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 denoted by a letter. 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
• 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's Sums
• Trigonometry
  • DeMoivre's Theorem
  • Roots of unity

Olympiad Topics

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