Download [PDF] 2019 UKMT Senior Kangaroo Questions Mathematical Challenge United Kingdom Mathematics Trust

File Information

Filename:	[PDF] 2019 UKMT Senior Kangaroo Questions Mathematical Challenge United Kingdom Mathematics Trust.pdf
Filesize:	264.25 KB
Uploaded:	09/08/2021 18:18:06
Keywords:	2019 ukmt senior kangaroo questions mathematical challenge united kingdom mathematics trust pdf
Description:	Download file or read online UKMT past exam paper senior mathematical challenge SMC senior kangaroo 2019 questions - United Kingdom Mathematics Trust
Downloads:	1

File Preview

Download Urls

Short Page Link

https://www.edufilestorage.com/3ey

Full Page Link

https://www.edufilestorage.com/3ey/PDF_2019_UKMT_Senior_Kangaroo_Questions_Mathematical_Challenge_United_Kingdom_Mathematics_Trust.pdf

HTML Code

<a href="https://www.edufilestorage.com/3ey/PDF_2019_UKMT_Senior_Kangaroo_Questions_Mathematical_Challenge_United_Kingdom_Mathematics_Trust.pdf" target="_blank" title="Download from eduFileStorage.com"><img src="https://www.edufilestorage.com/cache/plugins/filepreviewer/2434/pdf/150x190_middle_46f4e7862b1eb5bd4935adbbba5d79e8.jpg"/></a>

Forum Code

[url=https://www.edufilestorage.com/3ey/PDF_2019_UKMT_Senior_Kangaroo_Questions_Mathematical_Challenge_United_Kingdom_Mathematics_Trust.pdf][img]https://www.edufilestorage.com/cache/plugins/filepreviewer/2434/pdf/150x190_middle_46f4e7862b1eb5bd4935adbbba5d79e8.jpg[/img][/url]

[PDF] 2019 UKMT Senior Kangaroo Questions Mathematical Challenge United Kingdom Mathematics Trust.pdf | Plain Text

Senior Kangaroo Friday 29 November 2019 © 2019 UK 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 T0113 343 2339 enquiry@ukmt.org.uk www.ukmt.org.uk

Senior Kangaroo Friday 29 November 20191. What is the sum of all the factors of 144? 2.When I noticed that 24 = 42, I tried to find other pairs of numbers with this property. Trying 2and 16, I realised that 216 is larger than 162 . How many times larger is 216 ? 3. The two diagonals of a quadrilateral are perpendicular. The lengths of the diagonals are 14 and 30. What is the area of the quadrilateral? 4. The integer nsatisfies the inequality n+ (n + 1) + (n + 2) + · · · +(n + 20 )> 2019 . What is the minimum possible value of n? 5. Identical regular pentagons are arranged in a ring. The partially completed ring is shown in the diagram. Each of the regular pentagons has a perimeter of 65 . The regular polygon formed as the inner boundary of the ring has a perimeter of P. What is the value of P? 6. For natural numbers aand bwe are given that 2019 = a2 − b2 . It is known that a < 1000 . What is the value of a? 7. How many positive? integers nexist such that both n + 1 3 and 3n + 1are three-digit integers? 8. The function J(x ) is defined by: J (x ) =         4 + x forx≤ − 2, − x for−2 < x≤ 0, x forx> 0. How many distinct real solutions has the equation J(J (J (x ))) =0? 9. What is the smallest three-digit number K which can be written as K =ab + ba , where both aand bare one-digit positive integers? 10. What is the value of q 13 +p 28 +√ 281 ×q 13 −p 28 +√ 281 ×p 141 +√ 281 ? © 2019 UK Mathematics Trust www.ukmt.org.uk = 4 = 2 24 4 xy

Senior Kangaroo Friday 29 November 201911.In the triangle ABC the points M and N lie on the side AB such that AN=AC and B M =BC . We know that ∠M C N =43 °. Find the size in degrees of ∠AC B . 12. What is the value of A2 + B3 + C5 , given that: A = 3 p 16 √ 2 B = p 9 3 √ 9 C =[( 5 √ 2 )2 ]2 13. The real numbers aand b, where a > b , are solutions to the equation 32x − 10 ×3x+ 1 + 81 = 0. What is the value of 20a2 + 18 b2 ? 14. A number N is the product of three distinct primes. How many distinct factors does N 5 have? 15. Five Bunchkins sit in a horizontal field. No three of the Bunchkins are sitting in a straight line. Each Bunchkin knows the four distances between her and each of the others. Each Bunchkin calculates and then announces the total of these distances. These totals are 17, 43, 56, 66 and 76. A straight line is painted joining each pair of Bunchkins. What is the total length of paint required? 16. The real numbers xand ysatisfy the equations: x y − x= 180 andy+ xy = 208 . Let the two solutions be (x 1, y 1) and (x 2, y 2) . What is the value of x 1 + 10 y 1 + x 2 + 10 y 2? 17. In triangle ABC ,∠ B AC is 120 °. The length of AB is 123 . The point M is the midpoint of side BC. The line segments ABand AM are perpendicular. What is the length of side AC? 18. An integer is said to be chunky if it consists only of non-zero digits by which it is divisible when written in base 10. For example, the number 936 is Chunky since it is divisible by 9, 3 and 6. How many chunky integers are there between 13 and 113? © 2019 UK Mathematics Trust www.ukmt.org.uk A BC M N43 ◦

Senior Kangaroo Friday 29 November 201919.The square ABC D has sides of length 105 . The point M is the midpoint of side BC . The point N is the midpoint of B M . The lines B D and AM meet at the point P. The lines B D and ANmeet at the point Q. What is the area of triangle APQ? 20. Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct. What is the answer to 3 ACROSS? ACROSS DOWN 1. A composite factor of 1001 1. One more than a prime, one less than a prime 3. Not a palindrome 2. A multiple of 9 5. pq3 where p, q prime and p, q 4.p3 q using the same p, q as 5 ACROSS © 2019 UK Mathematics Trust www.ukmt.org.uk12 34 5