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Syllabus guide

IOQM Syllabus: Complete Topic Guide

Understand what students need to study, how the syllabus is organised and how to prepare with a structured plan.

Explore the IOQM CourseView the full syllabus
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FOR PARENTS

Turn the syllabus into a clear study plan

Use these three checkpoints first. The complete reference information remains available immediately below.

01

See the major areas your child needs to learn

02

Identify foundations before attempting advanced problems

03

Move from concepts to worksheets, papers and mock tests

IOQM PREPARATION ROADMAP

A clear journey from foundations to Olympiad depth

Follow the stages in order. Students should advance when the earlier ideas are understood and usable—not simply when a chapter has been read.

How to read this roadmap: HBCSE publishes broad areas rather than this stage-wise sequence. This is a suggested learning order; RMO and INMO stages are enrichment beyond core IOQM preparation.

  1. 01BEGIN HERE
    STAGE 1 · TOPICS 1–5

    Foundation Stage

    Build the essential language and habits needed for non-routine mathematics.

    1. 1Basic Number Theory
    2. 2Elementary Geometry
    3. 3Basic Counting
    4. 4Algebraic Identities and Polynomials
    5. 5Basic Inequalities
  2. 02IOQM FOCUS
    STAGE 2 · TOPICS 6–12

    IOQM Development Stage

    Develop the problem-solving tools most directly useful for IOQM preparation.

    1. 6Modular Arithmetic
    2. 7Pigeonhole Principle
    3. 8Combinations and Permutations
    4. 9Similarity, Circles and Area Methods
    5. 10Vieta’s Relations and Polynomial Divisibility
    6. 11AM–GM, Cauchy and SOS
    7. 12Mathematical Induction
  3. 03ADVANCE
    STAGE 3 · TOPICS 13–19

    RMO Development Stage

    Move from short competitive problems towards deeper reasoning and proof.

    1. 13Diophantine Equations
    2. 14Functional Equations
    3. 15Recurrence Relations
    4. 16Inclusion–Exclusion and Bijections
    5. 17Ceva, Menelaus, Power of a Point and Cyclic Geometry
    6. 18Advanced Polynomial Problems
    7. 19Advanced Inequality Techniques
  4. 04ENRICH
    STAGE 4 · TOPICS 20–26

    INMO Enrichment Stage

    Explore advanced techniques after the earlier stages are genuinely secure.

    1. 20Irreducibility of Polynomials
    2. 21Generating Functions
    3. 22Advanced Functional Equations
    4. 23Higher-Order Recurrences
    5. 24Jensen and Hölder Inequalities
    6. 25Advanced Concurrency and Circle Geometry
    7. 26Multi-topic Proof Problems
Not sure where your child should begin?Start with a mixed foundation assessment, then choose the earliest stage with visible gaps.
See the IOQM 2027 study plan →

DETAILED LEARNING MAP

Eight units. 122 subtopics.
One clear curriculum.

This detailed map turns the broad Mathematics Olympiad syllabus into teachable units, so parents can see what their child will learn and students can track progress topic by topic.

8 structured units122 detailed subtopics
01UNIT 1Polynomials16 subtopics
  1. 1.1Polynomial Functions
  2. 1.2Division in Polynomials
  3. 1.3Remainder Theorem
  4. 1.4Factor Theorem
  5. 1.5Fundamental Theorem of Algebra
  6. 1.6Polynomial Equations
  7. 1.7Rational Root Theorem
  8. 1.8Integer Root Theorem
  9. 1.9Vieta’s Relations
  10. 1.10Symmetric Functions
  11. 1.11Common Roots of Polynomial Equations
  12. 1.12HCF and Euclidean Algorithm for Polynomials
  13. 1.13Reducibility and Irreducibility of Polynomials
  14. 1.14Gauss Lemma
  15. 1.15Eisenstein’s Irreducibility Criterion
  16. 1.16Extended Eisenstein Criterion
02UNIT 2Inequalities16 subtopics
  1. 2.1Basic Rules of Inequalities
  2. 2.2Weierstrass’s Inequality
  3. 2.3Modulus Inequalities
  4. 2.4Triangle and Polygon Inequalities
  5. 2.5Sum of Squares Method
  6. 2.6Quadratic Inequalities
  7. 2.7Arithmetic Mean, Geometric Mean and Harmonic Mean
  8. 2.8Weighted Means
  9. 2.9Power Mean Inequality
  10. 2.10Rearrangement Inequality
  11. 2.11Chebyshev’s Inequality
  12. 2.12Cauchy–Schwarz Inequality
  13. 2.13Titu Andreescu’s Lemma
  14. 2.14Hölder’s Inequality
  15. 2.15Geometrical Inequalities
  16. 2.16Jensen’s Inequality
03UNIT 3Mathematical Induction9 subtopics
  1. 3.1Propositions and Mathematical Statements
  2. 3.2First (Weak) Principle of Mathematical Induction
  3. 3.3Working Rule for Induction
  4. 3.4Divisibility Problems
  5. 3.5Summation Identities
  6. 3.6Inequalities by Induction
  7. 3.7Transitive Comparison in Induction
  8. 3.8Second (Strong) Principle of Mathematical Induction
  9. 3.9Recursively Defined Sequences
04UNIT 4Recurrence Relations10 subtopics
  1. 4.1Introduction to Recurrence Relations
  2. 4.2Classification of Recurrence Relations
  3. 4.3First-Order Linear Recurrence Relations
  4. 4.4First-Order Non-Linear Recurrence Relations
  5. 4.5Second-Order Linear Homogeneous Recurrence Relations
  6. 4.6Characteristic Equation Method
  7. 4.7Repeated and Distinct Characteristic Roots
  8. 4.8Higher-Order Linear Homogeneous Recurrences
  9. 4.9Non-Homogeneous Linear Recurrence Relations
  10. 4.10Applications to Sequences and Olympiad Problems
05UNIT 5Functional Equations12 subtopics
  1. 5.1Functions: Domain, Codomain and Range
  2. 5.2Injective, Surjective and Bijective Functions
  3. 5.3Even, Odd, Monotonic and Periodic Functions
  4. 5.4Introduction to Functional Equations
  5. 5.5Standard Substitution Techniques
  6. 5.6Injectivity and Surjectivity Methods
  7. 5.7Fixed Points and Zeros
  8. 5.8Additive Functional Equations
  9. 5.9Multiplicative Functional Equations
  10. 5.10Jensen-Type Functional Equations
  11. 5.11Quadratic Functional Equations
  12. 5.12Iterative and Polynomial Functional Equations
06UNIT 6Number Theory20 subtopics
  1. 6.1Divisibility of Integers
  2. 6.2Euclid’s Division Lemma
  3. 6.3Greatest Common Divisor
  4. 6.4Least Common Multiple
  5. 6.5Bézout’s Identity
  6. 6.6Prime Numbers
  7. 6.7Fundamental Theorem of Arithmetic
  8. 6.8Number of Positive Divisors
  9. 6.9Sum and Product of Divisors
  10. 6.10Modular Arithmetic
  11. 6.11Complete and Reduced Residue Systems
  12. 6.12Fermat’s Little Theorem
  13. 6.13Euler’s Totient Function and Euler’s Theorem
  14. 6.14Wilson’s Theorem
  15. 6.15Chinese Remainder Theorem
  16. 6.16Scales of Notation and Number Bases
  17. 6.17Greatest Integer Function
  18. 6.18Linear Diophantine Equations
  19. 6.19Non-Linear and Exponential Diophantine Equations
  20. 6.20Pythagorean Triples and Integer Parametrisation
07UNIT 7Combinatorics18 subtopics
  1. 7.1Factorials
  2. 7.2Basic Counting Principles
  3. 7.3Permutations
  4. 7.4Combinations
  5. 7.5Bijection Principle
  6. 7.6Combinations with Repetition
  7. 7.7Circular Permutations
  8. 7.8Distribution of Distinct and Identical Objects
  9. 7.9Number of Integral Solutions
  10. 7.10Binomial Theorem
  11. 7.11Multinomial Theorem
  12. 7.12Generating Functions
  13. 7.13Applications of Recurrence Relations
  14. 7.14Principle of Inclusion and Exclusion
  15. 7.15Derangements
  16. 7.16Classical Occupancy Problems
  17. 7.17Dirichlet’s Pigeonhole Principle
  18. 7.18Double Counting and Extremal Counting
08UNIT 8Geometry21 subtopics
  1. 8.1Angles and Directed Angles
  2. 8.2Congruent Triangles
  3. 8.3Triangle Inequality
  4. 8.4Ratio and Proportion Theorem
  5. 8.5Area Lemma
  6. 8.6Midpoint Theorem
  7. 8.7Basic Proportionality Theorem
  8. 8.8Similar Triangles
  9. 8.9Pythagoras Theorem
  10. 8.10Quadrilaterals
  11. 8.11Concurrency and Collinearity
  12. 8.12Ceva’s Theorem
  13. 8.13Menelaus’ Theorem
  14. 8.14Triangle Centres
  15. 8.15Circles
  16. 8.16Power of a Point
  17. 8.17Cyclic Quadrilaterals
  18. 8.18Tangential Quadrilaterals
  19. 8.19Ptolemy’s Theorem
  20. 8.20Application of Trigonometry in Geometry
  21. 8.21Construction of Triangles

DETAILED SYLLABUS & PREPARATION ORDER

What should an IOQM student actually study?

IOQM rewards connected problem-solving, not memorising a short chapter list. Build the school-level foundation first, then develop depth in the four major Olympiad areas.

01

Number Theory

Divisibility, primes, remainders, congruences and integer equations.

02

Algebra

Equations, inequalities, polynomials, sequences and functional thinking.

03

Geometry

Triangles, circles, constructions, coordinates and proof-based relationships.

04

Combinatorics

Counting, arrangements, pigeonhole principle, recursion and elementary graphs.

START HERE

School mathematics is the foundation

The HBCSE syllabus page asks aspirants to be familiar with NCERT Mathematics topics from Classes VIII, IX and X. For younger students, readiness matters more than rushing into advanced tricks.

  • Arithmetic fluency, fractions, ratios and percentages
  • Integers, exponents, surds and basic inequalities
  • Linear and quadratic equations
  • Triangles, circles, mensuration and coordinate basics
  • Clear written reasoning and multi-step problem solving

TOPIC MAP

The concepts inside each major area

Use this as a preparation map. A student does not need every advanced theorem on day one.

01

Number Theory

Divisibility and the division algorithm; greatest common divisor and Euclidean algorithm; prime factorisation and the Fundamental Theorem of Arithmetic; congruences and linear congruences; Chinese Remainder Theorem; Fermat’s Little Theorem; Wilson’s Theorem; Euler phi function; Pythagorean triples and Diophantine equations.

Parent checkpoint: Can your child explain why a divisibility or remainder argument works?
02

Algebra

Inequalities; arithmetic, geometric and harmonic progressions; indices; systems of linear equations; theory of equations; binomial theorem and binomial coefficients; complex numbers; polynomials; functional equations and sequences.

Parent checkpoint: Look for flexible manipulation and pattern recognition—not only formula recall.
03

Plane Geometry

Triangles, quadrilaterals and circles; Euclidean constructions; concurrency and collinearity, including Ceva and Menelaus; trigonometric identities; sine and cosine rules; properties of polygons; coordinate geometry, vectors and related spatial ideas.

Parent checkpoint: Can your child draw a clean figure, identify relationships and justify each step?
04

Combinatorics

Basic enumeration; permutations and combinations in problem contexts; pigeonhole principle and its applications; recursion; invariants and elementary graph theory.

Parent checkpoint: Ask your child to describe what is being counted before choosing a formula.

PREPARATION ORDER

A practical path from foundation to exam readiness

  1. DiagnoseAttempt a mixed foundation worksheet or a previous IOQM paper without pressure.
  2. Build conceptsStudy one area at a time and solve progressively harder non-routine problems.
  3. Mix topicsPractise deciding which idea applies when the chapter name is not provided.
  4. SimulateUse timed papers and mock tests; review every error and maintain a revision list.

FOR PARENTS

Do not measure progress by “chapters completed” alone

Olympiad growth is visible when a student can stay with an unfamiliar problem, test examples, notice patterns, explain an idea and learn from an unsuccessful attempt.

A balanced week should include concept learning, untimed problem solving, review of written solutions and occasional timed practice.

COMMON QUESTIONS

What families usually ask

Is the IOQM syllabus simply “everything except Calculus”?

No. That description is too vague. HBCSE names four major areas and provides an indicative topic list, while also stating that the list is not exhaustive. Calculus is not named as a major area on that published list.

Can a Class 7 or 8 student begin IOQM preparation?

Yes, if the approach is foundation-first. Start with arithmetic, algebraic reasoning, elementary geometry and enjoyable problem solving before advanced Olympiad material.

Should my child finish the whole syllabus before solving papers?

No. Use selected past-paper problems alongside concept study. Full timed papers become more useful after the core foundation is reasonably secure.

Does completing every topic guarantee qualification?

No. IOQM tests application, choice of method, accuracy and composure. Preparation quality and thoughtful review matter more than checking off a list.

TRUSTED NEXT STEPS

Use official information and purposeful practice

Official HBCSE syllabusVerify the published scope ↗Official past papersPractise authentic problems ↗IOQM 2027 courseSee classes, practice and mentoring →
Ajeet Dubey, IOQM mentor and Mathematics Olympiad teacher

MENTOR PERSPECTIVE

“A syllabus is not a list to finish. It is a map for learning how to think.”

Ajeet Dubey · IOQM Mentor & Mathematics Olympiad Teacher, Mathiit

Meet your IOQM mentor →
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