Learning Path

Elementary Manipulation

Factoring, expanding equations and inequalities

Σ and Π Notation

Summation and product notation for sequences and series

Functions: Injectivity, Surjectivity

One-to-one, onto, and bijective function properties

Cauchy's Functional Equation

f(x+y) = f(x) + f(y) and its variations

Polynomials in One Variable

Vieta's relations, fundamental theorem of algebra

Polynomials in Multiple Variables

Symmetric polynomials, homogeneous polynomials

Complex Numbers & Trigonometry

Euler's formula, roots of unity, De Moivre's theorem

AM-GM and Cauchy-Schwarz

Fundamental inequalities for algebraic manipulation

Hölder and Jensen Inequality

Generalized inequalities for convex functions

Generating Functions

Power series for counting and recurrence solving

Calculus

Derivatives, integrals, optimization techniques

Basic Counting

2ⁿ, n!, binomial coefficients, permutations

Mathematical Induction

Strong and weak induction, well-ordering principle

Recursion & Recurrence Relations

Solving linear recurrences, characteristic equations

Pigeonhole Principle

If n+1 pigeons in n holes, one hole has ≥2 pigeons

Sets and Functions

Cardinality, injection/surjection counting arguments

Elementary Probability

Basic probability, conditional probability, independence

Expected Value & Linearity

E[X+Y] = E[X] + E[Y] even for dependent variables

Basic Graph Theory

Connectedness, degree, trees, Euler/Hamiltonian paths

Convex Hull

Smallest convex set containing given points

Advanced Graph Theory

Hall's lemma, Turán's theorem, matching theory

Triangle Centers

Incenter, centroid, orthocenter, circumcenter, excenters

Nine-Point Circle

Circle through midpoints, feet of altitudes, and more

Angle Chasing & Cyclic Quadrilaterals

Inscribed angles, tangent-chord angles, cyclic properties

Similar Triangles

AA, SAS, SSS similarity and ratio properties

Power of a Point

Radical axis, radical center, and circle intersections

Homothety

Scaling transformation centered at a point

Ceva and Menelaus Theorems

Concurrency and collinearity conditions for triangles

Trigonometry in Geometry

Law of sines, law of cosines, area formulas

Coordinate Systems

Cartesian, complex numbers, barycentric coordinates

Inversive Geometry

Circle inversion, preserving angles and circles/lines

Projective Geometry

Cross ratios, harmonic bundles, poles/polars, Pascal

Spiral Similarity

Rotation combined with scaling, complex multiplication

Conic Sections

Ellipse, parabola, hyperbola properties and tangents

Primes & Fundamental Theorem of Arithmetic

Unique prime factorization of positive integers

Modular Arithmetic

Fermat's little theorem, CRT, modular inverses, orders

Prime Factorization Exponents

νₚ(xy) = νₚ(x) + νₚ(y), p-adic valuation

Euclidean Algorithm

gcd(a,b) = gcd(a−b, b), extended Euclidean algorithm

Dirichlet's Theorem

Infinitely many primes in arithmetic progressions

Fermat's Two-Square Theorem

Odd prime p = x² + y² iff p ≡ 1 (mod 4)

Quadratic Residues & Reciprocity

Legendre symbol, quadratic reciprocity law

Prime Number Theorem

π(n) ~ n/ln(n), density of primes

AM-GM Inequality

Arithmetic mean ≥ geometric mean for positive reals

Cauchy-Schwarz Inequality

(Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²), Engel form

Rearrangement Inequality

Similarly sorted sequences maximize dot product

Triangle Inequality

|a + b| ≤ |a| + |b|, metric space properties

Weighted AM-GM

Weighted averages with positive weight sums

Jensen's Inequality

f(E[X]) ≤ E[f(X)] for convex functions

Hölder's Inequality

Generalization of Cauchy-Schwarz with p and q norms

Schur's Inequality

Symmetric inequality for three variables

SOS (Sum of Squares) Method

Writing expressions as sums of non-negative squares