The IOQM syllabus is comprehensive and covers advanced topics from five major areas of mathematics, generally aligning with Class 8 to 12 level, but with an Olympiad-level rigor:
1. Basic Mathematics:
* Number Systems (integers, rational, real, complex numbers)
* Basic Inequality Concepts (AM-GM, Cauchy-Schwarz, Rearrangement)
* Modulus & Greatest Integer Functions
* Logarithmic Functions
2. Number Theory:
* Prime Numbers, Prime Factorization
* Divisibility Rules, GCD, LCM, Euclidean Algorithm
* Modular Arithmetic (Congruences, Chinese Remainder Theorem)
* Diophantine Equations (Linear & Pell's)
* Number Bases (Binary, Octal, Hexadecimal)
* Arithmetic Functions (Euler's Totient \phi, Möbius \mu, \sigma Functions)
* Fermat's Little Theorem and Euler's Theorem
3. Algebra:
* Simplification & Factorization of Algebraic Expressions
* Inequalities (AM-GM, Cauchy-Schwarz, Rearrangement)
* Polynomials: Vieta's Formulas, Newton's Identities, Roots, Fundamental Theorem of Algebra, Factor Theorem, Rational Root Theorem
* Complex Numbers: Operations, Roots of Unity, De Moivre's Theorem
* Sequences & Series (AP, GP, Convergence, Infinite Series)
* Functional Equations (Cauchy, Jensen, etc.)
* Binomial Theorem and Combinatorics
* Quadratic Equations and Expressions
* System of Linear Equations
4. Combinatorics:
* Counting Principles (Multiplication, Addition, Inclusion-Exclusion)
* Permutations and Combinations
* Pigeonhole Principle
* Recurrence Relations (Homogeneous, Non-Homogeneous)
* Principle of Inclusion and Exclusion (PIE)
* Graph Theory: Basics, Trees, Eulerian & Hamiltonian Paths, Graph coloring, Connectivity
* Combinatorial Geometry & Theorems
* Generating Functions
* Combinatorial Identities (Catalan Numbers, Hockey Stick, Vandermonde's)
5. Geometry:
* Euclidean Geometry: Points, lines, planes, angles, triangles (congruence, similarity), quadrilaterals, circles (tangents, secants, angles, theorems), polygons
* Geometric Transformations: Reflection, rotation, translation, dilation, isometries, similarities
* Coordinate Geometry: Distance formula, slope and equations of lines, circles, conic sections
* Trigonometry: Sine, cosine, tangent, and their properties; trigonometric identities and equations; applications in geometry
* Locus problems
* Symmetry and tessellations
The IOQM exam aims to test problem-solving skills, mathematical reasoning, and creativity. While the syllabus broadly aligns with the NCERT curriculum for Classes 8-12, the problems are often more challenging and require a deeper understanding and application of concepts.
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