The Euclid Contest is one of the most rigorous and consequential mathematics competitions available to Canadian high school students. For students targeting the University of Waterloo’s Faculty of Mathematics — and for those aiming at the Canadian Mathematical Olympiad — working through Euclid past contests is the single most effective preparation method available.
But past papers used carelessly produce much less benefit than past papers used strategically. This guide covers where to find Euclid past contests, what each part of the contest actually requires, how to structure your practice sessions, how to use solutions correctly, and how to track your progress as the contest approaches.
For a complete overview of the Euclid Contest — what it is, who should write it, how it is scored, and what a strong result leads to — see our Euclid math contest guide.
Where to Find Euclid Past Contests
The CEMC website (cemc.uwaterloo.ca) is the authoritative source.
The Centre for Education in Mathematics and Computing maintains a complete archive of past Euclid Contest papers and full solutions going back many years. All materials are free to download.
Navigate to: CEMC → Contests → Euclid → Past Contests
Each year’s archive includes:
- The contest paper (English and French)
- Full solutions with complete worked arguments
- The results report (showing score distributions, average scores by part, and general performance data)
What the CEMC solutions provide:
CEMC solutions show the intended solution — often elegant and direct. They do not show alternative approaches, and they do not explain why the approach was chosen. For understanding how to think about problems (not just how to solve this specific problem), supplementing CEMC solutions with AoPS discussions and Think Academy instruction is valuable.
AoPS (artofproblemsolving.com):
The Art of Problem Solving forum has discussion threads for most recent Euclid papers. These show alternate solutions, common errors, and the reasoning behind different approaches — which is often more educational than the official solution alone.
Understanding the Euclid Contest Structure
Before working through past papers, understand exactly what each part demands — so you can target your preparation accordingly.
Contest overview:
| Feature | Detail |
|---|---|
| Duration | 2.5 hours |
| Total marks | 100 |
| Questions | 10 questions |
| Format | Part A (4 questions) + Part B (3 questions) + Part C (3 questions) |
| Calculator | Not permitted |
Part A (4 questions, ~4 marks each):
Part A questions are accessible — equivalent in difficulty to the harder questions on the Cayley or Fermat Contests. A student who has prepared seriously for the Euclid should aim to complete all four Part A questions correctly. These questions do not typically require full written proofs — a clear answer with working shown is sufficient.
Topics commonly in Part A:
- Algebra and equation solving
- Coordinate geometry
- Basic number theory
- Sequences and series
- Basic probability
Part B (3 questions, multi-part):
Part B questions are significantly harder and require complete written solutions. Each Part B question typically has two or three parts, where earlier parts build toward the later, harder parts. The approach matters here — work needs to be shown clearly and logically, not just the answer.
Topics commonly in Part B:
- Geometry proofs and calculations
- Algebraic manipulation
- Number theory
- Combinatorics
- Functions and polynomial properties
Part C (3 questions, multi-part proof):
Part C questions are the hardest in the contest. They require complete, rigorous mathematical proofs — typically 2–3 parts where the final part is genuinely difficult, even for very strong students. Many students leave Part C questions partially attempted or entirely unattempted. Partial credit is available for significant progress.
Topics commonly in Part C:
- Proof by induction, contradiction, or contrapositive
- Pigeonhole principle applications
- Advanced number theory (divisibility, modular arithmetic)
- Extremal combinatorics
- Geometric proofs using classical theorems
For proof techniques needed for Part C, see our math induction proof guide, proof by contradiction guide, contrapositive guide, and pigeonhole principle guide.
How to Use Euclid Past Contests Effectively
Working through past papers is necessary but not sufficient. The way you use them determines how much benefit you get.
Phase 1: Diagnostic (8–10 weeks before contest)
Purpose: Understand where you currently stand across all three parts.
Sit one complete past Euclid paper under full contest conditions:
- 2.5 hours, no calculator, no notes, no interruptions
- Write on paper as you would in the actual contest
- Do not look anything up mid-paper
After completing:
- Mark your own paper using the CEMC solutions
- Record your score for each part separately (not just the total)
- Categorise every question you did not fully solve: was it a topic gap (you didn’t know the content) or an approach gap (you knew the content but couldn’t see how to use it)?
This diagnostic tells you where to focus preparation time. A student who completes Part A fully but struggles on all Part B questions has a different preparation need from a student who solves some Part B problems but leaves Part A with careless errors.
Phase 2: Topic-Targeted Practice (6–8 weeks before contest)
Purpose: Address specific gaps identified in the diagnostic.
Do not work through past contests end-to-end during this phase. Instead:
Sort by topic, not by year.
Pull Part B and Part C questions from multiple past papers and group them by topic: geometry problems together, number theory problems together, induction proofs together. Work through the grouped problems, consulting solutions when stuck, and identify the patterns in how each topic appears.
This is how you build topic fluency, not by repeatedly sitting full contests.
Identify recurring question types. Across Euclid past contests going back 5–10 years, certain problem structures recur:
- A problem involving a clever substitution in an algebraic identity
- A number theory problem requiring modular arithmetic
- A geometry problem requiring a constructed auxiliary line
- A combinatorics proof using the pigeonhole principle or induction
Recognising these structures is the skill that makes the contest time-efficient. It develops from working through grouped, topic-sorted problems — not from random end-to-end paper sitting.
Work on proof-writing as a separate skill. Part C demands complete, rigorous proofs. Proof-writing is not just content knowledge — it is a specific expressive skill that improves with deliberate practice. Write out complete proof arguments even for questions you know how to solve. Show every logical step. Practise the language of mathematical proof: “Without loss of generality…”, “Assume for contradiction…”, “By the inductive hypothesis…”, “Therefore by [theorem name]…”
Phase 3: Timed Full Papers (3–4 weeks before contest)
Purpose: Build contest-condition fluency and time management.
Return to full-paper practice, but now with a clearer strategy:
Allocate time explicitly:
- Part A: aim for 30–35 minutes maximum (about 8 minutes per question)
- Part B: 60–70 minutes (20–25 minutes per question)
- Part C: remaining time — strategic, not exhaustive
Triage Part C intelligently. Part C(i) — the first part of each Part C question — is often significantly more accessible than Part C(ii) or C(iii). A student who spends 30 minutes attempting Part C(iii) from scratch and gets 0 marks would have been better served getting Part C(i) on all three questions for partial credit. Read all Part C problems before committing time to any of them.
Write partial solutions deliberately. If you cannot solve a problem fully, write what you can. The Euclid awards partial credit for significant progress — a clearly stated approach, a correct first step, a correct setup of the argument even without a complete conclusion. Leaving a page blank when you have partial understanding of a problem is a strategic error.
Phase 4: Review and Consolidation (Final week)
Purpose: Consolidate, not expand.
The final week before the Euclid is not the time to learn new topics. It is the time to:
- Review your own past solutions and solutions to problems you found hard
- Re-read the CEMC solutions for Part B and Part C questions you did not solve fully
- Work through 2–3 Part A problem sets to maintain fluency
- Practise proof-writing language and structure
- Rest adequately — mathematical performance degrades significantly with sleep deprivation
Do not sit a full timed paper in the final 3 days. The goal is confidence and composure, not new learning.
What a Strong Euclid Past Contest Score Looks Like
Understanding score distributions helps calibrate your practice goals.
Typical score ranges:
| Score | Approximate percentile |
|---|---|
| 90–100 | Top ~1% — CMO invitation territory |
| 75–89 | Top ~5–10% — strong result |
| 60–74 | Top 20–25% — competitive result |
| 40–59 | Average range |
| Below 40 | Below average |
These figures are approximations — exact distributions vary by year. Refer to the CEMC results reports for year-specific data.
What University of Waterloo considers:
The University of Waterloo uses Euclid scores as a significant input in admissions and scholarship decisions for its Faculty of Mathematics programmes. The Euclid Scholarship — Waterloo’s most prestigious mathematics entrance award — requires a very high Euclid score. A score in the 80s or 90s is in the range that attracts meaningful scholarship consideration. For the full context of what the Euclid leads to, see our Euclid math contest guide and Euclid mathematics contest guide.
Analysing Your Past Paper Results: A Framework
After each timed paper, use this framework to extract maximum learning:
1. Score by part, not just total. Record Part A score, Part B score, and Part C score separately. This tells you where your time should go in subsequent practice.
2. Categorise every non-full-credit question. For each question where you did not score full marks:
- Topic gap: did not know the relevant content or technique
- Approach gap: knew the content but could not see how to apply it to this problem
- Execution gap: identified the right approach but made an error in working it through
- Time gap: had the right approach but ran out of time
Each category has a different fix: topic gaps require content study; approach gaps require more varied problem exposure; execution gaps require careful checking habits; time gaps require pacing practice.
3. Read the solution for every non-full-credit question, even if you got partial marks. A partial solution that happened to score marks is not the same as a complete understanding. Read the full CEMC solution and ask: is the approach I used the same? If not, which approach is more elegant and why?
4. Re-attempt questions one week later. After reading the solution, come back to the question one week later without notes. Can you reproduce the argument? If not, the solution was understood in the moment but not retained.
Topic-by-Topic Guide to Euclid Past Contests
These are the topics that appear most consistently across Euclid past contests, with guidance on what each requires.
Algebra and Functions
What it looks like in past papers: Algebraic identities, clever substitutions, equations with multiple unknowns solved by symmetry or substitution, functional equations (find all f satisfying a condition).
What you need: Strong algebraic manipulation, comfort with completing the square, experience with substitution strategies. Functional equations are a specific technique — practise these separately from standard equation solving.
Representative past paper approach: Problems like “find all functions f: ℝ → ℝ satisfying f(x + y) = f(x) + f(y) for all x, y” require both mathematical insight and proof-level writing.
Geometry
What it looks like in past papers: Circle theorems, angle chasing, similar triangles, area calculations using multiple approaches, coordinate geometry combined with synthetic reasoning.
What you need: Solid knowledge of circle geometry (tangent-chord angles, power of a point, inscribed angle theorem), similar triangle properties, and the habit of drawing clear, labelled diagrams before attempting calculation.
Representative approach: Many Euclid geometry problems require constructing an auxiliary line or point — adding an element to the diagram that makes a relationship visible. This cannot be taught as a formula; it develops through exposure to many problems.
Number Theory
What it looks like in past papers: Divisibility proofs, modular arithmetic, Diophantine equations (integer solutions to equations), problems involving gcd and lcm.
What you need: Fluency with modular arithmetic (equivalences, properties mod n), the proof techniques from our proof by contradiction guide and contrapositive guide, and experience with proof-by-induction for divisibility results.
Combinatorics
What it looks like in past papers: Counting arguments, pigeonhole principle applications, graph-theoretic problems, existence proofs (show there exists a configuration with a given property).
What you need: The pigeonhole principle (see our pigeonhole principle guide), mathematical induction (see our math induction proof guide), and experience with careful case analysis.
Sequences and Series
What it looks like in past papers: Recursive sequences, properties of arithmetic and geometric sequences applied in non-standard contexts, sums of series in closed form.
What you need: Fluency with summation formulas from MCR3U (arithmetic and geometric series), the ability to identify closed forms for recursively defined sequences, and proof skills for establishing properties of all terms.
Euclid Past Contests: The Most Useful Years
All past Euclid papers are useful, but some deserve particular attention:
Most recent 5 years (highest priority): The most recent papers are the most representative of the current contest format and difficulty level. Work through these under the strictest timed conditions.
5–10 years ago (high priority for topic diversity): Papers from this range provide additional practice and expose you to topic variations not covered in the most recent years.
10–15 years ago (useful for topic-sorted practice): Older papers are useful for extracting specific topic practice — geometry problems, number theory problems — rather than for full-paper simulation.
Format note: The Euclid Contest format has evolved over the years. Check the CEMC website for any format changes and prioritise papers from years with the current format.
How Think Academy Supports Euclid Preparation
Think Academy Canada works with students preparing for the Euclid Contest at every stage of the pathway — from students who are targeting a strong Part A performance to those aiming for Part C solutions and CMO qualification.
Our Euclid preparation programme:
Diagnostic-led: every student begins with an assessment that identifies specific gaps across the Part A, Part B, and Part C topic areas — so preparation time is targeted at the problems most likely to improve the score.
Proof technique development: Part C requires complete, rigorous proofs. Think Academy builds proof-writing as a specific skill — not just content knowledge but the ability to construct and express a complete mathematical argument.
Past paper review: our instructors work through Euclid past contest solutions with students, explaining not just what the solution is but how to identify the right approach from the problem statement — the skill that cannot be extracted from reading solutions alone.
The CMO pathway: for students aiming beyond the Euclid toward the Canadian Mathematical Olympiad, Think Academy’s programme builds the mathematical depth that the CMO demands. See our Canadian Mathematical Olympiad guide.
Frequently Asked Questions
Where can I find Euclid past contests? On the CEMC website at cemc.uwaterloo.ca. Navigate to Contests → Euclid → Past Contests. All papers and full solutions are free to download.
How many past papers should I work through? For serious Euclid preparation, 8–12 full timed papers (supplemented by topic-sorted problem practice) over a 10–12 week preparation window is a reasonable target. Quality of review matters more than volume of papers attempted.
Should I time myself on every past paper? Not necessarily on every paper. During the topic-targeted phase (Phase 2), work without strict time pressure — understanding comes first. During the timed phase (Phase 3), strict contest conditions are essential. Most students benefit from mixing both approaches.
My score on Part A is strong but Part B is weak. What should I do? Focus on Part B problems from past papers, sorted by topic. Identify which Part B topic areas are weakest and work through multiple problems in those areas before returning to full papers. Part B problems require complete working shown — practise writing full solutions, not just getting the right answer.
I consistently score 0 on Part C. Should I focus on it? Focus on Part C(i) first — the first part of each Part C question is often significantly more accessible than later parts. Getting Part C(i) on all three questions adds meaningful marks. Full Part C(ii) and (iii) solutions require a level of mathematical depth that typically takes months of dedicated preparation to develop.
How long before the contest should I start using past papers? Begin with a diagnostic paper 10–12 weeks before the contest. This leaves enough time for topic-targeted work (Phase 2) before returning to full-paper practice (Phase 3) in the final 4 weeks.
See our related guides: Euclid math contest guide · Euclid mathematics contest guide · COMC math contest guide · Canadian Mathematical Olympiad guide · math induction proof guide · proof by contradiction guide · contrapositive math guide · pigeonhole principle guide · Canadian Senior Math Contest guide · math competitions in Canada
Euclid past contests are the best preparation available. Use them strategically.
