Unit 1: Arithmetic and Number Sense
Computation and estimation
Clever simplification and approximation
Fractions, decimals, and percents

Exponents and radicals
Laws of exponents and nested radicals
Rationalizing denominators

Word problems
Rates, mixtures, averages, and weighted averages
Divisibility and factors
Prime factorization
Number of divisors and sum of divisors

GCD and LCM
Euclidean algorithm

Modular arithmetic
Congruences and residues
Fermat’s Little Theorem and Euler’s Theorem (occasional)
Chinese Remainder Theorem
Units digits and cycles

Number bases

Diophantine equations
Linear Diophantine equations
Pythagorean triples
Digit problems

Floor and ceiling functions
Linear and absolute value
Systems of equations
Absolute value equations and inequalities

Quadratics
Completing the square, discriminant
Vieta’s formulas

Polynomials
Remainder and Factor Theorems
Rational Root Theorem
Vieta’s formulas for higher degrees
Fundamental Theorem of Algebra
Polynomial identities and root transformations

Functions
Composition and inverses
Transformations and symmetry
Functional equations
Even and odd functions, periodicity

Exponentials and logarithms
Logarithm properties and change of base
Exponential and logarithmic equations
Logarithmic inequalities

Sequences and series
Arithmetic and geometric sequences and series
Telescoping sums
Recursive sequences and closed formsInfinite series (geometric)

Rational functions and inequalities
Partial fractions (occasional)
Asymptotes and graphing
Basics
Operations, conjugates, and modulus
Powers of i

Polar form
Euler’s formula and De Moivre’s Theorem
Roots of unity

Geometry in the complex plane
Rotations, dilations, and distances
Loci of complex numbers
Factoring tricks
Simon’s Favorite Factoring Trick
Sum and difference of cubes

Inequalities
AM-GM and Cauchy-Schwarz (occasional)
Trivial inequality
Optimization with inequalities

Symmetric expressions
Manipulating sums and products of roots
Fundamentals
Unit circle and exact values
Radian measure
Graphs, amplitude, period, phase shift

Identities
Pythagorean, angle sum and difference
Double-angle and half-angle
Product-to-sum and sum-to-product

Equations and inverses
Solving trigonometric equations
Inverse trig functions

Triangle trigonometry
Law of Sines and Law of Cosines
Area as (1/2)ab sin C
Extended Law of Sines (circumradius)

Applications
Trig substitution
Roots of unity and polygons
Triangles
Pythagorean Theorem and similar/congruent triangles
Special right triangles
Heron’s formula
Angle Bisector Theorem, Stewart’s Theorem
Ceva’s and Menelaus’ Theorems (occasional)

Triangle centers
Centroid, incenter, circumcenter, orthocenter
Inradius and circumradius formulas
Euler line (occasional)

Circles
Inscribed and central angles
Power of a Point
Tangents and common tangent lengths
Cyclic quadrilaterals and Ptolemy’s Theorem

Polygons
Regular polygons
Areas and angle relationships

Strategies
Area ratios, mass points, and auxiliary lines
Transformations: reflection, rotation, spiral similarity (occasional)
Lines and distances
Slope, distance, midpoint
Distance from a point to a line

Conic sections
Circles, parabolas, ellipses, hyperbolas
Foci, directrix, and eccentricity

Areas and transformations
Shoelace formula
Reflections, rotations, and dilations

Vectors and matrices (occasional)
Dot product and basic vector geometry
Determinants for area

Coordinate bashing as a solution strategy
Surface area and volume
Prisms, pyramids, cones, spheres
Frustums and tetrahedra

3D reasoning
Cross sections and slicing
Space diagonals and distances
Inscribed and circumscribed spheres
Regular polyhedra
Counting principles
Casework and complementary counting
Overcounting and symmetry

Permutations and combinations
Arrangements with repetition
Circular permutations and derangements
Stars and bars
Binomial Theorem and combinatorial identities

Inclusion-exclusion (up to three or four sets)

Probability
Conditional probability and independence
Geometric probability
Expected value and linearity of expectation
Recursive probability and states

Advanced techniques
Pigeonhole Principle
Generating functions (rare)
Bijections and invariants
Statistics
Mean, median, mode, and variance
Effect of transforming data

Logic and games
Truth-teller and liar problems
Invariants and winning strategies
Draw and label a clear diagram
Try small cases and look for patterns
Use symmetry and clever substitution
Test answer choices to eliminate options
Use complex numbers or trig for geometry
Set up coordinates when synthetic methods stall
Use complementary counting
Check edge cases and extraneous solutions