Intermediate (AMC 10-12)

Strengthen problem-solving depth with multi-step AIME-style techniques, including functional equations, inequalities, advanced counting, and richer geometry.
This stage is about speed plus rigor: solve accurately under time pressure.

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Advanced Algebra

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Quadratics, systems, and polynomial identities for AMC 10/12.

Function Basics I

All about Polynomials and Ideas used in solving them.
Updated: 2 weeks ago

Function Basics II

All about Polynomials and Ideas used in solving them.
Updated: 2 weeks ago

Binomial Theorem I

Learn how the Binomial Theorem expands powers, explains binomial coefficients, and connects algebraic expressions with counting and probability.
Updated: 2 weeks ago

Binomial Theorem II

Learn how the Binomial Theorem expands powers, explains binomial coefficients, and connects algebraic expressions with counting and probability.
Updated: 2 weeks ago

Higher-Power Factorization and Symmetric Cubic Identities

Learn how to factor higher degree expressions and use cubic identities to simplify olympiad algebra problems.
Updated: 2 weeks ago

Sophie Germain Identity

Learn one of the most useful quartic factorisations in olympiad algebra and how to recognise when to apply it.
Updated: 2 weeks ago

Vieta's Formulas

Relating roots to coefficients for quadratics and cubics.
Updated: 2 weeks ago

Locating Roots

Learn how to determine where the roots of a quadratic equation lie using the discriminant, endpoint signs, and the vertex.
Updated: 2 weeks ago

Symmetric Polynomials

symmetric polynomials constructions, even symmetric polynomials, and their application.
Updated: 2 weeks ago

Sum Formulas for Powers

Common power sums used in algebra and counting.
Updated: 2 weeks ago

Newton's Sums

Recurrences that compute power sums from coefficients.
Updated: 2 weeks ago

Polynomial Manipulations

Shifting and inverting roots to simplify complex polynomial expressions.
Updated: 2 weeks ago

Vieta Jumping

Vieta jumping or root flipping, application to problems, and proof
Updated: 2 weeks ago

Telescoping Sums and Products

Use cancellation after algebraic rewriting.
Updated: 2 weeks ago

Functional Equations

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Standard substitutions and symmetry-based approaches.

Functional Equations I

Learn the main strategies for solving intermediate functional equations, including substitution, injectivity, surjectivity, fixed points, and guessing forms.
Updated: 2 weeks ago

Functional Equations II

Learn the main strategies for solving intermediate functional equations, including substitution, injectivity, surjectivity, fixed points, and guessing forms.
Updated: 2 weeks ago

Counting & Casework

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Multi-step counting with careful case analysis.

Advanced Counting

Inclusion-exclusion, bijections, derangements, Catalan numbers, Burnside's lemma, and recursive counting that are all advanced counting techniques on the AMC 12
Updated: 2 weeks ago

Block Walk Method

A USAMO guide module on using block walks in combinatorics and problem solving.
Updated: 2 weeks ago

Probability

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Conditional probability, geometric probability, and expected value.

Geometric Probability

A tool to deal with infinite probability problems using geometry
Updated: 2 weeks ago

Expected Value

An introduction to solving expected value problems
Updated: 2 weeks ago

Number Theory

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Modular arithmetic, CRT, and divisibility techniques.

Modular Arithmetic

Learn how congruences, residues, and modular techniques simplify olympiad problems involving divisibility, remainders, and cyclic behaviour.
Updated: 2 weeks ago

Chinese Remainder Theorem

Solving basic systems of linear congruences with pairwise coprime moduli and computing modular inverses.
Updated: 2 weeks ago

Euler's Totient Theorem

Fundamental theorem in modular arithmetic.
Updated: 2 weeks ago

Chicken McNugget Theorem

Determine the largest value that cannot be represented as a nonnegative combination of two coprime integers.
Updated: 2 weeks ago

Euclidean Geometry

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Circle theorems, power of a point, and cyclic quadrilaterals.

Euclidean Geometry

Arc and Chord Relations

Chord length, arc length, and angle theorems in circle geometry.
Updated: 2 weeks ago

Cyclic Quadrilaterals

Opposite angles and Ptolemy's theorem.
Updated: 2 weeks ago

Tangent Lines

Tangent-radius perpendicularity and equal tangents.
Updated: 2 weeks ago

Tangent Circles

External and internal tangency conditions.
Updated: 2 weeks ago

Descartes' Theorem

Tangents to circles, used in AMC10/12 and AIME.
Updated: 2 weeks ago

Power of a Point

Secants, tangents, and equal power products.
Updated: 2 weeks ago

Medians and the Centroid

Median properties and centroid ratios.
Updated: 2 weeks ago

Angle Bisectors

The angle bisector theorem and length facts.
Updated: 2 weeks ago

Altitudes and the Orthocenter

Altitude facts and area relationships.
Updated: 2 weeks ago

Law of Sines

Relate sides and angles using sine.
Updated: 2 weeks ago

Law of Cosines

Generalized Pythagorean theorem for any triangle.
Updated: 2 weeks ago

Coordinate Geometry

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Analytic methods, Shoelace formula, and coordinate-based proofs.

Shoelace Theorem I

Computing polygon areas from coordinates using the Shoelace (Surveyor's) Formula, with AMC/AIME-level applications.
Updated: 2 weeks ago

Shoelace Theorem II

Computing polygon areas from coordinates using the Shoelace (Surveyor's) Formula, with AMC/AIME-level applications.
Updated: 2 weeks ago

Inequalities

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Algebraic and geometric inequality techniques.

Inequalities Foundations

A comprehensive guide to interval bounding, absolute value geometry, quadratic optimization, and the QM-AM-GM-HM means chain.
Updated: 2 weeks ago

AM-GM Inequality

A standard tool for bounding sums and products.
Updated: 2 weeks ago

Cauchy-Schwarz Inequality

A core inequality for bounds and maximization.
Updated: 2 weeks ago

Sequences & Series

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Convergence, recursive formulas, and summation methods.

Sequences and Series I

A comprehensive deep-dive into AMC 10/12 sequence strategies, including structure recognition, advanced telescoping, fixed-point recurrences, polynomial summation, and Fibonacci identities.
Updated: 2 weeks ago

Sequences and Series II

Trigonometry

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Identities, equations, and unit circle applications.

Unit Circle and Angle Measurement

The geometric foundation of trigonometry
Updated: 2 weeks ago

Trigonometric Identities and Series I

Learn how product to sum identities, double angle formulas, triple angle formulas, and trigonometric series simplify olympiad style trigonometry problems.
Updated: 2 weeks ago

Trigonometric Identities and Series II

Learn how product to sum identities, double angle formulas, triple angle formulas, and trigonometric series simplify olympiad style trigonometry problems.
Updated: 2 weeks ago

Trig Equations and Quadratics

Solve basic and quadratic trig equations in a full cycle.
Updated: 2 weeks ago

Inverse Trigonometry

Definitions, principal values, and techniques
Updated: 2 weeks ago

Complex Numbers

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Algebraic and geometric interpretation of complex numbers.

Complex Numbers

Imaginary unit, algebra of complex numbers, conjugates, and division
Updated: 2 weeks ago

The Complex Plane

Geometry of complex numbers — the Argand plane, magnitude, vectors, and graphing
Updated: 2 weeks ago

Polar Form and De Moivre's Theorem

A training module on polar form, rotations, De Moivre's theorem, and complex bashing techniques.
Updated: 2 weeks ago

Roots of Unity

Geometry and sums of

n

th roots.
Updated: 2 weeks ago

Geometry Extensions

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Additional geometric structures.

Regular Polygons

Interior and exterior angles, diagonals, area, circumradius, inradius, and competition applications.
Updated: 2 weeks ago

Uncategorized

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

Barycentric Coordinates

Master the algebraic form of Mass Point Geometry. Learn to instantly solve complex cevian intersections, area ratios, and collinearity problems without construction lines.
Updated: 2 weeks ago

Geometry with Complex Numbers

Rotations, distances, and polygon criteria.
Updated: 2 weeks ago

Complex Numbers and Trig

Use

e^{i\theta}

to simplify trig products.
Updated: 2 weeks ago

Roots of Unity (Advanced)

Cyclotomic facts and power-sum filters.
Updated: 2 weeks ago

Symmetric Cubic Identity

The identity behind

x^3+y^3+z^3-3xyz

.
Updated: 2 weeks ago