Cayley Contest: Format, Eligibility, Scoring, and Preparation

The Cayley Contest is a 60-minute mathematics competition designed for students in Grade 10 (or its international equivalent), administered annually by the Centre for Education in Mathematics and Computing (CEMC) at the University of Waterloo, Canada. The contest consists of 25 multiple-choice questions testing problem-solving skills across algebra, geometry, number theory, and combinatorics. Students worldwide may participate through registered schools or online administration, with top performers earning certificates of distinction and consideration for invitation to the Canadian Mathematical Olympiad pathway.

What the Cayley Contest Is

The Cayley Contest forms part of the CEMC's suite of mathematics contests spanning Grades 7 through 12. Named after the British mathematician Arthur Cayley, the contest aims to develop mathematical problem-solving beyond routine curriculum work. Unlike timed tests of computational speed, the Cayley emphasizes reasoning: a student must often recognize which approach will work before calculating.

The contest serves three purposes. First, it gives students experience with non-routine problems that require insight rather than memorized procedures. Second, it provides schools with comparative data on student performance relative to an international cohort. Third, strong performance opens pathways to further competitions: students scoring in the top percentiles may be invited to write the Canadian Intermediate Mathematics Contest (CIMC) or, if eligible, the Galois Contest (Grade 10 students in the Fryer-Galois-Hypatia series).

The Cayley is not a qualification round in the strict sense—there is no cutoff score required to advance—but high scorers receive recognition through certificates and school honor rolls, and CEMC uses contest results to identify students for invitations to subsequent events.

Who Can Write the Contest

Any student in Grade 10 or below may write the Cayley Contest. Students in Grade 9 or lower are eligible and may find the contest challenging but accessible; students in Grade 11 or 12 should instead write the Fermat Contest (Grade 11) or the Euclid Contest (Grade 12).

There is no age restriction, only a grade-level guideline. Homeschooled students may participate if a supervising organization registers to administer the contest. International students are welcome: the contest is written in schools across more than 30 countries. The registration is handled by schools or authorized centers, not by individual students.

Students may write only one contest per year from the Grade 9–10 series (Pascal, Cayley, Fermat). A student in Grade 10 who writes the Cayley cannot also write the Pascal in the same year.

Contest Format and Scoring

The Cayley Contest contains 25 multiple-choice questions, each with five answer choices labeled (A) through (E). Students have 60 minutes to complete the paper. No calculators are permitted.

Questions are arranged in approximate order of difficulty, divided into three sections:

• Part A (Questions 1–10): Each correct answer earns 5 points. These problems test fundamental skills—simplifying expressions, solving linear equations, interpreting data—and most students answer many of these correctly.
• Part B (Questions 11–20): Each correct answer earns 6 points. These problems require multi-step reasoning: recognizing a pattern, combining geometric and algebraic ideas, or constructing a logical argument.
• Part C (Questions 21–25): Each correct answer earns 8 points. These are the most challenging problems on the paper, often requiring creative insight or elegant observation that shortcuts lengthy calculation.

Incorrect answers and unanswered questions earn 0 points. There is no penalty for guessing. The maximum possible score is 150 points: \( 10 \times 5 + 10 \times 6 + 5 \times 8 = 150 \).

Scores are reported as raw totals. CEMC publishes statistical summaries including mean scores, standard deviations, and percentile rankings. Certificates of distinction are awarded to students scoring above a threshold determined each year, typically around the 25th percentile from the top.

Important Dates and Administration

The Cayley Contest is usually held in late February or early March each year. Schools must register in advance, typically by mid-February, and pay a registration fee per student. The exact date, registration deadline, and fee structure are announced by CEMC several months before the contest.

Schools receive the contest papers in sealed packages and administer the exam on the scheduled date under supervised conditions. Students write the contest during school hours or in an arranged session. Papers are returned to CEMC for marking, and results are released online approximately six to eight weeks after the contest date.

In recent years, CEMC has offered an online proctored option for schools unable to administer the paper version. Details of online administration, including technical requirements and supervision protocols, are provided during registration.

What Topics Are Tested

The Cayley Contest draws from the standard Grade 9 and 10 mathematics curriculum but emphasizes problem-solving over rote computation. Topics include:

Algebra

Simplifying expressions, solving linear and quadratic equations, manipulating inequalities, working with systems of equations, and understanding functions. Problems may involve recognizing patterns in sequences or constructing algebraic models of word problems.

Geometry

Properties of triangles, quadrilaterals, and circles; angle relationships; area and perimeter; coordinate geometry including distance and midpoint; similar and congruent figures; and basic trigonometry (sine, cosine, tangent in right triangles). Some problems require visualization or constructing auxiliary lines.

Number Theory

Divisibility, prime factorization, greatest common divisors, least common multiples, properties of integers, and modular arithmetic concepts (though not always stated formally). Problems may ask about digit sums, counting solutions, or properties of specific numbers.

Combinatorics

Counting techniques, permutations and combinations (often without requiring formal notation), logical reasoning, and probability. Problems may involve counting paths, arrangements, or favorable outcomes in simple experiments.

Logic and Reasoning

Some problems test pure logical deduction: working backward from constraints, eliminating impossible cases, or constructing examples that satisfy given conditions.

The contest does not test calculus, vectors, complex numbers, or advanced trigonometry. However, it does reward mathematical maturity: the ability to see structure, test small cases, and reason carefully without formal proof.

How to Prepare Effectively

Effective preparation for the Cayley Contest builds problem-solving skill, not just content knowledge. Here is how to approach it:

Master the Fundamentals

Ensure fluency with algebraic manipulation, fraction arithmetic, and geometric formulas. If simplifying \( \frac{x^2 - 9}{x - 3} \) or calculating the area of a trapezoid requires conscious effort, practice these until they are automatic. This frees mental energy for the reasoning required in harder problems.

Work Through Past Papers

CEMC provides past Cayley papers with full solutions on their website, typically going back 15 years or more. Work through these under timed conditions: 60 minutes, no calculator, no interruptions. After each paper, study the solutions carefully—not just for the correct answer, but for the reasoning path. Ask: What did I miss? What observation would have shortened my work?

When reviewing Part C problems, note that the solutions often involve a clever insight that avoids lengthy calculation. For example, recognizing symmetry, using complementary counting, or spotting a telescoping pattern. These insights come from seeing many problems, not from memorizing techniques.

Practice Estimation and Answer Checking

Since the Cayley is multiple-choice, you can often eliminate wrong answers by estimating or testing boundary cases. If a problem asks for the number of solutions and you find three, check whether the answer choices suggest you might have missed one. If a geometric problem asks for an angle and your answer is 370°, you have made an error. Develop the habit of sanity-checking.

Study Worked Solutions Actively

When reading a solution, pause before the final step and try to complete it yourself. If a solution uses a method you did not consider, rework the problem using that method to internalize it. If you solved a problem correctly but slowly, compare your approach to the official solution to see if a shortcut exists.

Focus on Weak Areas

If you consistently struggle with geometry problems, dedicate extra time to geometry. Draw diagrams for every problem, label all given information, and look for congruent triangles or similar figures. If combinatorics is difficult, practice by solving counting problems from multiple sources until the patterns become familiar.

Time Management During the Contest

Do not spend 10 minutes on Question 1. Move through Part A quickly, aiming for accuracy but not perfection. Allocate more time to Part B and Part C. If a problem seems intractable after two minutes of thought, mark it and return later. The contest rewards points, not problem order.

Where to Find Past Papers and Practice Resources

The primary resource is the CEMC website, which hosts past Cayley Contest papers and solutions as downloadable PDFs. These are freely available and cover many years. The solutions are detailed and explain the reasoning at each step.

CEMC also publishes problem sets and resource materials for teachers and students, including problem-solving tips and common pitfalls. These are listed under the "Resources" section of the CEMC site.

For additional practice, consider:

• Pascal Contest papers: The Grade 9 contest in the same series. Problems are slightly easier on average but provide valuable practice with the same question style.
• Fermat Contest papers: The Grade 11 contest. Problems are somewhat harder but still accessible, especially Part A and Part B questions.
• CEMC problem sets: The CEMC publishes themed problem sets (e.g., geometry, counting) that cover typical contest topics.
• Other national contests: The Australian Mathematics Competition (AMC) Intermediate Division, the United Kingdom Mathematics Trust (UKMT) Intermediate Mathematical Challenge, and the American Mathematics Competitions (AMC 10) all test similar skills and provide additional practice.

Books such as The Art of Problem Solving series cover contest mathematics at this level, though they include some topics beyond the Cayley scope. Use them selectively, focusing on chapters aligned with the Cayley syllabus.

Common Mistakes to Avoid

Rushing Through Part A

Students sometimes make careless errors on early problems because they assume these are trivial. Read every problem carefully. A misread word or sign error costs the same 5 points as a difficult problem.

Spending Too Long on One Problem

If you do not see a path forward after a couple of minutes, move on. You can return to difficult problems after securing points elsewhere. Time spent stuck on Question 22 could have earned points on Questions 23, 24, or 25.

Ignoring the Answer Choices

The multiple-choice format is a tool. If a problem asks which of five numbers satisfies a condition, test each one rather than solving algebraically. If the choices are far apart, estimate rather than calculating exactly. Use the structure of the contest to your advantage.

Neglecting Diagrams

For geometry problems, always draw a diagram if one is not provided, and redraw the given diagram if it is not to scale. Label all known lengths, angles, and relationships. Many errors come from misinterpreting spatial relationships.

Forgetting Units or Constraints

If a problem asks for the number of integer solutions, do not report a non-integer. If a problem asks for an area, do not report a length. Read what the question asks for and check that your answer matches.

Stopping After Finding One Solution

Some problems ask "how many" or "which of the following." After finding one solution, check whether others exist. A common error is reporting the first solution found rather than the complete count.

Frequently Asked Questions

Can I use a calculator?

No. Calculators, smartphones, and all electronic devices are prohibited. Problems are designed so that arithmetic can be done by hand or, more often, simplified before calculation.

What if I am in Grade 9? Should I write the Pascal or the Cayley?

If you are in Grade 9, the Pascal Contest is the recommended contest. However, if you have covered the Grade 10 curriculum or want a greater challenge, you may write the Cayley instead. You cannot write both in the same year.

How do I register?

Registration is handled by schools, not individual students. Speak to your mathematics teacher or school administrator. If your school does not currently participate, they can register as a new contest center through the CEMC website.

What score is considered good?

This varies by year, but typically a score above 100 places a student in the top 25%, and a score above 120 places them in the top 10%. The mean score is usually around 70–80 points. Certificates of distinction are awarded to top performers, with the threshold announced after marking.

Does the Cayley help with university admissions?

Strong performance on the Cayley demonstrates mathematical ability and problem-solving skill, which can strengthen an application. However, it is one factor among many. The Cayley is more valuable as a learning experience and a stepping stone to further contests than as a direct admissions credential.

Can I write the contest online?

CEMC has offered online proctored administration in recent years. Check the CEMC website during the registration period for details on online options, technical requirements, and supervision protocols.

What happens after the Cayley?

Students who perform well may receive invitations to further contests such as the Canadian Intermediate Mathematics Contest (CIMC). The CIMC is a more challenging, written-response contest that serves as part of the pathway toward the Canadian Mathematical Olympiad (CMO). Invitations are based on contest scores and are sent by CEMC.

Are there other contests I should consider?

Yes. If you are interested in mathematics competitions, consider the Fryer Contest (Grade 9), the Galois Contest (Grade 10, written-response format), and eventually the Euclid Contest (Grade 12), which is widely recognized by universities. International students may also explore contests in their own countries, such as the AMC series in the United States, the UKMT challenges in the United Kingdom, or regional olympiads.