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The Journey
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The Journey

 

What the journey to mathematical olympiad world taught me




I was first introduced to the world of mathematical competitions in 5th grade. That time, I first participated in a math competition organized by PASKAL. I was thrilled. I enjoyed it so much coping with those hard problems, that at least, had nothing to do with the problems I encountered at school. These problems taught me a bunch about what mathematics really was, a science that promoted creativity and lateral thinking, just as much as it promoted logical and abstract thought. 

Anyways, as the best student in my school, I continued to participate in lots of competitions. Next year, I won 3rd place in a math competition, organized by ATA in Macedonia. While math started to become a wild obsession to me in the 7th grade, I managed to win again the 3rd place, thus I was confirmed as a Skopje contestant in my first pan-Albanian math competition. It was organized in Presevo. Wonderful experience. To encapsulate my primary-school experience in a few words, as obsession started to grow, at the same time my passion grew, and even though I had developed the intrusive habit of laziness, I still managed to win the 1st place in Skopje, and Macedonia (finally), so I managed to achive my long-awaited goal, and that's how the 2 year stand-by began to take over my math life.


I was finally re-introduced to math competitions again in the end of the 10th grade, when one of my friends started to tell me about his friend, a successful mathlete at the time. His name was Bojan Serafimov. It was this event, that I was introduced to the International Mathematical Olympiad, and the concept of International Science Olympiads (ISO). Out of curiosity, I googled his name, and this is how I came across his profile on pages like http://www.brilliant.org, and most importanly http://www.quora.com. In addition, I also found out about AoPS - http://www.artofproblemsolving.com, and also http://www.math.stackexchange.com

It was these three sites that brought a thoroughly, utterly different perspective in my life.

Brilliant, AoPS and Quora literally changed my entire lifestyle and thinking pattern.

What once revolved around useless things, surfing on the Net for countless hours, now turned into way more productive, constructive stuff.                                                        

I learnt lots of things, most notably

  • 1. The Power of Habit
  • 2. The Power of Focus
  • 3. Perserverance
  • 4. Impossible being a useless word most of the time.




Getting back to our point, I started using Quora in order to answer my most important queries and doubts at the time. 

That is how I brought in my life the concept of  Problem Solving.



Most important books



1. How to Solve It by George Polya
2. Art and Craft of Problem Solving by Paul Zeitz
3. Problem-Solving Strategies by Arthur Engel



Most important advice


1. How to practice efficiently => Deliberate Practice

2. “We are what we repeatedly do. Excellence, then, is not an act but a habit.” - Aristotle

3. DO PROBLEMS!

4. Probably the most important advice for a person aspiring to go to IMO would be to spend as much time as possible on AoPS, whether it is to improve his metacognitive skills, or solve some new problems. The reason is simple: being connected to AoPS brings one deeply into the Math Olympiad/ Problems world, so solving an IMO problem, winning first place in your national Olympiad, or being able to focus on the hot points of a problem, is not that much of a deal, and it never appears to be abstract, as it is for one that does the same job while searching in Quora, or StackExchange. 
RANKING of PURE MATH ATMOSPHERE: AOPS >> BRILLIANT > QUORA ~ STACKEXCHANGE.                     #MRWW

  • I don’t think one should ever stop learning theory, but always solve problems as well. The easiest way to remember an important theorem, concept, lemma, is to use it yourself in a number of problems.                                                                                                 David Popovic, IMO 2015 Bronze medallist
  • Never listen to people that tend to discourage you. Natural talent is very rare, most of the IMO contestants are undisputable result of

           1. Lots of deliberate practice

           2. Passion for math 

           3. Persistence

           4. RELEVANCE

  • The road to the IMO is long and requires effort. The key is to enjoy the journey. If you don’t like the road, you are never going to arrive to your destination.  Quora
  • The main reason why many people will discourage you in your quest for finding the way to IMO is, a lot of them have never experienced this pure mathematical world. Another reason is that some people found it terribly difficult, mainly because of their mindless practice, to get into IMO. Don't ever let their negativity affect you. The path to IMO is clear and certain if you follow some simple life advice and common sense. Their way is certainly NOT.
  • When you study a book or an article NEVER skip problems. You have to try all of them and solve at least 75% of them. NEVER skip them. You are trying to get better in problem solving, not theory memorizing.
  • You’ll want to do both theory and problem solving. There is tremendous synergy between the two: the theory will make more sense after you’ve tried a few problems yourself, and the problems will be easier when you know possible solution techniques.

  • I'll also equip you with Evan Chen's strategy: (You can clearly observe the Deliberate Practice mindset in 1,3,4,5,6 points of his strategy), DOC tool (Do, Observe, Correct), and Learning from your mistakes.

    1. Do lots of problems.
    2. Learn some standard tricks.
    3. Do problems which are just above your reach.
    4. Understand the motivation behind solutions to problems you do.
    5. Know when to give up.
    6. Do lots of problems.

  • Something that I'd like to use during my mathlete days would be to take a problem, study its solution so I know the motivation behind it, then I would add it into my toolbox. Then I'd try the problem myself. "How would I have thought of that?" would be the question while trying to understand the motivation. I would also try to determine the hot points, as well as understand the problem's nature. e.g.: If the problem is an inequality, I would determine the hot points of it, and then I would step in "My Inequality" world/ folder so that I could come up with interesting connections between the problem, the theory I already knew and also the tricks used for the similar or the same type of problems. Also, keep in mind, you should make great observations!

  • The following insightful comment introduced to me the idea of intuitive mathematics:

    I would say that the strategy you have described is not anywhere close to being the best strategy for olympiads, but it is quite possibly fairly close to the best FORMALIZABLE strategy. (In particular, this is what I think is commonly referred to as the “Chinese strategy.” On the other hand, in my opinion, the Chinese strategy is actually pretty terrible.)

    Your strategy mostly revolves around the problem-reflection cycle. I find the main issue with this to be that “motivation” is a very cheap form of understanding. The reason why the method works so well anyways is because olympiad math doesn’t have much depth, so this is not actually a huge issue.

    The main issue with “motivation” is that math is all about interconnections. With a “motivation”-based approach, understanding of interconnections is a secondary objective. The main objective a good sense of when each method is applicable. This is not a very deep skill.

    So what would I put as a bullet point? I would put the following as a bullet point: Whenever you see a problem you really like, store it (and the solution) in your mind like a cherished memory. (You really should have a strong emotional attachment to the problem for this to work.) The point of this is that you will see problems which will remind you of that problem despite having no obvious relation. You will not be able to say concretely what the relation is, but think a lot about it and give a name to the common aspect of the two problems. Eventually, you will see new problems for which you feel like could also be described by that name. Do this enough, and you will have a very powerful intuition that cannot be described easily concretely (and in particular, that nobody else will have).

    Here’s an example. A long time ago, I solved some ISL G2 which was actually combinatorial geometry. It had a solution with the extremal principle that I really liked, and I was pretty impressed with myself for coming up with it. I soon solved some other extremal principle combinatorial geometry and felt it was very similar. My “explanation” for it was “Well, you have points, and you have nearby points and faraway points, but you can’t have anything TOO near….” Of course, this is a meaningless explanation. But I basically started calling things along these lines “local-global principles”, which is not really a name anyone should endorse. And pretty quickly I could basically kill almost every combinatorics problem simply by understanding it in terms of my “local-global principles”. (In particular, I can give you an explanation of every combinatorics-themed chapter of Engel in terms of local-global principles.)

    Why does this process not count as formalizable? Because you really have to be genuine in your liking the problem and your wanting to give a name to the relation. Otherwise you might just say “oh they’re related by the extremal principle”, which is really not a relation at all. As I can’t give you a formalization of how to be genuine, I’m not going to count this as formalizable.

  • The “solving/creativity-based” structure of contests encourages people to focus much more on the “motivation-purely-for-solving” (I believe this is what you’re criticizing) rather than the “natural story/context/interconnections”, but I think this is just a (potentially easily-fixable) problem with current Olympiad culture.

    For example, if authors generally wrote on the story/context/interconnections behind their problems, e.g. how they came up with the problems, related ideas/themes, or even just liberal doses of vaguely-related links/references, I think everyone would gain a lot more out of contests. (This is something I’ve sort of tried to do in the comments sections of solutions to my problems, but still I always subconsciously leave out a lot of background.)  - Victor Wang

  • Sean Markan, a teacher that prepares students for AMC 10, 12 and AIME, on Olympiad problems gives the following advice:

    Read a lot of solutions. Learning to solve olympiad-level problems is hard, because one needs to be exposed to a large body of tricks and techniques before one can make much headway on them. These tricks are not taught in school, are not the same ones that appear on easier math contests, and you are very unlikely to discover them yourself. Therefore, you will need to acquire them by reading (or from a teacher who knows them). Also, while the basic tricks are often written down explicitly in book form (for example in Engel), most tricks are not, so there is no substitute for exposing yourself to a lot of actual solutions.

  • Markan also has his say about olympiad problems, talent, tricks and all stuff, and his opinion doesn't differ at all from mine (yay)

    Because olympiad tricks are obscure, the first time you try olympiad problems (and maybe the 20th time), you will do badly. That's normal and doesn't mean you're not smart enough for math olympiads. Don't get discouraged. Just keep reading solutions and figuring out the tricks. These tricks do exist and are learnable. Again, you need to read a lot of solutions before you will be able to construct them yourself. This is true of olympiads much more so than the easier contests. It also pays to be pretty aggressive in extracting general tricks: try not to extract merely the trick for the particular problem you're reading, but the whole way of thinking that led the solver to find it.

  • Carlos Shine, Brazilian IMO team coach suggests: (As I'm giving you more and more examples, you're finally realising that Deliberate Practice is the key ;))

    My piece of advice is that you should focus on working efficiently. Quality over quantity. Find some good material. Try to work on problems on a good level (just slightly above your level, but not so much that you are uncomfortable with it).
    The fallacy in your thinking is that hours of work is not proportional to amount of learning. It surely varies according to people. I don't think it's efficient to work on problems when you're tired, or even if you get stuck. If you do, you should take a break, do something else. If you're not tired, try a different problem; if you are, just do something outside of math; anyway, it's a sign that you need some fresh air. Don't be too hard on yourself!


  • One of the greatest intuitive mathletes, and undoubtly one of the most influential AoPSers, pythag011 in his blog Random Trivial Problems in a piece named "On Contest Math, Higher Math, Understanding, and Other Things" recommends:

    In olympiad training, you are given things called tools and told to apply them. And I think people tend to (or at least I did in my earlier years) believe that to solve olympiad problems, what you do is recognize what sorts of tools you could use and think of creative ways to use them. But that's definitely not how I solved problems on my good days (on my bad days, I usually was like "help I suck at math" and just tried random things which sometimes worked anyways.) The way I think of what I did (which is somewhat romanticized) is that I essentially thought in terms of theories instead of theorems and understood the relation of the problem to theories, and once I understood that relation well enough, the solution was completely obvious.​

    Some examples of what I mean: In many ways, number theory reflects the study of polynomials. Taking mod a prime corresponds to taking mod (x-a), or plugging in a, etc. So I often started number theory problems by thinking about how exactly the problem related to the polynomial part of number theory and how exactly it related to the "number theory" part of number theory, things like the Frobenius automorphism. For some problems, it becomes immediately clear under this viewpoint that the key point of the problem HAS to be the frobenius automorphism, and then it's usually a bit of toying with the problem until its clear exactly how its a Frobenius automorphism problem.

    I really think the hardest part of that process is the understanding of the theory. And I find that this process translates really well to higher math; this is very close to what I do whenever I learn a new theorem, or etc. There are of course differences. For one thing it's nearly impossible to successfully do this at the start. So you just have to never stop trying to understand the theorems at the very start, and once you reach the end, you'll probably be able to (and then you can understand the later theorems, etc...​
  • The two ways of looking at a problem are:

    There are two ways of looking at a problem: (oversimplification, but will suffice for the moment)

    1. Looking at it by itself
    2. Looking at it as a piece of a larger picture

    In general, 1 will have you solve the problem more quickly; 2 will have you learn much more though. (And learning intuition is largely doing 2.)

    Note that I'm not saying 2 is better, I'm just saying 2 is better for learning.

  • Looking at a problem by itself, in my humble opinion, should be exercised when training in AMC, AIME or National Competitions, while the second, more intuitive approach should be taken while solving USAMO, China, Iran, APMO, BMO or IMO problems, and more importantly in solving Russian problems, since they tend to be the more intuitive ones.

  • It's a great thing to know that math is all about connections**.

  • Something that helped me in honing my problem-solving skills was an answer I read on Quora, which suggested when approaching a math problem, never use the method or technique that first comes in mind!

  • The method I personally came up with, was to think of the problem as two parallell lines. The first, being the problem's statement and the second being the conclusion. The person who tries to solve the problem, should force connections between the two parallells. Given the progress obtained by observation, recognizing patterns, creating motivations, and most importantly working while being in the problem's nature (where all the tools, tricks, techniques and motivations are found) can ideally solve the problem 100% in a shorter period of time.

  • After seeing a solution to a problem which you were unable to solve on your own, ask yourself the following questions:

    What key insight did I miss which resulted in me not being able to solve the problem?
    How could I have seen from the problem statement that this key insight was needed?
    Are there gaps in my mathematical knowledge which resulted in me not being able to solve the problem? (If so, read up to fill those gaps.)

  • Even though we try tirelessly to emphasize the importance of intuition, you should remember one simple thing: Intuition is not proof. Just because something seems to make sense doesn’t mean that it is true. You can learn more about it here.

  • That's like ME to crush it. That's like ME. Thought motivation inspired by WWM of Lanny Basham.

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