IOQM study guide
IOQM · Work through your exam, one topic at a time.
IOQM, the Indian Olympiad Qualifier in Mathematics, is the first stage of India’s Mathematical Olympiad programme. It is conducted by the Mathematics Teachers’ Association (India) and tests mathematical problem solving; it is the qualifying stage for RMO. The exam is intended for students in classes 8–12 who meet the published eligibility conditions for the relevant year. For 2026, the class range is 8–12 and the date-of-birth window is 1 August 2007 to 31 July 2014; citizenship, residence and education conditions also apply. Topics on this exam: Number Theory, Algebra, Geometry, and Combinatorics. The IOQM study guide has 4 sections and about 1,056 words.
01Number Theory - IOQM Study Guide
Number Theory on IOQM tests exact integer reasoning: divisibility, residues, primes, gcd and lcm, and Diophantine constraints. The paper asks for a single integer answer in the range 00–99, so a residue or a small count is usually enough; a formal proof is not required, but every shortcut must be justified. Write divisibility through prime factorisation. If a c and b c with gcd(a,b)=1, then ab c.
291 words, about 1 min02Algebra - IOQM Study Guide
Algebra on IOQM tests structural manipulation rather than long computation: polynomials, systems, inequalities, sequences, and functional relations. The exam expects a compact identity, substitution, or inequality that turns the problem into one or two lines. For symmetric variables set p=a+b+c, q=ab+bc+ca, r=abc. Then a²+b²+c²=p²−2q and a³+b³+c³=p³−3pq+3r.
208 words, about 1 min03Geometry - IOQM Study Guide
Geometry on IOQM is synthetic and ratio-driven: angle chasing, cyclic quadrilaterals, similar triangles, circle theorems, and area comparisons. A well-chosen auxiliary line or a correct ratio usually beats coordinate computation. Angles in the same segment are equal; a quadrilateral is cyclic iff opposite angles are supplementary. Similar triangles: AA is enough; areas scale as the square of the linear scale factor.
301 words, about 2 min04Combinatorics - IOQM Study Guide
Combinatorics on IOQM tests counting with structure: arrangements, selections, inclusion–exclusion, pigeonhole, bijections, and simple recurrences. The answer is an exact count, and because IOQM answers are two-digit integers 00–99, a correct count must be converted carefully. Decide first whether order matters and whether repetition is allowed. Stars and bars: positive solutions of x1+...+xk=n number C(n−1,k−1); nonnegative solutions number C(n+k−1,k−1).
256 words, about 1 min