This section collects some materials used in the Treviso Territorial Olympic Stages, prepared by Prof. Maria Archetti (LSSA Max Planck). For more information about the stages, see the dedicated section here.
The notes introduce some fundamental tools used in Olympiad mathematics, particularly in algebra and number theory.
They are intended as study support and include theory, examples, and problems.
Number Theory
📑 Olympic Stage 2024 ▸ Number TheoryThese notes introduce several fundamental tools of number theory that frequently appear in Olympiad problems.
Main topics:
- Congruences and residue classes
- Operations with congruences
- Euler’s $\varphi$ function
- Fermat’s Little Theorem
- Euler–Fermat Theorem
- Chinese Remainder Theorem
- Applications to competition problems
The material shows how to use congruences to simplify computations and proofs, for example in the study of last digits of numbers or divisibility properties.
Olympiad Algebra
📑 Olympic Stage 2025 ▸ Algebra
These notes collect several algebraic techniques that are very useful in solving Olympiad problems.
Topics covered:
- Viète’s formulas
- Girard–Newton formulas
- Study of relations between the roots of a polynomial
- Fundamental inequalities
- Polynomial congruences
The techniques presented allow, for example, computing sums of powers of the roots or deducing properties of polynomials without explicitly determining their solutions.
Diophantine Equations and Number Theory Tools
📑 Olympic Stage 2026 ▸ Algebra and Number Theory
🧩 Training Exercises ▸ First Two Years 2026 🏁 Final Problems ▸ Olympic Stage 2026
These notes introduce several classical techniques for studying Diophantine equations, that is, equations in which integer solutions are sought.
Among the topics discussed:
- Linear Diophantine equations
- Euclidean algorithm
- Congruence method
- Nonlinear Diophantine equations
- Pythagorean triples
- Pell’s equation
The introductory exercises are designed as initial training and include proof-based problems and elementary number theory.
The final stage problems instead collect more challenging exercises, often similar in style to Olympiad competition problems, requiring the combined use of algebraic techniques and number theory.