The Euclid Mathematics Contest, organized annually by the Centre for Education in Mathematics and Computing (CEMC) at the University of Waterloo, is widely regarded as one of the most prestigious high school mathematics competitions in North America. Each April, tens of thousands of students from Canada and around the world take on 100 marks of carefully crafted problems that test creativity, rigor, and mathematical maturity.
What Makes the Euclid Contest Unique?
Unlike multiple-choice tests, the Euclid is a full-write-in contest. Every answer must be supported by clear, logical, step-by-step reasoning. This format mirrors how mathematics is practiced at the university level and beyond.
The University of Waterloo uses Euclid scores as a core component of admissions for the Faculty of Mathematics and the Faculty of Engineering. A score of 85 or above is typically needed to be competitive for programs like Computer Science and Software Engineering.
Exam Structure
Duration: 2.5 hours
Questions: 10 questions with multiple sub-parts (2 to 5 each)
Total Marks: 100
Calculators: Approved calculators permitted
Questions are arranged in increasing difficulty. The first few are accessible to any strong Grade 12 student; the final three or four require creative approaches and deep mathematical insight. Importantly, early parts of difficult questions are often easier than the question as a whole.
The Six Core Topic Areas
1. Algebra and Equations
Polynomial factoring, systems of equations, inequalities, and clever algebraic manipulation.
2. Functions
Polynomial, exponential, logarithmic, and rational functions. Function composition, inverses, and graph analysis.
3. Sequences and Series
Arithmetic and geometric sequences, telescoping series, recursive definitions, and summation notation.
4. Geometry
Analytic geometry, lines, circles, distance formulas, and optimization on the coordinate plane.
5. Trigonometry
The unit circle, trigonometric identities, solving equations, and applications to geometric problems.
6. Counting and Probability
Permutations, combinations, probability calculations, and careful case analysis.
A Sample Problem
Here is a representative problem inspired by past Euclid exams:
Problem: Let f(x) = ax² + bx + c be a quadratic function such that f(1) = 3, f(2) = 7, and f(3) = 13.
(a) Find the values of a, b, and c.
(b) Find the minimum value of f(x).
(c) Determine all values of x for which f(x) > 15.
This multi-part question is typical of the middle section. Part (a) tests setting up and solving a system of equations. Part (b) requires completing the square or using the vertex formula. Part (c) asks you to solve a quadratic inequality. Notice how the sub-parts build on each other — a deliberate design feature.
University Admissions Impact
For the University of Waterloo, the Euclid is a required component of applications to the Faculty of Mathematics and Faculty of Engineering. The Admission Average combines your Grade 12 top-six average with your Euclid score. For competitive programs, scores of 85-95+ are typically needed.
Other universities including the University of Toronto, UBC, McGill, and Western also value high Euclid scores for competitive programs.
Scholarship Opportunities
President's Scholarship of Distinction (Waterloo): $2,000 to $5,000+, typically requiring 90+
David Johnston Scholarship: Full tuition for four years for the top 10-20 national scorers
Faculty of Mathematics Entrance Scholarships: Dozens of awards based on Euclid performance
Ontario Scholar Recognition: $1,000 provincial grant for top scorers in Ontario
Preparation Strategy
Phase 1: Build Your Foundation (September to January)
Master the Grade 11 and 12 advanced mathematics curriculum. Practice writing clear, complete solutions.
Phase 2: Practice Past Papers (February to March)
The CEMC publishes past contests and official solutions on its website. Work through 5-10 years of past exams under timed conditions (2.5 hours, no notes). Review every mistake carefully.
Phase 3: Refine and Review (April, Before Contest)
Review notes, re-attempt difficult problems, focus on time management. Get adequate rest.
Key Tips for Contest Day
Write clearly and logically — clear methodology earns partial marks even if the final answer is wrong
Define your variables — start each sub-part with clear statements
Do not skip steps — show every significant algebraic manipulation
Attempt every question — partial work on difficult questions earns valuable marks
Manage your time — do not spend more than 15 minutes stuck on any single sub-part
Registration
Students cannot register individually. Registration is handled through schools. Talk to your math teacher or guidance counselor by January or February. Your school must register with the CEMC and order materials by late February or early March. International schools can also participate by contacting the CEMC directly.
Official Resources
CEMC Website: cemc.uwaterloo.ca
Past Contests Archive: Available on the CEMC website with full solutions
Final Thoughts
The Euclid Mathematics Contest rewards students who have built deep mathematical understanding over time. Start preparing early, practice with real past papers, and practice writing solutions that clearly communicate your reasoning. Whether you are aiming for Waterloo's Computer Science program, a full scholarship, or simply want to challenge yourself, the Euclid offers an experience that will sharpen your analytical skills and prepare you for success in university and beyond.
Good luck, and enjoy the challenge!
