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AMC · AIME · Euclid · COMC · Olympiad

Competition math teaches you how to find the solution.

School math hands you a method and asks you to apply it. A contest problem hands you nothing. We coach the thinking that gets you in — across the US and Canadian pathways, from a first AMC 8 to national selection.

Request a first sessionSee a session, in one problem

The difference

School math teaches you how to solve problems.Competition math teaches you how to find the solution.
Everything below follows from this.

Every question in a school textbook sits inside a chapter, and the chapter has already told you what to do. Contest problems arrive with no chapter. Nothing on the page says whether it wants a parity argument, a clever count, a construction or an inequality, and the real work of the first ten minutes is deciding. That decision — not the algebra that follows it — is what we coach.

It is also the part that transfers. A student who can look at an unfamiliar problem and generate three plausible lines of attack has something that outlasts any particular contest, any particular course, and any particular formula sheet.

Competition math and course tutoring are different products

Learn4Less does a great deal of ordinary course tutoring, and for most students it is the right choice. These are two different services with two different aims, not a good one and a weak one. Course tutoring makes the syllabus go well; competition math builds the thing that has no syllabus.

  • 01 — The question you are asked

    School & regular tutoring

    Here is a method. Apply it to twenty problems that look like this one.

    Competition math

    Here is a problem. Nothing you have been taught applies to it directly.

  • 02 — Where the difficulty lives

    School & regular tutoring

    In the execution — sign errors, algebra slips, running out of time.

    Competition math

    In the entry — deciding what to try before you have any evidence it will work.

  • 03 — What you practice

    School & regular tutoring

    Procedures, until they are automatic.

    Competition math

    Search: small cases, invariants, extremal choices, symmetry, contradiction.

  • 04 — Being stuck

    School & regular tutoring

    A signal that something went wrong. Ask for the next step.

    Competition math

    The normal state of the work. Stuck is where the learning is, so we do not rescue you out of it.

  • 05 — What a good answer looks like

    School & regular tutoring

    The right number.

    Competition math

    An argument another person can check — which is exactly why Fryer, Galois, Hypatia and Euclid are graded on completeness, clarity and presentation.

  • 06 — What transfers

    School & regular tutoring

    This unit's test.

    Competition math

    How you approach any unfamiliar problem, in any subject, for years.

Who this is for

Some students get much more out of this than others

This is for you if

  • You like hard problems. You would rather spend forty minutes on one problem you cannot do than an hour on thirty you can.
  • You have a contest in the calendar. There is a date — AMC 10, Cayley, Euclid, the COMC — and you want to walk in already familiar with the shape of what is coming.
  • School math has stopped asking anything of you. You finish the exercises early, your marks are fine, and almost nothing you are set requires you to think.
  • You want to reason better for its own sake. You are not chasing a medal; you want the kind of thinking that makes proofs, physics and computer science feel natural later on.
  • You have hit a wall on the back half of the paper. You clear the first fifteen questions comfortably and then stall, because the last ten are not testing anything anyone has taught you.

This is not the right fit if

  • The goal is a better report-card grade this term. That is regular tutoring, and we do that too — start there instead. See tutoring services
  • The student wants worked answers rather than to sit with a problem. Sessions are built around not being told, which is uncomfortable by design.
  • There is no room to work between sessions. An hour a week with nothing in between does not compound, and this is a subject that only pays off when it compounds.

The pathway

Two ladders, one destination

Two systems dominate North America, and students write them from all over the world. The American ladder runs through the AMC contests and is the most widely recognized internationally. The Canadian ladder is run by the CEMC at the University of Waterloo and by the Canadian Mathematical Society, and it is the one most BC schools already register for. Plenty of strong students write both, and there is no need to choose early — the mathematics is shared and only the format changes.

Registration and sitting dates move from year to year. Confirm those with your school or the organizer, not with us — we coach for these contests, we do not register students for them.

  1. Foundation

    Around Grades 6–9

    United States · MAA

    • AMC 8

      25 Q · 40 min · multiple choice · no calculator

      For most students the first time a question cannot be answered by copying a worked example.

    Canada · CEMC / CMS

    • Gauss

      25 Q · 1 h · multiple choice

      The CEMC's entry contest: a genuine first taste of contest thinking rather than a serious filter.

    • Pascal

      25 Q · 60 min · multiple choice

      Approachable at the front, deliberately awkward at the back.

    • FryerFull solution

      4 Q · 75 min · written solutions · out of 40

      The first contest that marks how well you explain yourself, not only what you found.

    • MathChallengers

      4 stages · Blitz · Bulls-Eye · Co-Op team · Face-Off

      The BC successor to MATHCOUNTS, and the only contest on this page with a team round.

  2. Intermediate

    Around Grades 9–11

    United States · MAA

    • AMC 10

      25 Q · 75 min · multiple choice · no calculator

      The main gateway to the AIME, and the paper most students target first.

    • AMC 12

      25 Q · 75 min · multiple choice · no calculator

      The senior paper, reaching into trigonometry, logarithms and complex numbers.

    Canada · CEMC / CMS

    • Cayley

      25 Q · 60 min · multiple choice

      The closest Canadian counterpart to the AMC 10 in level and feel.

    • Fermat

      25 Q · 60 min · multiple choice

      The last of the three multiple-choice papers before the senior contests.

    • GaloisFull solution

      4 Q · 75 min · written solutions · out of 40

      Fryer one year on, with the same requirement that the written argument stand alone.

    • HypatiaFull solution

      4 Q · 75 min · written solutions · out of 40

      The senior full-solution paper of the three, and good preparation for Euclid.

  3. Advanced

    Around Grades 10–12

    United States · MAA

    • AIME

      15 Q · 3 h · integer answers 000–999

      Every answer is a whole number, so there is nothing to guess between — you either have it or you do not.

    Canada · CEMC / CMS

    • EuclidFull solution

      10 Q · 2.5 h · short answer + written solutions

      Waterloo uses it in admissions and scholarship decisions, which makes it the most widely written serious contest in Canadian high schools.

    • CIMC / CSMCFull solution

      short answer + written solutions

      The Canadian Intermediate and Senior contests: more writing and less multiple choice than the Cayley and Fermat route.

    • COMCFull solution

      2.5 h · parts A / B / C

      The Canadian Open Mathematics Challenge — the official qualifier for the Canadian Mathematical Olympiad, and the natural target for a serious BC student.

  4. Olympiad

    By invitation

    United States · MAA

    • USAJMO / USAMOFull solution

      6 problems · 2 days × 4.5 h · full proof

      No multiple choice and no numerical answers — you are marked entirely on the argument.

    Canada · CEMC / CMS

    • CMO / CJMOFull solution

      full proof · held in March

      The last step before selection for Team Canada.

  5. International

    By selection

    Both pathways converge

    International Mathematical Olympiad

    6 problems · 2 days × 4.5 h · full proof · national teams of six

graded on completeness, clarity and presentation — not just the answer

From the 2025–26 season the US qualifying index is the AMC score plus 20 times the AIME score, up from 10 times — so each AIME question now counts for far more toward the next stage than it used to.

What you learn

Four subjects and a dozen strategies

The four subjects

Every contest on both ladders draws from the same four areas. The names are familiar. What they mean here is not.

  • S01

    Algebra

    School algebra is about solving for x. Contest algebra is about seeing structure: noticing that an expression is symmetric in its variables, that a system yields to adding all the equations rather than substituting one into another, that a polynomial is better understood through its roots than its coefficients. Calculus is almost never the intended route.

    Functional equations · Substitutions · Inequalities · Polynomials

  • S02

    Combinatorics

    There is no school subject called combinatorics, which is exactly why it is where most students lose the most marks. It is the mathematics of counting and arranging: how many ways, whether a configuration is possible at all, whether a process ever stops. It rewards constructions and clean case analysis rather than formulas.

    Counting · Constructions · Graphs · Games

  • S03

    Geometry

    School geometry is largely measurement and coordinates. Contest geometry is about configuration: angle chasing, similar triangles, cyclic quadrilaterals, power of a point, and the habit of adding the single line to a diagram that makes the problem collapse. The short solutions almost always come from seeing the picture properly first.

    Angle chasing · Similarity · Circles · Power of a point

  • S04

    Number theory

    Nearly absent from the BC curriculum, and unavoidable in competition. Divisibility, primes, remainders, modular arithmetic, greatest common divisors and Diophantine equations — the arithmetic of the whole numbers treated as a subject with theorems of its own. Most students improve here faster than anywhere else, precisely because so little of it depends on what they were taught before.

    Divisibility · Modular arithmetic · Primes · Diophantine equations

The strategies, named

Arthur Engel's Problem-Solving Strategies, the standard olympiad training text, is organized by strategy rather than by topic — chapters on invariance, coloring, the extremal principle, the pigeonhole principle. That is a deliberate choice and it is how we teach. A student who knows twenty theorems and no strategies stares at a blank page. A student who knows a dozen strategies always has somewhere to start.

  • Invariance Principle
  • Pigeonhole Principle
  • Extremal Principle
  • Coloring Proofs
  • Enumerative Combinatorics
  • Induction
  • Inequalities
  • Games and Strategy
  • Small Cases
  • Symmetry
  • Working Backward
  • Proof by Contradiction
  • Constructive Thinking

The eight marked in solid are chapter headings in Arthur Engel, Problem-Solving Strategies (Springer, Problem Books in Mathematics).

  • Invariance Principle

    Find a quantity the process cannot change, and use it to rule out every state that quantity forbids.

    Numbers on a board are repeatedly replaced by their difference — the parity of the total never moves, so the last number left is decided before you make a single move.

  • Pigeonhole Principle

    If you have more objects than boxes, some box holds at least two of them.

    Among any five whole numbers, two must leave the same remainder on division by four.

  • Extremal Principle

    Look at the largest, smallest, first or closest object in a configuration — it is usually forced to behave in a way the others are not.

    Assume a bad configuration exists, take the triangle of smallest area inside it, then build a smaller one — and the assumption dies.

  • Coloring Proofs

    Color the objects so that every legal move affects the colors in a fixed way, then count colors instead of tracking moves.

    A chessboard with two opposite corners removed cannot be tiled by dominoes: 30 squares of one color and 32 of the other are left, and every domino covers exactly one of each.

How a session runs

A session, in one problem

Sessions follow the same five-part arc every time, whatever the level. The order is the point. A student who is shown a solution learns that solution; a student who is led to a solution learns how solutions are found, and only the second is any use on a paper nobody has seen yet.

  1. 01

    Challenge

    The session opens with a problem you cannot immediately do. The problem is chosen carefully — hard enough to demand a real idea, close enough to what you know that the idea is within reach. You get time to sit with it before anyone says anything.

  2. 02

    Explore

    You try things. Small cases, a diagram, a guess, an approach that turns out to be a dead end. Nothing is corrected too quickly, because a line of attack that fails for an interesting reason very often points straight at the one that succeeds.

  3. 03

    Strategy

    Now we name what has been going on. This is an invariance problem, or a pigeonhole problem, or one that wants an extremal object — and the moment it has a name, it joins a category you can recognize on a future paper.

  4. 04

    Solve

    You write the solution out properly: every claim justified, every case covered, in the order a reader needs to meet them. On the full-solution contests this is worth marks directly.

  5. 05

    Extend

    Then we change the problem. Ten points instead of five. What if the square were a triangle. Extending a problem is how you find out whether you understood the idea or only the answer.

01 · Challenge AMC 10/12 · Cayley · early COMC

Difficulty

The last number on the board

The numbers 1,2,3,…,101, 2, 3, \ldots, 10 are written on a board. In one move you erase any two numbers aa and bb and write ∣a−b∣|a - b| in their place. After nine moves a single number is left.

Show that it cannot be 00, and determine exactly which values it can be.

  1. 02 · ExploreStop tracking the board

    You cannot follow the board itself — after three moves there are far too many possibilities to track, and the order of moves is not fixed.

    So look instead for something computed from the board that a move cannot disturb. Start with the total, and try a couple of moves on a smaller set to see what happens to it.

  2. 03 · StrategyName the thing that never changes

    This is the invariance principle. Rather than following the process, find a quantity the process is incapable of altering.

    The total on the board does change with every move — so ask a coarser question about it. What happens to its parity?

  3. 04 · SolveWrite the argument end to end

    For any two numbers aa and bb, the quantity (a+b)−∣a−b∣(a+b) - |a-b| equals either 2a2a or 2b2b, so it is always even. A move therefore changes the total on the board by an even amount, which means the parity of the total is an invariant.

    At the start the total is 1+2+⋯+10=551 + 2 + \cdots + 10 = 55, which is odd. After nine moves one number remains and that number is the total, so it is odd. In particular it cannot be 00.

    That settles the first half. For the second, note that ∣a−b∣≤max⁡(a,b)|a-b| \le \max(a,b), so no move ever produces a number larger than the largest already on the board. The largest starting number is 1010, so the final number is at most 1010; and it is non-negative, being an absolute value.

    The final number is therefore odd and lies between 11 and 1010, so it is one of 1,3,5,7,91, 3, 5, 7, 9 — and every one of those is genuinely reachable.

    To finish at 99: erase 1010 and 11 to write 99. Pair the remaining eight numbers as (2,3),(4,5),(6,7),(8,9)(2,3), (4,5), (6,7), (8,9), each giving 11. Combine those four 11s in pairs to get two 00s, combine those to get a single 00, and finish with ∣9−0∣=9|9 - 0| = 9. That is nine moves exactly. Replacing the first move with ∣10−7∣=3|10-7| = 3, ∣10−5∣=5|10-5| = 5 or ∣10−3∣=7|10-3| = 7 leaves eight numbers that again split into four consecutive pairs, so the same ending produces 33, 55 and 77.

    To finish at 11: pair (1,2),(3,4),(5,6),(7,8),(9,10)(1,2), (3,4), (5,6), (7,8), (9,10) to get five 11s, reduce four of them to a single 00, and finish with ∣0−1∣=1|0 - 1| = 1.

    So the last number is odd, and the complete set of possible values is {1,3,5,7,9}\{1, 3, 5, 7, 9\}.

    End of proof.

  4. 05 · ExtendWhere this goes next

    Three questions, in rising order of difficulty. We do not post the answers.

    1. Start from 11 to nn instead of 11 to 1010. Which values can the last number take?
    2. Replace ∣a−b∣|a-b| with a+ba + b. What is invariant now, and why is the problem suddenly boring?
    3. Replace it with ∣a−b∣|a - b| but start from nn copies of the same number. What changes?

    Any time a problem says "repeat this operation in any order until one thing is left", it is almost certainly asking what the operation cannot change.

We do not post the answers to 05. That is the part the student brings back.

Programs

Four ways to work

Competition coaching is quoted after we understand the student's level and target contests, because the scope varies a great deal between a weekly problem-solving track and an intensive block before an olympiad.

  • P01

    1-on-1 Coaching

    60–90 min · weekly · online or in person

    Students with a specific target, an unusual gap between their grade and their level, or a timetable that will not fit a group.

    • Weekly sessions built around problems selected for you
    • A short problem set between sessions
    • A written review after every contest you write
    Enquire

  • P02

    Small Groups

    2–4 students · matched by level

    Students who work better alongside peers at a similar level, and families who would rather share the cost.

    • A fixed weekly slot and a shared problem set
    • Time spent presenting solutions to each other
    • Small enough that everyone writes at the board
    Enquire

  • P03

    Competition Bootcamps

    short intensive block · ahead of a contest date

    Students with a contest coming up who want concentrated preparation rather than a long, slow build.

    • Full past papers written under real time limits
    • A debrief on every question missed
    • Targeted work on whichever subject is costing the most marks
    Enquire

  • P04

    Weekly Problem-Solving

    one problem a week · written feedback

    Students not aiming at any particular contest who simply want harder mathematics on a regular basis.

    • One problem worked through properly, start to finish
    • A second problem to take away
    • The least pressured way in, and it adds up across a year
    Enquire

Who you would be working with

Your coach

Doctorate
PhD in Mathematics, University of British Columbia
Teaching
13 years across school, college and university mathematics, plus competition preparation
Contest background
Competed as a student, then coached
Approach
Patience, clarity, and confidence that lasts past the contest

The interest in problem solving and logical thinking is long-standing and personal. Competing in mathematics contests as a student is what produced the step-by-step approach to unfamiliar problems that still shapes the teaching: take the problem apart, work out what it is actually asking, and build the argument in order rather than in hope.

The work spans the full range — students who need foundational support, students working at an advanced competitive level, and everyone between. The emphasis does not change across that range: patience, clarity, and confidence that outlasts the current unit.

What a student actually takes away

  • B01

    Comfort with not knowing

    Twenty minutes of being stuck stops feeling like failure and starts feeling like the ordinary first stage of solving something. That recalibration changes how a student approaches every difficult task afterward.

  • B02

    An argument someone else can check

    The CEMC's full-solution contests are marked on completeness, clarity and presentation as well as on being right — the guidance is blunt that a correct solution, poorly presented, will not earn full marks. That skill keeps paying out in lab reports and interviews long after the contest.

  • B03

    Genuine creativity

    Contest problems are solved by adding one line to a diagram, coloring a board, or looking at the quantity nobody asked about. Doing that weekly builds a habit of invention rather than a vague hope of inspiration.

  • B04

    Confidence that holds

    Confidence that came from solving something genuinely hard does not evaporate the morning of a school test. That is the difference between a student who has been reassured and one who knows what they can do.

We coach mathematical thinking, not contest tricks. If a student never sits a contest, the training still holds.

First-Session Fit Guarantee

We want your tutoring session to feel useful from the beginning. For your first session, if you feel within the first 30 minutes that the support is not the right fit, we can stop there and you will not be charged. If you choose to continue past the first 30 minutes, the full session is billed at the regular rate.

Questions

Common questions

What age or grade should a student start?

Around Grade 6 or 7 is a comfortable start, and Grade 4 or 5 works for a student who is genuinely keen. But there is no window that closes. A Grade 11 student who has never written a contest can still get a great deal out of the Euclid and the COMC, and the problem-solving habits are worth having whether or not a contest ever follows. What matters far more than age is whether the student is willing to sit with a problem they cannot immediately do.

Does my child need to be gifted, or already strong at math?

No. Contest mathematics depends less on prior schooling than almost anything else in the subject — number theory in particular is barely taught in BC schools, so very nearly everyone begins it from zero. What a student does need is some tolerance for being stuck, and that is built rather than inherited. We place students by what they can currently do rather than by a label, and Foundation work assumes nothing beyond ordinary school arithmetic and algebra.

Does the student have to actually enter a competition?

No, and a good number never do. Plenty of students come for the weekly problem-solving track because they want harder mathematics, not a result. Contests are a useful deadline and a useful measuring stick, but the reasoning, the writing and the persistence are the point, and none of the three require a contest hall.

Which contests do you prepare students for?

On the American side: AMC 8, AMC 10, AMC 12 and the AIME, plus proof work toward the USAJMO and USAMO. On the Canadian side: the CEMC ladder — Gauss, Pascal, Cayley, Fermat, Fryer, Galois, Hypatia, Euclid, CIMC and CSMC — along with the COMC, olympiad-level preparation toward the CMO and CJMO, and the BC-only MathChallengers. Many BC students write both systems, and there is no conflict in doing so.

Do you register students for contests?

No — we coach for them, but registration goes through the student's school or an approved test center. The AMC is written at a registered center, which is usually the school; if yours does not host it, the MAA publishes a tool for finding a nearby competition manager. CEMC contests are ordered by schools directly. Registration windows open months ahead and late entries are not accepted, so check early. We will happily tell you what to ask your school for.

Private sessions or a small group — which is better?

One-on-one is faster and fully adaptive, which matters most when a student is at an unusual level for their grade or is preparing for a specific contest under time pressure. A small group is better for motivation, and it supplies one thing a private session cannot: explaining your solution to a peer who is entitled to remain unconvinced. That is the most efficient way there is to discover a hole in your own argument. A fair number of students do a term of each.

How is this different from regular math tutoring?

Regular tutoring works from your course — your homework, your teacher's tests, the unit you are on now. It repairs what is broken and it is measured in grades. Competition coaching works from problems chosen for the idea they require, and the skill it builds is deciding what to do when nothing on the page tells you. We offer both, and if a student's real problem is next week's Pre-Calculus test, course tutoring is the honest recommendation.

Will this help or hurt school grades?

In our experience it helps, though indirectly and not straight away. Competition work makes ordinary school questions feel small, sharpens algebraic accuracy, and produces students who write clearer solutions and are therefore harder to deduct marks from. What it will not do is cover this week's chapter. If a student is behind in their course, fix that first and add competition work once there is room.

How much practice is expected between sessions?

For weekly problem-solving, one or two problems a week, taken seriously. For a student preparing for a specific contest, more like three to five hours a week, including at least one paper written under real time conditions. The number of hours matters less than what fills them: forty minutes of genuine struggle with a single problem is worth more than two hours of answering questions you already know how to do.

Where do you teach? Do I need to be in Vancouver?

No. Online sessions run anywhere in the world over a shared whiteboard, and we work with students well outside Canada — the contests themselves are written internationally. This subject suits an online format unusually well: the work is diagrams and written argument rather than a textbook, and everything from the session can be saved and sent afterward. In-person sessions are available in the Vancouver and UBC area subject to availability. Tell us your time zone in the enquiry and we will say honestly which slots work; students commonly switch between online and in person as the school year gets busier.

Can a student start part-way through the year?

Yes. There is no cohort to join and no syllabus to be behind on — the first session establishes where the student is and the problems are chosen from there. Starting a few weeks before a contest is fine as well, and a bootcamp block is designed for exactly that, though a longer run-up leaves room to build the strategies properly rather than rehearse them.

How much does competition coaching cost?

Competition coaching is quoted after we understand the student's level and target contests, because the scope varies a great deal between a weekly problem-solving track and an intensive block before an olympiad. Send an enquiry and we will confirm the rate and a suggested cadence before you commit to anything.

No payment to enquire

Tell us where the student is now

Grade, any contests already written and roughly how they went, and what you are aiming at. If none of that applies yet, say so — most students arrive with no contest history at all, and we will suggest a starting level.

First name is enough. If you are a parent, say so in the last box.

Where we will reply. No mailing list.

Target contests

Tap any that apply, or leave them all blank and we will suggest a starting point.

United States

Canada

Optional, but it is what shapes the first reply.

Online worldwide

Request a first sessionText