I am a freshman (math undergraduate) here in Argentina and I am deeply interested in mathematical olympiads but I really need some advice.
Right now, my problem solving skills are good but not that good: I can solve most of the problems in textbooks like Rudin's Principles of mathematical analysis, Halmos's finite dimensional vector spaces and Nathanson's Elementary methods in number theory (I've been working on this book the last two weeks) but I can't solve a single USAMO, IMO and even national contests problem (of course, the non-trivial ones).
I really would like to be able to solve problems from competitions like the USAMO (and every national contest), the IMO, the Putnam, the IMC, etc., but I don't know how I can prepare myself for that kind of competitions.
I've read that The art and craft of problem solving, Problem solving through problems, Putnam and beyond, Problem solving strategies and Winning solutions are great for general problem solving, but I don't know which one to choose ( I've done a few ¿easy? problems of each book).
Also, I don't know if it's better to start working on books by subject (e.g. A path to combinatorics for undergraduates, 104 number theory problems or even Elementary methods in number theory) instead of the ones above (general problem solving).
Right now I HAVE (and I want) to keep working on Rudin's Principles of mathematical analysis and Halmos's Finite dimensional vector spaces, so I cannot choose many olympiad books.
$\bullet$ What would be your advice?, should I pick a book which focuses on general problem solving (which one?) or should I pick a few books by subject (which ones?)?
¡Many thanks for your help! and I am sorry for the question but, I'm desperate (I really need some advice).
Best Answer
Well as a former math contestant I wanted to share my thoughts on this question.
First and foremost you have to understand that the competition and the result you'll achieve are by no means a measurement of your mathematical ability. A very nice example is a friend of mine. In my life I've participated in about 40 math contests (regional, county, national, international and even IMO). Since my friend was the same age as me, he also participated in all this contests and he was simply dominant. I mean of the 40 or so contests I've participated in he was better ranked in all of them, as far as I can remember. Actually there were around 10 instances when I was second only to him (few times we were tied, simply because both of us had perfect scores). So you would say that he was a better mathematician than me. Well, I believe that's not the case. We both chased a math degree on university and he had problems with topics that were not required for math competitions, such as calculus, differential equations... So at the end he decided to give up on math and he started studying computer science. So because of this I believe that he was the better math competitor of the two, but not the better mathematician. So the math competitions certainly help you to develop your math genius, but your success (or failure) there doesn't necessary translates to your further math career.
Since I come from a relatively small country, I noticed that we failed miserably at almost every international competition (both on individual and team scale). So I tried to investigate and find out why this is happening. Certainly the fact that the talent pool in my country is small plays a big role. On every international contest there are countries that have 100 to 500 times more population than my country, so I guess there is a greater probability that a math genius will be born in let say USA or China than in my country. But this wasn't the only reason. Compared to countries as big as mine, we still had very bad results. So I talked to other contestants and I asked them what's their formula to success. What I found out was astonishing.
Since our country is small, we believe that we have a deficit of talents, so we try to make up for it by practicing and practicing. So we learn a lot of theory and techniques. Actually when I asked some of the guys that won a gold or silver medal at the IMO, they had little or no clue about some techniques. That suprised me. But unlike in my country, where there are only 4 contests per year(regional, county, national and selection test), other countries have 15-20 contests or at least "friendly" contest (where students solve problems in an IMO athmosphere, but they receive no awards for the results). So I concluded that although I (simularly as every contestant in my country) had much more theoretical knowledges I failed to put it in practice or to implement it in a solution. But with the expereince they had others don't have this problem. So I would say that experience is as important as practice. So maybe the reason why you couldn't solve a Olympiad problem is because you didn't participate in math contest as a youngster.
Also what I found out is that most of the successful math contestant have a "competition mindset", i.e. they go on compeitions and they do their work for 4.5 hours. So as a competitor I wasn't able to establish this mindset in me, so what usualy was happening to me is that I was great on preparations camps or when I was practising and I was able to solve some tough IMO problems by myself. But when I was at the IMO I wasn't able to recreate the same success and I believe that was because of the pressure that I was feeling and of course the time constraint.
So since now occasionally I'm working with young math talents in the first few classes I teach them about this mental thing and I want them to get this "competition mindset", before the procede to learning techniques and "tricks" if they want to be successful on math contests.