UKMT Pathway · Senior Stage · The Crown
British Mathematical Olympiad — Round 1, Round 2, and the Pathway to Team UK at the IMO
The BMO is the senior-level Olympiad in the United Kingdom Mathematics Trust pipeline and the single mechanism by which the UK selects its six-member team for the International Mathematical Olympiad. Two written rounds, three and a half hours each, taken in November and January. This page covers format, eligibility, dates, marking, sample problems and how to prepare.
British Mathematical Olympiad by the numbers
in present form
each November
to Round 2
to Trinity Camp
selected
Overview
What is the British Mathematical Olympiad?
The British Mathematical Olympiad is the senior-level written Olympiad of the United Kingdom Mathematics Trust. It runs each academic year in two rounds — Round 1 in November and Round 2 in January — and sits at the top of a four-stage competition pipeline that begins with the Primary Kangaroo at age ten.
The BMO has been run in its present format since 1965, when the British Mathematical Olympiad Committee took over from the earlier IMO selection arrangements. It is organised by the BMO Subtrust of UKMT, and its papers are set by a committee of mathematicians drawn from UK universities and former Olympiad medallists. The competition is administered by the UKMT office in Leeds and the BMO Subtrust office at Trinity College, Cambridge.
The BMO’s defining feature is its problem style. Where the Senior Mathematical Challenge (SMC) sat in October is a twenty-five-question paper (22 multiple-choice plus 3 integer-answer) rewarding speed and recognition, the BMO is open-ended: every solution must be argued in full prose, with chains of reasoning the student is expected to construct from first principles. Marking is by a centralised panel and emphasises logical completeness over arithmetic.
Eligibility for BMO Round 1: students in Year 13 and below (S6 in Scotland, Year 14 in Northern Ireland) attending a school registered for UKMT competitions. Schools may also enter additional eligible students at their discretion. International students attending UK schools should consult their school’s mathematics department about discretionary entry. International students attending UK schools should consult their school’s mathematics department about discretionary entry.
From BMO Round 1, approximately the top one hundred performers are invited to sit Round 2 in late January. From Round 2, around 24 students are invited to the residential Trinity Camp held at Trinity College, Cambridge, over the Easter break. From the Trinity Camp, the final six-strong UK IMO team is selected each May, ready to compete at the International Mathematical Olympiad in July.
Format
Round 1 and Round 2 — Two Papers, One Pathway
Both BMO rounds are written, open-ended and sat over three and a half hours. They differ in problem count, audience size, and intent. Round 1 filters; Round 2 selects. The cohort that sits Round 2 is, by construction, the strongest assembly of British sixth-formers in mathematics in any given year.
The November Filter
- DateMid-November each year (specific Wednesday published by UKMT)
- Duration3 hours 30 minutes
- Problems6 open-ended problems, each worth 10 marks (60 total); Problem 1 is intentionally more accessible
- CalculatorCalculators and protractors not permitted; rulers, set squares and compasses are allowed
- EligibilityYear 13 and below (Y13 / S6 in Scotland / Y14 in Northern Ireland) at a school registered for UKMT competitions; must be eligible to represent the UK at IMO under UKMT\’s invitation guidelines
- MarkingCentralised by BMO Subtrust panel; full written solutions required
- Result tierDistinction, Merit, Qualification to Round 2
Round 1 is designed to be approachable to a strong SMC performer but rewards depth that the SMC does not test. The expectation is that a strong candidate will solve two problems completely and make substantial progress on a third. Cutoff to Round 2 fluctuates by year and paper difficulty but is typically in the upper third of total marks.
The January Selection
- DateLate January each year (Wednesday, by invitation only)
- Duration3 hours 30 minutes
- Problems4 open-ended problems, generally harder than Round 1
- CalculatorNot permitted
- EligibilityBy invitation only, based on Round 1 score
- MarkingCentralised; emphasis on rigour and elegance of argument
- Result tierDistinction (top tier), Merit, Qualification to Trinity Camp
Round 2 problems are written for a markedly stronger cohort. They tend to be harder both in mathematical content (more demanding number theory, geometry, combinatorics) and in the standard of proof expected. Top Round 2 scorers — typically the leading 24 — are invited to the Trinity College training camp at Easter.
Selection
Trinity Camp and the UK IMO Team Selection
From Round 2, the top performers are invited to a residential training camp at Trinity College, Cambridge, held over the Easter break. The camp combines structured teaching, problem-solving sessions and selection tests; from it, six students are chosen to represent the United Kingdom at the International Mathematical Olympiad in July.
BMO Round 1
The open written round. Six problems, 3 hr 30 min, marked centrally. The top hundred or so advance to Round 2.
BMO Round 2
By-invitation written round. Four problems, 3 hr 30 min. The strongest 24 papers are invited to the training camp.
Trinity Camp
Residential training week at Trinity College, Cambridge. Structured problem sets, lectures, selection tests across the week.
UK IMO Team
Final six selected based on combined Trinity Camp performance and Team Selection Test (TST) results — no interviews; selection is by test scores. They then represent the UK at the International Mathematical Olympiad.
Problem Style
A Representative Round 1 Problem
BMO Round 1 problems are typically one or two sentences long, deceptively short. Their difficulty lies in the chain of reasoning required, not in the volume of arithmetic. Below is a representative problem in the BMO Round 1 style, with notes on what a strong solution would establish.
Problem (in BMO style)
Let n be a positive integer. Suppose that n + 1 distinct positive integers are chosen from the set {1, 2, 3, …, 2n}. Prove that two of the chosen integers must differ by a multiple of n, and that two of them must be such that one divides the other.
What a strong solution would establish
A complete BMO Round 1 solution to a problem of this style would prove both claims separately, with each argument written as a self-contained chain of deductions. For the first part, a pigeonhole argument on residues modulo n is natural; the candidate must articulate why exactly n residue classes generate the required collision.
For the second part, the candidate would partition the integers from 1 to 2n into chains under the “divides” relation, observe that each chain corresponds to a unique odd number multiplied by powers of two, count the resulting chains (n of them), and apply pigeonhole again. Both arguments use the same combinatorial primitive but in different colours — a typical Round 1 design.
Marks would be awarded for completeness of argument, not for elegance alone. Markers look for: a clear statement of what is being proved; a self-contained chain of reasoning; correct logical closure. A “nearly right” solution receives partial credit; an “elegant but incomplete” solution typically receives less than a “long-winded but complete” one.
Preparation
How to Prepare for the British Mathematical Olympiad
There is no shortcut to BMO preparation. The most reliable approach is sustained problem-solving practice spread across multiple academic years, combined with targeted reading of the standard Olympiad texts and a regular diet of past papers. Below is the consensus reading list and a sketch of a realistic preparation calendar.
Reading List
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Mathematical Olympiad Problems and Solutions
The canonical annual collection of BMO Round 1 and Round 2 papers with full solutions. Published annually by UKMT. Essential reading.
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Problem Solving Strategies
The standard reference for Olympiad problem-solving techniques across number theory, algebra, geometry and combinatorics. Comprehensive and demanding.
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Mathematical Olympiad Challenges
A graded collection of problems with substantial commentary on solution methodology. Particularly strong on geometry and inequalities.
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Plane Euclidean Geometry: Theory and Problems
The standard British text on Olympiad geometry. Written by long-serving BMO problem-setters; covers the geometric techniques BMO papers most often demand.
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A Problem Seminar
A short, advanced collection emphasising elegance of method. Useful in the final months before BMO Round 2.
A Realistic Preparation Calendar
The standard British preparation pattern begins in Year 11, after the Maclaurin Olympiad results have been published. From there, two academic years of consistent problem-solving practice — typically two to four problems per week, with full written solutions — is the consensus baseline.
Year 11 — autumn term: work through Engel chapters on combinatorics and number theory; sit any remaining Maclaurin problems with full written solutions.
Year 12 — autumn term: sit Senior Mathematical Challenge in October as the BMO Round 1 qualifier; immediately begin sitting past BMO Round 1 papers under timed conditions; cycle through Gardiner and Bradley for Olympiad geometry.
Year 12 — November: sit BMO Round 1. Regardless of result, continue past-paper work over the Christmas holidays.
Year 13 — January: if invited to Round 2, focus on past Round 2 papers and Newman’s Problem Seminar; if not, return to Round 1 papers and aim again for Year 13’s sitting.
A school-based mentor — typically the mathematics department head — is the standard route for support. Where this is not available, our WhatsApp advisor can connect you with a structured preparation programme.
Frequently Asked
Eight Questions about the BMO We Receive Most
Eight questions we receive most often from students and parents preparing for BMO from outside the United Kingdom. Every answer is verified against the BMO Subtrust website at bmos.ukmt.org.uk and the UKMT competitions site.
- When does BMO Round 1 take place each year?
- BMO Round 1 is sat on a specified Wednesday in mid-November each academic year. The exact date is published by UKMT in advance of the autumn term and is fixed nationally — every entering school sits the paper on the same day. The paper runs from late morning to early afternoon, typically 9 am to 12:30 pm in the United Kingdom.
- How many problems are on the BMO Round 1 paper?
- BMO Round 1 consists of six open-ended problems, each worth 10 marks for a total of 60. Problem 1 is intentionally set to be more accessible than the rest, giving every entrant a realistic chance of scoring. The paper runs for three and a half hours, roughly 35 minutes per problem if attempted evenly; in practice, strong candidates aim to solve two problems completely and make substantial progress on a third.
- What’s the difference between BMO and the Senior Mathematical Challenge?
- The SMC is a multiple-choice paper of twenty-five problems, sat in October, designed to be accessible across a wide range of sixth-form ability levels. The BMO is open-ended, asks for full written solutions, and is set with a markedly smaller and stronger cohort in mind. Strong SMC performance is the gateway to BMO entry, but the two papers reward different skills: speed and recognition on one side, depth and patience on the other.
- Who is eligible to sit BMO Round 1?
- BMO Round 1 is open to students in Year 13 and below (S6 in Scotland, Year 14 in Northern Ireland) attending a school registered for UKMT competitions. In addition, schools may enter a limited number of additional students on a discretionary basis, subject to a per-student entry fee paid to the BMO Subtrust. International students attending UK schools should consult their school’s mathematics department about the discretionary entry route.
- How does BMO Round 2 work?
- Round 2 is sat by invitation only in late January, on a Wednesday published in advance by UKMT. It consists of four open-ended problems sat over three and a half hours, generally harder than Round 1. The strongest 24 or so Round 2 papers are invited to the Easter training camp at Trinity College, Cambridge. Round 2 is free for the ~100 qualifying scorers invited via Round 1. Schools may enter additional eligible students at their discretion for £50 per entry.
- What is the Trinity training camp?
- The Trinity training camp is a residential week of structured Olympiad teaching held at Trinity College, Cambridge, over the Easter break each year. Approximately 24 top Round 2 performers attend, working through problem sets, attending lectures from former Olympiad medallists and university mathematicians, and sitting selection tests across the week. The camp is fully funded by UKMT and the BMO Subtrust.
- How is the UK IMO team selected from BMO?
- After the Trinity Camp, around 11 students are retained in a UK IMO training squad. Further training and Team Selection Tests (TSTs) over the following weeks decide the final team of six (plus one reserve), based purely on test performance — there is no interview component. Selection is announced in May, after which the team enters a further training period before the International Mathematical Olympiad itself, held in July at a host country chosen on a rotating basis.
- Can international students based outside the UK sit BMO?
- Direct entry to BMO from a school outside the United Kingdom is not normally permitted — eligibility requires UK nationality or three years of UK education. However, students attending UK boarding schools or sixth-form colleges, including international students at such institutions, may be entered through the school’s standard or discretionary entry route. Students at non-UK schools who wish to pursue Olympiad mathematics should look at their own national equivalents (the AMC/AIME/USAMO pathway in the United States, for instance).
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