Mathematics
International Mathematical Olympiad
The world's most prestigious mathematics competition for school students.
What is IMO?
The International Mathematical Olympiad has run annually since 1959 and is the most prestigious mathematics competition for school students in the world. More than 100 countries send teams of up to six students to solve six exceptionally hard problems across two days.
An IMO medal — bronze, silver or gold — is recognised by every top university on earth as one of the strongest signals of mathematical ability that exists. Gold medalists are recruited directly by MIT, Cambridge, Oxford, Harvard, and leading technology companies.
Pakistan's pathway
Pakistan selects its IMO team through a national mathematics olympiad process, run under the Federal Board and referred to as the Pakistan Mathematical Olympiad (PMO) / National Mathematics Talent Contest (NMTC).
Stage 1
Registration
You need an aggregate of 60% or more in core subjects in your most recent examination. Registering completes this stage.Stage 2
Screening test
Held around March for the following year’s competition. Roughly 1,000–1,500 students sit it. It covers O Level and A Level mathematics plus introductory olympiad topics; around 50 students are shortlisted.
Format: an MCQ section (~50 marks, school syllabus plus introductory olympiad) and a subject section (~25 marks, olympiad problems — geometry appears most often).
Stage 3
Camps and selection tests
Shortlisted students attend training camps, online or physical. Each camp ends with an exam that narrows the field, repeating until six students remain: the Pakistan IMO team.
Eligibility
Under 20 on 30 June of the competition year. Open to Grade 9, 10 and 11 students (Matric/O Level) and FSc-1/AS students. A2 students may register if age-eligible and planning a gap year.
The four subjects
Geometry
The most heavily tested area in Pakistan's screening. Euclidean geometry, circles, triangles, transformations. Highly learnable with the right book: Evan Chen's covers what you need, and its first four chapters are enough for the screening test.
Number theory
Primes, divisibility, modular arithmetic, Diophantine equations. Appears in almost every IMO.
Combinatorics
Counting, probability, graph theory, combinatorial proof. Rewards creative thinking over memorised technique, and it is the area Pakistani students have usually seen least of.
Algebra
Functional equations, inequalities, polynomials.
Every IMO problem requires a proof.
Whatever the subject, you are proving a statement or proving your answer is complete. Writing proofs is not an extra skill on top of the mathematics — it is the mathematics.
Free book library
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Syllabus for IMO
Read this before opening anything else. It states exactly which topics IMO tests — algebra, number theory, combinatorics, geometry — so that every hour after it is aimed at the right target. Students who skip it routinely spend months preparing the wrong material.
A Step to Mathematical Olympiad
The right first book when moving from school mathematics to olympiad problems. It teaches the shift from mechanical calculation to creative reasoning at a manageable pace. Work every problem before moving on.
A Second Step to Mathematical Olympiad
The direct continuation of Book 2, in the same accessible style at higher difficulty. Start it immediately after finishing the first.
The Art of Problem Solving — Basics
The most widely used olympiad preparation series in the world, covering number theory, algebra, counting and geometry with hundreds of graded problems. Used by teams from the US, China and South Korea. Do not skip it.
The Art of Problem Solving — Beyond (Vol. 2)
The advanced companion to Basics: intermediate number theory, combinatorics and inequalities. Take it after Basics. Together, Books 4 and 5 are the backbone of preparation for students reaching national team level.
The Art and Craft of Problem Solving — Paul Zeitz
Unlike most olympiad books, this one is about what to do when you are completely stuck. Zeitz, a former US olympiad winner, lays out the strategies and habits of strong problem solvers. It changes how you approach every hard problem afterwards.
Introduction to Counting & Probability (AoPS)
Combinatorics is one of the four core areas and the one most Pakistani students have had least exposure to. This builds counting and probability from the ground up with olympiad-focused problems.
Introduction to Diophantine Equations
Diophantine equations appear at IMO nearly every year. This targets exactly that problem type, from foundations to competition difficulty. Take it before the heavier number theory books.
104 Number Theory Problems — Titu Andreescu
Andreescu coached multiple US gold-medal IMO teams. 104 problems from approachable to brutal, each with a full solution. Work every one — it is the most effective single number theory resource for IMO preparation.
Modern Olympiad Number Theory
Advanced treatment: lifting the exponent, orders and primitive roots, quadratic residues, p-adic valuations. Take it after Andreescu. Students aiming at national team selection should finish both.
Number Theory — Problems, Examples and Structures
A structural reference to use alongside Book 10 for extra problem variety and alternative explanations of the hard topics.
Euclidean Geometry in Mathematical Olympiads — Evan Chen
Widely regarded as the best olympiad geometry book written. Geometry appears at every IMO, and its first four chapters cover what Pakistan’s screening test asks for.
More free resources
Handouts, archives and communities
Also worth finding: Yufei Zhao’s olympiad handouts — search “Yufei Zhao olympiad handouts”.
How to study
Do not open the solution until you have finished the problem. If you are stuck for more than half an hour, leave it and return days later.
Do not rely on the camps. They are pitched at beginners. If you are aiming for the team, assume you will self-study the entire syllabus.
Two or three hours a day, consistently, is enough. Persistence and creative thinking beat volume.
Register free to get competition alerts and the full roadmap
Register freeStart with the syllabus.
It states exactly which topics IMO tests, so that every hour after it is aimed at the right target.
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