Advice for young mathematicians
Sylvester's Coin Theorem.
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
Maths Competition Books Many countries have maths enrichment organizations which publish relevant books. In Britain, this outfit is the United Kingdom Mathematics Trust and it has a publications site. I strongly recommend purchasing direct from this charity. There are many interesting texts there. I wrote A Mathematical Olympiad Primer available at that site. MOP is an elementary text aimed at people sitting BMO1 (or any similar first round national maths olympiad paper). You can now also find A Mathematical Olympiad Companion for people planning to sit BMO2 or any other final round national mathematical olympiad. Many people seek to get a geometry education: UKMT publishes excellent and appropriate texts: for example Christopher Bradley and Tony Gardiner's Plane Euclidean Geometry and Gerry Leversha's Crossing the Bridge and The Geometry of the Triangle.
However, it may not be sensible to purchase these materials if you come from a country where hard currency is very expensive. There are sometimes materials made available by maths competition enthusiasts in your own country, so try to find out if there are such people, and get in contact with them. If you do have access to hard currency, and you are sufficiently anglophone, then you may wish to consider the texts available through the publishing arms of the Mathematical Association of America and the Australian Mathematics Trust.
There are also free books on the internet which you can download. I wrote a foreword for the excellent text Infinity by the prolific problem composer Hojoo Lee, and two collaborators Tom Lovering and Cosmin Pohoata. There is also Kiran Kedlaya's beautiful Geometry Unbound.
If you would like to recommend that I make a link to some free internet resources, please make a suggestion in email with "link recommendation" in the subject line.
Other useful resources include
|AoPS, that is over 25000 past olympiad problems!||Brilliant.org||NIMO||Cut the knot||Wolfram Mathworld|
|MacTutor history of mathematics||Yufei Zhao's handouts||IMOmath||Evan Chen's MIT propaganda|
|Crux Mathematicorum||Mathematical Reflections (AwesomeMath)||Mathematical Excalibur||Chapters of an Olympiad Combinatorics book|
|Cody Johnson's notes on algorithms||Canadian materials||Po-Shen Loh's talks||Art of Problem Solving (front)|
|en francais Kortchemski's page||en francais Animath materials||y en espanol Geometry in Spanish||y en espanol Mexican Olympiad Prepárate material|
There are international maths competions with permanent websites:
Here is specific advice about learning Euclidean Geometry.
Dr Geoff Smith MBE Email: G.C.Smith@bath.ac.uk Department of Mathematical Sciences Tel: +44 (0)1225 386182 (direct) University of Bath Fax: +44 (0)1225 386492 Bath BA2 7AY Room: 4W2.16 England UKMT vice chair of the trust, chair of the BMO, president of the IMO board and @GeoffBath advisory board member of EGMO