British Mathematical Olympiad, Round 1 (BMO 1)

This is a 3½-hour paper with 6 problems (the first being intended to be more accessible than the rest), taken by students in their own schools. Selection is based on performance in the UK Senior Mathematical Challenge (UKSMC). Students achieving a certain score in UKSMC (this score varying from year to year depending on overall performance) and eligible to represent the UK at the IMO are entered automatically for BMO 1. Teachers are invited to use their judgement in entering other students. See BMO entry information for more details. Typically 1300 students sit BMO 1. A team of around 60 markers gathers in December to mark all the scripts over a 3-day period. The top 100 students receive prizes. See also notes on BMO marking (PDF).

A Mathematical Olympiad Primer, with hints and solutions for BMO1 problems from 1996–1997 to 2010–2011, may be purchased from UKMT; A Mathematical Olympiad Primer II, covering the period from 2011–2012 to 2022–2023, is also available; video solutions for BMO1 are also available from 2005–2006 onwards. Other UKMT publications of use for BMO preparation include Plane Euclidean Geometry: Theory and Problems, especially chapters 3–7, and Crossing the Bridge for further geometry preparation. Serious geometers may also be interested in New Problems in Euclidean Geometry.

Past papers:

• A PDF file with all BMO problems from 1993 onwards (both BMO1 and BMO2) is also available.
• For older past papers (BMO 1965–1991, BMO 1 1992–1996), with hints and outline solutions 1975–1996, see Tony Gardiner, The Mathematical Olympiad Handbook: An introduction to problem solving based on the first 32 British Mathematical Olympiads 1965–1996 (OUP, 1997).
• An extra more accessible question was added to the start of the paper in 2005; papers here from 1992 to 2004 consist of five questions without this more accessible question, while the style of pre-1992 papers varied; pre-1975 papers are especially different in style from the recent papers.
• Papers from 1965 to 1972 not currently available (although the problem statements are in Tony Gardiner’s book); please let us know if you can help provide copies or scans of the original papers.
• 1973
• 1974
• 24 March 1975
• 24 March 1976
• 17 March 1977
• 21 March 1978
• 14 March 1979
• 13 March 1980
• 19 March 1981
• 18 March 1982
• 10 March 1983
• 13 March 1984
• 5 March 1985
• 4 March 1986
• 20 March 1987
• 20 November 1987
• 13 December 1988
• 17 January 1990
• 16 January 1991
• 15 January 1992
• 13 January 1993
• 19 January 1994
• 18 January 1995
• 17 January 1996
• 15 January 1997
• 14 January 1998
• 13 January 1999
• 12 January 2000
• 17 January 2001
• 5 December 2001
• 11 December 2002
• 3 December 2003
• 1 December 2004 (last five-question paper); histogram of distribution of marks
• 30 November 2005 (first six-question paper); list of prizewinners
• 1 December 2006
• 30 November 2007
• 4 December 2008
• 3 December 2009
• 2 December 2010
• 2 December 2011
• 30 November 2012
• 29 November 2013
• 28 November 2014
• 27 November 2015
• 2 December 2016; report for schools
• 1 December 2017; report for schools
• 30 November 2018; report for schools
• 29 November 2019; markers’ report
• 26 November 2020; markers’ report
• 2 December 2021; markers’ report
• 16 November 2022; markers’ report
• 15 November 2023; markers’ report
• 20 November 2024; markers’ report

British Mathematical Olympiad, Round 2 (BMO 2)

This is a 3½-hour paper with 4 problems, taken by students in their own schools. Based on performances in BMO 1, up to 100 students (who are eligible to represent the UK at the IMO) are invited to sit BMO 2, with selection taking account of age and other factors; others may enter on payment of the entry fee. Marking is carried out by around 20 people.

BMO2 solutions are included in A Mathematical Olympiad Companion, available from the UKMT.

Past papers:

• This paper was known as the Further International Selection Test (FIST) from 1972 to 1991. The number of questions and rules for marking varied.
• 17 May 1972
• Paper from 1973 not currently available; please let us know if you can help provide this.
• 1974
• 5 May 1975
• 5 May 1976
• 11 May 1977
• 12 May 1978
• 10 May 1979
• 8 May 1980
• 14 May 1981
• 13 May 1982
• 11 May 1983
• 23 March 1984
• 15 March 1985
• 14 March 1986
• There was no FIST paper in 1987, with BMO and correspondence training leading instead to a Reading Selection Test held at the Reading training camp to select the team.
• 5 February 1988
• 1 March 1989
• 7 March 1990
• 28 February 1991
• 13 February 1992
• 11 February 1993
• 24 February 1994
• 16 February 1995
• 15 February 1996
• 27 February 1997
• 26 February 1998
• 25 February 1999
• 23 February 2000
• 27 February 2001
• 26 February 2002
• 25 February 2003
• 24 February 2004
• 1 February 2005
• 31 January 2006
• 30 January 2007
• 31 January 2008
• 29 January 2009
• 28 January 2010
• 27 January 2011
• 26 January 2012
• 31 January 2013
• 30 January 2014
• 29 January 2015
• 28 January 2016
• 26 January 2017
• 25 January 2018
• 24 January 2019
• 30 January 2020
• 28 January 2021; solutions
• 27 January 2022; solutions
• 25 January 2023
• 24 January 2024