The 2026 Euclid Mathematics Contest was successfully held on April 1st in non-North American regions (including China). As the University of Waterloo's most influential math competition, this year's paper showed significant changes in content distribution, question structure, and difficulty gradient, with a particularly higher demand placed on algebraic abilities. This article provides an accurate review and efficient preparation guide based on the latest test conditions.
I. Key Changes in the 2026 Euclid Contest
1. Major adjustments in content distribution:
- The algebra section has reached its highest proportion in recent years: functions, equations, polynomials, sequences, and other algebraic content run throughout the paper, particularly dominating questions 7–10.
- Permutations & combinations and analytic geometry are significantly reduced: traditional high-frequency topics such as complex combinatorial counting and conic section problems in coordinate systems have almost disappeared.
- Changes in the assessment of trigonometry: no standalone trigonometric equation problems appeared; instead, they were integrated into composite function contexts, emphasizing conceptual understanding over mechanical solving.
Trend interpretation: Euclid is shifting from "broad coverage" to "deep exploration," with algebraic thinking and logical rigor becoming the keys to high scores.
II. Question-by-Question Style Analysis
First 5 Questions: Steady with Changes, Foundation is King
Following the style of previous years, they mainly tested:
- Algebraic operations and proportional applications (e.g., speed, concentration problems)
- Parabola vertex and axis of symmetry properties
- Prime factorization and base-10 expansion
- Basic probability (classical probability, independent events)
Although they are foundational, computational details are prone to errors; you must secure full marks on the first 5 questions.
Questions 6–8: Sudden Style Shift, Difficulty Spike
Question 6:
Combines logarithms and composite functions: no longer directly asks to solve logarithmic equations, but requires understanding the structure and reasoning using logarithmic monotonicity and domain restrictions.
Skills required: awareness of function composition + flexible use of logarithmic properties.
Question 7:
A traditional algebra synthesis problem involving inequalities and parameter discussions, but no traps; it's a "conquerable" medium-level problem.
Question 8 (the biggest difficulty this year!):
- Part (a): Tests the Inscribed Angle Theorem and the Right Triangle Altitude Theorem; no tangents are examined, and auxiliary lines need to be constructed.
- Part (b): Uniquely combines the triangle inequality and the floor function, requiring students to have relatively rich experience in handling the floor function.
Overall difficulty: Significantly higher than previous year's question 8, becoming a key differentiator between Distinction (top 25%) and Honor Roll (top 5%).
Questions 9–10: Final Tough Problems Remain, Algebra and Combinatorics as Main Tracks
Question 9 (Algebra final problem):
- Part (a): Simple graph analysis, easy points.
- Part (b): Tests understanding of polynomial roots, requiring a good grasp of Vieta's formulas and their applications.
- Part (c): Requires the ability to analyze solutions of quadratic Diophantine equations, demanding a relatively high level of number theory.
Question 10 (Combinatorics final problem):
- Part (a): Small-scale enumeration, intuitive.
- Parts (b) and (c): Relatively traditional permutations and combinations problems; constructing a recursive function for n solves them.
Although it is combinatorics, the thinking is more algebraic, reflecting the trend of "algebraization of combinatorics."
III. Difficulty and Cutoff Predictions
| Indicator | 2026 Trend |
|---|---|
| Overall difficulty | Higher than 2024–2025, especially questions 8 and 9 raising the bar |
| Distinction cutoff | Expected around 68 points |
| Honor Roll cutoff | Expected around 90 points |
IV. Preparation Suggestions for 2027 and Beyond
1. Solidify the Six Fundamental Modules
Ensure no blind spots in:
- Exponents and logarithms (including applications in composite functions)
- Functions and systems of equations (including cubic equations, Vieta's formulas)
- Trigonometry (properties, graphs, identities)
- Sequences (arithmetic/geometric, recursion, summation)
- Counting and probability (permutation and combination models, expectation)
- Elementary number theory (divisibility, modular arithmetic, Diophantine equations)
2. Strengthen Two Core Areas of Attack
| Area | Key Focus Points |
|---|---|
| Plane Geometry | - Inscribed angles, tangent-chord angles, Altitude Theorem - Auxiliary line construction (perpendiculars, medians, similar triangles) - Integration with algebra (e.g., coordinate method, vectors) |
| Counting & Probability | - Recursive modeling (state transitions) - Inclusion-Exclusion Principle, introduction to generating functions - Problems integrating with sequences/inequalities |
3. Special Plan for Final Problems
Questions 7–8: Practice 2 medium-to-high difficulty problems each week, focusing on multi-knowledge point integration.
Questions 9–10: Develop strategies based on target awards:
- Pursuing Distinction: Ensure parts (a) and (b) of question 9 and parts (a) and (b) of question 10 are correct.
- Pursuing Honor Roll: Must conquer question 8(b), question 9(c), and question 10(c); systematic study of elementary number theory and advanced recursion techniques is recommended.
