**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

- 2019 Bath, United Kingdom
- 2020 St Petersburg, Russian Federation
- 2021 Washington DC, United States of America
- 2022 Norway
- 2023 Tokyo, Japan
**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 materialThere are international maths competions with permanent websites:

- IMO official (records, rules, procedures)
- IMO Foundation (the public face of the IMO)
- UK IMO register reports, recent ones contain idiosyncratic Balkan Maths Olympiad, EGMO and IMO diaries.
- The Romanian Master of Mathematics
- The European Girls' Mathematical Olympiad (EGMO).
- The Benelux Mathematical Olympiad.
- The Pan African Mathematical Olympiad.

Here is specific advice about learning Euclidean Geometry.

**Contact Information:**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