Waterloo Math Contests: How to Prepare for Euclid, Fermat and Cayley
Every spring, tens of thousands of students across Canada write one of the Waterloo Math Contests, which the Centre for Education in Mathematics and Computing (CEMC) at the University of Waterloo organizes. For students in advanced math streams or at private schools, these contests carry weight. Euclid scores count toward some Waterloo entrance scholarship decisions, top performers receive Certificates of Distinction, and a good contest record shows scholarship committees and admissions offices something clear.
Even so, many high achievers underestimate how far contest math is from regular coursework. The Cayley, Fermat and Euclid test reasoning and creative problem-solving in ways that challenge even students with straight-A math marks if they have not prepared specifically. This guide explains how to prepare for each level so your child can arrive confident and perform well.
How are the Waterloo Math Contests organized?
The Waterloo Math Contests are a set of yearly competitions run by CEMC, each aimed at one grade. The main ones for high school students are:
| Contest | Grade Level | Format | Duration |
|---|---|---|---|
| Cayley | Grade 10 | 25 multiple-choice questions in 3 parts | 60 minutes |
| Fermat | Grade 11 | 25 multiple-choice questions in 3 parts | 60 minutes |
| Euclid | Grade 12 | 10 multi-part questions, short answer and full solution | 2.5 hours |
Cayley and Fermat share one format: 25 questions in three parts (A, B and C), marked out of 150, with harder questions worth more points. Students in the top 25% of all participants get a Certificate of Distinction. The Euclid is a different kind of test. It asks for short answers and written, justified solutions to 10 multi-part questions and draws on the whole Grade 12 curriculum and more.
According to CEMC, 23,985 students wrote the Euclid in 2026, in Canada and abroad, many from private schools and enriched programs where strong math results are expected.
What sets the Euclid apart from Cayley and Fermat?
The Euclid demands full written solutions with mathematical justification, not only correct answers. That is the biggest difference from the Cayley and Fermat, which are entirely multiple choice.
Students who have only trained for multiple-choice contests are often surprised by the Euclid. Partial credit counts: a well-organized but unfinished attempt at a hard question can earn more marks than a lucky guess on a multiple-choice paper. The skills involved are closer to olympiad math, namely clear logic, systematic case analysis and the ability to build a coherent argument under time pressure.
For Grade 12 students applying to competitive university programs, particularly Waterloo's mathematics programs, the Euclid can matter for entrance scholarships. Students who put 8-12 weeks into structured Euclid preparation tend to do better than those who rely on strength in the curriculum alone.
Which topics should students study for each contest?
The Waterloo Math Contests build on curriculum content but reward creative use over rote recall. These are the most heavily tested areas.
Core topics for Cayley and Fermat
- Algebra and arithmetic: percentages, ratios, equations, sequences and pattern recognition
- Number theory: divisibility, factors, remainders and prime numbers
- Geometry: angle relationships, area calculations and coordinate geometry
- Combinatorics: counting principles and basic probability
- Multi-step word problems: problems that need logical reasoning across several steps
Core topics for the Euclid
- Trigonometry: laws of sines and cosines, solving triangles and trig identities
- Functions and polynomials: roots, transformations and functional equations
- Sequences and series: arithmetic and geometric sequences
- Number theory: modular arithmetic, Diophantine equations and prime factorization
- Analytic geometry: circles, parabolas and where curves intersect
- Combinatorics and probability: including multi-step counting arguments
None of the Waterloo Math Contests requires calculus. Students need not be enrolled in MCV4U (Calculus and Vectors) to do well on the Euclid, although strong algebra and mathematical maturity are essential.
How many hours of preparation does a strong score take?
For Cayley and Fermat, students with a solid classroom math base usually need 6-10 hours of focused contest practice to reach Certificate of Distinction level. For the Euclid, 15-30 hours spread over 8-12 weeks is a realistic goal for a student aiming for a strong score above 70 out of 100.
The most effective method is to work through past papers, free on the CEMC website at cemc.uwaterloo.ca, and then go over every missed question in detail. Checking the answers is not enough. Building the problem-solving toolkit that contest math requires you to understand why an approach worked and to name the technique behind it.
A weekly Euclid plan could look like this:
- Weeks 1-3: Use 2-3 past papers to find weak topics, and concentrate review on the areas where the most marks were lost.
- Weeks 4-7: Practise by topic (for example, one week on number theory and one on geometry), solving 5-10 problems per session.
- Weeks 8-10: Write full timed practice papers in exam conditions, then review the solutions in detail.
- Final 1-2 weeks: Do a light review of any remaining gaps and prepare mentally for test day.
Which resources work best for contest preparation?
Start with the CEMC website, which has past papers and complete worked solutions going back many years. For students aiming at top scores, Art of Problem Solving (AoPS) textbooks on algebra, number theory and combinatorics are widely used by those preparing for the AMC, Euclid and olympiad-level contests.
A tutor with direct contest experience can speed up progress considerably, especially for the Euclid, where the open-ended format calls for feedback on solution structure and mathematical writing, not just numerical correctness. Many students prepare for the Euclid and the AMC 10/12 at the same time, since problem-solving skills carry over well between the two. You can find contest math tutoring from Bosen Academics under our tutoring subjects.
Start early for the best results
Students who do best on the Waterloo Math Contests, the Euclid especially, treat contest preparation as its own skill set and begin building it well before contest day. Past papers, targeted topic review and steady problem-solving practice are the base of strong performance at every level.
If your child is aiming for a strong Euclid or Fermat result, or applying to programs where contest math is considered, book a free consultation with Bosen Academics to work with tutors who specialize in advanced math and contest preparation.
Frequently asked questions
When are the Waterloo Math Contests held each year?
Cayley and Fermat are usually written in late February, and Euclid in early April, typically the first or second week. Students register through their school's math department, and the ordering deadlines are early (in 2027, February 9 for Cayley and Fermat in Canada and March 11 for Euclid). Home-schooled students or those whose school does not take part can often write at a nearby participating school.
Does the Euclid score affect university admissions?
For some purposes it matters. The University of Waterloo does not require a contest for admission to the Faculty of Mathematics, but writing the Euclid (or the Canadian Senior Mathematics Contest) is required for some entrance scholarship decisions, and Ontario applicants must write the Euclid at their regular school to be considered. A score above the Certificate of Distinction cutoff (66 out of 100 in 2026) is a strong result.
Can Grade 11 students write the Euclid?
Yes. It is designated for Grade 12 but open to any grade, and some advanced Grade 11 students write it a year early for practice. Most students do better writing the Fermat in Grade 11 and then concentrating on Euclid in Grade 12, once they have covered the curriculum.
How is the Euclid scored?
It is marked out of 100, with 10 questions worth 10 marks each, most with several parts of rising difficulty. Marks are given for incomplete but sound work, so clear, organized reasoning counts even without a finished solution. The 2026 average was 52.2, so a score above the 66 needed for a Certificate of Distinction is genuinely competitive.
What score earns a Certificate of Distinction on the Euclid?
The certificate goes to the top 25% of participants. The cutoff varies with how hard the contest was, and in 2026 it was 66 out of 100. Scores well above the cutoff put students among the very top performers.
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