[PDF] UKMT - JMC - Junior Mathematical Challenge 2015 - Questions.pdf | Plain Text

19. One of the following cubes is the smallest cube that can be written as the sum of three positivecubes. Which is it? A 27 B 64 C 125 D 216 E 512 20. The diagram shows a pyramid made up of 30 cubes, each measuring . 1m× 1m× 1m What is the total surface area of the whole pyramid (including its base)? A B C D E 30m2 62m2 72m2 152m2 180m2 21. Gill is now 27 and has moved into a new flat. She has four pictures to hang in a horizontal rowon a wall which is 4800 mm wide. The pictures are identical in size and are 420 mm wide.Gill hangs the first two pictures so that one is on the extreme left of the wall and one is on theextreme right of the wall. She wants to hang the remaining two pictures so that all fourpictures are equally spaced. How far should Gill place the centre of each of the two remainingpictures from a vertical line down the centre of the wall? A 210 mm B 520 mm C 730 mm D 840 mm E 1040 mm 22. The diagram shows a shaded region inside a regular hexagon. The shaded region is divided into equilateral triangles. What fraction of the area of the hexagon is shaded? A B C D E 3 8 2 5 3 7 5 12 1 2 23. The diagram shows four shaded glass squares, with areas 1 cm 2, 4 cm 2, 9 cm 2 and 16 cm 2, placed in the corners of a rectangle. The largest square overlaps two others. The area of the region inside the rectangle but not covered by any square (shown unshaded) is cm 2. 1.5 What is the area of the region where squares overlap (shown dark grey)? A B C D E 2.5cm 2 3cm 2 3.5cm 2 4cm 2 4.5cm 2 24. A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers thatare both multiples of 45? A 9180 B 9090 C 9000 D 8910 E 8190 25. The four straight lines in the diagram aresuch that . The sizes of , and are , and . VU = VW •UXZ •VYZ •VZX x°y° z° Which of the following equations gives in terms of and ? x y z U X W Y Z V x° y° z° A B C x=y-z x=180-y-z x= y- z 2 D E x= y+z-90 x= y-z 2 UKMT UKMT UKMT UK J UNIOR M ATHEMATICAL C HALLENGE THURSDAY 30th APRIL 2015 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 . 4. Candidates in England and Wales must be in School Year 8 or below. Candidates in Scotland must be in S2 or below. Candidates in Northern Ireland must be in School Year 9 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 JMC is about solving interesting problems, not about lucky guessing. The UKMT is a registered charity http://www.ukmt.org.uk

1. Which of the following calculations gives the largest answer? A B C D E 1™2+3+4 1+2™3+4 1+2+3™4 1+2™3™4 1™2™3+4 2. It has just turned 22:22. How many minutes are there until midnight? A 178 B 138 C 128 D 108 E 98 3. What is the value of ? 12 345 1+2+3+4+5 A 1 B 8 C 678 D 823 E 12 359 4. In this partly completed pyramid, each rectangle is to befilled with the sum of the two numbers in the rectanglesimmediately below it. What number should replace ?x A 3 B 4 C 5 D 7 E 12 105 47 31 13 9 x 5. The difference between of a certain number and of the same number is 3. What is that number? 1 3 1 4 A 24 B 36 C 48 D 60 E 72 6. What is the value of in this triangle?x A 45 B 50 C 55 D 60 E 65 x° 110° 120° 7. The result of the calculation 123 456 789 × 8 is almost the same as 987 654 321 except thattwo of the digits are in a different order. What is the sum of these two digits? A 3 B 7 C 9 D 15 E 17 8. Which of the following has the same remainder when it is divided by 2 as when it is dividedby 3? A 3 B 5 C 7 D 9 E 11 9. According to a newspaper report, ”A 63-year-old man has rowed around the world withoutleaving his living room.• He clocked up 25 048 miles on a rowing machine that he receivedfor his 50th birthday. Roughly how many miles per year has he rowed since he was given the machine? A 200 B 500 C 1000 D 2000 E 4000 10. In the expression each is to be replaced by either + or . 1 ¸ 2 ¸ 3 ¸ 4 ¸ × What is the largest value of all the expressions that can be obtained in this way? A 10 B 14 C 15 D 24 E 25 11. What is the smallest prime number that is the sum of three different prime numbers? A 11 B 15 C 17 D 19 E 23 12. A fish weighs the total of 2 kg plus a third of its own weight. What is the weight of the fish in kg? A B 3 C 4 D 6 E 8 213 13. In the figure shown, each line joining two numbers is to be labelled with the sum of the two numbers that are at its endpoints. How many of these labels are multiples of 3? A 10 B 9 C 8 D 7 E 6 1 2 3 4 5 6 7 8 14. Digits on a calculator are represented by a number of horizontal andvertical illuminated bars. The digits and the bars which represent them areshown in the diagram. How many digits are both prime and represented by a prime number ofilluminated bars? A 0 B 1 C 2 D 3 E 4 15. Which of the following is divisible by all of the integers from 1 to 10 inclusive? A B C D E 23×34 34×45 45×56 56×67 67×78 16. The diagram shows a square inside an equilateral triangle. What is the value of ? x+y A 105 B 120 C 135 D 150 E 165 x° y° 17. Knave of Hearts: ”I stole the tarts.• Knave of Clubs: ”The Knave of Hearts is lying.• Knave of Diamonds: ”The Knave of Clubs is lying.• Knave of Spades: ”The Knave of Diamonds is lying.• How many of the four Knaves were telling the truth? A 1 B 2 C 3 D 4 E more information needed 18. Each of the fractions and uses the digits 1 to 9 exactly once. The first fraction simplifies to . What is the simplified form of the second fraction? 2637 18 459 5274 36 918 1 7 A B C D E 1 8 1 7 5 34 9 61 2 7

1. Which of the following calculations gives the largest answer? A B C D E 1™2+3+4 1+2™3+4 1+2+3™4 1+2™3™4 1™2™3+4 2. It has just turned 22:22. How many minutes are there until midnight? A 178 B 138 C 128 D 108 E 98 3. What is the value of ? 12 345 1+2+3+4+5 A 1 B 8 C 678 D 823 E 12 359 4. In this partly completed pyramid, each rectangle is to befilled with the sum of the two numbers in the rectanglesimmediately below it. What number should replace ?x A 3 B 4 C 5 D 7 E 12 105 47 31 13 9 x 5. The difference between of a certain number and of the same number is 3. What is that number? 1 3 1 4 A 24 B 36 C 48 D 60 E 72 6. What is the value of in this triangle?x A 45 B 50 C 55 D 60 E 65 x° 110° 120° 7. The result of the calculation 123 456 789 × 8 is almost the same as 987 654 321 except thattwo of the digits are in a different order. What is the sum of these two digits? A 3 B 7 C 9 D 15 E 17 8. Which of the following has the same remainder when it is divided by 2 as when it is dividedby 3? A 3 B 5 C 7 D 9 E 11 9. According to a newspaper report, ”A 63-year-old man has rowed around the world withoutleaving his living room.• He clocked up 25 048 miles on a rowing machine that he receivedfor his 50th birthday. Roughly how many miles per year has he rowed since he was given the machine? A 200 B 500 C 1000 D 2000 E 4000 10. In the expression each is to be replaced by either + or . 1 ¸ 2 ¸ 3 ¸ 4 ¸ × What is the largest value of all the expressions that can be obtained in this way? A 10 B 14 C 15 D 24 E 25 11. What is the smallest prime number that is the sum of three different prime numbers? A 11 B 15 C 17 D 19 E 23 12. A fish weighs the total of 2 kg plus a third of its own weight. What is the weight of the fish in kg? A B 3 C 4 D 6 E 8 213 13. In the figure shown, each line joining two numbers is to be labelled with the sum of the two numbers that are at its endpoints. How many of these labels are multiples of 3? A 10 B 9 C 8 D 7 E 6 1 2 3 4 5 6 7 8 14. Digits on a calculator are represented by a number of horizontal andvertical illuminated bars. The digits and the bars which represent them areshown in the diagram. How many digits are both prime and represented by a prime number ofilluminated bars? A 0 B 1 C 2 D 3 E 4 15. Which of the following is divisible by all of the integers from 1 to 10 inclusive? A B C D E 23×34 34×45 45×56 56×67 67×78 16. The diagram shows a square inside an equilateral triangle. What is the value of ? x+y A 105 B 120 C 135 D 150 E 165 x° y° 17. Knave of Hearts: ”I stole the tarts.• Knave of Clubs: ”The Knave of Hearts is lying.• Knave of Diamonds: ”The Knave of Clubs is lying.• Knave of Spades: ”The Knave of Diamonds is lying.• How many of the four Knaves were telling the truth? A 1 B 2 C 3 D 4 E more information needed 18. Each of the fractions and uses the digits 1 to 9 exactly once. The first fraction simplifies to . What is the simplified form of the second fraction? 2637 18 459 5274 36 918 1 7 A B C D E 1 8 1 7 5 34 9 61 2 7

19. One of the following cubes is the smallest cube that can be written as the sum of three positivecubes. Which is it? A 27 B 64 C 125 D 216 E 512 20. The diagram shows a pyramid made up of 30 cubes, each measuring . 1m× 1m× 1m What is the total surface area of the whole pyramid (including its base)? A B C D E 30m2 62m2 72m2 152m2 180m2 21. Gill is now 27 and has moved into a new flat. She has four pictures to hang in a horizontal rowon a wall which is 4800 mm wide. The pictures are identical in size and are 420 mm wide.Gill hangs the first two pictures so that one is on the extreme left of the wall and one is on theextreme right of the wall. She wants to hang the remaining two pictures so that all fourpictures are equally spaced. How far should Gill place the centre of each of the two remainingpictures from a vertical line down the centre of the wall? A 210 mm B 520 mm C 730 mm D 840 mm E 1040 mm 22. The diagram shows a shaded region inside a regular hexagon. The shaded region is divided into equilateral triangles. What fraction of the area of the hexagon is shaded? A B C D E 3 8 2 5 3 7 5 12 1 2 23. The diagram shows four shaded glass squares, with areas 1 cm 2, 4 cm 2, 9 cm 2 and 16 cm 2, placed in the corners of a rectangle. The largest square overlaps two others. The area of the region inside the rectangle but not covered by any square (shown unshaded) is cm 2. 1.5 What is the area of the region where squares overlap (shown dark grey)? A B C D E 2.5cm 2 3cm 2 3.5cm 2 4cm 2 4.5cm 2 24. A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers thatare both multiples of 45? A 9180 B 9090 C 9000 D 8910 E 8190 25. The four straight lines in the diagram aresuch that . The sizes of , and are , and . VU = VW •UXZ •VYZ •VZX x°y° z° Which of the following equations gives in terms of and ? x y z U X W Y Z V x° y° z° A B C x=y-z x=180-y-z x= y- z 2 D E x= y+z-90 x= y-z 2 UKMT UKMT UKMT UK J UNIOR M ATHEMATICAL C HALLENGE THURSDAY 30th APRIL 2015 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 . 4. Candidates in England and Wales must be in School Year 8 or below. Candidates in Scotland must be in S2 or below. Candidates in Northern Ireland must be in School Year 9 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 JMC is about solving interesting problems, not about lucky guessing. The UKMT is a registered charity http://www.ukmt.org.uk