IOQM Syllabus
Syllabus for IOQM is everything except Calculus but there are certain concept which every students should know while preparing Mathematics Olympiad.
Please refer following chapter and concept inside it
- Number System
- Basic Inequality
- Log Concept
- Modulus Concept
- Greatest Integer
- Prime Numbers:
- Prime factorization
- Prime counting functions
- Sieve methods (e.g., Eratosthenes’ sieve)
- Properties of prime numbers
- Divisibility:
- Divisibility rules
- Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
- Euclidean algorithm
- Modular Arithmetic:
- Congruences and modular arithmetic
- Residues and non-residues
- Chinese Remainder Theorem
- Diophantine Equations:
- Linear Diophantine equations
- Pell’s equation
- Fermat’s Last Theorem
- Number Bases:
- Binary, octal, hexadecimal, and other bases
- Base conversion
- Arithmetic Functions:
- Euler’s totient function (φ)
- Mobius function (μ)
- Number of divisors function (σ)
- Sum of divisors function (σ)
- Fermat’s Little Theorem and Euler’s Totient Theorem
Basic Algebraic Manipulations:
- Simplification of algebraic expressions
- Factorization of polynomials
- Solving algebraic equations
Inequalities:
- Arithmetic Mean-Geometric Mean (AM-GM) inequality
- Cauchy-Schwarz inequality
- Rearrangement inequality
Polynomials:
- Fundamental theorem of algebra
- Vieta’s formulas
- Newton’s identities
- Eisenstein’s criterion
Complex Numbers:
- Operations with complex numbers
- De Moivre’s Theorem
- Roots of unity
Sequences and Series:
- Arithmetic progressions
- Geometric progressions
- Convergent and divergent series
- Infinite series summation (e.g., geometric series)
Inequalities:
- Arithmetic Mean-Geometric Mean (AM-GM) inequality
- Cauchy-Schwarz inequality
- Jensen’s inequality
Functional Equations:
- Cauchy’s functional equation
- Jensen’s functional equation
- Other functional equations
Binomial Theorem and Combinatorics:
- Binomial coefficients
- Multinomial coefficients
- Combinatorial identities
Polynomial Equations:
- Roots and coefficients of polynomial equations
- Factor theorem
- Rational root theorem
Inequalities:
- Triangle inequalities
- Holder’s inequality
- Muirhead’s inequality
Counting Principles:
- Multiplication principle
- Addition principle
- Inclusion-Exclusion principle
Permutations and Combinations:
- Arrangements (permutations)
- Selections (combinations)
- Combinatorial identities
Pigeonhole Principle:
- Dirichlet’s principle
- Application in solving problems
Recurrence Relations:
- Linear recurrence relations
- Homogeneous and non-homogeneous recurrences
- Solving recurrence relations
Principle of Inclusion and Exclusion:
- Solving problems with PIE
- Counting problems with constraints
Graph Theory:
- Basics of graph theory
- Graph coloring
- Trees and spanning trees
- Connectivity and Eulerian graphs
- Hamiltonian cycles and paths
Combinatorial Geometry:
- Geometric counting problems
- Theorems like the Sylvester-Gallai theorem
Generating Functions:
- Generating functions for combinatorial sequences
- Operations on generating functions
Combinatorial Identities:
- Vandermonde’s identity
- Hockey stick identity (Combinatorial sum)
- Catalan numbers and other combinatorial sequences
- Euclidean Geometry:
- Points, lines, and planes
- Angle measurement and properties
- Congruence and similarity of triangles
- Quadrilaterals (properties and theorems)
- Circles (tangents, secants, angles, and theorems)
- Polygons (properties and interior/exterior angles)
- Geometric Transformations:
- Reflection, rotation, translation, and dilation
- Isometries and similarities
- Symmetry and tessellations
- Coordinate Geometry:
- Distance formula
- Slope and equations of lines
- Midpoint formula
- Conic sections (parabola, ellipse, hyperbola)
- Trigonometry:
- Sine, cosine, tangent, and their properties
- Trigonometric identities and equations
- Applications in geometry