ukmt problem solving book

UKMT Senior Challenge Questions

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United Kingdom Mathematics Trust

On this page you will find a collection of problems adapted from the UKMT Mathematical Challenges.

The questions have been grouped into two sets of quizzes:

1. Themed Quizzes

These consist of sets of four or five questions grouped by topic. You may choose to use these at the end of teaching a topic unit to provide some challenge, as starters or plenaries to promote positive discussion, as group work tasks, or as extension material for some students in your class whilst you work on the fundamentals with others.

2. Mathematical Challenge Practice Quizzes

These are sets of five questions that each cover a random selection of topics. There are three challenge levels for these Practice Quizzes:

Bronze Taken from questions 1 to 8 on the original Maths Challenge Papers Silver Taken from questions 9 to 16 on the original Maths Challenge Papers Gold Taken from questions 17 to 25 on the original Maths Challenge Papers

Each quiz is taken from a range of years, so you can use these alongside the paper versions of the Mathematical Challenges. You may choose to use these in the build up to a Challenge, or make use of them throughout the year as a way of helping students develop problem solving and analytical skills.

The UKMT questions and quizzes may be used in class, or set for students to answer on our system. The advantage of the latter is that they will be marked for you, and you can access detailed insights into your students’ performance and possible misconceptions.

All these questions have been adapted from Junior, Intermediate and Senior Mathematical Challenge problems. However, they have not been labelled as such, and hence can be used by students of all ages to challenge them to think mathematically!

The UK Mathematics Trust is a registered charity that aims to advance the education of young people in mathematics.

The UK Mathematics Trust is a registered charity that aims to advance the education of young people in mathematics.They run solo and team competitions, organise enrichment activities for students, publish books, and train the UK team for the International Mathematical Olympiad. Find out more about the UKMT on their website website .

• Junior Themed
• Junior Practice
• Intermediate Themed
• Intermediate Practice
• Senior Themed
• Senior Practice

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Mathematics, Learning and Technology

Ukmt maths challenges.

The  UK Maths Challenges provide a wonderful library of multiple choice problems; you can practice for the UK Maths Challenges with past papers for the Junior , Intermediate , or Senior Challenges. Questions and full solutions are provided.

As well as a full solution that does not use the multiple-choice answers further ideas are given for investigation.

Maths Challenge questions are an excellent resource at any time – not just for preparation for the competitions, particularly with the increased requirement for problem-solving skills at all levels. On the outstanding Diagnostic Questions site you can choose Junior Maths Challenge questions by topic by choosing the Themed Quizzes option. These quizzes consist of sets of four or five questions grouped by topic. (Log in to Diagnostic Questions to use the link.)

Further outstanding resources from UKMT include the Team Challenge resources .

The UK Maths Challenges   are included on the Problems & Activities page .

For more sources of multiple-choice questions see this post .

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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 25 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) and guidance for BMO preparation .

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.
• 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

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 who entered BMO 1 may enter on payment of the entry fee. Marking is carried out by around ten people, under the direction of the current IMO Team Leader.

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

• 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.
• 11 May 1977
• 12 May 1978
• 10 May 1979
• 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

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• Mathematics
• Mathematical Foundations

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The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind Paperback – 1 Nov. 2018

Purchase options and add-ons.

’Be warned: cracking puzzles releases a very addictive drug.’ – Marcus du Sautoy

Have you ever wanted to be a puzzle pro or logical luminary? Well, look no further!

The perfect way to liven up your day, The Ultimate Mathematical Challenge has over 365 puzzles to test your wits and excite your mind. From starter puzzles to perplexing Olympiad problems designed to stretch even the strongest mathematicians, this book is the ideal forum to get your brain into gear and feed it with the challenges it craves.

Specially curated from the UK Mathematics Trust’s catalogue of puzzles, most of these problems can be tackled using no more than a little numerical knowledge, logical thinking and native wit. Including interludes of crossnumber conundrums and shuttle challenges, space for your working out and a handy glossary for those obscure mathematical terms, this book has everything you need to solve captivating problems all year round.

Do you have what it takes to conquer The Ultimate Mathematical Challenge ?

• Print length 288 pages
• Language English
• Publisher HarperCollins
• Publication date 1 Nov. 2018
• Dimensions 15.3 x 2.5 x 23.4 cm
• ISBN-10 0008316406
• ISBN-13 978-0008316402
• See all details

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Product description

‘In this book you can immerse yourself in the joys of […] the amazing buzz you get when you’ve struggled to crack a challenging puzzle and then suddenly you see a clever way to unlock the enigma’ – Marcus du Sautoy

‘As a school maths teacher, I thrive on writing puzzles for my students. Now teacher becomes student, as I roll up my sleeves and get stuck into these puzzles. Many of my students have participated in UKMT maths challenges, so it's a chance for everyone to put their grey matter to the test! A must have book for anyone that wants to stretch their mind’ – Bobby Seagull, author of The Life-Changing Magic of Numbers

About the Author

The UK Mathematics Trust is well-known for its challenging and entertaining Maths problems circulated throughout numerous schools and colleges in the country. They are a registered charity responsible for the organisation and running of a wide range of mathematical competitions, including both individual and team challenges, the British Mathematical Olympiad, as well as UKMT Summer Schools and Mathematical Circles. Their puzzles and solutions are written by over 400 volunteers across the UK.

Product details

• Publisher ‏ : ‎ HarperCollins (1 Nov. 2018)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 288 pages
• ISBN-10 ‏ : ‎ 0008316406
• ISBN-13 ‏ : ‎ 978-0008316402
• Dimensions ‏ : ‎ 15.3 x 2.5 x 23.4 cm
• 5 in Mathematical Logic (Books)
• 7 in Recreational Mathematics
• 10 in Mathematical Games

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The uk mathematics trust.

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