Among internationally recognized math competitions, the Euclid Mathematics Contest, organized by the University of Waterloo, is known as the "TOEFL of the math world." It is not only an "entry ticket" to the University of Waterloo's Faculty of Mathematics but also an important benchmark for top G5 universities such as Oxford and Cambridge to select science and engineering talent.
To break into the top 5% or even the top 1% globally in the Euclid contest, a solid foundation in regular math classes is far from enough. You need a precise "knowledge point map" and a flexible set of "exam hall game theory."
I. Core Knowledge Points: "Compulsory Courses" for High Scorers
The distribution of questions in Euclid follows a highly consistent pattern: Questions 1-6 are basic and intermediate levels, questions 7-8 are the "divide," and questions 9-10 are the zone where top-tier competitors contend. To consistently achieve a high score of 85+, you must master the following modules:
1. Advanced Algebra and Functions (Algebra & Functions)
Algebra is the lifeblood of the Euclid contest, permeating almost all of the problems.
Core points: Polynomial division, the Remainder Theorem, advanced applications of Vieta's formulas.
High score requirements: Proficiency in manipulating logarithms and exponential equations, especially in comprehensive problems involving complex function graph transformations and optimization (using discriminants or the AM-GM inequality).
2. Analytic Geometry and Trigonometric Functions (Geometry & Trigonometry)
Euclid has a strong preference for combining circle properties with trigonometric functions.
Core points: Law of Sines, Law of Cosines, sum-to-product formulas.
High score requirements: Proficiency in using analytic methods (coordinate systems) to solve plane geometry problems. You must also be familiar with "contest-level" geometry concepts like the Power of a Point theorem and Ptolemy's theorem, which are often needed for geometry problems in questions 8-9.
3. Combinatorics and Probability (Counting & Probability)
This module is highly flexible in its questioning.
Core points: Stars and bars method, the Inclusion-Exclusion Principle, conditional probability.
High score requirements: The ability to handle complex dynamic counting problems, especially probability models that often appear in the last two questions and require solving with recurrence relations.
4. Elementary Number Theory (Number Theory)
Although not as deep as in the AMC, it appears every year.
Core points: Divisibility analysis, congruences, greatest common divisor and least common multiple, prime factorization.
High score requirements: Mastery of the Euclidean algorithm and its inverse operations, and understanding fundamental number theory tools such as Bézout's identity.
II. Exam Hall Strategies: How to Maximize Your Score
The Euclid contest is 150 minutes long, consists of 10 long-answer questions, and is worth a total of 100 points. Its unique feature is that it includes both "answer-only" questions and "full-solution" questions.
1. Master the Scoring Logic of "Full-Solution" Questions
The final few questions in Euclid typically require you to write out a complete proof or solution steps.
Strategy: Examiners award points step-by-step. Even if you don't get the final answer correct, if your logical framework is correct, you've written down the core equations, and you've proven the key propositions, you can still earn 60%–80% of the points. Never leave a blank space. Even if you only write down the given conditions and your line of reasoning, do so.
2. Adhere to a Strict "Writing Standard"
As the exam is manually graded, clear logical expression is crucial.
Strategy: Use connective words like "Because/Therefore" to outline the logical chain. For geometry problems, be sure to briefly describe how you constructed your auxiliary lines. For algebra problems, clearly define your variables. A neat, presentable answer sheet can significantly improve the "impression score" from the grader.
3. Apply the "Asymmetrical Principle" of Time Allocation
Questions 1-6 (within 45 minutes): Aim for "zero errors." These questions form the foundation for winning any award. Answer them quickly and accurately, and don't waste too much scrap paper on them.
Questions 7-8 (45 minutes): Approach these steadily. These two questions determine whether you will receive a certificate (Distinction). Reserve enough time for a thorough logical breakdown.
Questions 9-10 (60 minutes): These are for deep strategic play. They usually have 2-3 sub-questions, with the first often being a guide. Even if you can't solve the final part, make sure to securely earn the points from the first part.
4. Make Good Use of "Specialization" and "Estimation"
When dealing with multiple-choice or fill-in-the-blank style questions, if the algebraic expression is too abstract:
Strategy: Try substituting special values (like 0, 1, or extreme geometric positions) to find patterns. This "informal" method is very efficient for the first half of the Euclid contest.
III. Preparation Advice for the 2027 Season
Practice with past papers is the only shortcut: The style of Euclid questions is very consistent. It is recommended to go through the past 10 years' past papers at least twice, especially questions 7-10.
Strengthen your English expression: Many students are strong in math but cannot accurately describe their logical reasoning in English. It is recommended to read the Full Solutions officially released by the University of Waterloo and mimic their style of expression.
Mindset building: The Euclid contest is a contest of "patience." 150 minutes is a long time, and the last two questions may leave you completely puzzled. However, staying calm and continuously trying to find a breakthrough by analyzing small-scale data is often the key to success.
