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Vesna Kadelburg

British Mathematical Olympiad Committee Email: vka@stephenperse.com

1UKMT Activities

Maths Challenges: Junior, Intermediate and Senior

Follow on rounds:

Junior Mathematical Olympiad - JMO;

Intermediate Mathematical Olympiad and Kangaroo (suite of 3 Olympiad and 2 multiple choice European Kangaroo papers) IMOK;

British Mathematical Olympiad Round 1 BMO1 and the new Senior Kangaroo (the first Senior Kangaroo took place last academic year held on the same day as BMO1 - not multiple choice but single answers marked by a machine).

Over 600,000 pupils from more than 4000 schools take the Challenges each year.

A. Maths Challenges (and mention follow-on rounds)How they work, why to do them, and how to get involvedWhat the questions are testing we will try some laterHow to use the questions in the classroom its not just for competitions!

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UKMT ActivitiesTeam Challenges: Team Challenge (years 8 and 9) and Senior Team Challenge (years 12 and 13), a series of regional events run throughout the UK (Feb, March and April for TMC, Oct and Nov for STMC) with a National Final held in London for both events. Teams of four students, a range of tasks for the day, very popular and great fun.

Primary Team Maths ResourcesNew on UKMT website, can be used in the classroom, or as a competition within a school or for a group of schools (e.g. secondaries with feeder schools), also useful in secondary school (probably lower ability for revision)

Mathematical Olympiad for Girls (UKMOG)Inaugural event hosted by Murray Edwards College in 2012, same format as the International Olympiad (two papers over two days). Aim to encourage and inspire girls, meet other female mathematicians.

B. Team CompetitionsOpportunity for team work for the whole class or year groupOpportunity for the most successful to represent the schoolCan provide focus for Maths ClubsCan be used with an individual class

C. Primary school resourcesThese are new, so talk about a bit more about where to find them and how to use them. Emphasise possibility for use in Y7/8. Will show some examples later.

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UKMT ActivitiesMentoring schemes for able pupils, ran externally or in schools

Summer Schools: week-long residential schools each year for able pupils in the UK . Five schools this year.

Training and selection of the team to represent the UK at the International Mathematical Olympiad (BMO1 leads to BMO round 2, and training events in Oxford, Hungary, Trinity College Cambridge, and Oundle School). Also UK teams at the Balkans Mathematical Olympiad, Romanian Masters in Mathematics and the European Girls Mathematical Olympiad.D. Mentoring and summer schoolsEmphasise that although senior and advanced mentoring is by invitation only, junior and intermediate mentoring can be organised within the school and is open to anyone. There will be an optional session on mentoring later today.

4UKMT Activities

Publications - past papers, and UKMT publications. UKMT are European agents for the Art of Problem Solving books. New books: - A problem solvers handbook by Andrew Jobbings - The Geometry of the triangle by Gerry LevershaAll books on sale here today.

Testbase. A CD ROM containing 825 questions from the Junior, Intermediate and Senior challenges from 1997 to 2008 and can perform searches on Challenge, topic, difficulty of the questions and allows people to create their own question sheets from them.

UKMT Yearbook sent to all schools which participate in the Challenges.

G. PublicationsOn sale today (no p&p) all 10, Gerrys GoT 15 (special conference price)Past Challenge and Olympiad questions, including TestBase.Bookscompetition preparation (general and by topic) mention new onesGeneral mathematical interest (e.g. starting point for extended project, individual or group presentations, posters)UK agents for Art of Problem Solving

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UKMT ActivitiesTeacher Meetings

Mathematical Circles: Two-day long non residential events packed with mathematics beyond the curriculum, aimed at Year 10 pupils. These are new events and well be organising more of them in the next couple of years, UKMT will contact your school when there is one in your area.

F. Teacher meetings

E. Mathematical CirclesTwo-day non-residential courses for Year 10 pupils with strong interest in mathematics. These are new events and well be organising more of them in the next couple of years, UKMT will contact your school when there is one in your area.

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UKMT website: www.ukmt.org.uk .

Extended challenge solutions, including some extension material.

Resources website: www.ukmt-resources.org.uk

All Challenge papers since 2004; many other resources.

The Millfield Team Competition(with thanks to Ceri Fides)I collect about 15 questions on a particular topic using the testbase software (or you could even just use a whole IMC or SMC paper) and then print out sets of questions on coloured paper. Each team gets one set of questions. Each question has three boxes at the bottom for answers (for this reason I normally remove the multiple choice option and just look for numerical answers. The questions are each worth different amounts of points and the team to answer a question first gets double points. The points for each question and current leader board are displayed throughout the lesson. The teacher just sits at the desk and students bring questions up. If they are correct hold onto the slips and if they are incorrect cross out one of the boxes and hand the slip back. When the teacher gets a chance they can record the scores on the score board (this is where the colours are invaluable as you can see which team handed in what and I keep all the slips in a pile so I can see which came in first).2. Using materials in the classroomShow different ways the materials can be used in the classroom, illustrating each use with questions from a different level.(a) Group competitions (taking one Q at a time)Example: Intermediate worksheet on number from TestBase (in their pack I have it printed out)Show Ceris score sheet

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TEAMQu1234567891011121314Pts33334457555689TotalA6336881014121618104B33334925C336618D3610928The first team to complete a question will get double pointsRelay competitionPrint out each question on a separate piece of paper, and make as many copies as there are teams (teams of 3-4 work well, or 4-6 if split into two groups)One member of each team comes to the front desk to collect Q1On completing Q1 they bring the answer to the front. If its correct sore 2 points and get Q2.If incorrect they can have another go for 1 point. After that give them Q2.Ask them to put the team number on each sheet and keep the sheets, put 2 or 1 for scores, then add up points at the end.Alternative: Each team splits into two groups who answer alternate questions and then take the next question to the other team.

(b) Relays (two pairs doing different questions)Example from Primary Team Challenge Relay Round (from UKMT website)

10Mini RelayEach team consists of either two or four pairs (or threes).With four pairs, each pair gets one question. With two pairs, one gets 1&3, the other 2&4.Questions 2, 3 and 4 use the answer from the previous question, but some working can be done before the answer is received.The pairs pass the answer sheet until all four questions have answers. The last pair then bring the sheet to the front desk.If all four answers are correct award 12 points. Otherwise award 3 points for each correct answer until the first wrong one. Mark the wrong answer and return the sheet, so they can have another go (starting with the pair who gave the first incorrect answer).The team to get four correct answers first gets additional 3 points.(See Primary or Senior Team Challenge materials on the website)Question 4T stands for the number you will receive.

I travel T km to work and it takes me 40 minutes.

What is my average speed in kilometres per hour?

Write down your answer.Split into four groups; each group to copy their question and do it; then read out answers from 1 to 4.12Question 3T stands for the number you will receive.

Work out:1/5 of T plus 1/6 of (T + 1)

Pass on your answer.Question 2T stands for the number you will receive.

A triangle has one angle of 2T 20 degrees and one angle of 3T + 10 degrees.

How many degrees are there in the third angle?

Pass on your answer.

Question 1The sum of two whole numbers is 30 and their difference is 20.

Pass on the larger of these two numbers.From Primary Resources

UKMT website: www.ukmt.org.uk Crossnumbers(c) CrossnumbersExample from Primary Resources (UKMT website)

Can be used with two teams (team of 4-6, 2 groups of 2-3 is best). Or each pair can have both sets of clues and they have to figure out in what order to do them. Good for practising number facts, strategy and planning.16

Head to HeadAnother round from a team challenge two teams guessing alternately.

Show how different types of questions can be adapted to various levels, and also just do some interesting questions. We will actually work through the questions together.(i) Head-to-HeadUsed in some team challenge competitions (give concrete examples from website)Allow people to suggest answers, but can have some volunteers as a team.

17Head to Head2 7 12 9 5 42 150 999 5 15 251985 30111 1999f(n) = 2n + 1

Try this one together first.Can also play with the whole class, using whiteboards to vote for the next number.18Head to Head2 5 10 4 7 30 0.5 200 7 28 103199033.2552 40 003f(n) = n2 + 3

Use to practice particular number bonds / facts. You can see how it can easily be adapted to any level and circumstance: competition, extension or revision.19Head to Head4 6 14 34 15 11 98 128 2 3 7 17 11 7 5 2f(n) = largest prime factor of n

This one should make them think how you cant always spot a pattern by looking at the first 3-4 numbers in the sequence. Can use when talking about sequences and patterns in early secondary, or as an into to a story about the search for prime numbers (Mersenne primes: people thought that all numbers of the form 2^p 1 were prime, but this is not true (e.g. p = 11), but by the early 1900s only numbers below 250 had been checked, and not much more could be done without computers. It is not known whether there are infinitely many Mersenne primes, and about 50 have been found so far, the largest has about 17million digits). Also can use to motivate the need for proof (e.g. Induction in FP1)20Head to Head8 12 16 24 40 6 99 5000 2 3 4 4 2 9 6 70f(n) = integer part of square root of n

Another one just to practice number facts, and also think about the idea that a function does not need to be defined by a formula (or have a continuous graph)21Head to Head6 12 16 1 3 8 11 20 3 6 835 65 6Six Twelve Sixteen One Three Eight Eleven Twenty f(n) = number of letters in the word n

And this is a classic22Same total problems

(ii) Same totals illustrate different levels, from just playing with numbers to algebraic proofs

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985*Seb has been challenged to place the numbers 1 to 9 inclusive in the nine regions formed by the Olympic rings s that there is exactly one number in each region and the sum of the numbersin each ring is 11. The diagram shows part of his solution.What number should replace * ?

A 6B 4C 3D 2E 1.

IMC 2012 Q8

We can immediately place 2, 4 and 3 in the spaces shown. So the values of c, d, e are 1, 6, 7 in some order. If c = 1 then d = 6, but then e would need to equal 2, not 7. If c = 6, then d =1, e = 7 and this is a valid solution. Finally, if c = 7 then d would need to equal 0, which is not possible. So in the only possible solution, * is replaced by 6.

c + d = 7, d + e = 8So c is one less then e, and the only possibility is c = 6, e = 7.

985c24de325122001 JMO QB1

The numbers from 1 to 7 inclusive are to be placed, one per square, in the figure on the right so that the totals of the three numbers in each of the three straight lines is the same.In how many different ways can this be done if the numbers 1 and 2 must be in the positions shown?

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When the three equal totals are added up, the numbers 2 and x appear twice and all other number appear once. Now the sum of the numbers from 1 to 7 inclusive is 28 and therefore the sum of the three equal totals will be 28 + x + 2 = 30 +x. This must be a multiple of 3 and hence x must also be a multiple of 3,i.e. x = 3 or 6. If x =3, the total of each line = 33/ 3 = 11 and therefore y = 6 and z = 7. The two remaining squares are filled with 4 and 5 so this may be done in two different ways.If x = 6, the total of each line = 36 /3 = 12 and therefore y = 4 and z = 5. The two remaining squares are filled with 3 and 7 so this may also be done in two different ways.There are, therefore, four different ways of completing the grid: 12x1235764123476512635471237763yz272006 JMO QB6

The numbers 1 to 7 are to be placed in the seven regions formed by three overlapping circles, with 6 in the central region, so that no regionis empty and the total of the numbers in each circle is T. What values of T are possible?

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acbfedLet the numbers inside the regions be a, b, c, d, e, f as shown.Then: a + d + e + 6 = T; b + d + f + 6 = T; c + e + f + 6 = T.Adding these equations gives a + b + c + 2d + 2e + 2f + 18 = 3T. Now a, b, c, d, e, f are 1, 2, 3, 4, 5, 7 in some order, so a + b + c + d + e + f = 22.Therefore 22 + d + e + f + 18 = 3T, that is 40 + d + e + f = 3T. The minimum value of d + e + f = 1 + 2 + 3 = 6 and its maximum value = 4 + 5 + 7 = 16.

So 46 3T 56 Since T is a positive integer, it cannot, then, take any values other than 16, 17 or 18.If T = 16, then d + e + f = 8 and the task may be completed with a = 7, b = 4, c = 3, d = 1, e = 2, f = 5. If T = 17, then d + e + f = 11 . However, a + d + e + 6 = T = 17, so a + d + e = 11.This requires a to equal f, which is impossible, so T cannot be 17.If T = 18, then d + e + f = 14 and the task may be completed with a = 5, b = 2, c = 1, d = 3, e = 4, f = 7.The only possible values of T, then, are 16 and 18.292004 JMO QB6

The diagram must be completed so that each square contains a different whole number from 1 to 12 inclusive and also so that the four numbers in the set of squares along each edge have the same total.In how many different ways can the diagram be completed correctly?

56312

302004 JMO QB6

The total of the numbers in the four rows = 78 + 5 + 6 + 12 + x= 101 + x

So 101 + x is a multiple of 4.

Possible values of x are 3, 7, 11 but 3 is allocated.

If x = 7 then each row totals 27.

If x = 11 then each row totals 28.56312

x312004 JMO QB6

If x = 7 then each row totals 27, so y = 12.(Impossible as 12 already allocated.)

56312

7y2004 JMO QB6

If x = 11 then each row totals 28, so y = 9.

56312

11y2004 JMO QB6

56312

119Numbers remaining: 1, 2, 4, 7, 8, 10Total 11Total 11

Total 10

822004 JMO QB6

56312

119Numbers remaining: 1, 4, 7, 10Total 11Total 11

Total 10

28Number of different ways = 16

Proving its impossible(iii) ImpossibilitiesCan also be done as group projects with posters produced at the end(i) mutilated chessboard(ii) tetrominoesfirst find all possible tetromionoes(iii) finding a path

36Two opposite corner squares are removed from a chessboard. Can the resulting board be covered by 2x1 dominoes?TETROMINOESHow many different tetrominoes are there?

38Can the seven tetrominoes be used to fill the 7 x 4 rectangle?

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42ABFind a route from room A to room B which goes through every other room exactly once.

UKMT website: www.ukmt.org.uk .

Extended challenge solutions, including some extension material.

Resources website: www.ukmt-resources.org.uk

All Challenge papers since 2004; many other resources.47