Fermat Contest: Format, Eligibility, Scoring, and Preparation

The Fermat Contest is a 75-minute mathematics competition consisting of 25 questions, administered each February by the Centre for Education in Mathematics and Computing (CEMC) at the University of Waterloo. The contest targets grade 11 students (or equivalent year groups internationally) and tests problem-solving ability across algebra, geometry, number theory, and combinatorics. Students write the contest at their own schools under teacher supervision, and results are used by many universities as supplementary admission information.

Who Can Write the Fermat Contest

The Fermat Contest is designed for students in grade 11, though eligibility rules allow flexibility. Any student may write the contest regardless of grade level, but the problems are calibrated for students who have completed two years of secondary school mathematics. Younger students who write the contest compete in the same pool as grade 11 students; there are no separate divisions by age or grade.

International students follow equivalent year groups: Year 12 in the UK, Class 11 in India, or students aged 16-17 in most systems. Home-schooled students may participate if a supervising adult can administer the contest according to CEMC rules.

Students may write only one contest from the CEMC series in a given year. The series includes the Gauss (grades 7-8), Pascal (grade 9), Cayley (grade 10), Fermat (grade 11), Euclid (grade 12), Fryer (grade 9), Galois (grade 10), and Hypatia (grade 11). A student in grade 11 could choose between the Fermat and Hypatia, but not both.

Contest Format and Scoring

The contest consists of 25 multiple-choice questions divided into three sections of increasing difficulty:

• Part A: Questions 1–10, each worth 5 points
• Part B: Questions 11–20, each worth 6 points
• Part C: Questions 21–25, each worth 8 points

The maximum possible score is 150 points. Each question has five answer choices labeled (A) through (E). There is no penalty for incorrect answers, so students should answer every question even if guessing.

Calculators are not permitted. Students may not use any computational devices, formula sheets, or reference materials. Scratch paper, rulers, compasses, and protractors are allowed.

The progression in difficulty is deliberate. Part A questions typically require straightforward application of formulas or single-step reasoning. Part B questions demand multi-step problem-solving or insight into underlying structure. Part C questions often require creative approaches, pattern recognition, or combining ideas from different mathematical domains.

Consider a typical difficulty progression. A Part A question might ask for the value of \(x\) when \(3x + 7 = 22\). A Part B question might present a geometric configuration requiring the Pythagorean theorem combined with properties of similar triangles. A Part C question might ask: "How many positive integers \(n < 1000\) satisfy \(n^2 \equiv 1 \pmod{24}\)?" requiring understanding of modular arithmetic and systematic casework.

When the Contest Is Held

The Fermat Contest is held annually in February. Schools select a specific date within a designated writing period, typically spanning several days in mid-February. The exact dates vary by year.

Registration deadlines typically fall in late January or early February, approximately two to three weeks before the contest window opens. Schools must register students before the deadline; individual students cannot register directly.

Results are released approximately six to eight weeks after the contest date, usually in April. The CEMC publishes results online, and participating schools receive detailed score reports for their students.

How Registration Works

Registration occurs through schools, not individual students. A teacher or administrator at the student's school must register as a contest supervisor, order the contest materials, and arrange for proctoring on the contest day.

The school pays a registration fee per student. The fee structure varies by country and institution type. Schools outside Canada may face different fee schedules or administrative requirements.

Once registered, the school receives the contest materials—question booklets, answer sheets, and supervisor instructions—by mail or secure download, depending on the school's location. The supervisor administers the contest according to CEMC rules: 75 minutes of working time, no electronic devices, and strict supervision to maintain integrity.

Students without access to a participating school should contact the CEMC directly to explore options. Some regions have community organizations or tutoring centers that serve as satellite test centers.

What Topics the Contest Covers

The Fermat Contest assumes knowledge of mathematics through grade 11, but the emphasis is on problem-solving rather than curriculum coverage. The main topic areas include:

Algebra: Solving equations and inequalities, manipulating expressions, working with polynomials, understanding functions and their properties, exponential and logarithmic relationships, and systems of equations. Questions might involve factoring, completing the square, or recognizing algebraic identities.

Geometry: Properties of triangles, circles, quadrilaterals, and polygons; angle relationships; similarity and congruence; coordinate geometry; area and perimeter calculations; and the Pythagorean theorem. Three-dimensional geometry appears occasionally but is less common than plane geometry.

Number Theory: Divisibility rules, prime factorization, greatest common divisors, least common multiples, modular arithmetic (though usually at an intuitive level), and properties of integers. Questions often involve finding patterns in sequences or counting integers with specific properties.

Combinatorics and Counting: Basic counting principles, permutations and combinations (though formal notation is often unnecessary), logical reasoning, and systematic casework. These questions test organizational thinking and the ability to enumerate possibilities without overcounting.

Trigonometry appears occasionally but is not a major focus. Calculus is never required. The contest rewards students who can think flexibly, recognize patterns, and apply elementary techniques in novel contexts rather than those who have memorized advanced formulas.

For example, a question might state: "The integers \(a\), \(b\), \(c\) satisfy \(a + b + c = 30\) and \(a^2 + b^2 + c^2 = 350\). What is \(ab + bc + ca\)?" This requires recognizing the algebraic identity \((a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca)\), then solving: \(900 = 350 + 2(ab+bc+ca)\), giving \(ab+bc+ca = 275\). The technique is elementary, but the insight to apply it is not obvious.

How to Prepare Effectively

Effective preparation for the Fermat Contest combines topic review, problem-solving practice, and timed simulation.

Master foundational techniques: Ensure fluency with algebraic manipulation, factoring, equation-solving, and geometric properties. Many students lose points not because problems are conceptually difficult, but because algebraic errors accumulate. Practice simplifying expressions like \(\frac{x^2 - 9}{x^2 + 6x + 9}\) instantly: factor to \(\frac{(x-3)(x+3)}{(x+3)^2} = \frac{x-3}{x+3}\).

Work through past papers systematically: Past Fermat papers are the best preparation resource. Work through complete papers under timed conditions, then review solutions carefully. For each problem, ask: What was the key insight? What technique did I miss? Could I have eliminated wrong answers faster?

When reviewing Part C problems, don't just check your answer. Understand why the correct approach works and why plausible alternatives fail. If a problem asks for the number of integer solutions to \(x^2 + y^2 \leq 25\), systematically enumerate by cases (\(x = 0, \pm1, \pm2, \ldots\)) rather than guessing. Count carefully: for \(x = 0\), \(y\) ranges from \(-5\) to \(5\) (11 values); for \(x = \pm1\), \(y^2 \leq 24\) so \(y \in \{-4, \ldots, 4\}\) (9 values each, 18 total); continue this logic.

Practice mental arithmetic: Since calculators are forbidden, comfort with arithmetic is essential. Practice computing \(17 \times 23\) mentally: \((20-3)(20+3) = 400 - 9 = 391\). Or \(\frac{144}{36} = 4\) instantly. Speed on Part A and early Part B questions creates time for harder problems.

Study worked solutions, not just answers: When you encounter a problem you cannot solve, study the solution actively. Cover the solution, attempt the problem again, then check. Repeat until you can solve it independently. This builds pattern recognition.

Simulate test conditions: At least two weeks before the contest, take a full past paper under strict time limits. No pauses, no calculator, no notes. This reveals time-management weaknesses and identifies topics needing review.

Learn strategic guessing: With no penalty for wrong answers, never leave a question blank. On Part C questions, eliminate obviously incorrect answers first. If a problem asks for a count and the choices are 12, 24, 36, 48, 60, check divisibility or parity constraints to eliminate options.

Past Papers and Practice Resources

The CEMC provides past Fermat Contest papers with full solutions on their official website. Papers are available as PDF downloads, typically covering the most recent 10-15 years. These are the primary preparation resource and should form the core of any study plan.

Each past paper includes the question booklet and a separate solution document. The solutions show complete working and often explain the reasoning behind key steps. Students should work through problems independently before consulting solutions.

The CEMC also publishes past papers for related contests (Pascal, Cayley, Euclid). The Pascal and Cayley contests cover earlier material but use a similar format, making them useful for building foundational skills. The Euclid contest is more difficult and includes full-solution problems, offering a challenge for students who finish Fermat preparation early.

Problem books focusing on olympiad-style mathematics can supplement Fermat preparation, but choose carefully. The Fermat Contest emphasizes breadth and speed over depth. Books like The Art and Craft of Problem Solving by Paul Zeitz or Problem-Solving Strategies by Arthur Engel contain relevant techniques but include many problems too difficult or specialized for the Fermat. Use them selectively for topics like counting or number theory.

Online resources vary in quality. Some websites aggregate past problems with community solutions, but these solutions may contain errors or use overly complicated methods. Always cross-reference with official CEMC solutions.

How Results and Recognition Work

After the contest, the CEMC scores all papers and generates several types of recognition:

Certificates of Distinction: Awarded to top-scoring students, typically those scoring in the top 25% of all participants. The exact cutoff varies by year depending on overall performance.

School Champion certificates: Given to the highest-scoring student at each participating school.

Regional and national rankings: The CEMC publishes lists of high-scoring students by region and overall. These lists include student names and scores (with permission).

Universities in Canada, particularly the University of Waterloo, consider Fermat results during admissions. A strong Fermat score can supplement an application, especially for mathematics, engineering, or computer science programs. Some scholarship programs explicitly reference CEMC contest performance.

The recognition is meaningful but should be understood in context. The Fermat Contest is one of many mathematics competitions worldwide. It does not directly qualify students for international olympiads (that pathway runs through the Canadian Mathematical Olympiad). Its value lies in demonstrating problem-solving ability and mathematical interest to universities and in providing students with a benchmark for their skills.

Common Questions About the Contest

Can I write the contest if my school doesn't participate? Contact the CEMC to inquire about satellite test centers or alternative arrangements. Some regions have community organizations that administer the contest.

Is the Fermat Contest harder than the Euclid? No. The Euclid Contest, designed for grade 12 students, is generally more difficult. It includes full-solution questions requiring written proofs, whereas the Fermat is entirely multiple-choice.

What score is considered good? This depends on context, but as a rough guide: scoring above 100 typically places a student in the top 25%; above 120 places them in the top 10%. The maximum score is 150, but perfect scores are rare.

Should I guess on questions I don't know? Yes, always. There is no penalty for incorrect answers, so leaving a question blank guarantees zero points while guessing gives a 20% chance of earning points.

How much time should I spend on each question? With 75 minutes for 25 questions, you have an average of 3 minutes per question. Part A questions should take 1-2 minutes, Part B questions 3-4 minutes, and Part C questions 5-7 minutes. If stuck on a question beyond its time budget, make your best guess and move on.

Do I need to know calculus? No. The Fermat Contest never requires calculus. All problems can be solved using algebra, geometry, and logical reasoning accessible to grade 11 students.

Can I use a protractor or ruler? Yes. Geometric tools like rulers, protractors, and compasses are permitted. Calculators and electronic devices are not.

What happens if I'm sick on contest day? The contest is administered by your school on a specific date within the writing period. If you cannot write on your school's chosen date, ask your contest supervisor if an alternative date within the writing period is possible. Make-up exams outside the official window are generally not permitted.

The Fermat Contest rewards careful preparation, clear thinking, and mathematical curiosity. Students who work systematically through past papers, master foundational techniques, and practice under timed conditions will find themselves well-prepared for both the contest and for deeper mathematical study.