The first round of the BMO is held in December, and from 2006 is an open entry competition, costing £17 to enter. However, this fee is waived for those who achieve the qualifying mark in the Senior Mathematical Challengeand are British citizens, or will have studied for 3 full years of full-time secondary education in the UK by the time they leave school. The paper lasts 3½ hours, and consists of six questions, each worth 10 marks. Candidates are encouraged to write full proofs to the questions they attempt, as a full answer to a question is worth many more marks than incomplete answers to several questions. This is because of the marking scheme: an answer is marked on either a "0+" or a "10-" mark scheme, depending on whether the answer looks generally complete or not. So if an answer is judged incomplete or unfinished, it is awarded a few marks for progress and relevant observations, whereas if it is presented as complete and correct, marks are deducted for faults, poor reasoning, or unproven assumptions. As a result, it is quite uncommon for an answer to score a middling mark. Roughly 1300 students sit the BMO1 paper each year, and many of these attain a very low score as a percentage of the total number of marks available—the median score in 2004 was approximately 5 or 6. However, the paper is meant to be a way of selecting the best young mathematicians in the country, and is therefore very difficult. In addition to the British students, there is a history of about 20 students from New Zealand being invited to take part. Generally only about 30–40 students will score more than 50% in BMO1, with perhaps only 1 or 2 scoring above 90%. In 2005, UKMT changed the system and added an extra easier question meaning the median is now raised. In 2008, 23 got over 40 out of 60 and around 50 got over 30. 31 was the mark needed to get into the BMO2. For the December 2010 paper, the mark needed for a year 13 pupil to automatically qualify for BMO2 was 28; it was 22 for a year 12 pupil and 17 for a pupil in year 11 and below. From the results of BMO1, 100 students are chosen to be invited to sit BMO2—generally from amongst the top scorers, although younger students scoring slightly less than their older counterparts are often chosen also. It is possible to pay £22 to sit BMO2, but the candidate must have sat BMO1.
BMO2 is normally held in early February, and is significantly more difficult than BMO1. BMO2 also lasts 3½ hours, but consists of only four questions, each worth 10 marks. Like the BMO1 paper, it is not designed to test knowledge of advanced mathematics, but rather the candidate's ability to apply basic ideas to unusual problems. Half of the candidates score less than 10, and it is rare for someone to score more than 35. Twenty of the top scorers from BMO2 are subsequently invited to the training camp at Trinity College, Cambridge for the first stage of the IMO UK team selection.
