Maths Olympiad Concepts: Theorems, Examples and Practice

Complete and reduced residue systems

Use representatives without confusing them with the residue classes.

Diophantine Equations by Factorisation

Find all integer solutions and prove completeness.

Also find: integer solutions; Diophantine equations by factorization.

Divisibility Rules and Prime Factors

Use divisibility as a statement about an integer multiplier.

Euclidean division and remainders

Represent an integer with a valid bounded remainder.

Fermat, Euler, Wilson and Chinese Remainder Theorems

Select a theorem from its hypotheses and explain when it is unavailable.

Also find: Fermat’s little theorem; Euler’s theorem; Euler’s totient function; Wilson’s theorem; Chinese remainder theorem (CRT).

Floor, Ceiling and Greatest Integer Functions

Use the defining interval to control a floor expression.

Also find: greatest integer function; least integer function; fractional part.

Fundamental Theorem of Arithmetic

Use prime-exponent parity to justify perfect-power conclusions.

Also find: unique factorization theorem; prime factorisation theorem.

GCD, HCF and the Euclidean Algorithm

Use common divisors and linear combinations to explain the Euclidean algorithm.

Also find: highest common factor (HCF); greatest common factor (GCF); Bézout’s identity.

Modular Arithmetic and Congruences

Choose a useful modulus and justify cancellation or impossibility.

Also find: congruences modulo n; modular congruence.

Number and Sum of Divisors

Translate divisor questions into independent choices of prime exponents.

Also find: divisor-counting function; sum-of-divisors function.

Number Bases: Conversion and Digit Problems

Translate a digit string into a polynomial in the base.

Parity: Odd and Even Numbers

Use odd and even numbers to prove impossibility.

Prime numbers

Prove compositeness or an infinitude claim using a clear contradiction or factorisation.

Algebraic identities and factorisation

Factor expressions and verify each identity by expanding back.

Also find: factorization; algebraic factorisation.

Arithmetic and Geometric Sequences and Series

Compute terms, state index ranges and explain a finite-sum method.

Complex Numbers and Conjugates

Interpret complex roots before the advanced algebra branches.

Fractions, Ratios and Exact Arithmetic

Calculate accurately and retain exact values until approximation is requested.

Geometry Basics: Lines, Angles and Ratios

Read only the given geometric facts and use a consistent ratio convention.

Laws of Exponents, Surds and Logarithms

Transform expressions with all domain restrictions visible.

Mathematical language and sets

Read a statement, name its domain, and distinguish for every from there exists.

Mathematical Proof: Direct Proof and Contradiction

Write a connected argument with a reason for each decisive step.

Quadratic Equations and the Quadratic Formula

Solve equations while preserving and checking the full solution set.

Trigonometry foundations

Use right-triangle ratios, angle units and trigonometric identities with valid domains.

Common roots of polynomials

Reduce the degree of an equation satisfied by a shared root.

Irreducible polynomials

Prove irreducibility with a valid criterion over a specified field.

Polynomial functions

Recognise a polynomial and explain how its degree can change under addition.

Polynomial Long Division

Write and check P = DQ + R with the correct remainder degree.

Polynomial roots and multiplicity

Use the theorem with the correct coefficient field and multiplicities.

Remainder Theorem and Factor Theorem

Choose a substitution or a new polynomial that turns given values into factors.

Solving polynomial equations

Solve structured equations without losing or inventing roots.

Symmetric polynomials

Rewrite symmetric expressions in a smaller set of quantities.

Vieta’s Formulas: Roots and Coefficients

Find root expressions without calculating each root separately.

Also find: Vieta’s theorem; Viète’s formulas.

Absolute-value inequalities

Translate absolute values into distance or justified cases.

AM–GM Inequality and Harmonic Mean

Choose the terms in a mean inequality and check attainable equality.

Also find: AM-GM inequality; arithmetic mean–geometric mean inequality; AM-GM-HM inequality.

Cauchy–Schwarz inequality

Recognise useful squares or denominators and state equality correctly.

Also find: Cauchy inequality; Cauchy–Bunyakovsky–Schwarz inequality; Titu’s lemma (Engel form).

Chebyshev’s Sum Inequality

Recognise when ordering permits a product-of-averages comparison.

Also find: Chebyshev sum inequality; Chebyshev inequality for ordered sequences.

Hölder’s inequality

Select matching factors and exponents instead of recalling a formula blindly.

Also find: Holder inequality; Hoelder inequality.

Inequalities in geometry

Check both the algebraic bound and the geometric attainability.

Jensen’s inequality

Establish convexity on the relevant interval before applying Jensen.

Also find: Jensen inequality; convexity inequality.

Power mean inequality

Choose a useful pair of means with valid inputs and exponents.

Also find: generalised mean inequality; RMS-AM-GM-HM.

Rearrangement inequality

Prove why matching orders changes a sum of products.

Solving Inequalities: Rules and Sign Changes

Transform inequalities while preserving equivalent conditions.

Sum of Squares Method for Inequalities

Turn a desired bound into a sum of nonnegative terms.

Weierstrass product inequalities

State the domain and prove a product bound from a two-factor case.

Weighted AM–GM Inequality

Use nonnegative weights with their sum explicitly normalised.

Also find: weighted arithmetic–geometric mean inequality.

Mathematical Induction: Steps and Solved Examples

Write a complete induction proof with a valid starting index.

Strong induction

Choose a sufficient induction hypothesis and justify every smaller case used.

Also find: complete induction.

Why Mathematical Induction Works

Distinguish a conjecture from a proof covering all admissible integers.

First-order linear recurrences

Transform a first-order recurrence and verify its explicit formula.

Also find: first-order difference equations.

Higher-order linear recurrences

Explain how the order and root multiplicities control the general form.

Non-homogeneous recurrences

Choose a valid particular form and adjust it when it overlaps the homogeneous part.

Also find: inhomogeneous recurrence relations; particular solutions.

Nonlinear recurrences and substitutions

Find a substitution suggested by the recurrence rather than guessing at random.

Recurrence Relations: Order and Types

Classify a recurrence to select an appropriate method.

Recursive sequences

Compute terms and explain why initial conditions matter.

Second-order linear recurrences

Solve a second-order recurrence and justify the resulting expression.

Functions: Inverse, Composition and One-to-One

Use function properties precisely without assuming a formula.

Also find: injective and surjective functions; inverse functions.

Solving functional equations

Find all admissible functions and verify the complete family in the original equation.

Binomial Theorem and Generating Functions

Explain what each factor and exponent counts.

Circular permutations

Specify which configurations count as identical before dividing.

Also find: circular arrangements; rotational symmetry.

Combinations and binomial coefficients

Distinguish a selection from an arrangement and justify the overcount factor.

Combinations with Repetition: Stars and Bars

Count repeated selections with clearly stated object and box types.

Also find: multisets; stars-and-bars method.

Counting Integer Solutions with Stars and Bars

Convert a constrained sum into a counting model with valid bounds.

Also find: nonnegative integer solutions; bounded stars and bars.

Counting Principles: Addition and Multiplication

Define the counted object and choose disjoint cases or sequential choices.

Counting with bijections

Prove a map is both one-to-one and onto.

Also find: bijective proof; one-to-one correspondence.

Counting with recurrences

Derive a recurrence from a disjoint partition of objects.

Derangements: Formula, Recurrence and Examples

Count permutations that avoid every original position.

Also find: permutations with no fixed points; subfactorial.

Distributing Objects into Boxes

Choose the model from distinguishability and occupancy conditions.

Also find: balls into boxes; distribution problems.

Dividing Objects into Groups

Count partitions with correct group labels and size symmetries.

Factorials

Explain the empty arrangement and simplify only valid factorial expressions.

Inclusion–Exclusion Principle

Explain the alternating correction by tracking one object’s multiplicity.

Permutations and arrangements

Choose a representation that enforces the restriction without duplicate counting.

Also find: permutations with repeated letters; restricted permutations.

Pigeonhole principle

Define useful boxes and justify why a collision forces the conclusion.

Also find: Dirichlet’s box principle; pigeon hole principle.

Invariants and colouring

Find a property unchanged by every legal move, then use it to prove impossibility or compute a final value.

The extremal principle

Choose a smallest, largest or best object, justify its existence, and use it to prove a claim or a sharp bound.

Lattice paths and blocked routes

Count restricted grid routes using checkpoints, blocked edges, disjoint cases and a last-step recurrence.

Prefix sums and divisibility

Find and count consecutive blocks using boundary sums, remainders, parity and the pigeonhole principle.

Double counting: pairs and incidence

Count the same collection in two ways to solve membership, pairing and coloured-relationship problems.

Angle Chasing in Triangles and Parallel Lines

Find an angle using stated incidence and justified equalities.

Also find: angle sum property; angles on parallel lines.

Ceva, Menelaus and Triangle Concurrency

Distinguish concurrent lines from collinear points and select a suitable criterion.

Also find: Ceva’s theorem; Menelaus’ theorem; centroid and orthocentre.

Cyclic and tangential quadrilaterals

Prove cyclicity or tangency before applying its consequences.

Also find: Ptolemy’s theorem; Pitot’s theorem; cyclic quadrilateral.

Parallel lines and angle bisectors

Use parallelism or an angle bisector to obtain a correctly oriented ratio.

Also find: basic proportionality theorem; intercept theorem; angle bisector theorem.

Power of a Point, Circle Chords and Tangents

Combine angle and length properties while checking the point’s position.

Also find: intersecting chords theorem; tangent–secant theorem; radical axis.

Pythagorean, Apollonius and Stewart’s Theorems

Choose a length identity from a verified geometric configuration.

Also find: Pythagoras’ theorem; Pythagorean theorem; Apollonius’ theorem; Stewart’s theorem.

Quadrilaterals and their diagonals

Use the defining properties of each quadrilateral without assuming extra ones.

Similar triangles

Identify a correspondence and distinguish length scaling from area scaling.

Also find: triangle similarity; AA, SAS and SSS similarity.

The midpoint theorem

Prove a midpoint or a parallel segment from the appropriate direction of the theorem.

Triangle area ratios

Translate a length ratio into an area ratio using an identified common altitude.

Triangle congruence

Prove a useful triangle correspondence before transferring equalities.

Also find: triangle congruency; SSS, SAS, ASA and AAS; RHS or HL congruence; CPCTC.

Triangle Construction with Ruler and Compass

Give construction steps and prove they produce exactly the required possibilities.

Also find: straightedge-and-compass construction.

Triangle inequalities

Use strict triangle inequalities and explain geometric equality limits.

Trigonometry in geometry

Select a trigonometric relation that simplifies the geometry and check angle domains.