COMC Preparation: Canadian Open Challenge Guide

COMC preparation hinges on a single number that most students never work out: Part C is worth forty of the eighty marks. Everything about how a student should train follows from that.

The paper, by the numbers

The Canadian Open Mathematics Challenge is run by the Canadian Mathematical Society, which calls it Canada's premier national mathematics competition. It is open to students in Grade 12 and below, with the CMS recommending Grade 8 and up, and it is written internationally rather than only in Canada.

The structure is two and a half hours for 80 marks in three parts. Part A is four questions at 4 marks each. Part B is four questions at 6 marks each. Part C is four full-solution problems at 10 marks each.

In Parts A and B the final answer is what is required, although work shown can earn partial marks when the answer is wrong. In Part C the complete argument must be written out.

Half the paper sits in four problems

Any serious COMC preparation plan starts here. Parts A and B together are 40 marks across eight questions. Part C is 40 marks across four.

So a single Part C problem is worth more than two Part A questions, and a student who leaves Part C untouched has capped themselves at 50 per cent before writing anything.

This happens constantly. Students see four longer problems demanding proof, judge them out of reach, and spend the remaining time rechecking Parts A and B. They have protected a handful of marks and forfeited forty.

Often those problems are not out of reach. They usually need one idea rather than advanced machinery, and a student who has practised starting — writing down what is given, trying a small case, drawing the configuration — finds a reasonable share of them open up.

Writing a solution that earns the marks

Part C marks the argument, not the answer. A student who reaches the right number without justifying it can score very little; a clearly presented partial attempt that never reaches the answer can score well.

What a full-mark solution contains is not mysterious: state what is assumed, justify the steps that are not self-evident, handle the awkward case rather than the convenient one, and finish by reaching the conclusion explicitly rather than implying it.

Students almost never write this way without being taught, because school examinations rarely ask for it. Good COMC preparation marks written work to contest standard from early on, which is uncomfortable at first and decisive by the end.

Time as a strategic decision

With 40 marks concentrated in four long problems, COMC preparation has to treat time as a decision: dividing two and a half hours evenly across twelve questions is usually wrong.

Students who secure Parts A and B efficiently and then commit the bulk of the remaining time to Part C outscore students of equal ability who simply work through in order and run out of clock.

The harder discipline is abandoning a Part C problem that is not yielding, in order to start another. That decision is far easier settled in advance than at minute ninety with adrenaline running.

What to cover

Algebra underpins everything: factoring, surds, inequalities, systems, worked to the fluency the paper assumes rather than the fluency school requires. Our algebra course follows that progression.

Number theory covers divisibility, modular arithmetic, primes and digit problems, taught as competition problems from the start. A student who has only met modular arithmetic as a clock analogy will not get far with a problem built on it.

Geometry works triangles, circles, similar figures and areas, with attention to the configurations that recur year after year.

Combinatorics covers systematic counting, pigeonhole, invariants and recursion, where a clear idea beats heavy calculation. Our combinatorics course covers these methods in the depth Part C rewards.

Where a strong result leads

This is the question families ask first. Top scorers are invited to the Canadian Mathematical Olympiad or the Junior CMO, and the CMO is the qualifier for Canada's IMO team. A group of leading students is also selected to write the APMO as Team Canada.

Qualifying scores are set each year and move, so we do not publish a target mark. Anyone quoting a fixed cut-off is quoting one particular year, and planning around it is a mistake.

The habit that unlocks Part C

Students who score on Part C are rarely the ones who know more theorems. They are the ones who have practised starting.

Faced with a full-solution problem, the trained response is mechanical rather than inspired: write down what is given, in symbols. Try the smallest case. Draw the configuration if there is one. Ask what is being conserved. Guess the answer and test whether it fits.

None of that requires recognising the problem, which is the point. A student with five reliable opening moves can begin any Part C problem, and beginning is most of the difficulty. Very often the first or second move reveals the structure and the rest follows.

Students who wait until they recognise a problem before starting will leave Part C blank most years, regardless of how much theory they have covered.

How COMC compares with the alternatives

Against the AMC 10, the COMC is slower and asks for writing. AMC rewards recognising a type and executing quickly; COMC rewards sustained argument on its most valuable section.

For students on the Indian olympiad pathway, COMC is unusually good practice. RMO and INMO are written-proof examinations and most students arrive there having only ever produced answers. Part C gives that experience at lower stakes and earlier.

The CEMC written-solution contests serve a similar purpose and are worth doing alongside.

What a year of preparation looks like

Expect the early months to feel slow. A student new to writing solutions produces work they consider finished and gets it back with half of it questioned — not because the mathematics is wrong, but because steps were assumed.

By the third month most students begin pre-empting those objections themselves, and the writing stops being an afterthought. That is when Part C becomes available to them, and that is when the score changes.

Dates, qualifying scores and arrangements for international candidates are set by the CMS and change between cycles. Check cms.math.ca for the current year.