British Mathematical Olympiad
The British Mathematical Olympiad asks students to write complete solutions. Round 1 follows the Senior Mathematical Challenge, and qualifiers can be invited to Round 2. At Exact Science we teach students to find an approach, write it clearly and revise the argument after feedback.
Write one proof and revise it
Try question 1 from BMO1 on 19 November 2025. Give yourself 45 minutes for the attempt and write-up, then compare your reasoning with the markers’ report.
Open the November 2025 BMO1 paperWork out what needs proving
Write down what the question asks you to establish for every prime and exponent. Test a few small cases. Look for a way to use the divisibility conditions to express the three numbers using fewer unknowns.
Write your argument in full
State your assumptions and justify the step that rules out or constructs each case. If you’re stuck, write down precisely what you have proved and where the gap starts.
Check coverage, then revise
Use the markers’ report below to see whether your reasoning covers all the required cases. Put the report aside and rewrite your weakest step in your own words.
How to improve before the paper
BMO1 · choose a problem you can finish
Scan the paper, then start where you can name a plausible method. Leave time to write up a complete solution from rough working and check boundary cases before moving on.
BMO2 · find what the whole grid tells you
Try question 3 from the January 2026 Round 2 paper. Before attempting to fill the grid, work out the range and total of the row and column sums. Ask what those facts would force if all 60 sums were distinct.
Ask for feedback on a specific step
Show a reader your own working and ask which line first needs more justification. Rewrite from that point, then try a fresh problem that needs the same idea.
Prepare with Exact Science
Bring an attempted proof to a trial lesson. We can discuss the ideas you tried, find the first gap in the argument and see what you can do with a hint. Weekly group or one-to-one training gives you practice explaining and revising your own solutions.
What comes after this paper?
UKMT invites eligible qualifiers from the Senior Mathematical Challenge to BMO1. Schools can check discretionary entry for BMO1 with UKMT. BMO2 is the invitation-only National Final, with invitations based on Round 1 results.
Frequently asked questions
Can I enter BMO1 through my school?
Ask your school to check UKMT’s eligibility, qualification and discretionary entry rules for Round 1.
UKMT Round 1 dates and entryCan a school enter me for BMO2?
The BMO National Final is by invitation following BMO1. UKMT does not accept discretionary entries for the final.
UKMT National Final detailsHow much of my reasoning should I write down?
Both rounds require full written solutions. Write enough for a reader to check why each step follows and why your conclusion covers every case the problem asks for.
Official BMO papers and guidanceFind the right level for your preparation
Try a lesson and get a recommendation on what to work on first.