Structure

The BMO is the highest-level secondary-school mathematics competition in the UK, run by UKMT, and is the pathway for selecting the British team for the International Mathematical Olympiad (IMO). Note: this is an academic competition, not an entrance examination—it is not used for university application, but a Gold certificate in the SMC, and especially an excellent result in BMO1, is a highly weighty academic signal in applications for mathematics and related subjects, and its problem-solving and proof style is also highly consistent with Oxford and Cambridge interviews. Progression pathway: Senior Maths Challenge → BMO1 → BMO2 → national team training camp.

The competition is a tiered series: Senior Maths Challenge (10, multiple-choice/numerical questions, which this platform's SMC question bank is benchmarked against) → BMO Round 1 (11, 6 proof questions, which this platform's long questions are benchmarked against) → BMO Round 2 (the following year, 1, IMO-level difficulty, by invitation) → national team training and the IMO. The SMC allows ordinary calculators (calculators are banned in some years; the rules of that year apply); the BMO is pure handwritten proof and calculators must not be used. Marking characteristics: a full score on the BMO is extremely difficult, and one complete proof written properly often beats six half-finished ones—presentation and rigour are themselves important scoring points.

The SMC awards Gold / Silver / Bronze proportionally, and issues BMO1 invitations according to the cut-off scores; BMO1 awards prizes according to performance and selects candidates to enter BMO2. On this platform's question bank: SMC short questions have been converted into multiple-choice questions to support automatic marking and instant feedback (those that were originally 000–999 numerical-answer questions are noted in the question stem, and all have been verified by computation); BMO1-style long proof questions retain their complete format, with guiding sub-questions and full solutions for self-assessment.

• ▸SMC is usually registered by the school for all students at 10; BMO1 can be entered by qualifying through SMC, or by the school paying for direct entry (about £40 per person)—communicate with the school mathematics teacher as early as possible.
• ▸The first mindset hurdle in BMO: a full score is not the goal. One complete, rigorous solution is better than six half-finished ones — marks are awarded for the rigour of the proof.
• ▸Be sure to do official past papers (free download at ukmt.org.uk): this question bank provides topic-based training and immediate feedback, while past papers provide exam feel and proof writing practice.
• ▸The last 3 questions of the SMC are 000–999 with numerical answers (this platform presents them as multiple-choice and has noted this): highest difficulty; attempt them only if you can, and do not exhaust all your time on them.
• ▸For proof questions, write 'complete in both directions': you must both give the solution/construction and argue that there are no others, or when equality holds in the inequality — omitting this half often loses marks.
• ▸Have the usual weapons ready: for number theory use difference of squares plus parity/congruence; for combinatorics first look for an invariant or a monotone quantity; for inequalities think of AM–GM / Cauchy–Schwarz and analyse the equality case separately.

Dates and arrangements change. Check the current official information before registering.