How to Prepare for the British Mathematical Olympiad: Writing Proofs, Practising Problems and Getting Feedback

A student can achieve an excellent result in the Senior Maths Challenge and still find their first British Mathematical Olympiad paper uncomfortable. They may have the right idea and even the right answer. The harder question is whether another person would accept their argument as a complete proof.

Olympiad mathematics is about learning to communicate mathematical reasoning, and that requires feedback.

A student often cannot see the assumption they have left unstated or the case they have overlooked. Reading a published solution can help, but their own valid approach may look quite different from it. They need to learn how to examine the reasoning they have actually written.

At Exact Science, we combine challenging problems, discussion with an experienced coach and feedback on students' own arguments. We have also developed an AI-assisted mathematics tutor on problems.cc, designed around our teaching methodology, so students can continue examining and improving their proofs between group sessions.

This guide explains how those parts fit together and how to use the BMO past papers and practice sets on problems.cc. The final-three-weeks section is specifically for BMO1 on 18 November 2026; the practice method applies beyond this sitting.

What changes from the Senior Maths Challenge?

The Senior Mathematical Challenge rewards choosing and finding answers under time pressure. BMO asks students to write full mathematical solutions: what follows from the conditions, why each important step is valid and why the conclusion covers the whole problem.

For the current season:

Competition	Time	Format
Senior Mathematical Challenge, 2026 trial	90 minutes	22 multiple-choice questions, followed by three numerical answers from 000 to 999.
BMO Round 1	3 hours 30 minutes	Six problems requiring full written mathematical solutions.
BMO Round 2	3 hours 30 minutes	Four problems requiring full written mathematical solutions.

The 2026 SMC was not entirely multiple choice. UKMT changed the final three questions to numerical answers for this year's trial. The important transition to BMO is the requirement to present a complete argument.

The UK sitting dates for the 2026-27 season are BMO1 on Wednesday 18 November 2026 and BMO2 on Wednesday 20 January 2027. These details come from UKMT's SMC page, BMO1 page, BMO2 page and the BMOS calendar, checked on 28 October 2026. Use the official pages for subsequent seasons.

The BMOS entry policy describes both BMO rounds as open-entry competitions for students in full-time secondary education, taken at a registered UKMT exam centre. Arrange entry through your school or centre. Automatic qualification and discretionary entry have eligibility and fee rules which your centre should check. BMO2 is a harder round; attending preparation lessons does not automatically qualify a student for it.

For a shorter introduction, see our companion BMO preparation guide on Olympiads.co.uk, with the wider selection route in its UKMT, BMO and IMO pathway guide.

Why a correct answer may still leave an incomplete proof

Consider an introductory example: find all positive integer solutions of $x^2-y^2=15$.

A student tests values and finds $(4,1)$ and $(8,7)$. Both pairs work. But the word all leaves another task: explaining why there cannot be any other solutions.

Factorising gives $(x-y)(x+y)=15$. The original equation implies $x>y>0$, so both factors are positive integers, with $x+y>x-y$. The factor pairs $(x-y,x+y)$ can therefore only be $(1,15)$ and $(3,5)$. Solving these gives precisely the two pairs found earlier.

The extra reasoning makes the search complete. This example is simpler than a typical BMO problem, but the habit matters throughout Olympiad mathematics:

• In geometry, an angle relationship may look convincing in a diagram without being justified.
• In number theory, several successful examples may suggest a pattern without proving it.
• In combinatorics, a count may omit a case or include the same arrangement twice.

UKMT's BMO1 instructions emphasise full written solutions and the value of completing a proof. Learning to find and repair gaps belongs in preparation from the beginning.

Why proof-writing needs knowledgeable feedback

A student asking “Is my solution correct?” may mean several different things. Does the answer work? Is the method sound? Has every required claim been proved? Could a careful reader follow the argument without supplying missing steps?

Reading an Olympiad proof is a specialised skill. An excellent school maths teacher may have limited experience with Olympiad marking conventions, unusual number-theory arguments or unfamiliar geometric constructions. Checking an alternative proof requires mathematical judgement about the argument itself, even when it does not follow the published method.

An experienced Olympiad coach can help students recognise the difference between a promising observation and a justified conclusion. Discussion also reveals the broader issue: whether a student needs a new technique, clearer notation or a better way of organising cases.

Independent practice continues between lessons. If a student writes an argument on Tuesday and their next group session is on Sunday, they need a productive way to investigate it in the meantime. Opening a full solution immediately can end the exploration before they have understood their own gap.

That is one reason we developed the AI-assisted tutor on problems.cc: to help students examine their written reasoning while they are practising, then bring the questions that remain to their coach.

How to practise with the problems.cc AI assistant between lessons

Use the cycle Attempt → Write → Review → Improve → Repeat. The aim is to develop your own ideas and become better at checking and communicating them.

1. Choose a problem and attempt it independently

Open the BMO1 archive, choose a paper and read the questions. Pick a problem with a plausible starting point. For an initial focused session, 30-60 minutes of sustained exploration can be useful.

Keep your rough work, including approaches that fail. Try small cases, draw a diagram or rewrite the conditions, then ask what those observations suggest. Use recent papers for full mocks because older papers can have different formats.

2. Write your best complete argument

Before asking for feedback, turn the best approach into a proof. Introduce notation clearly, use the problem's conditions explicitly and justify the important implications.

If you cannot finish a step, identify it honestly. “I think these triangles are similar, but I cannot justify the second angle equality” gives a reader much more to work with than a diagram and an unexplained conclusion.

3. Ask the assistant to review the whole proof

Sign in to problems.cc, open the individual problem and put your reasoning into its conversation or written-answer box. Explain which parts you believe are complete and where you are unsure.

Our tutor is designed around the teaching principles we use at Exact Science: examining the student's actual argument, questioning unsupported steps and encouraging the student to do the mathematical work.

Useful requests include:

Can you review my proof and identify any steps I have not justified? Please do not give me the full solution yet.

Have I considered every possible case, or have I made an assumption that needs proving?

Ask about the whole argument, including whether it establishes everything the question asks for. A correct final answer or a sound intermediate step is not enough if the rest of the problem remains untreated.

4. Repair the argument and continue the discussion

Read the feedback critically. Try to repair the first gap yourself, then explain your revision in the conversation. If you need help, ask for a smaller hint and attempt to use it before requesting another.

Sometimes the repair is one sentence explaining why a factor is positive. Sometimes it reveals that your method needs a new idea. If you disagree with a comment, explain why you think your step is valid and examine the disagreement carefully.

AI feedback can miss a gap or question a valid argument. Check mathematical claims against a reliable solution or an experienced teacher or coach. Any platform assessment is practice feedback, not an official BMO mark or a prediction of UKMT scoring.

5. Reconstruct the proof and try a related problem

Close the feedback and write the argument from a blank page. Return to it a few days later, then attempt a related problem independently.

Understanding a proof you have just read is different from recognising the idea in a new question. The next attempt is where you discover what you have learnt.

Keep a short note of the first step you missed and any disagreement you could not resolve. Those notes give you something specific to discuss in the next group session.

Focused practice across the main mathematical areas

After a serious attempt, ask what stopped you. Did you lack a fact, miss a connection, leave a case untreated or struggle to explain an idea? Choose the next practice session around that diagnosis.

The Practice by skill section on the BMO1 page brings related past problems together. These topic sets are available:

Practice set	What to practise explaining
Equations in Integers	Why a bound, factorisation or case split covers every possible integer solution.
Divisibility	Why one expression must divide another, using the stated conditions.
Modular Arithmetic	How remainders restrict possibilities or rule them out.
Counting	What is being counted, with checks for omissions and double-counting.
Constraint Problems	Which arrangements remain possible as the conditions interact.
Recurrence Relations	Why a sequence behaves as claimed, beyond its first few terms.

These are focused collections, not a complete BMO syllabus or a prediction of the next paper. The hub also lists method-based sets; its method labels are partly informed by AI-checked solution drafts and may be refined. Human mathematical review and a teacher's judgement still matter when choosing a sequence for a student.

Keep breadth across four areas:

Area	Useful foundations
Number theory	Divisibility, primes, greatest common divisors, parity, remainders and equations in integers.
Algebra	Factorisation, rearrangement, inequalities, sequences and useful substitutions.
Geometry	Congruence, similarity, circles, area ratios and justified auxiliary constructions.
Combinatorics	Systematic counting, the pigeonhole principle, invariants, extremal arguments and complete case analysis.

For geometry, combine past problems with the Geometry Problems library. Choose a chapter to address a gap, then return to a competition question. Our BMO geometry guide explains a route through those foundations.

The final three weeks before BMO1 2026

On 28 October 2026, there are 21 days until BMO1 on 18 November. The immediate task is to improve problem selection, complete proofs and the gaps your attempts expose.

Period	Focus	Something concrete to finish
28 October-3 November	Diagnose with a selection of sustained BMO1 attempts.	One carefully written proof and a short list of gaps.
4-10 November	Target those gaps with related problems on problems.cc, including geometry where needed.	Revised arguments and a fresh problem attempted without help.
11-17 November	Sit an unseen past paper under the full time limit, then review it.	A realistic selection strategy and corrected versions of incomplete proofs.

Use the assistant during review, after an independent attempt. Keep it closed during a mock paper so the attempt shows what you can do under exam conditions.

Allow time for schoolwork and rest. In the final days, revisit familiar ideas and your own corrected proofs instead of trying to collect new advanced techniques.

Preparing over a longer period, including BMO2

With more time, alternate focused sets, mixed questions and occasional full papers. A coach can help choose the balance from your attempts and the difficulties that keep recurring.

Keep some papers unseen for timed practice. Use the assistant during review and keep it closed during a mock, so the attempt shows what you can do independently.

If you already solve several BMO1 problems on your own, the BMO2 archive offers harder work. Choose questions appropriate to your current level and check entry arrangements separately with your centre. Continuing after November is worthwhile whether your next target is BMO2 or a future BMO1 sitting.

Combining group lessons with independent practice

During our small-group lessons at Exact Science, students discuss approaches, learn techniques, compare arguments and receive feedback from an experienced Olympiad coach. Between sessions, they can continue working on problems.cc and use the AI-assisted tutor to examine their own proofs.

The roles fit together:

Who	Contribution
The coach	Teaches approaches to unfamiliar problems, reads arguments in context and guides longer-term mathematical development.
The AI-assisted tutor	Helps the student investigate questions about their written reasoning during independent practice between lessons.
The student	Develops the ideas, checks the feedback, repairs gaps and completes the proof independently.

The purpose is to make independent practice more productive. Students can bring an improved argument, a specific obstacle or an unresolved question to the next lesson.

We are planning further small-group BMO and Olympiad problem-solving sessions covering number theory, algebra, geometry and combinatorics. For students taking BMO1 on 18 November 2026, any sessions before that date can support the remaining preparation. Lessons continuing afterwards develop proof-writing for future challenges, including BMO2 where appropriate.

See our timetable for upcoming groups and availability. If a suitable group is not listed, use Help me choose and tell us about the student's experience and aims.

Where to start today

Choose one past BMO1 problem, attempt it independently and write the most complete proof you can. Ask for feedback on the argument, improve it and try a related problem.

You can browse problems without an account. A free account lets you submit written work and save progress; use the features and allowances shown for your account. Official papers and BMOS video solutions remain useful reference points after an attempt.

Bring what you learn back to the next discussion with your coach. A useful preparation session should change something about how you approach, explain or check the next problem.