Advanced (AIME)

Tackle olympiad-level ideas and harder AIME/USAMO crossover topics such as advanced number theory, geometry transformations, and deep algebraic methods.
Expect longer chains of reasoning and proof-oriented structure.

This section is currently under development. The content you see here is filler for now.

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Strong Number Theory

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Lifting the exponent, orders, and deeper divisibility arguments.

Strong Number Theory

Orders, lifting exponents, and tougher Diophantine equations for AIME.
Updated: 2 weeks ago

Vectors

Lifting the exponent, orders, and deeper divisibility arguments.

Introduction to Vectors

Covers vectors, vector components, and vector geometry — including operations, linear combinations, magnitude and direction, unit vectors, and applications in science.
Updated: 2 weeks ago

Advanced Counting

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Generating functions, bijections, and harder combinatorial arguments.

Advanced Counting

Bijections, recursion, and invariants for AIME-level counting.
Updated: 2 weeks ago

Calculus

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Differentiation, Integration, and Differential Equations

Introduction to Derivatives

Introduction to Derivatives
Updated: 2 weeks ago

Rules of Differentiation

Rules of Differentiation
Updated: 2 weeks ago

Chain Rule

The Chain Rule for Differentiation
Updated: 2 weeks ago

L'Hôpital's Rule

Using L'Hôpital's Rule to evaluate limits in indeterminate form
Updated: 2 weeks ago

Implicit Differentiation

Differentiating equations without explicitly solving for y
Updated: 2 weeks ago

Minima, Maxima, and Optimization

Finding and classifying extrema of functions, and applying these techniques to real-world optimization problems
Updated: 2 weeks ago

Taylor Series

Taylor Series and their applications in physics and mathematics
Updated: 2 weeks ago

Intuition Behind Integrals

Building intuition for integrals as sums of infinitesimals, with applications in physics
Updated: 2 weeks ago

Fundamental Theorem of Calculus

A complete guide to the Fundamental Theorem of Calculus — formal definition, intuition, computing integrals as reverse derivatives, and practice problems with solutions.
Updated: 2 weeks ago

Integration Rules

A comprehensive reference for fundamental integration rules — constant, exponential, power, sum, difference, reciprocal, trigonometric, and inverse trigonometric rules, special integrals, definite integral properties, and commonly used JEE results, each illustrated with worked examples.
Updated: 2 weeks ago

Substitutions

A guide to u-substitution and trigonometric substitutions — why they work, step-by-step procedure, a table of trig substitution forms, and challenging practice problems with solutions.
Updated: 2 weeks ago

Integration by Parts

A guide to integration by parts — derivation of the formula, selecting u and dv, applying the technique multiple times, reduction formulas, and three worked problems with full solutions.
Updated: 2 weeks ago

Trigonometric Substitution

A focused guide to trigonometric substitution — when to use it, the three common substitution forms, and four fully worked examples covering the standard integral types.
Updated: 2 weeks ago

Integral Averages

How to take the average of a function across a range using integral averages — definition, formula derivation, and practice problems.
Updated: 2 weeks ago

Gaussian Integrals

A guide to Gaussian integrals — motivation from probability distributions, the polar coordinate trick, expectation values, and related higher-order Gaussian integrals with worked examples.
Updated: 2 weeks ago

Polynomials & Roots

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Root bounding, irreducibility, and polynomial manipulation.

Polynomials and Root Techniques

Vieta's formulas, factor theorem, and root manipulation.
Updated: 2 weeks ago

Advanced Geometry

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Projective methods, inversions, homothety, and spiral similarity.

Angle Chasing Mastery

Systematic angle relationships in complex diagrams.
Updated: 2 weeks ago

Advanced Geometry

Cevians, homothety, and power of a point for AIME.
Updated: 2 weeks ago

Homothety and Spiral Similarity

Scaling and rotation transformations for advanced geometry.
Updated: 2 weeks ago

Sphere Geometry

Surface area, volume, and planar cross-sections.
Updated: 2 weeks ago

Trig in Contests

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Contest-specific trig tricks and identities.

Trigonometry in Contests

Identities, sine and cosine rules, and trig substitution.
Updated: 2 weeks ago

Inequalities (AIME Level)

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Multi-variable and advanced inequality methods.

Inequalities (AIME Level)

Cauchy, Jensen (intro), and strategic substitutions.
Updated: 2 weeks ago

Generating Functions

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Using formal power series to solve counting problems.

Generating Functions (Intro)

Translate counting into algebra using power series.
Updated: 2 weeks ago

Functional Equations (AIME)

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Harder functional equations requiring injectivity and Cauchy-type analysis.

Functional Equations (AIME Level)

More sophisticated substitutions and invariants in FE problems.
Updated: 2 weeks ago