The Euclid Mathematics Contest, hosted by the University of Waterloo's CEMC, is for global high school students in grades 10–12. Known for its rigorous logic, process-oriented approach, and absence of obscure questions, it serves as a key reference for top universities like MIT, Waterloo, and the University of Toronto to assess students' mathematical potential. Its difficulty lies between the Australian AMC and AMC12/AIME, making it very friendly for domestic high school students—no calculus, no multiple-choice, emphasizing logical expression over tricky techniques.
The following constructs an efficient preparation path from four dimensions: topic distribution, difficulty gradient, question trends, and sprint focus.
I. Topic Distribution: Four Major Modules, Clear Weighting
| Module | Proportion | Test Focus | Difficulty Level |
|---|---|---|---|
| Algebra | ≈40% | • Function properties & graph transformations • High-degree polynomial factorization (symmetric, cyclic) • Sequences & recurrence (arithmetic/geometric, second-order linear) • Inequalities (absolute value, rational) |
Foundation + Advanced, largest scoring section |
| Geometry | ≈30% | • Plane geometry (triangles, circles, similarity, congruence) • Analytic geometry (locus, vectors, parametric equations) • Power of a point, Ptolemy's theorem, chord-tangent angle |
Often integrated with algebra, auxiliary line construction is key |
| Number Theory | ≈15% | • Divisibility, prime factorization • Congruence, modular arithmetic • Fermat's Little Theorem (for simplifying calculations) |
Often appears in medium/hard problems, frequently combined with algebra/combinatorics |
| Combinatorics & Probability | ≈15% | • Counting principles (addition/multiplication) • Inclusion-exclusion principle, recursive counting • Classical probability, conditional probability |
High frequency in final problems, emphasizing rigorous thinking |
Key Features:
No beyond-syllabus content: Entirely based on high school mathematics extensions.
Strong integration trend: Problems 6–10 often cross 2–3 modules (e.g., "solving geometry problems using sequences").
II. Difficulty Gradient: Three-Tier Structure, Precise Stratification
| Question # | Difficulty | Tested Abilities | Target Strategy |
|---|---|---|---|
| 1–4 | Foundation | • Formula application • Simple reasoning • Calculation accuracy |
Must get all correct! Target accuracy ≥95% → Foundation for aiming for Distinction (~70 points) |
| 5–8 | Medium (Core Score-Differentiation Zone) | • Multi-knowledge point integration • Hidden condition identification • Numerical-graphical combination ability |
Focus on breaking through! → Determines if you can reach top 25% → top 5% |
| 9–10 | Final (High-Score Differentiator) | • Abstract modeling • Constructive proof • Rigorous logical chain |
Process marks > answer marks → Even if unsolved, writing a reasonable approach can earn 3–6 points. |
III. Recent 5-Year Question Trends
Foundation Problems are more "Flexible": The first 7 problems no longer mechanically apply formulas but test depth of understanding through variations (e.g., deducing a function's equation from its graph).
Cross-Module Integration is Normalized:
Example 2023 Q9: Using recurrence sequences to find the perimeter limit of an inscribed polygon (Algebra + Geometry + Limit concept).
Example 2022 Q8: Congruence equations + Combinatorial counting (Number Theory + Combinatorics).
Increase in Real-World Context Modeling: E.g., "cell tower coverage area," "loan repayment model," testing the ability to abstract mathematical structure from real problems.
Final Problems Emphasize "Solvability" over "Obscurity": Although difficult, they provide clear logical steps (sub-questions A→B→C), encouraging students to demonstrate their thought process.
IV. Four Key Focus Areas for the Sprint Phase (With Past Paper Strategy)
1. Algebra and Equations
Core Skills: High-degree polynomial factorization (including symmetric forms), functions, inequalities & graph transformations (involving absolute value, rational inequalities), sequences & recurrence (focus on arithmetic, geometric, second-order linear).
Past Paper Focus: Practice questions 1, 3, 6 from 2019–2024, mastering techniques like "completing the square," "splitting/adding terms," "symmetric substitution."
2. Geometry (Constructive Ability is Decisive)
High-Frequency Models: Cyclic quadrilaterals (opposite angles supplementary, Ptolemy's theorem), midpoint of a chord + perpendicular bisector of chord theorem, finding loci in coordinate systems (parametric method, elimination).
Training Suggestion: Hand-draw 1 geometry problem daily, force yourself to draw auxiliary lines, summarize "when to draw perpendiculars? When to connect to the circle's center?"
3. Number Theory & Combinatorics (Breakthrough for Final Problems)
Practical Strategies:
For integer solution problems → Try mod 3 / mod 4 analysis to narrow the scope.
Complex counting → Enumerate small cases (n=1,2,3) to find patterns.
Probability problems → Clearly define the sample space + favorable events.
4. Past Paper Practice + Process Refinement
Timed Mock Exams: 2 sets of recent 5-year past papers weekly, strict 150-minute limit, simulating exam rhythm.
Mistake Classification:
| Error Type | Countermeasure |
|---|---|
| Calculation Error | Strengthen scratch work standards, verify each step. |
| Wrong Approach | Redo + compare with official solution, extract the "key to solving." |
| Knowledge Gap | Return to module-specific training. |
2026 Season Euclid Mathematics Contest registration is open!