Past Papers · Books · Courses · Mentors
UKMT and BMO Resources — Past Papers, Reading List, and Preparation Pathways
A curated index of the preparation resources used by serious UKMT competitors: official past-paper archives from UKMT and the BMO Subtrust, the consensus reading list across Junior through Senior Olympiad levels, and the structured online and in-person preparation pathways available to international students.
Past Papers
Official Past-Paper Archives
UKMT and the BMO Subtrust publish full past-paper archives for every competition in the pipeline, with mark schemes and worked solutions. The BMO archive at bmos.ukmt.org.uk runs back to the 1960s; the other competitions are typically archived from the late 1990s onwards.
| Competition | Archive scope | Source |
|---|---|---|
| Stage 01 — Primary | ||
| Primary Kangaroo | Recent years with full solutions | ukmt.org.uk |
| Stage 02 — Junior | ||
| JMC | 1998 onwards, with mark schemes | ukmt.org.uk |
| Junior Kangaroo | Recent years | ukmt.org.uk |
| JMO | 1989 onwards, with full solutions | ukmt.org.uk |
| Stage 03 — Intermediate | ||
| IMC | 1997 onwards | ukmt.org.uk |
| Grey + Pink Kangaroo | Recent years | ukmt.org.uk |
| Cayley · Hamilton · Maclaurin | 2004 onwards, with full mark schemes | ukmt.org.uk |
| Stage 04 — Senior | ||
| SMC | 1997 onwards | ukmt.org.uk |
| A. J. Senior Kangaroo | Recent years | ukmt.org.uk |
| MOG | 2011 onwards, with full solutions | ukmt.org.uk |
| BMO Round 1 + Round 2 | 1965 onwards, complete archive with full solutions and shortlists | bmos.ukmt.org.uk |
Reading List
The Consensus Olympiad Reading List
Nine books that form the canonical reading list for serious UKMT Olympiad preparation across Junior, Intermediate and Senior levels. The list reflects the consensus among UK Olympiad trainers and the BMO Subtrust problem-setting committee.
The canonical annual collections of past papers and solutions for JMO, Cayley, Hamilton, Maclaurin, and BMO. Published annually by UKMT. Essential reading at every Olympiad level.
The standard reference for Olympiad problem-solving techniques across number theory, algebra, geometry and combinatorics. Comprehensive and demanding; the canonical Engel chapters are essential for Maclaurin and BMO Round 1 preparation.
The standard British text on Olympiad geometry. Written by long-serving BMO problem-setters; covers the geometric techniques BMO and Intermediate Olympiad papers most often demand. Essential for Cayley onwards.
A graded collection of problems with substantial commentary on solution methodology. Particularly strong on geometry and inequalities. The standard transition text from Maclaurin to BMO Round 1.
A short, advanced collection emphasising elegance of method. Useful in the final months before BMO Round 2 and Trinity Camp.
A capstone problem collection for the strongest senior Olympiad cohort. Used by Trinity Camp trainers; readable cover-to-cover only by serious BMO Round 2 candidates.
The classical second-pass on Olympiad-level Euclidean geometry. Picks up where Gardiner and Bradley leave off, with material on projective and inversive methods used in the hardest BMO Round 2 problems.
A standalone number theory companion for serious Olympiad preparation. Useful for BMO Round 2 candidates working through modular arithmetic, divisibility, and elementary Diophantine problems.
The accessible reference for Olympiad-level combinatorics — pigeonhole, bijections, generating functions, graph theory at introductory level. Used as a Year 11–12 combinatorics text by many UK Olympiad coaches.
Courses & Mentors
Online Courses and Preparation Pathways
Three online platforms and one structured mentor pathway are the standard routes used by UK Olympiad students beyond schoolbook reading and past-paper practice. All are accessible to international students at UK schools; some are accessible from outside the UK.
UKMT Mentoring Scheme
The UKMT runs a free mentoring scheme for students who have qualified for the Olympiads at each stage. Mentors are university mathematicians and former Olympiad medallists who provide problem sets and written feedback over the academic year. Apply through UKMT after invitation to JMO, Cayley/Hamilton/Maclaurin, or BMO.
Art of Problem Solving (AoPS)
The Art of Problem Solving online platform offers structured Olympiad preparation courses across the JMO through BMO Round 2 range. Used widely by US Olympiad students; increasingly used by UK and international students at UK boarding schools. Cohort-based courses with weekly problem sets and discussion.
School-based mentoring
Many UK boarding schools and selective sixth-form colleges run their own structured Olympiad mentoring programmes, often led by mathematics department heads or former Olympiad medallists in residence. Speak to your school’s mathematics department about availability and the appropriate level for your year group.
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Get a Tailored Reading List for Your Year Group and Olympiad Target
The WhatsApp advisor can help match the reading list above to your year group, your prior Olympiad results, and your target competition. Tailored preparation schedules from Year 7 JMO through to BMO Round 2 are available. Written exchanges in English or Chinese welcomed.