Preparing for the Waterloo Math Contests: Cayley, Fermat and Euclid
For students in advanced math programs at private schools across Canada, writing a Waterloo math contest is a real academic milestone. A family might want to strengthen a University of Waterloo engineering application, build a contest record for competitive admissions, or simply see their child stretch beyond the classroom. The Centre for Education in Mathematics and Computing (CEMC) contest series is among the most respected challenges open to students in grades 7 through 12. Preparation is what separates a participation certificate from a distinction. This article covers what families should know about the Cayley, Fermat and Euclid contests and how to get ready for each.
Who writes the Waterloo math contests, and at which level?
The Waterloo math contests are a series of yearly math competitions run by the University of Waterloo's CEMC. Students in Canada, the United States and other countries can enter, and they write through participating schools. Each contest is pitched at one grade level:
| Contest | Grade Level | Format | Duration |
|---|---|---|---|
| Gauss | 7 and 8 | Multiple choice, 25 questions | 60 minutes |
| Pascal | 9 | Multiple choice, 25 questions | 60 minutes |
| Cayley | 10 | Multiple choice, 25 questions | 60 minutes |
| Fermat | 11 | Multiple choice, 25 questions | 60 minutes |
| Euclid | 12 | Short answer and full solution, 10 questions | 2.5 hours |
Students in enriched and accelerated math streams at schools such as Upper Canada College, Crescent School and the University of Toronto Schools regularly take part. Any student in a gifted, enriched or advanced math program is a strong candidate for the contest at their grade, and high achievers often write one level above their grade to gain experience early.
What are the formats and scoring of the three contests?
Knowing how scoring works is the first step toward a focused preparation plan.
The Cayley and Fermat contests use the same format: 25 questions in three sections, for a maximum of 150 points. Part A has 10 multiple-choice questions worth 5 points each, Part B has 10 multiple-choice questions worth 6 points each, and Part C has 5 questions worth 8 points each, where the answer is a whole number from 0 to 99. There is no penalty for a wrong answer, and in the 2026 contests each unanswered question (up to 10) earned 2 points. A Certificate of Distinction goes to the top 25%: in 2026 the cutoffs were 106 on the Cayley and 102 on the Fermat.
The Euclid contest is a different kind of challenge. It is an exam of 10 questions answered over 2.5 hours, graded by human markers out of 100 points, with short-answer and full-solution parts. Partial credit is given, so students who show clear mathematical reasoning earn marks even if they never reach a final answer. A score above the Certificate of Distinction cutoff (66 in 2026) is a strong result, particularly for applicants to the University of Waterloo. For Grade 12 students targeting competitive math, engineering or computer science programs, the Euclid is often the most strategically important contest they will write.
When is the right time to start preparing?
A window of three to four months is ideal. Cayley and Fermat are usually held in February and the Euclid in April, so serious preparation should begin by October or November for Cayley and Fermat, and by January at the latest for the Euclid.
Working with students at top private schools and gifted programs across Canada shows that a focused student who commits two to three hours a week can make significant, measurable progress within six to eight weeks. Students aiming for the highest percentile scores on the Euclid should start building problem-solving depth at least six months ahead, especially if they have little contest experience.
Which topics does each contest level require?
Across all Waterloo contests the core areas are algebra, geometry, number theory and combinatorics. In more detail:
- Algebra: systems of equations, sequences and series, polynomial manipulation, and exponential and logarithmic functions
- Geometry: properties of triangles and circles, coordinate geometry, and properties of 3D figures
- Number theory: divisibility rules, prime factorization, modular arithmetic and Diophantine equations
- Combinatorics and counting: permutations, combinations, basic probability and the pigeonhole principle
Cayley and Fermat draw on curriculum common to all Canadian provinces, so students in strong academic programs will recognize most of the concepts from class. The real challenge is applying them creatively against the clock. The Euclid, by contrast, reaches past the standard high school syllabus into geometric proofs and advanced number theory. A student hoping to score 70 or higher on the Euclid usually needs exposure to reasoning techniques that go well past what most Grade 12 classrooms cover.
What is the best way to practise?
The most effective strategy is steady, deliberate work with past contest papers. CEMC posts every past Waterloo contest paper free on its website (cemc.uwaterloo.ca), along with full solutions, and most serious preparation plans are built around that archive.
A tried-and-tested four-step cycle:
- Write a full past paper under timed, exam-like conditions without interruptions
- Mark it honestly and flag every question you got wrong or guessed
- Study the official CEMC solution for each mistake, paying attention to the reasoning strategy and not only the final answer
- Retry similar questions from other years before moving to the next paper
Experienced contest educators say the most common error is reviewing solutions passively. Reading a solution and truly understanding it are different things. What drives real improvement by exam day is being able to rebuild the key steps from scratch on a blank page.
Students preparing for the Euclid also need to practise writing clear, structured solutions. Human markers reward logical progression, so a correct answer with no supporting work earns very few marks. Presenting solutions clearly is as important as mastering the math behind them.
Does the Euclid score matter for university admissions?
Yes, especially for the University of Waterloo, though not as an admission requirement. Its Faculty of Mathematics is among the most competitive in Canada. Waterloo does not require a contest for admission, but writing the Euclid or the Canadian Senior Mathematics Contest is required for some entrance scholarship decisions, and its Admission Information Form lets applicants list contest results.
Outside Waterloo, a strong Euclid score adds weight to university applications in two ways. For Canadian programs, it shows mathematical ability that goes beyond grades. For US applications to engineering and quantitative programs at schools such as MIT, Carnegie Mellon or selective liberal arts colleges, Canadian math contest results are seen as a credible sign of mathematical talent, especially when paired with AMC or AIME results.
A student aiming for a top math, engineering or computer science program should regard the Euclid as a meaningful part of their academic profile, not an optional extra.
If your child is preparing for a Waterloo math contest and you want structured coaching from educators who understand advanced academic programs, book a free consultation with Bosen Academics. You can also browse our full range of tutoring subjects.
Frequently asked questions
What is a good score on the Euclid?
The 2026 average was 52.2, and the cutoff for a Certificate of Distinction (the top 25%) was 66. A score well above 66 is exceptional. The maximum is 100 points, and very few students reach it in any year. Waterloo does not require the Euclid for admission, but it counts toward some entrance scholarship decisions.
Can students in the United States write the Waterloo math contests?
Yes. CEMC contests are open to students internationally, including in the US, who write through a school that has registered with CEMC. Families should check with the school each fall, because registration deadlines usually close several weeks before the contest date.
How does the Euclid differ from the AMC 12?
The AMC 12 is a 25-question multiple-choice test finished in 75 minutes. The Euclid is a 10-question exam of short-answer and full-solution parts taking 2.5 hours, and it gives partial credit for written reasoning, so it tests problem-solving more deeply. Students preparing for both should practise each format separately.
Should a Grade 11 student write the Fermat or the Euclid?
Most Grade 11 students write the Fermat, the contest meant for their grade. Strong students in enriched or accelerated programs sometimes write the Euclid a year early. That can help those planning to write it in Grade 12 for admissions, because early exposure lowers first-attempt nerves.
How many hours of preparation does the Euclid take?
A student new to contest math should plan on 40 to 60 hours of focused practice over three to four months to reach a competitive score. Students with earlier AMC or CEMC contest experience may need 20 to 30 focused hours to develop Euclid-specific skills, especially full-solution writing.
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