The Euclid Mathematics Contest, organized by the University of Waterloo in Canada, is one of the world's most influential math competitions for high school students. Known for its standardized problems, moderate difficulty, and strong alignment with university curricula, it has become a key academic credential for applications to Canada's top 3 universities (U of T, UBC, Waterloo) and prestigious STEM schools in the UK and US (such as CMU, Cornell, Imperial College London). In the 2025–2026 season, Euclid has maintained its characteristic high level of knowledge stability, but the trend toward compound problems has significantly increased—final challenging questions often integrate cross-module content such as "sequences + geometry" or "trigonometry + algebra," placing higher demands on students' ability to integrate knowledge.
This article systematically explains the six core modules of Euclid, recent trends in question design, high-scoring preparation strategies, and final-sprint suggestions to help you prepare efficiently and aim for the top 5% (Distinction) or even top global rankings.
I. Core Characteristics of the Euclid Contest
Exam Date: Annually in April (April 2, 2026)
Exam Duration: 2.5 hours
Question Format: 10 long-answer questions (some with sub-parts), total 100 points
Language: English (mathematical symbols are universal, low language barrier)
Target Grade: Grade 12 and below (equivalent to China's Grade 11–12)
Value for Applications:
University of Waterloo (Mathematics/CS): Euclid scores directly used for scholarship assessment.
University of Toronto, UBC: Strongly encouraged to submit, especially for Mathematics, CS, and Engineering programs.
UK / US universities: Serves as a powerful supplement to AMC/AIME, demonstrating mathematical breadth.
II. Six Core Modules & Recent Question Trends
| Module | Weight | Key Focus | 2025–2026 New Trends |
|---|---|---|---|
| 1. Algebra & Equations | 30%–40% (largest weight) | Polynomial factorization, exponential & logarithmic operations, inequalities, function properties | Triple fusion problems involving "functions + exponents + inequalities"; high-degree symmetric factorization appears more frequently |
| 2. Geometry | ≈30% | Plane geometry (circles, triangles), analytic geometry (coordinate systems, vectors) | Increased graphical complexity; requires synthesis of Pythagorean theorem, similarity, power of a point; constructing auxiliary lines is key |
| 3. Trigonometry | 10%–15% | Identity transformations, graph analysis, applications of law of sines/cosines | Often combined with algebra (e.g., solving trigonometric equations); an area of strength for Chinese students |
| 4. Probability & Combinatorics | 10%–15% | Counting principles, conditional probability, recursive counting | Proportion increasing; often appears in Questions 5–7 or as final problems; frequently intersects with number theory/sequences |
| 5. Number Theory | 5%–10% | Prime numbers, divisibility, factorization, modular arithmetic | Problem contexts more complex (e.g., palindromic numbers + prime determination); mostly appears in Questions 9–10 |
| 6. Sequences & Recursion | Frequent in final problems | Arithmetic/geometric sequences, term derivation, summation, second-order linear recurrence | Appears in the last four questions; often as the final challenge; some years integrated with geometry/combinatorics |
III. Four High-Score Preparation Modules (Essential for Distinction)
1. Algebra & Equations (Over 1/3 of Total Points)
Key Focus:
Factorization of high-degree polynomials (techniques for symmetric/palindromic polynomials)
Absolute value / fractional inequalities
Exponential and logarithmic equations (change of base, same-base conversion)
Function graph transformations (translation, scaling, reflection)
Past Paper Focus:
Questions 1, 3, 6 from 2019–2024. Master techniques like "splitting and adding terms" and "completing the square".
2. Geometry (25% of Points, Often a Score Differentiator)
Core Tools:
Similar triangles, power of a point theorem, Ptolemy's theorem
Coordinate methods (setting points, vectors, parametric equations)
Auxiliary line models (midpoints of chords, cyclic quadrilaterals, angle bisectors)
Strategy: If stuck with pure geometry, switch decisively to analytic geometry (coordinate system calculations).
3. Number Theory & Combinatorics (Most Flexible)
High-Frequency Topics:
Modular equations (ax ≡ b mod m)
Fermat's Little Theorem for simplifying large exponent modular arithmetic
Inclusion–exclusion principle, recursive counting (e.g., staircase climbing variants)
Problem-Solving Techniques:
"Enumerate small cases → identify pattern → generalize proof" — especially effective for difficult combinatorics problems.
4. Sequences & Recursion (Staple of Final Problems)
Essential Models:
Arithmetic / geometric sequence summation
Second-order linear homogeneous recurrence (characteristic equation method)
Fractional recurrence, periodic sequences
Final-Problem Strategy:
Even if you cannot fully solve, write down the recurrence relation + first few terms + conjecture the general term to earn step-by-step points.
IV. Sprint Phase Preparation Suggestions
1. Timed Mock Exams with Past Papers
Work through past 5 years' papers (2021–2025) strictly within 2.5 hours.
Focus on time allocation: first 6 questions ≤60 minutes; reserve ample time (≥90 minutes) for the last 4 questions.
2. Categorized Error Review
Create an error log and categorize mistakes:
Calculation errors: → Strengthen scratch paper discipline
Conceptual errors: → Supplement solution models (e.g., "seeing a circle and a tangent → recall the tangent-chord theorem")
Knowledge gaps: → Return to textbooks / topic-specific drills
3. Specialized Final-Problem Practice
Intensively study 1–2 final problems (Questions 9–10) per week; summarize:
Common integration patterns (e.g., "sequences + inequalities")
Standardized solution flows (Assume → Derive → Prove → Answer)
Techniques for earning step-by-step points (even if the final answer is wrong, a clear process can still earn ≥60% of the points)
4. Train Standardized Solution Writing
Euclid awards points step by step. Ensure you:
Clearly state assumptions and definitions
Maintain logical coherence throughout derivations
Box the final answer
