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UKMT UKMT UKMT UK I NTERMEDIATE M ATHEMATICAL C HALLENGE THURSDAY 6TH FEBRUARY 2014 Organised by the United Kingdom Mathematics Trust and supported by 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 . 4. Candidates in England and Wales must be in School Year 11 or below. Candidates in Scotland must be in S4 or below. Candidates in Northern Ireland must be in School Year 12 or below. 5. Use B or HB pencil only . Mark at most one of the options A, B, C, D, E on the Answer Sheet for each question. Do not mark more than one option. 6. Do not expect to finish the whole paper in 1 hour . Concentrate first on Questions 1-15. When you have checked your answers to these, have a go at some of the later questions. 7. Five marks are awarded for each correct answer to Questions 1-15. Six marks are awarded for each correct answer to Questions 16-25. Each incorrect answer to Questions 16-20 loses 1 mark. Each incorrect answer to Questions 21-25 loses 2 marks. 8. Your Answer Sheet will be read only by a dumb machine . Do not write or doodle on the sheet except to mark your chosen options . The machine 'sees' all black pencil markings even if they are in the wrong places. If you mark the sheet in the wrong place, or leave bits of rubber stuck to the page, the machine will 'see' a mark and interpret this mark in its own way. 9. The questions on this paper challenge you to think , not to guess. You get more marks, and more satisfaction, by doing one question carefully than by guessing lots of answers. The UK IMC is about solving interesting problems, not about lucky guessing. The UKMT is a registered charity http://www.ukmt.org.uk

1. What is 25 % of ?34 A B C D 1 E 3 3 16 1 4 1 3 2. Which is the smallest positive integer for which all these are true? (i) It is odd. (ii) It is not prime. (iii) The next largest odd integer is not prime. A 9 B 15 C 21 D 25 E 33 3. An equilateral triangle is placed inside a larger equilateral triangle so that the diagram has three lines of symmetry. What is the value of ?x A 100 B 110 C 120 D 130 E 150 x° 4. You are given that is an even integer and is an odd integer. Which of these is an odd integer ? m n A B C D E 3m + 4n 5mn (m + 3n)2 m3n3 5m + 6n 5. A ship's bell is struck every half hour, starting with one bell at 0030, two bells (meaning the bell is struck twice) at 0100, three bells at 0130 until the cycle is complete with eight bells at 0400. The cycle then starts again with one bell at 0430, two bells at 0500 and so on. What is the total number of times the bell is struck between 0015 on one day and 0015 on the following day? A 24 B 48 C 108 D 144 E 216 6. The shape shown on the right was assembled from three identical copies of one of the smaller shapes below, without gaps or overlaps. Which smaller shape was used? A B C D E 7. Just one positive integer has exactly 8 factors including 6 and 15. What is the integer? A 21 B 30 C 45 D 60 E 90 8. A large cube is made by stacking eight dice. The diagram shows the result, except that one of the dice is missing. Each die has faces with 1, 2, 3, 4, 5 and 6 pips and the total number of pips on each pair of opposite faces is 7. When two dice are placed face to face, the matching faces must have the same number of pips. What could the missing die look like? A B C D E

9. At the age of twenty-six, Gill has passed her driving test and bought a car. Her car uses litres of petrol per 100 km travelled. How many litres of petrol would be required for a journey of km? p d A B C D E pd 100 100p d 100d p 100 pd p 100d 10. The diagram shows five touching semicircles, each with radius 2. What is the length of the perimeter of the shaded shape? A B C D E 5p 6p 7p 8p 9p 11. Not all characters in the Woodentops series tell the truth. When Mr Plod asked them, ”How many people are there in the Woodentops family?•, four of them replied as follows: Jenny: ”An even number.• Willie: ”An odd number.• Sam: ”A prime number.• Mrs Scrubitt: ”A number which is the product of two integers greater than one.• How many of these four were telling the truth? A 0 B 1 C 2 D 3 E 4 12. The diagram shows an isosceles right-angled triangle divided into strips of equal width. Four of the strips are shaded. What fraction of the area of the triangle is shaded? A B C D E 11 32 3 8 13 32 7 16 15 32 13. How many numbers can be written as a sum of two different positive integers each at most 100? A 100 B 197 C 198 D 199 E 200 14. This year the Tour de France starts in Leeds on 5 July. Last year, the total length of the Tour was 3404 km and the winner, Chris Froome, took a total time of 83 hours 56 minutes 40 seconds to cover this distance. Which of these is closest to his average speed over the whole event? A 32 km/h B 40 km/h C 48 km/h D 56 km/h E 64 km/h 15. Zac halves a certain number and then adds 8 to the result. He finds that he obtains the same answer if he doubles his original number and then subtracts 8 from the result. What is Zac's original number? A B C D E 823 913 923 1013 1023

16. The base of a triangle is increased by 25% but the area of the triangle is unchanged. By what percentage is the corresponding perpendicular height decreased? A B C D E 1212% 16% 20% 25% 50% 17. How many weeks are there in 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 minutes? A 1 B 2 C 3 D 4 E 5 18. Consider looking from the origin (0, 0) towards all the points , where each of and is an integer. Some points are hidden , because they are directly in line with another nearer point. For example, (2, 2) is hidden by (1, 1). (m,n) m n How many of the points (6, 2), (6, 3), (6, 4), (6, 5) are not hidden points? A 0 B 1 C 2 D 3 E 4 19. Suppose that . What is the value of ? 8m = 27 4m A 3 B 4 C 9 D 13.5 E there is no such m 20. The diagram shows a regular pentagon and five circular arcs. The sides of the pentagon have length 4. The centre of each arc is a vertex of the pentagon, and the ends of the arc are the midpoints of the two adjacent edges. What is the total shaded area? A B C D E 8p 10p 12p 14p 16p 21. In King Arthur's jousting tournament, each of the several competing knights receives 17 points for every bout he enters. The winner of each bout receives an extra 3 points. At the end of the tournament, the Black Knight has exactly one more point than the Red Knight. What is the smallest number of bouts that the Black Knight could have entered? A 3 B 4 C 5 D 6 E 7 22. The positive integers , and are all different. None of them is a square but all the products , and are squares. What is the least value that can take? a b c ab ac bc a + b + c A 14 B 28 C 42 D 56 E 70 23. A sector of a disc is removed by making two straight cuts from the circumference to the centre. The perimeter of the sector has the same length as the circumference of the original disc. What fraction of the area of the disc is removed? A B C D E p - 1 p 1 p p 360 1 3 1 2 24. How many 4-digit integers (from 1000 to 9999) have at least one digit repeated? A B C D E 62 × 72 52 × 72 52 × 82 42 × 82 42 × 92 25. The diagram shows two concentric circles with radii 1 and 2 units, together with a shaded octagon, all of whose sides are equal. What is the length of the perimeter of the octagon? A B C 8 2 8 3 8 3p D E 2 5 + 2 2 8 5 - 2 2