A mathematics-focused student in Ontario could take Calculus & Vectors, Advanced Functions, Computer Science, and Physics—the first two courses should be straightforward for this student.

Though Computer Science and Physics are distinctly different from the mathematics courses, these are still directly useful for a mathematics student to learn—the problem-solving skills should also carry over. Key mathematical discoveries have been inspired by problems in computer science and physics, and many rigorous university-level mathematics books still draw from problems in these fields to motivate certain problems. At the least, they are less laboratory-heavy than Biology and Chemistry (the student could still attempt these subjects, though, and choose to omit the grades for university admissions).

That leaves a couple of other classes—or just one if English is required, as noted by another commenter. My school offered subjects like Grade 12 Drama, Visual Arts, and Music, where much of the grading was effort-based. In my school, most students in my classes saw these courses as a break from other intensive courses, with grades not being as much of a concern. This would allow the student to avoid using a grade for History, Economics, French (or another foreign language), or another subject.

The English requirement would then be a difficult challenge for the mathematics-focused student. I wish I could speak more about what it was like for most of my classmates who went on to study engineering, as many of them took the standard English course (I took a more demanding version of the course, due to personal interest). My classmates at the time did not seem to have an issue with university admissions to competitive programs despite not enjoying the subject at the time, but the other commenter makes a good point that minimum grades for admission standards have increased greatly since then.

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As an aside: a past classmate later poked fun some years later about the Ontario government's naming for math courses. He said, "there's Grade 11 Functions... and then in Grade 12, there's Advanced (!) Functions." He was brilliant at every high school subject, and was also very personable. The last I heard, he now works at the Citadel hedge fund.

> then in Grade 12, there's Advanced (!) Functions."

Then in university, there’s Elementary Functional Analysis!

The first math course as math major in university involved proving that a+b=b+a and that a+(b+c)=(a+b)+c. It's quite fun to go from 'advanced' calculus during the final classes of high school, to "OK, let's consider the expression 1+1=2. What does it actually mean and why is it true?"

> The first math course as math major in university involved proving that a+b=b+a and that a+(b+c)=(a+b)+c.

Really? What were the axioms?

Basically the Peano axioms

How did that proof work for rational numbers?

It's been 20+ years, and I don't remember the exact steps the course went thought. But basically we started with defining N using the Peano axioms (although I don't recall the name 'Peano' being mentioned) and proving some basic rules of addition and equality. Then we defined subtraction, multiplication and inverse elements, and constructed Z and Q. From there you get to algebraic groups, and from there you can hand wave a lot of details.

As I said it was literally one of the first math courses we did, so it wasn't super rigorous with all the technical and logical details.

Interesting. So it seems that from reading this, Canada, or at least Ontario, has a lot more electives at grade 12 than the US. So this might be more like the UK with their A-levels. In the US a highly motivated student might take say 4-5 AP courses, which would essentially count as their freshman year at university. But they would be unlikely (short of going to a few dedicate highschools) of having more than maybe 2 math courses in a single year.

You can get tested into qualification, by most colleges. Colleges (which includes Universities, State schools, etc) aren't robotic. This is part of the responsibility of the administration. You may get delayed a semester or a year, with a little community college gatekeeping. Dedicated students can always get into a University.