Download [PDF] 2012 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 2ND FEBRUARY 2012 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. How many of the following four numbers are prime? 3 33 333 3333 A0 B1 C2 D3 E 4 2. Three positive integers are all different. Their sum is 7. What is their product? A12 B 10 C 9 D8 E 5 3. An equilateral triangle, a square and a pentagon all have the same side length. The triangle is drawn on and above the top edge of the square and the pentagon is drawn on and below the bottom edge of the square. What is the sum of the interior angles of the resulting polygon? ABCDE10×180°9×180°8×180°7×180°6×180° 4. All four digits of two 2-digit numbers are different. What is the largest possible sum of two such numbers? A 169 B 174 C 183 D 190 E 197 5. How many minutes will elapse between 20:12 today and 21:02 tomorrow? A 50 B 770 C 1250 D 1490 E 2450 6. Triangle is isosceles and right-angled.QRS Beatrix reflects the P-shape in the side to get an image.QR She reflects the first image in the side to get a second image. QS Finally, she reflects the second image in the side to get a third image.RS What does the third image look like? ABCDE P P P P P 7. The prime numbers and are the smallest primes that differ by 6. What is the sum of and ? pq pq A 12 B 14 C 16 D 20 E 28 8. Seb has been challenged to place the numbers 1 to 9 inclusive in the nine regions formed by the Olympic rings so that there is exactly one number in each region and the sum of the numbers in each ring is 11. The diagram shows part of his solution. What number goes in the region marked * ? A6 B 4 C 3 D2 E 1 9 8 * 5 9. Auntie Fi's dog Itchy has a million fleas. His anti-flea shampoo claims to leave no more than 1% of the original number of fleas after use. What is the least number of fleas that will be eradicated by the treatment? A 900 000 B 990 000 C 999 000 D 999 990 E 999 999

10. An ‘abundant’ number is a positive integer , such that the sum of the factors of (excluding itself) is greater than . What is the smallest abundant number? NN NN A5 B6 C10 D12 E 15 11. In the diagram, is a parallelogram; ; and .PQRS∠QRS=50° ∠SPT=62°PQ=PT What is the size of ?∠TQR AB C DE84°90°96°112°124° 50 o 62 o PQ R S T 12. Which one of the following has a different value from the others? A 18% of £30 B 12% of £50 C 6% of £90 D 4% of £135 E 2% of £270 13. Alex Erlich and Paneth Farcas shared an opening rally of 2 hours and 12 minutes during their table tennis match at the 1936 World Games. Each player hit around 45 shots per minute. Which of the following is closest to the total number of shots played in the rally? A 200 B 2000 C 8000 D 12 000 E 20 000 14. What value of makes the mean of the first three numbers in this list equal to the mean of the last four?x 15 5x7917 A 19 B 21 C 24 D 25 E 27 15. Which of the following has a value that is closest to 0? ABCDE 1 2+ 1 3× 1 4 1 2+ 1 3 ÷ 1 4 1 2× 1 3 ÷ 1 4 1 2− 1 3 ÷ 1 4 1 2− 1 3× 1 4  7KHGLDJUDPVKRZVDODUJHHTXLODWHUDOWULDQJOHGLYLGHGE\WKUHHVWUDLJKWOLQHV LQWRVHYHQUHJLRQV7KHWKUHHJUH\UHJLRQVDUHHTXLODWHUDOWULDQJOHVZLWKVLGHV RIOHQJWKFPDQGWKHFHQWUDOEODFNUHJLRQLVDQHTXLODWHUDOWULDQJOHZLWKVLGHV RIOHQJWKFP :KDWLVWKHVLGHOHQJWKRIWKHRULJLQDOODUJHWULDQJOH" $ FP % FP & FP ' FP ( FP  7KHILUVWWHUPRIDVHTXHQFHRISRVLWLYHLQWHJHUVLV7KHRWKHUWHUPVLQWKHVHTXHQFHIROORZ WKHVHUXOHV LIDWHUPLVHYHQWKHQGLYLGHLWE\WRREWDLQWKHQH[WWHUP LIDWHUPLVRGGWKHQPXOWLSO\LWE\DQGVXEWUDFWWRREWDLQWKHQH[WWHUP )RUZKLFKYDOXHVRI LVWKH WHUPHTXDOWR "nnthn A 10 only B 13 only C 16 only D 10 and 13 only E 13 and 16 only 18. Peri the winkle starts at the origin and slithers anticlockwise around a semicircle with centre (4, 0). Peri then slides anticlockwise around a second semicircle with centre (6, 0), and finally clockwise around a third semicircle with centre (3, 0). Where does Peri end this expedition? A (0, 0) B (1, 0) C (2, 0) D (4, 0) E (6, 0)

19. The shaded region shown in the diagram is bounded by four arcs, each of the same radius as that of the surrounding circle. What fraction of the surrounding circle is shaded? ABCDE it depends on the radius of the circle 4 π−11− π 4 1 2 1 3  $UHFWDQJOHZLWKDUHD KDVVLGHVLQWKHUDWLR:KDWLVWKHSHULPHWHURIWKH UHFWDQJOH" FP  $ FP % FP & FP ' FP ( FP  7KHSDUDOOHORJUDP LVIRUPHGE\MRLQLQJWRJHWKHUIRXU HTXLODWHUDOWULDQJOHVRIVLGHXQLWDVVKRZQPQRS What is the length of the diagonal ?SQ AB C DE 7 83 6 5 P Q R S 22. What is the maximum possible value of the median number of cups of coffee bought per customer on a day when Sundollars Coffee Shop sells 477 cups of coffee to 190 customers, and every customer buys at least one cup of coffee? A 1.5 B 2 C 2.5 D 3 E 3.5 23. In triangle , ; ; ; is the foot of the perpendicular from to and .PQR PS=2SR=1∠PRQ=45°T PQS∠PST=60° What is the size of ?∠QPR AB C DE45°60°75°90°105° 60o 45o 2 1 P Q R S T 24. All the positive integers are written in the cells of a square grid. Starting from 1, the numbers spiral anticlockwise. The first part of the spiral is shown in the diagram. What number will be immediately below 2012? A 1837 B 2011 C 2013 D 2195 E 2210 …3231 17 16 15 14 13 30 18 5 4 3 12 29 19 6 1 2 11 28 20 7 8 9 10 27 21 22 23 24 25 26 25. The diagram shows a ceramic design by the Catalan architect Antoni Gaudi. It is formed by drawing eight lines connecting points which divide the edges of the outer regular octagon into three equal parts, as shown. What fraction of the octagon is shaded? AB C DE 1 5 2 9 1 4 3 10 5 16