Foundations (AMC 8)

Build your core toolbox first: algebra, geometry, counting, and number theory fundamentals that show up across AMC and early AIME problems.
Focus on understanding patterns and writing clean solutions before moving on.

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Number Theory Basics

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Fluency with arithmetic and foundational number theory.

Arithmetic and Number Theory Basics

Divisibility, primes, gcd, and modular arithmetic intuition for AMC 8 and AMC 10.
Updated: 5 days ago

Linear Diophantine Equations

Solving equations of the form ax + by = c using the Euclidean algorithm and Bezout's Identity.
Updated: 5 days ago

Chicken McNugget Theorem

Determine the largest value that cannot be represented as a nonnegative combination of two coprime integers.
Updated: 5 days ago

Algebra Basics

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Linear equations, factoring patterns, and core manipulation skills.

Algebra Basics

Linear equations, factoring, and manipulating expressions with confidence.
Updated: 5 days ago

Exponent Rules

Laws of exponents for simplifying and solving equations.
Updated: 5 days ago

Quadratic Identities

Difference of squares and binomial expansions.
Updated: 5 days ago

Quadratic Formula and Discriminant

Solving quadratics and classifying roots
Updated: 5 days ago

Systems of Equations

Substitution, elimination, and symmetric tactics.
Updated: 5 days ago

Substitution Techniques

Replace repeated structure to lower degree and simplify.
Updated: 5 days ago

Simon's Favorite Factoring Trick

Adding constants to factors multi-variable equations
Updated: 5 days ago

Cubic Factorizations

Sum and difference of cubes plus binomial cubes.
Updated: 5 days ago

Algebra Tools

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Vieta's formulas and polynomial root transformations.

Vieta's Formulas

Relating roots to coefficients for quadratics and cubics.
Updated: 5 days ago

Polynomial Manipulations

Shifting and inverting roots to simplify expressions.
Updated: 5 days ago

Averages and Sequences

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Means, arithmetic and geometric sequences, and telescoping sums.

Mean, Median, Mode, and Harmonic Mean

Core averages and how to interpret them in contest problems.
Updated: 5 days ago

Arithmetic Sequences

Constant differences, term formulas, and sum tricks.
Updated: 5 days ago

Geometric Sequences

Constant ratios, finite sums, and convergence.
Updated: 5 days ago

Telescoping Sums and Products

Use cancellation after algebraic rewriting.
Updated: 5 days ago

Intro Inequalities

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AM-GM, Cauchy-Schwarz, and foundational bounding techniques.

AM-GM Inequality

A standard tool for bounding sums and products.
Updated: 5 days ago

Cauchy-Schwarz Inequality

A core inequality for bounds and maximization.
Updated: 5 days ago

Inequalities Foundations

A comprehensive guide to interval bounding, absolute value geometry, quadratic optimization, and the QM-AM-GM-HM means chain.
Updated: 5 days ago

Counting Fundamentals

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Permutations, combinations, inclusion-exclusion, and pigeonhole.

Counting Fundamentals

Rule of product, basic permutations and combinations, and careful casework.
Updated: 5 days ago

Pascal's Triangle & Binomial Theorem

Pascal's triangle structure and its symmetry, hockey stick identity, and basic binomial expansion.
Updated: 1 hour ago

Inclusion-Exclusion

Basic Two and three set inclusion exclusion, Venn diagram interpretation, and methods of solveing "at least one" problems.
Updated: 5 days ago

Pigeonhole Principle

The pigeonhole principle, and its applications to counting, number theory, and combinatorial geometry
Updated: 5 days ago

Stars and Bars

Distributing identical objects into distinct "boxes", positive integer solutions, and distributions that are constraint
Updated: 5 days ago

Geometry Basics

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Triangles, similarity, circles, and core area formulas.

Geometry Basics

Angle chasing, triangle facts, area, and circles.
Updated: 5 days ago

Triangle Angle Sum

Interior and exterior angle facts for triangles.
Updated: 5 days ago

Triangle Area Formulas

Base-height, Heron, and trigonometric area.
Updated: 5 days ago

Special Triangles

Equilateral and isosceles triangle properties.
Updated: 5 days ago

Right Triangles and Pythagorean Triples

Pythagorean theorem and common triples.
Updated: 5 days ago

Special Right Triangles

30-60-90 and 45-45-90 ratios.
Updated: 5 days ago

Medians and the Centroid

Median properties and centroid ratios.
Updated: 5 days ago

Angle Bisectors

The angle bisector theorem and length facts.
Updated: 5 days ago

Altitudes and the Orthocenter

Altitude facts and area relationships.
Updated: 5 days ago

Triangle Similarity

Basic Proportionality (Thales)

Parallel lines divide sides proportionally.
Updated: 5 days ago

Law of Sines

Relate sides and angles using sine.
Updated: 5 days ago

Law of Cosines

Generalized Pythagorean theorem for any triangle.
Updated: 5 days ago

Inscribed and Central Angles

Inscribed angle theorem and Thales' theorem.
Updated: 5 days ago

Special Quadrilaterals

Parallelograms, rectangles, rhombi, and trapezoids.
Updated: 5 days ago

Coordinate Geometry Basics

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Distance, midpoint, slope, and line equations.

Coordinate Geometry Basics

Distance, midpoint, and slope.
Updated: 5 days ago

Line Equations

Point-slope, slope-intercept, and perpendicular slopes.
Updated: 5 days ago

Word Problems

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Translating stories into equations using rates, mixtures, and ratios.

Word Problems

Translate stories into equations, use ratios, rates, and mixtures.
Updated: 5 days ago

Intro to Probability

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Sample spaces, complements, conditional probability, and expected value.

Introduction to Probability

Basic probability, complements, and simple counting-based models.
Updated: 5 days ago

Uncategorized

Auto-discovered modules not yet placed in a syllabus chapter.

Triangle Congruence & Similarity

The foundations of geometric deduction, covering SSS/SAS/AA criteria, the right triangle altitude theorem, and area scaling.
Updated: 5 days ago