The syllabus for Mathematical Olympiad (regional, national and international) is pre-degree college mathematics. The difficulty level increases from RMO to INMO to IMO. A broad syllabus for RMO and INMO is:
Algebra : Basic set theory, principle of mathematical induction, inequalities(AMGM and Cauchy-Schwarz) , theory of equations(remainder theorem, relation between roots and coefficients, symmetric expressions in roots, statement of Fundamental theorem of algebra and its applications), functional equations;
Geometry: Similarity, congruence, concurrence, collinearity, parallelism and orthogonality, tangency, concyclicity, theorems of Appollonius, Ceva, Menelaus and Ptolemy, special points of a triangle such as circum-centre, in-centre, excentres, ortho-centre and centroid;
Combinatorics : Basic counting numbers such as factorial, number of permutations and combinations, cardinality of a power set, problems based on induction and bijection techniques, existence problems, pigeonhole principle;
Number theory: Divisibility, gcd and lcm, primes, fundamental theorem of arithmetic(canonical factorisation), congruences, Fermat's little theorem, Wilson's theorem, integer and fractional parts of a real number, Pythagorean triplets, polynomials with integer coefficients.
