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    BMO Round 2 Past Papers & Solutions

    British Mathematical Olympiad Round 2

    BMO2 past papers with solutions and IMO-squad-level proof coaching — four hard problems, three and a half hours, no calculators.

    Eligibility
    Invited BMO Round 1 high scorers
    Format
    4 problems requiring full written proofs
    Duration
    3 hours 30 minutes
    Scoring
    10 marks per problem (40 max)
    UKMT British Mathematical Olympiad

    BMO Round 2 is genuine IMO-preparation territory. Four problems, each of which can absorb an hour of thought before the key idea appears, and where partial progress must still be written up carefully to earn anything. Working past BMO2 papers teaches you to spend time productively on a problem you cannot yet solve — to extract structure, test small cases and record partial results — which is exactly the skill the IMO squad selection is looking for.

    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 2 2025BMO Round 2 2024BMO Round 2 2023BMO Round 2 2022BMO Round 2 2021BMO Round 2 2020BMO Round 2 2019BMO Round 2 2018BMO Round 2 2017BMO Round 2 2016BMO Round 2 2015BMO Round 2 2014BMO Round 2 2013BMO Round 2 2012BMO Round 2 2011BMO Round 2 2010

    Stuck on a BMO Round 2 question?

    Send us the question number. A mentor who has sat and medalled in BMO Round 2 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

    Advanced geometry

    Projective ideas, radical axes, spiral similarity, inversion basics.

    Number theory

    Order arguments, lifting the exponent, Diophantine analysis.

    Combinatorics

    Extremal and algorithmic arguments, graph reasoning, strategy games.

    Inequalities & algebra

    Smoothing, SOS, functional equations over the reals.

    Exam craft

    Time allocation across four problems and salvaging partial credit.

    Where it leads: Leads to the UK IMO training camps and eventual IMO team selection. See our coaching programme.

    Book personalised BMO Round 2 classes

    Past papers show you the gap; a mentor closes it. Our BMO Round 2 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 2 course

    Want the full BMO Round 2 paper pack and score guide?

    Tell us your target and we will send the BMO Round 2 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

    How hard is BMO2 compared with BMO1?
    Substantially harder. BMO1 rewards a strong student who has practised; BMO2 problems are designed so that solving two of the four is an excellent performance for most participants.
    Do I get credit for a partial solution?
    Some, but olympiad marking is famously unforgiving: a solution that is 90% complete with a genuine gap usually scores low. Writing up what you have proved precisely, and stating clearly what remains, is worth practising.

    Related paper libraries

    BMO Round 1 past papersUKMT SMC 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.