Download [PDF] 2018 UKMT Senior Kangaroo Questions Mathematical Challenge United Kingdom Mathematics Trust

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Senior Kangaroo Friday 30 November 2018 Organised by the United Kingdom Mathematics Trust a member of the Association Kangourou sans Frontières England & Wales: Year 13 or below Scotland: S6 or below Northern Ireland: Year 14 or below Instructions 1.Do not open the paper until the invigilator tells you to do so. 2.Time allowed: 60 minutes. No answers, or personal details, may be entered after the allowed time is over. 3.The use of blank or lined paper for rough working is allowed; squared paper ,calculators and measuring instruments are forbidden . 4. Use a B or an HB non-propelling pencil to record your answer to each problem as a three-digit number from 000 to 999. Pay close attention to the example on the Answer Sheet that shows how to code your answers. 5. Do not expect to finish the whole paper in the time allowed. The questions in this paper have been arranged in approximate order of difficulty with the harder questions towards the end. You are not expected to complete all the questions during the time. You should bear this in mind when deciding which questions to tackle. 6. Scoring rules: 5 marks are awarded for each correct answer; There is no penalty for giving an incorrect answer. 7. The questions on this paper are designed to challenge you to think, not to guess. You will gain more marks, and more satisfaction, by doing one question carefully than by guessing lots of answers. This paper is about solving interesting problems, not about lucky guessing. Enquiries about the Senior Kangaroo should be sent to: UK Mathematics Trust, School of Mathematics, University of Leeds, Leeds LS2 9JT T 0113 343 2339 enquiry@ukmt.org.uk www.ukmt.org.uk

Senior Kangaroo Friday 30 November 20181.My age is a two-digit number that is a power of 5. My cousin’s age is a two-digit number that is a power of 2. The sum of the digits of our ages is an odd number. What is the product of the digits of our ages? 2. Let Kbe the largest integer for which n200

Senior Kangaroo Friday 30 November 20188.An integer xsatisfies the inequality x 2 ≤ 729 ≤ − x3 .P and Q are possible values of x. What is the maximum possible value of 10(P −Q)? 9. The two science classes 7A and 7B each consist of a number of boys and a number of girls. Each class has exactly 30 students. The girls in 7A have a mean score of 48. The overall mean across both classes is 60. The mean score across all the girls of both classes is also 60. The 5 girls in 7B have a mean score that is double that of the 15 boys in 7A. The mean score of the boys in 7B is µ. What is the value of 10µ? 10. The function SPF (n ) denotes the sum of the prime factors of n, where the prime factors are not necessarily distinct. For example, 120=23 × 3× 5, so SPF (120 )= 2+ 2+ 2+ 3+ 5= 14 . Find the value of SPF(2 22 −4). 11. A sequence U 1, U 2, U 3, . . . is defined as follows: • U 1= 2; • if U nis prime then U n+ 1 is the smallest positive integer not yet in the sequence; • if U nis not prime then U n+ 1 is the smallest prime not yet in the sequence. The integer kis the smallest such that U k+ 1 − U k> 10 . What is the value of k× U k? 12. The diagram shows a 16 metre by 16 metre wall. Three grey squares are painted on the wall as shown. The two smaller grey squares are equal in size and each makes an angle of 45°with the edge of the wall. The grey squares cover a total area of Bmetres squared. What is the value of B? 13. A nine-digit number is odd. The sum of its digits is 10. The product of the digits of the number is non-zero. The number is divisible by seven. When rounded to three significant figures, how many millions is the number equal to? © UK Mathematics Trust www.ukmt.org.uk

Senior Kangaroo Friday 30 November 201814.A square ABC D has side 40 units. Point F is the midpoint of side AD . Point G lies on C F such that 3 CG =2G F . What is the area of triangle BCG? 15. In the sequence 20 ,18 ,2,20 , − 18 , . . . the first two terms a 1 and a2 are 20 and 18 respectively. The third term is found by subtracting the second from the first, a3 = a 1 − a 2 . The fourth is the sum of the two preceding elements, a 4 = a 2 + a 3. Then a 5 = a 3 − a 4, a 6 = a 4 + a 5, and so on. What is the sum of the first 2018 terms of this sequence? 16. A right-angled triangle has sides of integer length. One of its sides has length 20. Toni writes down a list of all the different possible hypotenuses of such triangles. What is the sum of all the numbers in Toni’s list? 17. Sarah chooses two numbers a and bfrom the set {1,2,3, . . . , 26 }. The product ab is equal to the sum of the remaining 24 numbers. What is the difference between aand b? 18. How many zeros are there at the end of 2018! 30! ×11! ? 19. Shan solves the simultaneous equations xy = 15 and (2 x − y)4 = 1 where xand yare real numbers. She calculates z, the sum of the squares of all the y-values in her solutions. What is the value of z? 20. Determine the value of the integer ygiven that y= 3x2 and 2 x 5 = 1 1 − 2 3 + 1 4 − 5 6 − x © UK Mathematics Trust www.ukmt.org.uk