UK JUNIOR MATHEMATICAL CHALLENGE THURSDAY 26th APRIL 2012 Organised by the United Kingdom Mathematics Trust from the School of Mathematics, University of Leeds RULES AND GUIDELINES (to be read before starting) 1. Do not open the paper until the Invigilator tells you to do so. 2. Time allowed: 1 hour. No answers, or personal details, may be entered after the allowed hour is over. 3. The use of rough paper is allowed; calculators and measuring instruments are forbidden.
In the diagram PQ = PR = RS. What is the size of angle x? J54°K72°L90°M108° N144°
Classroom use? Use questions to create a mini competition
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).
Question 6 ( 4 points) Gladys plants 60 gladioli bulbs. When they flower, she notes that half are yellow; one third of those which are not yellow are red; and one quarter of those which are neither yellow nor red are pink. The remainder are white. What fraction of the gladioli are white?
TEAM Qu1234567891011121314 Pts33334455555689Total A63368810 121618104 B33334 925 C336 6 18 D36 10 928 The first team to complete a question will get double points
Intermediate Mathematical Olympiad and Kangaroo (IMOK) Olympiad Cayley Paper Thursday 15th March 2012 All candidates must be in School Year 9 or below (England and Wales), S2 or below (Scotland), or School Year 10 or below (Northern Ireland). READ THESE INSTRUCTIONS CAREFULLY BEFORE STARTING 1. Time allowed: 2 hours. 2. The use of calculators, protractors and squared paper is forbidden. Rulers and compasses may be used. 3. Solutions must be written neatly on A4 paper. Sheets must be STAPLED together in the top left corner with the Cover Sheet on top. 4. Start each question on a fresh A4 sheet. You may wish to work in rough first, then set out your final solution with clear explanations and proofs. Do not hand in rough work. 5. Answers must be FULLY SIMPLIFIED, and EXACT. They may contain symbols such as,fractions, or square roots, if appropriate, but NOT decimal approximations. 6. Give full written solutions, including mathematical reasons as to why your method is correct. Just stating an answer, even a correct one, will earn you very few marks; also, incomplete or poorly presented solutions will not receive full marks. 7. These problems are meant to be challenging! The earlier questions tend to be easier; the last two questions are the most demanding. Do not hurry, but spend time working carefully on one question before attempting another. Try to finish whole questions even if you cannot do many: you will have done well if you hand in full solutions to two or more questions.
What do these two questions have in common? Two different rectangles are placed together, edge to edge, to form a large rectangle. The length of the perimeter of the large rectangle is ⅔ of the total perimeter of the original two rectangles. Prove that the final rectangle is in fact a square. Teams A, B, C and D competed against each other once. The results table was as follows. (a) Find (with proof) which team won in each of the six matches. (b) Find (with proof) the scores in each of the six matches. TeamWinDrawLossForAgainst A30051 B11122 C02156 D01236
Answer 1 Both are questions from the Year 9 Olympiad paper from 2009
Answer 2 Both are questions chosen by the Welsh government for their publication “Developing higher order mathematical skills” as exemplars for what KS3 pupils can achieve and to support preparation for the 2012 GCSE where applications of mathematics and solving multi- step problems will be important
B5 Calum and his friend cycle from A to C, passing through B. During the trip he asks his friend how far they have cycled. His friend replies “one third as far as it is from here to B”. Ten miles later Calum asks him how far they have to cycle to reach C. His friend replies again “one third as far as it is from here to B”. How far from A will Calum have cycled when he reaches C?
Solution Let the points at which Calum asked each question be P and Q. Let the distances AP and QC be x miles and y miles respectively. Then the distance from P to B is 3x miles and the point P must lie between A and B. Similarly the distance from B to Q is 3y miles. Thus. A____P____________B_______________Q_____C x 3x 3y y Since they have cycled 10 miles between P and Q, we know that 3x + 3y = 10, and so AC = 4x + 4y = 4/3 (3x + 3y) = 4/3 × 10 = 13⅓ miles.
B1 In the Mathstown greengrocer’s shop: 9 apples cost the same as 6 bananas, and 4 bananas cost the same as 3 coconuts. At these same prices, A apples cost the same as 8 coconuts. Pass on the value of A. B3 T is the number that you will receive. A regular T-sided polygon has all of its interior diagonals drawn in. The polygon has D different lengths of interior diagonal. Pass on the value of D. B2 T is the number that you will receive. Five numbers (in no particular order) are: 10, 4, 20, 15, T Their mean is a and their median is b. Pass on the value of 2a – b. B4 T is the number that you will receive. Sophie cycles the first 150 metres of her race in 8T seconds, then cycles the remaining 250 metres in 12T seconds. Calculate her average speed for the whole 400 metre race, in metres per second.