Formula bank

Algebra65

• Arithmetic SeriesAMC 8$S_n = \frac{n(a_1 + a_n)}{2}$
• AM-GM InequalityAMC 10$\frac{a+b}{2} \ge \sqrt{ab}$
• Simon's Favorite Factoring TrickAMC 10$xy + ax + by + ab = (x+b)(y+a)$
• Vieta's Formulas (Quadratic)AMC 10$r_1 + r_2 = -\frac{b}{a}, \qquad r_1 r_2 = \frac{c}{a}$
• Geometric SeriesAMC 10$S_n = a\cdot\frac{1-r^n}{1-r}, \qquad S_\infty = \frac{a}{1-r}\ \ (|r|<1)$
• Sums of PowersAMC 10$\sum k = \tfrac{n(n+1)}{2},\quad \sum k^2 = \tfrac{n(n+1)(2n+1)}{6},\quad \sum k^3 = \left(\tfrac{n(n+1)}{2}\right)^2$
• Remainder TheoremAMC 10 Full version
• Factor TheoremAMC 10 Full version
• Trivial InequalityAMC 10 Full version
• Logarithm IdentitiesAMC 10 Full version
• Conjugate Root TheoremAMC 10 Full version
• Triangle Inequality for Absolute ValuesAMC 10 Full version
• Sum of Reciprocals of RootsAMC 10 Full version
• Telescoping FractionsAMC 10 Full version
• Rational Root TheoremAMC 10 Full version
• Difference of RootsAMC 10 Full version
• DiscriminantAMC 10 Full version
• Partial FractionsAMC 10 Full version
• Power Sums from a+b and abAMC 10 Full version
• Sum & Difference of CubesAMC 10 Full version
• Common FactorizationsAMC 10 Full version
• Fundamental Theorem of AlgebraAMC 12 Full version
• Location of RootsAMC 12 Full version
• Rearrangement InequalityAMC 12 Full version
• Floor Function PropertiesAMC 12 Full version
• Polynomials with Transformed RootsAMC 12 Full version
• Bernoulli's InequalityAMC 12 Full version
• Nesbitt's InequalityAMC 12 Full version
• Denesting RadicalsAMC 12 Full version
• Elementary Symmetric SumsAMC 12 Full version
• Weighted AM-GMAMC 12 Full version
• Change of BaseAMC 12 Full version
• Harmonic SequencesAMC 12 Full version
• Titu's LemmaAMC 12 Full version
• Cramer's RuleAMC 12 Full version
• Minkowski's InequalityAMC 12 Full version
• Complex NumbersAMC 12 Full version
• Vieta's Formulas (Any Degree)AMC 12 Full version
• Sum of Three Cubes IdentityAMC 12 Full version
• Sum and Difference of PowersAMC 12 Full version
• QM-AM-GM-HM ChainAMC 12 Full version
• Vieta's Formulas (Cubic)AMC 12 Full version
• Schur's InequalityAIME Full version
• Karamata's InequalityAIME Full version
• Cauchy's Functional EquationAIME Full version
• Linear RecurrencesAIME Full version
• Vieta JumpingAIME Full version
• Hermite's IdentityAIME Full version
• Sum of Squares of RootsAIME Full version
• Lagrange InterpolationAIME Full version
• The uvw MethodAIME Full version
• Cauchy-Schwarz InequalityAIME Full version
• Depressed CubicAIME Full version
• Lagrange's IdentityAIME Full version
• Chebyshev's Sum InequalityAIME Full version
• Hölder's InequalityAIME Full version
• Maclaurin's InequalityAIME Full version
• Jensen's InequalityAIME Full version
• Newton's InequalityAIME Full version
• Aczél's InequalityAIME Full version
• Brahmagupta–Fibonacci IdentityAIME Full version
• Power Mean InequalityAIME Full version
• Sophie Germain IdentityAIME Full version
• Muirhead's InequalityAIME Full version
• Binet's FormulaAIME Full version