README.md

List of mathematics topics in competitive programming, organized from basic to advanced:

Basic Mathematics Concepts:

1. Arithmetic
   • Basic Operations (Addition, Subtraction, Multiplication, Division)
   • Order of Operations (PEMDAS/BODMAS)
2. Number Properties
   • Even and Odd Numbers
   • Prime Numbers and Composite Numbers
   • Divisibility Rules
3. Fractions and Decimals
   • Conversion between Fractions and Decimals
   • Simplification of Fractions
4. Percentages
   • Calculation of Percentages
   • Percentage Increase/Decrease
5. Ratios and Proportions
   • Basic Ratios
   • Proportional Relationships
6. Exponents and Radicals
   • Laws of Exponents
   • Simplifying Radicals
7. Basic Algebra
   • Solving Linear Equations
   • Solving Inequalities
   • Quadratic Equations (Factoring, Quadratic Formula)
8. Basic Geometry
   • Angles (Types and Properties)
   • Triangles (Types and Properties)
   • Circles (Radius, Diameter, Circumference)
   • Perimeter and Area of Basic Shapes

Intermediate Mathematics Topics:

9. Advanced Algebra
   • Polynomials
   • Systems of Equations
   • Sequences and Series (Arithmetic and Geometric)
10. Advanced Geometry
    • Coordinate Geometry (Distance, Midpoint, Slope)
    • Trigonometry (Sine, Cosine, Tangent)
    • Properties of Special Triangles (Equilateral, Isosceles, Right)
    • Properties of Circles (Chord, Tangent, Sector)
11. Matrices and Determinants
    • Matrix Operations (Addition, Multiplication)
    • Determinants and Inverses
    • Systems of Equations using Matrices
12. Vectors
    • Vector Operations (Addition, Subtraction, Scalar Multiplication)
    • Dot Product
    • Cross Product
13. Probability and Statistics
    • Basic Probability (Independent and Dependent Events)
    • Combinations and Permutations
    • Mean, Median, Mode, and Range
    • Standard Deviation and Variance
14. Discrete Mathematics
    • Set Theory (Union, Intersection, Difference)
    • Logic and Propositional Calculus
    • Graph Theory (Basic Concepts, Eulerian and Hamiltonian Paths)
    • Counting Principles (Pigeonhole Principle, Inclusion-Exclusion)
15. Calculus
    • Limits and Continuity
    • Derivatives (Basic Rules, Chain Rule, Product Rule)
    • Integrals (Definite and Indefinite)
    • Applications of Derivatives and Integrals (Optimization, Area under Curves)

Advanced Mathematics Concepts:

16. Advanced Probability and Statistics
    • Bayes’ Theorem
    • Probability Distributions (Binomial, Poisson, Normal)
    • Hypothesis Testing and Confidence Intervals
17. Advanced Geometry
    • Conic Sections (Parabolas, Ellipses, Hyperbolas)
    • Transformations (Translation, Rotation, Scaling, Reflection)
    • Geometric Proofs and Theorems
18. Advanced Calculus
    • Multivariable Calculus (Partial Derivatives, Multiple Integrals)
    • Differential Equations (First Order, Second Order)
    • Series and Sequences (Convergence Tests, Power Series)
19. Linear Algebra
    • Eigenvalues and Eigenvectors
    • Diagonalization of Matrices
    • Vector Spaces and Subspaces
    • Linear Transformations
20. Complex Numbers
    • Arithmetic of Complex Numbers
    • Polar Form and De Moivre’s Theorem
    • Roots of Complex Numbers
21. Number Theory
    • Modular Arithmetic
    • Prime Factorization and Divisors
    • Diophantine Equations
    • Fermat’s Little Theorem, Euler’s Theorem
22. Combinatorics
    • Permutations and Combinations
    • Generating Functions
    • Recurrence Relations
    • Graph Coloring and Planarity
23. Game Theory
    • Zero-Sum Games
    • Nash Equilibrium
    • Combinatorial Games (Nim, Grundy Numbers)
24. Optimization
    • Linear Programming
    • Integer Programming
    • Convex Optimization
25. Mathematical Logic and Proofs
    • Direct and Indirect Proofs
    • Proof by Contradiction
    • Mathematical Induction
    • Propositional and Predicate Logic

Specialized and Hybrid Techniques:

26. Fourier Analysis
    • Fourier Series
    • Fourier Transform
    • Applications in Signal Processing
27. Numerical Methods
    • Root Finding (Bisection Method, Newton-Raphson Method)
    • Numerical Integration (Trapezoidal Rule, Simpson’s Rule)
    • Numerical Solutions to Differential Equations (Euler’s Method)
28. Algorithmic Mathematics
    • Complexity Analysis (Big O Notation)
    • Algorithm Design Techniques (Greedy, Divide and Conquer, Dynamic Programming)
    • Data Structures (Arrays, Trees, Graphs, Hash Tables)
29. Cryptography
    • Basic Encryption and Decryption Algorithms
    • RSA Algorithm
    • Elliptic Curve Cryptography
30. Mathematical Modeling
    • Formulating Real-World Problems
    • Solving Models using Mathematical Techniques
    • Interpreting and Validating Models