This advanced mathematical olympiad exam is designed for serious mathematics competitors. It features 20 carefully selected problems covering the core areas of mathematical olympiads: Number Theory (divisibility, modular arithmetic, prime numbers), Combinatorics (counting principles, generating functions), Algebra (polynomials, functional equations, systems), Geometry (classical theorems, coordinate geometry), and Inequalities (AM-GM, Cauchy-Schwarz, Jensen). Each problem requires deep mathematical insight and sophisticated problem-solving techniques.