(Cross-posted in MSE 1301476.)

I want to start to do Olympiad type questions but have absolutely no knowledge on how to solve these apart from my school curriculum. I'm 16 but know maths up to the 18 y/o level. I think I will start learning the theory of the topics (Elementary Number Theory, Combinatorics, Euclidean Plane Geometry) then going on to trying the questions, but I need help in knowing what books to use to learn the theory. I have seen several list (such as http://www.artofproblemsolving.com/wiki/index.php/Olympiad_books) but does anyone know which ones are the best for my level of knowledge.

P.S. I live in UK if that matters.

  • 2
    $\begingroup$ Would this be better suited for Math Stack Exchange since you're asking for resources? $\endgroup$ Commented May 27, 2015 at 19:15
  • $\begingroup$ @CameronWilliams I have posted it there right now too. math.stackexchange.com/questions/1301476/… Should I delete this post? $\endgroup$
    – MKu
    Commented May 27, 2015 at 19:48
  • $\begingroup$ I'd say not. You might get some good resources here too. $\endgroup$ Commented May 27, 2015 at 19:49
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    $\begingroup$ "18 y/o level in UK" is a meaningless descriptor of your mathematical abilities for many readers of this site. If you want recommendations based on your level of knowledge you should try to be more clear in describing precisely what you have learned (subject area/ textbook used/ keywords from the contents). $\endgroup$ Commented May 28, 2015 at 12:04
  • $\begingroup$ But if your intent is to train for the Math Olympiads, and you haven't done anything like it before, perhaps a good start would be to read some of the collected problem books with solutions from the various olympiads (USAMO or the USSR ones from the 80s and 90s are a good place to start; the questions then were slightly easier to play with and understand, and you quickly get a feel of what general flavour to expect). $\endgroup$ Commented May 28, 2015 at 12:11

2 Answers 2


Since this question has been posted in Educators stack exchange, I would try to structure my response in a way that addresses different learning approaches.

Approach - 1 : Learn techniques first, apply learnt techniques

This is the most traditional approach. Let us take the example of number theory. The chapters from a typical number theory book look like this - Introduction, Prime numbers, Congruences, ...

You start with the prime numbers chapter and go through every method, technique and understand the applications,then start solving exercise problems belonging to that particular chapter. Then move on to the next chapter.

While this approach is good, I am a bit concerned about the way it affects the development of your problem solving abilities. Some of these chapters may take days to complete and the exercises at the back may focus exclusively on a few select topics.

Approach - 2: Learn enough theory; work on problems; go back and learn techniques

Learn enough theory to understand problems. For instance, you can't solve congruence equations without having knowledge about congruences. Grappling with problems at an early stage is very advantageous. After you spend sufficient time on a problem and are unable to get any ideas, go for the solution. This is a good way to track your progress. Initially the progress may be very slow and frustrating, every problem that you try may elude a solution. But gradually you would start doing better.

My suggestion would be to look at a book of problems very early on to see where you stand. If you are able to follow the solutions without too much difficulty, this would be a good indication of your aptitude for olympiad math. This is a short book by Titu Andreescu that has a good collection of problems and enough theory to back them up.


A few other resources that might help:



Caveat: Though hard work is very important, a certain level of problem solving ability and intuition are indispensable when it comes to solving Olympiad mathematics. You develop them as you work. You are up against some of the most competent problem solvers. If you are doing good enough at school, I think you must certainly go for Olympiad mathematics.


The best single book for preparing for mathematical olympiads in my opinion is

Arthur Engel, Problem-Solving Strategies (1998).

It has chapters on the invariance principle, colouring proofs, the extremal principle, the box principle, enumerative combinatorics, number theory, inequalities, induction, sequences, polynomials, functional equations, geometry, games, and further strategies.

Engel was leader of the German team at the International Mathematical Olympiad. Sadly he passed away a little more than a year ago. You can read more about him on his Wikipedia page here.


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