Download [PDF] 2011 UKMT UK IMC Questions Intermediate Mathematical Challenge United Kingdom Mathematics Trust

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UKMT UKMT UKMT UK I NTERMEDIATE M ATHEMATICAL C HALLENGE THURSDAY 3RD FEBRUARY 2011 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 the value of ?4.5×5.5+4.5×4.5 A 36.5 B 45 C 50 D 90 E 100 2. To find the diameter in mm of a Japanese knitting needle, you multiply the size by 0.3 and add 2.1. What is the diameter in mm of a size 5 Japanese knitting needle? A 3.6 B 7.4 C 10.8 D 12 E 17.1 3. The consecutive digits 1, 2, 3, 4 in that order can be arranged to make the correct division, . One other sequence of four consecutive digits makes a correct division, . What is the value of in this case? 12÷3=4p,q,r,s ‘pq’÷r=ss A4 B5 C6 D7 E 8 4. The angles of a triangle are in the ratio 2:3:5. What is the difference between the largest angle and the smallest angle? ABCDE9°18°36°45°54° 5. The diagram shows a rectangle placed on a grid of squares.1cm×1cm What is the area of the rectangle in ?cm 2 A 15 B C 30 D 36 E 4522 1 2  :KHQ,JODQFHGDWP\FDUPLORPHWHULWVKRZHGDSDOLQGURPLFQXPEHU7ZRGD\VODWHU ,QRWLFHGWKDWLWVKRZHGWKHQH[WSDOLQGURPLFQXPEHU+RZPDQ\PLOHVGLGP\FDUWUDYHOLQ WKRVHWZRGD\V" $  %  &  '  (   :KDWLVWKHYDOXHRI LQWKLVGLDJUDP"x A 30 B 35 C 40 D 45 E 50 70° 80° x° x° x° 8. A square piece of card has a square of side 2 cm cut out from each of its corners. The remaining card is then folded along the dotted lines shown to form an open box whose total internal surface area is 180 .cm 2 What is the volume of the open box in ?cm 3 A 100 B 128 C 162 D 180 E 200 9. In the diagram, is a straight line. What is the value of ?XY x A 170 B 160 C 150 D 140 E 130 °40°80° 150° x° XY

10. Merlin magically transforms a 6 tonne monster into mice with the same total mass. Each mouse has a mass of 20g. How many mice does Merlin make? A 30 B 300 C 3000 D 30 000 E 300 000 11. What is the value of ?19 1 2 ×20 1 2 A 250 B C D 395 E380 1 4 390 1 4 399 3 4 12. What is the sum of the first 2011 digits when is written as a decimal? 20÷11 A 6013 B 7024 C 8035 D 9046 E 10057 13. The three blind mice stole a piece of cheese. In the night, the first mouse ate of the cheese. 1 3 Later, the second mouse ate of the remaining cheese. Finally, the third mouse ate of what was then left of the cheese. 1 3 1 3 Between them, what fraction of the cheese did they eat? ABCDE 16 27 17 27 2 3 19 27 20 27 14. The number 6 lies exactly halfway between 3 and . Which of the following is not halfway between a positive integer and its square?3 2 A3 B10C15D21E30 15. The equilateral triangle has sides of length 1 and lies on the line . The triangle is rotated clockwise around until lies on the line . It is then rotated similarly around and then about as shown in the diagram. ABC AB X Y BBC XY C A A XYA A BB B C CC What is the length of the path traced out by point during this sequence of rotations? C ABCD3E 4π 32 3 8π 3 2π 3 16. The diagram shows an L-shape divided into squares. Gwyn cuts the shape along some of the lines shown to make two pieces, neither of which is a square. She then uses the pieces to form a rectangle.1×1 2×6 What is the difference between the areas of the two pieces? A0 B 1 C 2 D3 E 4 17. A shop advertised “Everything half price in our sale”, but also now advertises that there is “An additional 15% off sale prices”. Overall, this is equivalent to what reduction on the original prices? A 7.5% B 35% C 57.5% D 65% E 80%

18. The diagram contains six equilateral triangles with sides of length 2 and a regular hexagon with sides of length 1. What fraction of the whole shape is shaded? AB C DE 1 8 1 7 1 6 1 5 1 4  +DUURJDWHLVNPGXHQRUWKRI/HHGV