Syllabus for Maths Olympiad

One can never give complete syllabus as such, but here is a broad idea of syllabus. 

The syllabus for Mathematical Olympiad (regional, national and international) consists of pre-degree college mathematics. The difficulty level increases from RMO to INMO to IMO. 

Broadly the syllabus for RMO and INMO is:

 Algebra (basic set theory, principle of Mathematical Induction,inequalities (AM-GM and Cauchy-Schwarz), Theory of equations (remainder theorem, relation between roots and coefficients, symmetric expressions in roots, applications of the 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 circumcentre, in-centre, ex-centres, 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 PHP) ;  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).