International Mathematical Olympiad Syllabus: Topics Guide

Table of Contents

• IMO Syllabus at a Glance

• Key Points to Know Before You Start

• School Mathematics You Must Know First

• Algebra Topics

• Plane Geometry Topics

• Combinatorics Topics

• Number Theory Topics

• Syllabus Across IOQM, RMO, INMO and IMO

• How Topics Show Up in Problems

• Is This the Same as the School-Level Olympiad Syllabus?

• Recommended Books

• How to Cover the Syllabus

• Common Mistakes to Avoid

The International Mathematical Olympiad syllabus is different from the syllabus of any school board or entrance exam. There is no official list of chapters. The IMO draws its problems from four areas of school-level mathematics: algebra, combinatorics, geometry and number theory. What makes the problems hard is how they are posed, not how much advanced theory they need.

This page turns that idea into a practical checklist. It follows the topic list published by the Homi Bhabha Centre for Science Education (HBCSE), which runs India's Mathematical Olympiad programme, and adds guidance on where to start and how the four areas connect across IOQM, RMO, INMO and the IMO.

A note on the syllabus: the IMO does not publish a topic-wise syllabus. The lists below come from HBCSE's indicative syllabus for the Indian stages that lead to the IMO. HBCSE itself says the list is not exhaustive and that RMO and INMO problems may include topics not named in it. Use it as a working checklist, not as a guarantee.

IMO Syllabus at a Glance

Key Points to Know Before You Start

• No calculus needed: Calculus is never required at the IMO, and the IOQM guidance also says calculus and statistics are excluded. A student with basic school mathematics should be able to understand every problem, even when the solution is hard.

• Difficulty rises by stage: HBCSE states that the level goes up from RMO to INMO to the IMO. The same four areas appear throughout, but the problems become more sophisticated.

• Problem solving beats coverage: Olympiads reward clear reasoning and creativity more than extensive knowledge. Finishing a long topic list matters less than solving many varied problems.

• Proofs matter from RMO onwards: The IOQM asks for numerical answers. RMO, INMO and the IMO require complete written proofs.

• The list is indicative: New ideas, such as extra techniques in geometry or combinatorics, appear in papers from time to time.

School Mathematics You Must Know First

HBCSE advises every Olympiad aspirant to be familiar with all topics in the NCERT mathematics books for Classes 8, 9 and 10 before moving to the topics below. These chapters are the foundation, and weak basics are the most common reason students stall.

Algebra Topics

Algebra problems in olympiads usually ask you to prove an inequality, find all functions with a given property, or work with polynomials and sequences. HBCSE lists the following topics.

• Inequalities: Standard inequalities and the technique of proving that one expression is always at least as large as another.

• Progressions: Arithmetic, geometric and harmonic progressions, along with their means.

• Theory of indices: Laws of exponents and their use in simplifying and bounding expressions.

• Systems of linear equations and theory of equations: Relations between roots and coefficients, and solving equations with constraints.

• Binomial theorem: Expansions and properties of binomial coefficients.

• Complex numbers: Algebra and geometry of complex numbers, including roots of unity.

• Polynomials in one and two variables: Factor and remainder ideas, root properties and identities.

• Functional equations and sequences: Finding functions or sequences that satisfy given rules.

Start with inequalities and polynomials, because these appear most often in the early stages. Functional equations look intimidating, but they respond well to a few standard substitutions that you learn through practice.

Plane Geometry Topics

Geometry problems often hide a simple fact behind a crowded diagram. The topics below are the usual tools.

Begin with angle chasing, similar triangles and circle theorems. Add trigonometry and coordinate or vector methods once you can solve classical problems by pure geometry, because these methods give you a second way to attack a stuck problem.

Combinatorics Topics

Combinatorics has the least formal theory and the widest variety of problems. HBCSE names a short list.

• Basic enumeration: Counting arrangements and selections without double counting.

• Pigeonhole principle: Proving that something must exist by comparing the number of objects and boxes, including clever applications.

• Recursion: Setting up and solving relations that describe how a count grows.

• Elementary graph theory: Vertices, edges, degrees, paths, cycles and trees used to model a problem.

This is the area where students often feel they have nothing to study. In practice, the best method is to solve many different problems and note each trick you meet: invariants, extremal arguments, colouring, and induction.

Number Theory Topics

Number theory is the most structured of the four areas, and HBCSE gives a precise sequence of topics.

Master divisibility and modular arithmetic first. Most olympiad number theory problems become manageable once you are comfortable working modulo small numbers and with prime factorisation.

Syllabus Across IOQM, RMO, INMO and IMO

The four areas are the same at every stage. What changes is the format and the depth expected.

Because one skill carries across all stages, you do not need to study separately for each exam. Study the four areas once, then adjust your practice to the format of the next stage.

How Topics Show Up in Problems

Knowing a topic name is only the first step. In an Olympiad paper, the topic is rarely announced. A problem about a grid of coloured squares may really be a pigeonhole argument. A problem about a strange-looking sum may be a polynomial question in disguise. A geometry problem with many circles may fall apart once you spot one pair of similar triangles.

This is why practice matters more than coverage. After each problem you solve, write one line stating which idea unlocked it, for example "invariant under swapping" or "consider remainders modulo 7". After a few weeks, this list becomes your personal syllabus, and it is more useful than any printed one.

Is This the Same as the School-Level Olympiad Syllabus?

No. A school-level International Mathematics Olympiad run for Classes 1 to 12 follows the school curriculum of the student's class. The syllabus on this page is for the global International Mathematical Olympiad and the Indian stages that lead to it, which test problem solving well beyond the textbook. If you are preparing for the school-level exam, use the syllabus published by its organiser instead.

Recommended Books

HBCSE lists the following books for the Olympiad topics above. You do not need all of them. Pick one good book per area and work through it carefully.

HBCSE also recommends solving the question papers of earlier years, which are available on its Olympiads website at the official past papers page.

How to Cover the Syllabus

The plan below assumes a student starting from Class 8 or 9 level mathematics. Adjust the timing to your own pace, since progress depends on practice rather than the calendar.

1. Phase 1, foundation (first 6 to 8 weeks): Revise NCERT mathematics of Classes 8 to 10. Fix gaps in algebra identities, triangles, circles and divisibility.

2. Phase 2, one area at a time (next 3 to 4 months): Study number theory and algebra first, then geometry, then combinatorics. For each area, read a short theory section and solve 15 to 20 problems before moving on.

3. Phase 3, mixed practice: Solve past IOQM and RMO papers by topic, then in full under time limits. Mark every problem you could not solve.

4. Phase 4, proof writing: From RMO preparation onwards, write out complete solutions and get them checked. A correct idea with an incomplete proof loses marks.

5. Phase 5, advanced problems: After the early stages, move to INMO and international-level problems, and keep a notebook of techniques that worked.

Common Mistakes to Avoid

• Reading theory without solving problems: Olympiad skill comes from struggling with problems. Spend most of your time on questions, not on notes.

• Neglecting combinatorics: It has no thick textbook, so students skip it. It still appears in every paper.

• Skipping the school base: Gaps in Class 8 to 10 mathematics slow down every advanced topic.

• Ignoring written solutions: From RMO onwards, unclear writing costs marks even when the idea is right.

• Giving up on a problem too early: Allow a problem real thinking time, then read the solution and note what you missed.

• Trusting unofficial syllabus lists blindly: Some pages mix this exam with the school-level olympiad. Check the official HBCSE syllabus page for the Indian stages.