Competitive Mathematics: Algebra

Basic Concepts:
- Variables, constants, coefficients.
- Operations: addition, subtraction, multiplication, division.
Equations:
- Linear equations: (ax + b = c).
- Quadratic equations: (ax^2 + bx + c = 0), solved by factoring, completing the square, or using the quadratic formula.
Inequalities:
- Solving inequalities and understanding solution sets.
- Graphical representation on number lines.
Functions:
- Definition and notation.
- Types: linear, quadratic, polynomial, exponential, logarithmic functions.
Polynomials:
- Operations: addition, multiplication, division, factoring.
- Remainder and factor theorem.
Systems of Equations:
- Solving using substitution and elimination methods.
- Matrix representation and row reduction for larger systems.
Sequences and Series:
- Arithmetic and geometric sequences.
- Sum formulas and applications.
Complex Numbers:
- Definitions: (a + bi), where (i^2 = -1).
- Operations and polar form representation.
Exam Strategies:
- Time management and problem selection.
- Practice with previous competition problems.

Divisibility:
- Rules of divisibility for numbers (2, 3, 5, 10, etc.).
- Prime numbers: definition and importance.
Greatest Common Divisor (GCD) and Least Common Multiple (LCM):
- Methods to calculate using prime factorization and Euclidean algorithm.
Modular Arithmetic:
- Definition: (a \equiv b \mod m).
- Applications in problem-solving, congruences, and residues.
Prime Factorization:
- Unique factorization theorem.
- Applications in simplifying fractions, solving equations.
Fermat's Little Theorem:
- Statement: If (p) is prime, then (a^{p-1} \equiv 1 \mod p) for any integer (a).
Euler's Totient Function:
- Definition and computation of (\phi(n)).
- Applications in cryptography.
Diophantine Equations:
- Definition of equations that seek integer solutions.
- Techniques for solving linear Diophantine equations: existence and methodology.
Exam Strategies:
- Familiarity with classic problems and theorems.
- Focus on problem simplification and modular techniques.

Variables, Constants and Coefficients are fundamental building blocks for algebraic expressions
Linear Equations are solved by isolating the variable, using addition, subtraction and division
Quadratic Equations have the form (ax^2 + bx + c = 0), and can be solved through factoring, completing the square, or the quadratic formula
Inequalities are used to represent relationships where one side is greater than or less than the other, the solution set is displayed on a number line
Functions are relationships between input and output values, and are often expressed using equations
Polynomials are expressions consisting of variables and coefficients, operations include addition, multiplication, division, factoring, and using the remainder and factor theorem
Systems of Equations can be solved using substitution, elimination, matrix representation or row reduction
Sequences and Series are ordered lists of numbers, arithmetic sequences have a common difference, geometric sequences have a common ratio, and there are specific formulas to calculate their sums
Complex Numbers involve the imaginary unit (i), where (i^2 = -1) and can be represented in both rectangular and polar form
Exam Strategies for Algebra include effective time management, practicing previous competition problems, and focusing on the key concepts

Divisibility Rules help determine whether a number is divisible by another number, for instance, a number divisible by 2 is an even number
Prime Numbers are whole numbers greater than 1 that are only divisible by 1 and themselves, they play a crucial role in number theory
Greatest Common Divisor (GCD) and Least Common Multiple (LCM) are used to simplify fractions and find common divisors
Modular Arithmetic deals with remainders after division, denoted by (a \equiv b \mod m), and has applications in solving congruences
Prime Factorization is a unique representation of a number as a product of prime numbers, commonly used to simplify fractions and solve equations
Fermat's Little Theorem states that (a^{p-1} \equiv 1 \mod p) if (p) is prime and (a) is any integer, and has applications in cryptography
Euler's Totient Function denoted by (\phi(n)), calculates the number of positive integers less than (n) that are relatively prime to (n), and has applications in cryptography
Diophantine Equations are equations where only integer solutions are sought, linear Diophantine equations can be solved using specific techniques
Exam Strategies for Number Theory involve familiarity with classic problems and theorems, focusing on simplifying problems using modular techniques and understanding the connections between different topics.