IOQM syllabus: what the olympiad qualifier tests and how to prepare

Where IOQM sits in the olympiad ladder

India picks its team for the International Mathematical Olympiad through a series of rounds run by the Homi Bhabha Centre for Science Education (HBCSE), part of TIFR. The Mathematics Teachers' Association (India) conducts the first round across the country.

1. IOQM: open to school students, held at centres nationwide, usually in early September.
2. RMO (Regional Mathematical Olympiad): for students selected on IOQM scores; a proof based paper held in November in the latest cycle.
3. INMO (Indian National Mathematical Olympiad): a tougher proof based paper, held in January in the latest cycle.
4. IMO Training Camp: for INMO awardees, from which the final team is chosen.

The selection rules from IOQM to RMO, including how many students move up from each region and class, are published every year on the HBCSE olympiad site.

Paper pattern and marking

Each answer is a two digit integer filled on an OMR sheet, so there are no options to guess from and no partial credit. Errors carry no penalty, which means every question should get some answer before time ends. Half the marks sit in the last ten problems, but those are also the hardest; a solid score usually comes from clearing most of the first twenty and picking off a few five markers.

IOQM syllabus by area: algebra and number theory

Algebra

• Manipulating expressions, factorisation of polynomials, identities
• Quadratic equations and expressions: roots, discriminant, relations between roots and coefficients
• Systems of linear equations
• Inequalities: AM GM, Cauchy Schwarz and simple rearrangement arguments
• Finite series: arithmetic and geometric progressions, telescoping sums
• Complex numbers at a basic level, and simple functional equations

Number theory

• Divisibility, primes, GCD and LCM, the Euclidean algorithm
• Modular arithmetic and congruences; Fermat's little theorem and Euler's theorem
• Number of divisors and sum of divisors; Euler's totient function
• Linear Diophantine equations and equations solved by factoring
• Digits, bases and last digit problems

Geometry and combinatorics topics

Geometry

• Triangles: congruence, similarity, angle bisector theorem, centres of a triangle
• Circles: chords, tangents, cyclic quadrilaterals, power of a point
• Areas and lengths, including trigonometric area formulas and the sine and cosine rules
• Coordinate geometry of lines and circles
• Polygons and basic solid figures

Combinatorics and probability

• Rules of sum and product, arrangements and selections
• Inclusion and exclusion, the pigeonhole principle
• Binomial coefficients and identities
• Recurrences for counting problems
• Elementary probability and elementary graph theory

HBCSE's own description lists integer arithmetic, geometry, quadratics, trigonometry, coordinate geometry, linear systems, permutations and combinations, polynomial factorisation, inequalities, elementary combinatorics, probability, number theory, finite series, complex numbers and elementary graph theory. The finer sub topics above are a guide to what those headings usually mean in olympiad problems.

Five practice problems like the early IOQM questions

Below are five original warm ups pitched at the level of questions 1 to 10. Every answer is a whole number under 100.

1. Find the remainder when 7 raised to the power 100 is divided by 13.
2. How many positive integers less than 100 have exactly three positive divisors?
3. In triangle ABC, AB = 13, BC = 14 and CA = 15. Find the area of the triangle divided by 4.
4. If x + 1/x = 5, find the value of x squared + 1/x squared.
5. In how many ways can 6 identical sweets be shared among 3 children so that each child gets at least one?

Answers: 9, 4, 21, 23 and 10. The first needs Fermat's little theorem, the second the insight that only squares of primes work, the third Heron's formula, the fourth an identity, and the fifth stars and bars.

A preparation path from school maths to olympiad level

Board level maths is the floor, not the ceiling. Students who do well usually spend months on problem solving rather than on new chapters.

• Foundation (2 to 3 months): make sure NCERT topics on number systems, polynomials, quadratic equations, triangles, circles, progressions, and permutations and combinations are fully secure, including the class 11 chapters on binomial theorem and complex numbers.
• Olympiad toolkit (3 to 4 months): learn the standard results in each area one at a time: modular arithmetic, AM GM, power of a point, pigeonhole, inclusion and exclusion.
• Past papers (last 2 months): solve old IOQM and PRMO papers under a 3 hour clock and note which area costs you the most marks.

IOQM preparation also strengthens the reasoning needed for the JEE Advanced syllabus later, though olympiad problems rely far less on calculus.

Who can take IOQM and where to register

Students from class 8 to class 12 can register. Each year the eligibility notice sets a birth date limit and other conditions, such as rules for students who have finished class 12, so read it before registering. Registration happens through the official IOQM portal linked from the HBCSE olympiad site, and centres are listed there region by region.

Past papers and solutions are also shared through the official olympiad pages. For other entrance and competitive exams for school students, browse the entrance exams section.