Euclid Contest Preparation: Waterloo Full Solutions

Euclid contest preparation differs from almost every other mathematics competition in one respect that students consistently underestimate: a substantial part of the paper is marked on how you write, not only on what you conclude.

Why Euclid is worth taking seriously

Euclid is the senior contest in the University of Waterloo CEMC series, written in the final years of school. Below it sit Pascal, Cayley and Fermat, and below those the Gauss contest for junior students.

What separates Euclid from the rest of the series, and from most mathematics competitions generally, is that it is considered in University of Waterloo admission and scholarship decisions. A contest that influences a university application occupies a different category from one that produces a certificate.

The exact weight is set by the university and varies by programme, so check the Waterloo admission pages directly rather than relying on any coaching site's summary, including this one.

Presentation is marked, and CEMC says so

CEMC is explicit about how full-solution answers are assessed: for completeness, clarity and presentation style, with the stated position that a correct solution which is poorly presented will not earn a full score.

That sentence should shape the whole of a student's Euclid contest preparation. It means marks are available to be lost by students who have done the mathematics correctly, which is an uncomfortable and entirely fixable problem.

In practice, a full-mark solution states what is assumed, justifies the steps that are not self-evident, handles the cases the problem actually contains, and reaches its conclusion explicitly. Students almost never do this naturally, because school examinations rarely require it.

The adjustment from multiple choice

A student arriving from AMC-style contests finds this the hardest change.

There, reaching the right number is the entire job, and the working is private. Here the working is the submission. A student who solves a problem elegantly in their head and writes two lines has given the marker nothing to award.

The converse is equally important and more encouraging: a clearly presented partial attempt that never reaches the final answer can earn a good deal. Students who learn this stop leaving hard problems blank, which is where a surprising share of the available marks sit.

What the paper covers

The content is the senior school syllabus worked to contest depth.

Algebra: equations and systems, polynomials and their roots, sequences and series, and the manipulation that must be automatic. Our algebra course covers this progression including the polynomial structure Euclid problems lean on.

Functions: domain and range, composition and inverses, transformations, exponentials and logarithms.

Trigonometry: identities, equations, and trigonometry used as a tool inside geometry problems rather than as a topic in isolation.

Analytic and plane geometry: coordinates, circles, triangles, areas, and the configurations that recur. The geometry course covers these in depth.

Counting and number theory appear less heavily than in olympiad papers but reliably enough that neglecting them costs marks.

How to prepare for the writing specifically

This is the part of Euclid contest preparation that distinguishes a good programme from a generic one, and it is simple to describe and uncomfortable to do.

Written solutions should be marked to contest standard from the first week, not from the final month. Students find the early feedback deflating: work they considered finished comes back with half of it questioned, not because the mathematics is wrong but because steps were assumed or a case was ignored.

By about the sixth week most students begin pre-empting the objections themselves, and the writing stops being an afterthought. That shift is the entire return on the exercise.

Starting the series early

A student who first meets the Waterloo problem style at Euclid is learning the format at the exact moment it counts towards an application.

The CEMC problem voice is distinctive: carefully worded, never deliberately confusing, and dependent on reading precisely what was said. A student used to skimming for numbers misreads these problems, because the condition that matters is usually stated plainly and in passing.

Writing Pascal, Cayley or Fermat in earlier years removes that unfamiliarity entirely. By Euclid the student is simply sitting another paper in a series they understand, and can spend their attention on the mathematics.

Managing the paper on the day

Because part of the paper asks for full solutions and part does not, time management on Euclid is a real decision rather than an afterthought.

The short-answer questions should be secured efficiently; they are marks every prepared student can collect and spending too long polishing them is the commonest way to run short later. The full-solution problems need genuine time, both to solve and, separately, to write.

That second cost is the one students forget. A problem solved in eight minutes may take another six to write properly, and a student who allocates only the solving time will finish with arguments that are correct in their head and incomplete on the page.

We practise this split explicitly in the closing weeks: solve, then write, with the writing timed as its own task.

How Euclid fits with everything else

For a student also preparing for the Indian olympiad pathway, Euclid is unusually useful practice. RMO and INMO are written-proof examinations, and most students arrive there having never been marked on presentation. Euclid provides that experience at lower stakes.

Against the AMC 12, the two are complementary rather than competing: AMC trains speed and recognition, Euclid trains sustained argument. Students who do both are stronger at each.

Dates, registration and arrangements for international candidates are set by CEMC each year. Check cemc.uwaterloo.ca and ask your school for the current cycle before planning around it.