Worksheets

Homework problems that cover mental math, a niche yet useful skill for various competitions.

Homework problems on rigid transformations — reflections, rotations, and translations — and their applications to competition geometry.

Practice problems on logarithms covering log identities, change of base, and exponential-to-log conversion for competition settings.

Coordinate geometry problems on minimum path lengths, reflections across lines, and shortest-distance constructions on the Cartesian plane.

Mental math techniques including the nearest-multiple-of-10 method for rapid multiplication, useful for timed competition rounds.

Homework problems that cover Ptolemy's theorem, a rather fundamental theorem in algebra concerning cyclic quadrilaterals.

Problem set applying Ptolemy's Theorem to cyclic quadrilaterals and regular polygons to find diagonal and side lengths.

Homework problems on Stars and Bars, one of the most common ways to solve combinatorics problems.

Counting problems solved using the Stars and Bars technique, covering distributions of identical objects into distinct bins.

Learn the basics of bases, which can be used to simplify a lot of difficult problems into much easier ones.

A comprehensive problem set that takes you through the basics and use cases of the nine-point circle.

Problems on triangle centers — incenter, circumcenter, and nine-point circle — using both coordinate and classical geometric approaches.

Homework set on invariants, often used to quickly find patterns and answers in seemingly complex or unsolvable problems.

Problems on mathematical invariants — quantities preserved under repeated operations — used to prove impossibility results and find closed-form answers.

Learn about and solve questions on modular arithmetic and Euler's totient theorem.

Problems applying Fermat's Little Theorem and Euler's totient function to compute large modular exponentiations efficiently.

Homework set with problems on Pascal's triangle, a powerful tool in combinatorics, algebra.

Problems exploring Pascal's Triangle, binomial coefficients, and their symmetry properties and applications to counting.

Homework set with problems on complex numbers, often necessary for understanding the solutions to many different problems.

Problem set on complex number arithmetic, modulus, conjugates, and their applications in competition problems.

Learn about various use cases of the geometric series and its summations.

Problems on geometric series including finding common ratios, evaluating infinite sums, and applying the series formula to word problems.

Proof by induction problems establishing base cases and inductive steps to prove divisibility results, closed-form formulas, and inequalities.

Discover how to prove various different problems using the method of induction, often necessary in proof-based competitions.

Learn the basics of game theory through the game of Nim and finding winning strategies.

Game theory problems based on Nim-style coin removal games, practicing the analysis of winning and losing positions under optimal play.

Homework set with problems on factorials and their prime factorizations.

Problems on factorials, trailing zeros, and finding the largest prime power dividing n! using Legendre's formula.

Homework set with problems on Simon's Favorite Factoring Trick, one of the tools used to solve Diophantine equations.

Introduction to Simon's Favorite Factoring Trick for factoring and solving integer equations of the form xy + ax + by + c = 0.

Kickoff problem set covering a mixed range of competition topics including geometric series, number theory, and combinatorics.

Problems on expected value and basic probability including dice outcomes, investment scenarios, and real-world applications.

Coordinate geometry problems covering perpendicular bisectors, parallelogram coordinates, and point relationships on the Cartesian plane.

Problems on converting between number bases and working with base representations, including large numbers in non-decimal bases.

Problems on prime numbers including finding primes with specific sum or product properties.

Problems proving and applying divisibility rules for integers 1–10, with applications to checking divisibility of large numbers.

Problems on linear Diophantine equations, determining when integer solutions exist and finding them using the extended Euclidean algorithm.

Problems applying Euler's totient function to compute modular exponentiations and find totient values, including cases where the base and modulus are not coprime.

Introduction to modular arithmetic through clock problems, last-digit computations, and finding patterns in powers of integers.

Problems on greatest common divisors and least common multiples, including the GCD-LCM product identity and sum minimization.