu/oz/turgor/archives.php

This is truly the best advice ever about Competition math on AoPS. Helped a ton!

Finally, someone revives a useful thread that I otherwise might not have seen.

there should be a combined motivation sticky at the top

Sir Evan Chen,
I have also books like AoPS volume 1 and 2 and precalculas.I also have books like challenge and thrill of pre-college mathematics(indian authors) and Mathematical Circles.
I am also preparing for olympiads.I just wanted some seriour advice from you about:
1.How much time should I spend on any problem or in preparation?
2.What to do when you get frustrated within 5 minutes you try to solve a problem?
3.My friends are going to coaching classes for competitions like IMO,IIT(advance and mains;It is an engineering college in India where the exam is really really tough),etc. but I have not joined any.Will I be left behind...or am I doing the right thing by preparing by myself?
Please answer
Also anyone could answer my questions.Please answer.
Thank you very much!

@Pks: Read BOGTRO's post on Learn how to learn

Hi @v_Enhace and all AoPsers:
@v_Enhace
When you say "do a lot of hard problems" which country's problems you meant?
Can you please say all such sources that has a great help to improve bronze-silver to silver-gold?
(In IMO/USAMO)
Also If I dont know some topics like "Cyclotomic Polynomials" or "Algebraic NT" can I try for hard problems?

And the last question :
How many hard problems for a day and how much time to think about one problem?
Thanks !!!

Ugh ,so training olympiad x hours a day?
İn school days İ have at most 5 or 6 hours to train olympiad.

Before a contest, GET EXCITED

Kayak was being sarcastic - there is no good answer to your question.
There is no 'correct' amount of problems to do, and there is not 'correct' amount of time to spend on a problem.