Algebra

In mathematics, algebra is the study of examining, manipulating, and solving equations, inequalities, and other mathematical expressions. Algebra revolves around the concept of the variable, an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with algebraic manipulation.

Introductory Topics

• Algebraic expressions
  • Algebraic manipulation
  • Substitution
• Complex numbers
• Equations
  • Linear equations (1 and 2 variables)
  • Systems of equations
• Factoring
• Functions
  • Exponentials
  • Graphing
    • Lines
    • Quadratics
    • Basic Transformations
  • Logarithms
  • Square root function
  • Polynomials
    • Quadratics
• Inequalities
  • Trivial inequality
  • Arithmetic Mean-Geometric Mean for n=2
• Word problems

Intermediate Topics

• Equations
  • Systems of equations
• Factoring
  • Simon's Favorite Factoring Trick
• Functions
  • Exponentials
  • Greatest integer function
  • Graphing
  • Least integer function
  • Logarithms
  • Polynomials
  • Functional Equations
• Inequalities
  • Trivial inequality
  • Cauchy-Schwarz Inequality
  • Arithmetic Mean-Geometric Mean Inequality
  • Triangle Inequality
• Newton sums
• Trigonometry
  • DeMoivre's Theorem
  • Roots of unity

Olympiad Topics

• Inequalities
  • Arithmetic Mean-Geometric Mean Inequality
  • Cauchy-Schwarz Inequality
  • Chebyshev's Inequality
  • Geometric inequalities
  • Hölder's inequality
  • Isoperimetric inequalities
  • Jensen's Inequality
  • Minkowski Inequality
  • Muirhead's Inequality
  • Power mean inequality
  • Rearrangement Inequality
  • Schur's Inequality
  • Smoothing
  • Triangle Inequality
  • Trigonometric inequalities
  • Trivial inequality

More Advanced Topics in Algebra

• Group theory
• Ring theory
• Field theory
• Galois theory
• Homological algebra
• Lie theory
• Algebraic geometry