IOQM case studies
IOQM · Put your knowledge to work in context.
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. The IOQM case studies are 5 cases and about 1,512 words.
1Divisor Count and the Missing Exponent Pattern
In a number-theory mock, the problem asks for the least positive integer N with exactly 12 positive divisors, divisible by 6, and not divisible by 9. To satisfy divisibility by 6 but not 9, they put exponent 1 on 3, exponent 2 on 2, and exponent 1 on 5, obtaining N=2^2·3·5=60.
360 words, about 2 min2AM-GM Equality Confusion with a Fixed Product
In an algebra worksheet, positive real numbers x,y,z satisfy x+y+z=6 and xyz=4. A candidate writes that AM-GM gives xyz ≤ ((x+y+z)/3)^3 = 8, with equality only at x=y=z=2.
339 words, about 2 min3The Ambiguous SSA Triangle in an Angle-Chase
In a geometry mock, triangle ABC has AB=10, AC=7, and ∠ABC=30°. A candidate applies the sine rule as 7/sin30°=10/sin C, obtains sin C=5/7, and takes C≈45.6°.
270 words, about 1 min4Exactly One of Two Grid Points
A combinatorics mock asks for the number of paths from (0,0) to (5,5) using only unit right and up steps that pass through exactly one of the points P=(2,2) and Q=(3,3). A candidate computes paths through P as C(4,2)C(6,3)=120 and paths through Q as C(6,3)C(4,2)=120.
250 words, about 1 min5Non-Coprime Congruences and the Wrong Modulus
In a number-theory warm-up, students solve the system x≡5 (mod 6), x≡8 (mod 9). A candidate notices gcd(6,9)=3, and since 5≡8≡2 (mod 3), they decide the system is consistent.
293 words, about 1 min