Free practice resources
Past papers & award boundaries.
Choose a competition below. Use the boundary for the year of the paper you attempt: thresholds and scoring rules can change. Keep some unseen papers for timed practice, then revisit difficult questions after reviewing the solutions.
JMC — Junior Mathematical Challenge
An introduction to competition problem solving. Tuition develops careful reading, number sense, geometry and the habit of testing simple cases.
60 minutes · 25 multiple-choice questions · 135 marks
Year 8 and below in England, Wales and overseas; S2 and below in Scotland; Year 9 and below in Northern Ireland. Official competition information
Source: UKMT award thresholds. Scores are minimum marks for the listed year.
IMC — Intermediate Mathematical Challenge
Build a wider problem-solving toolkit and become more confident with the later questions, including geometry, counting, number theory and efficient algebra.
60 minutes · 25 multiple-choice questions · 135 marks
Year 11 and below in England, Wales and overseas; S4 and below in Scotland; Year 12 and below in Northern Ireland. Official competition information
Source: UKMT award thresholds. Scores are minimum marks for the listed year.
SMC — Senior Mathematical Challenge
Develop the fluency and judgement needed for demanding challenge questions, with preparation for BMO proof problems where appropriate.
90 minutes · 25 questions · 125 marks
Year 13 and below in England, Wales and overseas; S6 and below in Scotland; Year 14 and below in Northern Ireland. Candidates must be full-time students at secondary school or FE college. Official competition information
SMC format from 2026: 22 multiple-choice questions and three questions with a three-digit answer. The 2023–2025 papers below use the earlier multiple-choice format. See the current format.
Source: UKMT award thresholds. Scores are minimum marks for the listed year.
*Free BMO1 entry also depends on UK IMO eligibility; discretionary paid entry may be available. See the official eligibility guidance in the BMO section.
AMC 10 — American Mathematics Competition
Build confidence with algebra, geometry, number theory and counting, alongside efficient approaches to multiple-choice problems.
75 minutes · 25 multiple-choice questions
US grade 10 and below, under 17.5 years on competition day. Official AMC information
Past papers and solutions
AMC 10 problems and solutions — AoPS archive. Community-maintained solutions; choose the year and A or B paper within the archive.
Official free sample: 2022 AMC 10A — Paper (PDF) · Solutions (PDF).
MAA student resources and official practice collections
AIME qualification cutoffs
These are qualification cutoffs, not school grades. Historical A and B thresholds can differ; use the correct paper and year.
2026 AMC → 2027 AIME: MAA lists qualifying scores of 100 on AMC 10 or 85 on AMC 12, subject to its eligibility rules, including a minimum age of 13 for AIME in 2027. International students should follow their country or region’s programme policies. Read the current MAA guidance.
AMC 12 — American Mathematics Competition
Develop a broader high-school problem-solving toolkit, including trigonometry, with practice that combines mathematical depth and careful use of time.
75 minutes · 25 multiple-choice questions
US grade 12 and below, under 19.5 years on competition day. Official AMC information
Past papers and solutions
AMC 12 problems and solutions — AoPS archive. Community-maintained solutions; choose the year and A or B paper within the archive.
MAA student resources and official practice collections
AIME qualification cutoffs
These are qualification cutoffs, not school grades. Historical A and B thresholds can differ; use the correct paper and year.
Additional AIME eligibility rules apply, including age and international programme policies. Current MAA guidance.
AIME — American Invitational Mathematics Examination
Advanced problem-solving tuition for students moving on from AMC 10 or AMC 12. Lessons develop algebra, geometry, number theory, counting and probability, with a focus on finding a route through longer problems and carrying an argument through accurately.
15 integer-answer problems · Answers from 0 to 999
Past papers and solutions
AIME problems and solutions — AoPS archive. Choose a year and AIME I or AIME II paper in the community-maintained archive.
Preparing for 2027: the US/Canada competition uses two 90-minute parts, with eight problems followed by seven. Older AIME I and II papers remain useful for practice, but use the current timing for final preparation. International students should check their local organiser’s arrangements. Official AIME format and eligibility.
Qualification and scores
AIME does not have UKMT-style Bronze, Silver and Gold boundaries. The AMC scores needed for entry are listed in the AMC sections above; progression to USAJMO or USAMO follows the relevant year’s selection rules.
BMO — British Mathematical Olympiad
BMO preparation focuses on proof. Lessons develop ways to explore a problem, identify a promising argument and turn it into a complete written solution, with detailed discussion of gaps and unjustified steps.
British Mathematical Olympiad Round 1
3½ hours · 6 proof problems · 60 marks
Years below refer to the November sitting. The official BMO1 filenames use the following year: for example, the 2025 paper is named “bmo1-2026.pdf”.
Source: UKMT BMO1 award thresholds. See the full tables for any medal or follow-on qualification thresholds.
British Mathematical Olympiad Round 2
3½ hours · 4 proof problems · 40 marks
Source: UKMT BMO2 award thresholds.
Free BMO entry requires the relevant qualifying score and UK IMO eligibility. BMO2 thresholds also vary by year group; discretionary paid entry may be available through a registered centre.
Resources checked 10 September 2026. UKMT and MAA links lead to the organisers’ material; AoPS is a community archive. For entry, scoring and qualification, always consult the current organiser guidance linked above.