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19. The island of Nogardia is inhabited by dragons, each of which has either six, seven or eightlegs. Dragons with seven legs always lie; dragons with an even number of legs always tell thetruth. One day four dragons met.The blue one said, “We have 28 legs altogether.”The green one said, “We have 27 legs altogether.”The yellow one said, “We have 26 legs altogether.”The red one said, “We have 25 legs altogether.”Which of the following statements is true?A the red dragon definitely has 6 legs B the red dragon definitely has 7 legsC the red dragon definitely has 8 legsD the red dragon has either 6 or 8 legs, but we can’t be sure whichE the red dragon has 6, 7, or 8 legs, but we can’t be sure which20. The diagram shows a square with sides of length 2. Four semicirclesare drawn whose centres are the four vertices of the square. Thesesemicircles meet at the centre of the square, and adjacent semicirclesmeet at their ends. Four circles are drawn whose centres lie on theedges of the square and which each touch two semicircles. What is thetotal shaded area?16A 4π(3 − 2 2) B 4π 2 C D E9 π π 42 π21. The first three terms of a sequence are 1, 2, 3. From the fourth term onwards, each term iscalculated from the previous three terms using the rule “Add the first two and subtract the third.”So the sequence begins 1, 2, 3, 0, 5, −2, 7, … . What is the 2010th term in the sequence?A −2006 B −2004 C −2002 D −2000 E some other number22. A single natural number is written on each edge of a pentagon so that adjacent numbers neverhave a common factor greater than 1 and non-adjacent numbers always have a common factorgreater than 1. Which of the following could be one of the numbers?A 1 B 8 C 9 D 10 E 1123. How many 3-digit integers have the property that their middle digit is the mean of the othertwo digits?A 9 B 12 C 16 D 25 E 4524. An oval is constructed from four arcs of circles. Arc PQ isthe same as arc RS, and has radius 1 cm. Arc QR is thesame as arc PS. At the points P, Q, R, S where the arcstouch, they have a common tangent. The oval touches themidpoints of the sides of a rectangle with dimensions 8 cmby 4 cm. What is the radius of the arc PS, in cm?A 6 B 6.5 C 7 D 7.5 E 825. A bar code of the type shown is composed of alternate strips of black andwhite, always beginning and ending with a black strip. Each strip in the barcode has width either 1 or 2, and the total width of the bar code is 12. Two barcodes are different if they read differently from left to right. How manydifferent bar codes of this type can be made?A 24 B 132 C 66 D 116 E 144P1Q8SRdiagram not to scale4UKMTUKMTUKMTEUROPEAN ‘KANGAROO’ MATHEMATICAL CHALLENGE‘PINK’Thursday 18th March 2010Organised by the United Kingdom Mathematics Trust and theAssociation Kangourou Sans FrontièresKangaroo papers are being taken by over 5.5 million studentsin 46 countries in Europe and beyond.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 areforbidden.4. Candidates in England and Wales must be in School Year 10 or 11.Candidates in Scotland must be in S3 or S4.Candidates in Northern Ireland must be in School Year 11 or 12.5. Use B or HB pencil only. For each question, mark at most one of the options A, B, C,D, E on the Answer Sheet. Do not mark more than one option.6. Five marks will be awarded for each correct answer to Questions 1 - 15.Six marks will be awarded for each correct answer to Questions 16 - 25.7. 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.8. The questions on this paper challenge you to think, not to guess. Though you will notlose marks for getting answers wrong, you will undoubtedly get more marks, and moresatisfaction, by doing a few questions carefully than by guessing lots of answers.Enquiries about the European Kangaroo should be sent to: Maths Challenges Office,School of Maths Satellite, University of Leeds, Leeds, LS2 9JT.(Tel. 0113 343 2339)http://www.ukmt.org.uk

1. What is the result of dividing 20102010 by 2010?A 11 B 101 C 1001 D 10001 E not an integer2. Ivan, Tibor and Alex sat a test and achieved 85%, 90% and 100% respectively. Tibor scoredjust one more mark than Ivan. How many marks did Alex get?A 5 B 17 C 18 D 20 E 253. Four cubes, each with surface area 24 cm 2 , are placed together to forma cuboid as shown. What is the surface area of this cuboid, in cm 2 ?A 24 B 32 C 64 D 92 E 964. A rectangular strip of paper is folded in half three times, with each fold line parallel to theshort edges. It is then unfolded so that the seven folds up or down can all be seen. Which ofthe following strips, viewed from a long edge, could not be made in this way?ABCDE5. Six points are marked on a sheet of squared paper as shown. Which ofthe following shapes cannot be made by connecting some of thesepoints using straight lines?A parallelogram B trapezium C right-angled triangleD obtuse-angled triangle E all the shapes A – D can be made6. Brigitte plans to visit Verona. Starting and finishing at Verona train station, she wants to crosseach of the five famous bridges across the river Adige at least once, without crossing anyother bridge. Brigitte realises that there are only certain possibilities for the number of timesshe would cross the river. Which of the following is possible?A 4 B 5 C 6 D 7 E 97. The diagram shows a square PQRS and two equilateral triangles RSUand PST. PQ has length 1. What is the length of TU?3A 2 B C 3 D 5 − 1 E 6 − 128. Today is my teacher’s birthday. He says that the product of his age in years and his father’sage in years is 2010. In which year was my teacher born?A 1943 B 1953 C 1980 D 1995 E 20059. In the diagram, angle PQR is 20° , and the reflex angle at P is330° . The line segments QT and SU are perpendicular. Whatis the size of angle RSP?A 10° B 20° C 30° D 40° E 50°QS20°QRRTUUPSTdiagram not to scaleP330°10. A positive integer is called ‘jumpy’ if the sum of its digits is 2010 and the product of its digitsis 2. How many ‘jumpy’ integers are there?A 2010 B 2009 C 2008 D 1005 E 100411. Today's date is Thursday the 18th of March, which is an even day of the month. In a certainmonth, three Thursdays fell on even days. What day of the week was the 21st day of thatmonth?A Monday B Tuesday C Wednesday D Thursday E Friday12. A circle of radius 4 cm is divided into four congruent parts by arcs ofradius 2 cm as shown. What is the length of the perimeter of one of theparts, in cm?A 2π B 4π C 6π D 8π E 12π13. The scatter graph shows the distance run and time taken by fivestudents during a training session. Who ran with the fastestaverage speed?A Alicia B Bea C Carlos D Dani E Ernesto14. A triangular piece of paper is folded along the dotted lineshown in the left-hand diagram to form the heptagonshown in the right-hand diagram. The total area of theshaded parts of the heptagon is 1 cm 2 . The area of theoriginal triangle is 1½ times the area of the heptagon.What is the area of the original triangle, in cm ?DistanceODaniBeaCarlosErnestoAliciaTime2 diagram not to scaleA 2 B 3 C 4 D 5 E more information needed15. In a supermarket trolley park, there are two lines of tightly-packed trolleys.The first line has ten trolleys and is 2.9 m long. The second line has twentytrolleys and is 4.9 m long. What is the length of one trolley, in m?A 0.8 B 1 C 1.1 D 1.2 E 1.416. The diagram shows a large equilateral triangle divided into 36 smallequilateral triangles, each with area 1 cm 2 . What is the area of theshaded triangle, in cm 2 ?A 11 B 12 C 13 D 14 E 1517. The diagram shows a trapezium FGHI with FG parallel to IH.GH and FI both have length 2.The point M is the midpoint of FI and ∠HMG = 90° .What is the length of the perimeter of the trapezium?A 5 B 6 C 7 D 8 E impossible to determineMIHFGdiagram not to scale18. How many integers n, between 1 and 100 inclusive, have the property that n n is a squarenumber?A 99 B 55 C 50 D 10 E 5