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Advice for young mathematicians

From time to time I am approached by students interested in advice about becoming more effective contestants in mathematics olympiads. Here it is.

Do lots and lots, and then more, past papers. Begin with national mathematical olympiads, starting with the less difficult papers. Now, I am not going to risk insulting any countries by saying that their national maths olympiads are easy. Work it out for yourself. Countries which have small populations, and no great tradition of success in maths competitions, will generally have easier questions. When you become very good at those, then move on to hard national maths olympiad problems and the less demanding international competitions.

Whether you should learn lots of university level mathematics while you are at school is moot. Personally, I do not recommend it. It is more fun to learn things with other people, so why not leave the university syllabus pretty much alone until you get there? You will need to learn a bit of elementary number theory for international maths competitions (Chinese Remainder, Pell's equation and maybe Quadratic Reciprocity). Learn enough Euclidean Geometry to be on good terms with Simson's Theorem, Ceva's theorem, excircles, Menelaus's theorem, power of a point and the radical axis theorem. For combinatorics, the pigeon-hole principle (also called the Dirichlet principle) and double-counting are all you usually need (together with ingenuity). For inequalities, know the famous ones, and practise using them on real competition problems. The famous inequalities are discussed in my UKMT book, a Mathematical Olympiad Companion (UKMT) and quoting from there: The list of inequalities here is perhaps a little more comprehensive than it needs to be. The basic inequalities are: sums of squares are positive, AM-GM, Rearrangement and Cauchy-Schwarz. The HM-GM-AM-RMS chain probably comes next. Then look at power means, Jensen's inequality and Holder's inequality. Finally, international competitors must keep a grip on Muirhead and Schur's inequalities, otherwise three variable symmetric inequalities will start to appear in mathematics competitions again, as welcome as Calystegia sepium in your garden.

I am often approached by students from developing countries. Sometimes students complain that there is no satisfactory educational or training regime in my country. Please check that this is true! The IMO contact person in your country may tell you otherwise. In the worst case, where there is no competent organization providing free (or nearly free) assistance to young mathematicians, then you will have to help yourself. Try to locate other young people in your country who are interested in mathematics, and work together. Fortunately there is a vast collection of free resources on the internet: over 25 thousand past problems from maths competitions are available at the extensive Art of Problem Solving site, and if you explore, you will find discussions of solutions. Don't look up the solutions too quickly (be prepared to spend many hours thinking about each problem). If you want to start on some problems which are less demanding than a full national maths olympiad, here are plenty of British Maths Olympiad round 1 problems. The round 2 problems are more challenging.

The forthcoming IMOs will be