COMC Canadian Open Mathematics Challenge Practice Test PDF (Free Printable 2026)

COMC Canadian Open Mathematics Challenge Practice Test PDF

The Canadian Open Mathematics Challenge (COMC) is the first major competition in the Canadian mathematical olympiad pathway, open to any student in Canada up to Grade 12. Organized by the Canadian Mathematical Society (CMS), the COMC serves as a qualifying round for the Canadian Mathematical Olympiad (CMO) and, through it, Canada's International Mathematical Olympiad (IMO) team selection. A strong COMC result also opens the door to the Repêchage competition and invitations to math camps.

Preparing for the COMC requires more than computational fluency — it demands mathematical creativity, proof-writing skill, and the ability to solve problems you have never seen before. Working through printed practice tests by hand, without hints, is one of the most effective training methods available. Download our free PDF and use it alongside the official past papers on the CMS website.

COMC Content Areas and Key Topics

Algebra

Algebra questions on the COMC range from accessible manipulation tasks in Part A to intricate functional equations in Part C. Core skills include polynomial manipulation (factoring, expanding, synthetic division), working with rational expressions and algebraic fractions, and completing the square to analyze quadratic forms. Systems of equations appear regularly — both linear systems that reward substitution or elimination and nonlinear systems that require clever substitution or parametric approaches. Functional equations, where the unknown is a function rather than a number, are a COMC signature: typical forms include f(x + y) = f(x) + f(y) or f(f(x)) = x, and solving them rigorously requires careful case analysis and verification.

Number Theory

Number theory problems reward students who are comfortable with divisibility arguments, prime factorization, and modular arithmetic. Expect questions that ask you to find all integers satisfying a divisibility condition, prove that an expression is always divisible by a given integer, or determine the last two digits of a large power (which requires modular arithmetic). Diophantine equations — equations in integers — appear in Parts B and C and often combine number theory with algebraic insight. Knowing Fermat's Little Theorem and the Chinese Remainder Theorem at a conceptual level is helpful for the hardest problems.

Geometry

COMC geometry problems span classical Euclidean proofs, circle theorems, and coordinate geometry. You should be able to use the properties of parallel lines, similar triangles, and congruent triangles fluently. Circle theorems — inscribed angle theorem, power of a point, Ptolemy's theorem — are tested regularly in Parts B and C. Triangle centers (centroid, circumcenter, incenter, orthocenter) and their properties appear both as standalone topics and as tools inside larger problems. Coordinate geometry questions sometimes require you to find areas using the shoelace formula or locate intersection points of loci defined by geometric conditions.

Combinatorics

Combinatorics is frequently the most accessible subject for students with strong algebraic instincts. Counting problems in Part A test permutations and combinations directly. Parts B and C require the inclusion-exclusion principle (counting objects with overlapping properties), the pigeonhole principle (proving that some configuration must exist), and bijective proofs (showing two counts are equal by constructing an explicit correspondence). Graph theory concepts — paths, cycles, colorings — occasionally appear in the hardest Part C problems.

Inequalities

Inequality problems reward students who know a handful of classical results. The AM-GM inequality (arithmetic mean ≥ geometric mean) is the most frequently applied tool on Canadian competitions; it handles optimization problems and bounds elegantly. The Cauchy-Schwarz inequality appears in more advanced Part C problems involving sums of fractions or dot products. When working with inequalities, always verify that equality conditions are achieved — a proof that only establishes a bound without demonstrating that the bound is tight is incomplete.

The COMC to CMO Pathway

The COMC is a national contest written by thousands of students each year. The top 50 or so scorers receive invitations to write the Canadian Mathematical Olympiad (CMO), a six-problem proof-based exam with a five-hour time limit. CMO results, combined with performance at competitions such as the Asian Pacific Mathematics Olympiad (APMO), determine Canada's six-member IMO team. Students interested in the full pathway should also explore the CMS's online problem sets and consider attending regional math circles or the Canadian Mathematical Society's summer programs for high performers.

How to Approach COMC Practice

The COMC rewards problem-solving instinct built through repeated exposure to competition-style problems. Unlike school exams, where knowing the right formula is often enough, COMC problems require you to combine ideas from different areas of mathematics in unexpected ways. The best preparation strategy is to work through problems you cannot immediately solve — sit with a hard problem for 20 to 30 minutes before consulting a solution. That productive struggle builds the mathematical intuition that distinguishes high scorers.

After working through our practice PDF, review the official CMS past papers, which date back several decades. Solutions are available on the CMS website and from Art of Problem Solving (AoPS). For each problem you miss, write a short summary of the key insight so you can recognize similar patterns in future competitions.