When it comes to international math competitions, many parents immediately think of high-difficulty, high-investment contests like AMC and AIME, and assume that their child—"not a competition-oriented student"—probably stands no chance. But in fact, the Euclid Mathematics Contest is precisely the opposite: it does not demand advanced problem-solving tricks or raw talent; it rewards consistency and a solid grasp of fundamentals. If you are still weighing whether to enroll your child in an international math competition, or worry that heavy preparation might not yield an award, Euclid may be the most suitable choice.
I. Three Groups of Students Who Are Especially Suited for Euclid
1. Students with a strong foundation in school mathematics, but without traditional competition training
The Euclid contest does not blindly pursue advanced competitive techniques; instead, it places greater emphasis on mathematical fundamentals and logical reasoning ability. For students in Grade 10 or 11 who perform well in regular math courses, think clearly and logically, but lack systematic competition training, Euclid offers an excellent entry point.
2. Students with good English skills who intend to study in North America (especially Canada)
Euclid scores are highly regarded by North American universities and serve as a "hard currency" for STEM applications. For students targeting the University of Waterloo's Faculty of Mathematics and Faculty of Engineering in particular, Euclid results are a core reference in admission decisions and entrance scholarship awards.
3. Students seeking high cost-effectiveness and a quick boost to their competition profile
Compared to high-intensity contests such as AMC12 and AIME, Euclid is notably more moderate and stable in difficulty—overall positioned between AMC10 and AMC12. The preparation cycle is short and the return on investment is exceptionally high: the top 25% globally receive a Certificate of Distinction. For students with strong math test-taking skills, just a short period of targeted training—adapting to North American mathematical logic and English expression—can yield impressive results with relatively low time investment, adding significant weight to college applications.
II. Difficulty Positioning: Between AMC10 and AMC12, Top 25% Earn an Award
Many parents ask: just how difficult is Euclid? Objectively speaking, Euclid's difficulty falls between the AMC10 and AMC12.
The contest is scored out of 100 points, with a duration of 2.5 hours. It consists of 10 long-answer questions, each containing 2–3 sub-questions, progressing from easy to challenging.
More importantly, the award threshold is relatively accessible—scoring in the top 25% earns a Certificate of Distinction. For Chinese students, whose mathematical test-taking ability is generally strong, achieving this certificate is not particularly difficult after short-term training to become familiar with North American problem-solving logic and response conventions.
Looking at historical cutoff scores, Euclid's scoring system is remarkably stable. Based on data from 2019 to 2026, the fluctuations in the average score and the top-25% cutoff have consistently remained within 1–3 points, ensuring both the value and fairness of the awards remain intact:
Certificate of Distinction (Top 25% globally): 66 points
Global average score: 52.2 points
School Medal Champion (School Champion Award): 52 points
The slight decline in the 2026 cutoff is not because the questions became easier, but rather a natural consequence of adjustments in the question design style.
The 2026 Euclid contest has revealed clear trends in question design. Future test-takers should prepare accordingly:
Signal 1: Emphasis on understanding, not rote techniques
Questions are moving away from the direct application of single知识点 (knowledge points) and instead require students to truly grasp mathematical definitions and derivation processes, rather than relying on template-based problem-solving. This means that the strategy of "brute-force practice of question types" is becoming less effective—understanding the essence of concepts is the key to scoring.
Signal 2: Cross-topic integration
Euclid is breaking down the barriers between algebra, geometry, probability, and other modules. The proportion of questions that combine multiple knowledge areas has increased significantly. For example, there are now more questions that integrate analytic geometry with probability, closer to the thinking patterns of university-level mathematics. Students need to develop the ability to "call upon knowledge across modules."
Signal 3: Geometry questions are moving away from fixed models
Common fixed geometric models found in traditional competitions are being reduced. Instead, students are expected to perform rigorous derivations by setting up coordinate systems and using proportional relationships. This places higher demands on spatial imagination and the completeness of logical chains.
IV. Advice for Parents and Students
If you match any of the following profiles, Euclid is worth prioritizing:
Above-average math grades at school, but no competition background
Targeting Canadian or North American undergraduate programs and needing academic profile enhancement
Limited time, hoping to exchange 3–6 months of preparation for a highly recognized certificate
In terms of preparation strategy, avoid the old "drilling endless problems" approach. Instead:
Focus on understanding core concepts and derivations rather than memorizing problem templates
Practice cross-topic questions to build flexible thinking
Get familiar with the English mathematical expression and North American style of problem presentation
Work through past papers to understand the progression of difficulty within each long-answer question
One of the most valuable resources available to students preparing for the Euclid Mathematics Contest is the collection of official solutions published by the Centre for Education in Mathematics and Computing. These solutions, freely available on the CEMC website, provide far more than just the answers to past contest questions. They offer a window into the mindset, methodology, and communication standards expected of top-performing students. Yet many students treat these solutions as mere answer keys, checking their work quickly and moving on without extracting the full learning value.
Official Euclid solutions are one of the most powerful learning tools available
In this article, we provide a comprehensive guide to mastering the art of reading and learning from Euclid solutions. We will explore how to analyze official marking schemes, identify key problem-solving strategies, understand the communication standards expected by markers, and use solutions to systematically improve your performance. Whether you are just beginning your Euclid preparation or are refining your skills for contest day, this guide will help you transform past solutions from simple answer keys into powerful learning tools.
Understanding how solutions are structured helps you learn more effectively
Before diving into how to learn from Euclid solutions, it is important to understand how they are structured and what they are designed to communicate. Official Euclid solutions are written by experienced mathematicians and educators, and they follow a consistent format that reflects the marking standards of the contest.
Each solution typically begins by restating the problem or the key information given, then proceeds through a series of logical steps to reach the final answer. Along the way, the solution will define variables, state theorems or formulas being used, and explain the reasoning behind each step. The solution concludes with a clear statement of the final answer, often boxed or highlighted for emphasis.
Importantly, official solutions are written to demonstrate ideal mathematical communication. They show not just what to do, but how to present your work in a way that is clear, logical, and easy for a marker to follow. This is crucial, because the Euclid awards marks not just for correct answers but for the quality of your mathematical reasoning and presentation.
The Four-Pass Method for Studying Solutions
To extract the maximum learning value from Euclid solutions, we recommend a four-pass approach. Each pass serves a different purpose and helps you develop different aspects of your problem-solving abilities.
The first pass is the attempt pass. Before looking at the official solution, attempt the problem on your own under timed conditions. Write out your solution as completely as you can, even if you are not confident in your approach. This pass is essential because it forces you to engage actively with the problem and identify your own strengths and weaknesses. Without this attempt, you will not have the context needed to appreciate the insights in the official solution.
Comparing your solution to the official one reveals valuable insights
The second pass is the comparison pass. After completing your attempt, carefully read the official solution and compare it to your own work. Look for differences in approach, methodology, and presentation. Ask yourself: Did the official solution use a different technique than I did? Are there steps in the official solution that I missed or could have presented more clearly? Did I make any errors in reasoning or calculation? This pass helps you identify specific areas where your problem-solving skills need improvement.
The third pass is the analysis pass. Go back through the official solution line by line and analyze every step. For each step, ask yourself: Why did the author take this step? What theorem or principle justifies this step? Could this step have been done differently? What would have happened if I had taken a different approach? This deep analysis helps you understand not just the mechanics of the solution but the underlying problem-solving strategies that make the solution work.
The fourth pass is the re-attempt pass. A week or two after studying the solution, attempt the problem again without looking at the official solution. This time, try to incorporate the insights you gained from your analysis. Write out your solution as completely and clearly as you can, using the techniques and presentation standards you observed in the official solution. This pass tests whether you have truly internalized the lessons from the solution and can apply them independently.
Identifying Key Problem-Solving Strategies
Official solutions reveal the strategies that top students use
As you work through official Euclid solutions, you will notice that certain problem-solving strategies appear repeatedly. These are not just tricks or shortcuts; they are fundamental approaches that reflect deep mathematical understanding. By identifying and internalizing these strategies, you can significantly improve your ability to tackle unfamiliar problems on the contest.
One common strategy is working backwards from the desired result. In many Euclid problems, especially those involving proofs or optimizations, it is helpful to start by considering what the final answer should look like and then work backwards to determine the steps needed to reach it. Official solutions often demonstrate this approach implicitly, so pay attention to how the solution is structured and whether working backwards would have been a natural way to discover the approach.
Another important strategy is breaking complex problems into simpler sub-problems. Many Euclid questions, particularly in the later parts of the contest, involve multiple steps or require you to combine several concepts. Official solutions often demonstrate how to decompose a complex problem into manageable pieces, solve each piece separately, and then combine the results. Learning to recognize when and how to apply this strategy is essential for success on the harder questions.
A third strategy is looking for patterns and generalizations. In problems involving sequences, series, or combinatorics, official solutions often reveal patterns that simplify the problem or lead to a more elegant solution. By studying how these patterns are identified and exploited in official solutions, you can develop your own pattern recognition skills and learn to apply them in new contexts.
Learning Communication Standards from Official Solutions
One of the most valuable aspects of official Euclid solutions is that they demonstrate the communication standards expected by markers. The Euclid is a full-write-in contest, meaning that marks are awarded not just for correct answers but for the clarity, logic, and completeness of your written solutions. Official solutions show you exactly what this looks like in practice.
Official solutions teach you how to communicate mathematics effectively
Pay close attention to how variables are defined in official solutions. Notice that every variable is clearly defined when it is first introduced, often with a phrase such as "Let x represent the number of..." This clarity makes it easy for the marker to follow the solution and understand what each variable represents. In your own solutions, you should follow this standard and define all variables explicitly.
Also observe how theorems and formulas are stated before they are used. Official solutions typically name the theorem or formula being applied, such as "By the Pythagorean theorem..." or "Using the quadratic formula..." This not only justifies the step but also demonstrates your understanding of the underlying mathematics. In your own solutions, you should state the theorems and formulas you use, even if they seem obvious.
Finally, notice how the final answer is presented in official solutions. The final answer is often boxed, underlined, or otherwise highlighted to make it easy for the marker to find. The solution also typically includes a brief statement that connects the final answer back to the original problem, such as "Therefore, the value of x is 5." In your own solutions, you should follow this convention and make sure your final answer is clearly stated and easy to find.
Building a Solution Journal
To systematize your learning from official Euclid solutions, we recommend maintaining a solution journal. This journal serves as a personal reference that captures the key insights, strategies, and techniques you learn from each solution you study.
For each official solution you study, create an entry in your journal that includes the following elements. First, write a brief summary of the problem and the main approach used in the official solution. Second, note any new techniques, strategies, or insights that you learned from the solution. Third, identify specific mistakes or weaknesses in your own attempt that the official solution helped you address. Fourth, record any key formulas, theorems, or patterns that you want to remember for future reference.
Review your solution journal regularly, especially in the weeks leading up to the contest. This review will help reinforce the lessons you have learned and keep key strategies fresh in your mind. Over time, your solution journal will become a valuable personal resource that reflects your growth as a problem solver.
Common Mistakes Students Make When Reading Solutions
While official Euclid solutions are invaluable learning tools, students often make certain mistakes that limit their effectiveness. Being aware of these mistakes can help you avoid them and get more out of your solution study sessions.
Effective solution study leads to mastery over time
One common mistake is reading solutions too quickly. Many students glance at the official solution, see that their answer matches, and move on without carefully analyzing the approach. This is a missed opportunity. Even if your answer is correct, the official solution may use a more elegant or efficient approach than you did, or it may reveal insights that you can apply to future problems. Take the time to read each solution carefully and understand every step.
Another mistake is failing to attempt the problem first. Some students go straight to the official solution without trying the problem on their own. This approach severely limits the learning value of the solution. Without first struggling with the problem yourself, you will not have the context needed to appreciate the insights in the official solution, and you will not be able to identify your own weaknesses. Always attempt the problem before reading the solution.
A third mistake is not re-attempting the problem later. Studying a solution once is not enough to internalize the lessons. You need to revisit the problem after some time has passed and attempt it again without looking at the solution. This re-attempt tests whether you have truly learned from the solution and can apply the insights independently. Make re-attempting problems a regular part of your study routine.
Using Solutions to Identify Weak Areas
Official Euclid solutions are not just tools for learning new techniques; they are also powerful tools for identifying your weak areas. As you work through solutions, pay attention to the types of problems and steps that give you difficulty. These patterns will reveal the areas where you need to focus your study.
For example, if you consistently struggle with problems involving trigonometric identities, this indicates that you need to strengthen your knowledge of trigonometry. If you find that you often miss steps in algebraic manipulation, this suggests that you need to work on your algebraic fluency. If you have difficulty understanding the reasoning in geometric proofs, this points to a need for more practice with geometric reasoning.
Keep a running list of the weak areas you identify as you study solutions. Use this list to guide your study plan, focusing on the topics and skills where you need the most improvement. Over time, as you address these weak areas, you will see your overall performance improve.
The Long-Term Benefits of Solution Study
Studying official Euclid solutions is not just about preparing for a single contest. The skills and insights you develop through this process have long-term benefits that extend far beyond the Euclid. You will develop stronger problem-solving skills, better mathematical communication abilities, and a deeper understanding of the mathematical concepts that appear in the contest.
These skills will serve you well not only in future mathematics competitions but also in your university studies and your career. The ability to analyze complex problems, identify key strategies, and communicate your reasoning clearly is valuable in any field that requires analytical thinking. By mastering the art of reading Euclid solutions, you are investing in skills that will benefit you for years to come.
So the next time you sit down to study for the Euclid, do not just work through problems and check your answers. Dive deep into the official solutions, analyze them carefully, and extract every lesson they have to offer. Your future self will thank you for the effort.
