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    BMO Round 1 Past Papers & Solutions

    British Mathematical Olympiad Round 1

    BMO1 past papers and solutions with mentor-marked proof feedback — the first UK round where writing the argument matters as much as finding it.

    Eligibility
    Qualifiers from the Senior Mathematical Challenge (and eligible younger students)
    Format
    6 problems requiring full written proofs
    Duration
    3 hours 30 minutes
    Scoring
    10 marks per problem, awarded all-or-nothing in spirit — partial credit is rare
    UKMT British Mathematical Olympiad

    BMO Round 1 is the moment UK olympiad maths stops being about answers and starts being about proofs. Six problems in three and a half hours sounds generous until you discover that a solution you can see in your head still takes forty minutes to write rigorously. Past papers are the only realistic preparation: the syllabus is 'anything', but the styles — a geometry configuration, a number-theory construction, a combinatorial invariant, an inequality — repeat reliably. Our mentors mark your write-ups the way BMO markers do, so you learn where marks quietly disappear.

    Past papers by year

    Each year page lists the official paper source, the answer key location, difficulty notes and how to review your solutions.

    BMO Round 1 2025BMO Round 1 2024BMO Round 1 2023BMO Round 1 2022BMO Round 1 2021BMO Round 1 2020BMO Round 1 2019BMO Round 1 2018BMO Round 1 2017BMO Round 1 2016BMO Round 1 2015BMO Round 1 2014BMO Round 1 2013BMO Round 1 2012BMO Round 1 2011BMO Round 1 2010

    Stuck on a BMO Round 1 question?

    Send us the question number. A mentor who has sat and medalled in BMO Round 1 will reply with a step-by-step walkthrough and the faster competition method.

    Free to ask. No spam, ever — we only reply about your question.

    What the paper tests

    Olympiad geometry

    Angle chasing, cyclic quadrilaterals, similar triangles, configuration hunting.

    Number theory

    Divisibility arguments, modular constraints, constructions and infinite descent.

    Combinatorics

    Invariants, colouring, extremal arguments, pigeonhole and induction.

    Algebra & inequalities

    AM–GM, Cauchy–Schwarz, functional equations, polynomial arguments.

    Proof writing

    Structuring a complete argument, justifying every claim, avoiding hand-waving.

    Where it leads: The strongest BMO1 scripts are invited to BMO Round 2 and then the UK IMO training squad. See our coaching programme.

    Book personalised BMO Round 1 classes

    Past papers show you the gap; a mentor closes it. Our BMO Round 1 mentors are medal winners who walk through every solution 1-to-1, then set targeted practice on the archetypes you lost marks on.

    View the BMO Round 1 course

    Want the full BMO Round 1 paper pack and score guide?

    Tell us your target and we will send the BMO Round 1 paper list, the official scoring rules and a preparation plan built around the questions you keep losing marks on.

    Free to ask. No spam, ever — we only reply about your question.

    Frequently asked questions

    Where can I find BMO1 past papers and solutions?
    UKMT publishes BMO Round 1 papers together with official solution booklets and markers' reports on its website. We link to the official archive and add mentor-marked feedback on your own written solutions.
    What is a good BMO1 score?
    Scores are out of 60. Solving two problems completely is a solid result for a first attempt, and distinction and BMO2 qualification thresholds are set each year based on the field.
    How do I prepare for BMO1?
    Work old problems slowly and write every solution out in full, then have it marked by someone who knows olympiad standards. Reading a solution and thinking 'I could have done that' is the classic trap — the marks come from the writing.

    Related paper libraries

    BMO Round 2 past papersUKMT SMC past papersAMC 12 past papers

    Preparing beyond past papers? See our 1-to-1 Maths Olympiad tutoring hub for the full UKMT, BMO, AMC and AIME pathway with verified expert mentors.