Download [PDF] 2013 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 7TH FEBRUARY 2013 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. Which of the following is divisible by 6? A one million minus one B one million minus two C one million minus three D one million minus four E one million minus five 2. A machine cracks open 180 000 eggs per hour. How many eggs is that per second? A 5 B 50 C 500 D 5000 E 50 000 3. How many quadrilaterals are there in this diagram, which is constructed using 6 straight lines ? A4 B5 C7 D8 E 9  $VWDQGDUGSDFNRISXPSNLQVHHGVFRQWDLQVVHHGV$VSHFLDOSDFNFRQWDLQVPRUH VHHGV5DFKHOERXJKWDVSHFLDOSDFNDQGRIWKHVHHGVJHUPLQDWHG+RZPDQ\SXPSNLQ SODQWVGLG5DFKHOKDYH" $  %  &  '  (   7KHQRUWKHUQZKHDWHDULVDVPDOOELUGZHLJKLQJOHVVWKDQDQRXQFH6RPHQRUWKHUQZKHDWHDUV PLJUDWHIURPVXE6DKDUDQ$IULFDWRWKHLU$UFWLFEUHHGLQJJURXQGVWUDYHOOLQJDOPRVWNP 7KHMRXUQH\WDNHVMXVWRYHUZHHNV5RXJKO\KRZIDUGRWKH\WUDYHOHDFKGD\RQDYHUDJH" $ NP % NP & NP ' NP ( NP  :KLFKRIWKHIROORZLQJKDVWKHOHDVWYDOXH" $%&'( −0 1 21−1 2 32−2 3 43−3 4 54−4 5 7. The faces of a regular octahedron are to be painted so that no two faces which have an edge in common are painted in the same colour. What is the smallest number of colours required? A2 B3 C4 D6 E 8 8. Jim rolled some dice and was surprised that the sum of the scores on the dice was equal to the product of the scores on the dice. One of the dice showed a score of 2, one showed 3 and one showed 5. The rest showed a score of 1. How many dice did Jim roll? A 10 B 13 C 17 D 23 E 30 9. Jane has 20 identical cards in the shape of an isosceles right-angled triangle. She uses the cards to make the five shapes below. Which of the shapes has the shortest perimeter? A % & ' (  LVDUHJXODUSHQWDJRQDQG LVDQHTXLODWHUDOWULDQJOHVXFKWKDW LVLQVLGH  ABC DE BC F F ABC DE What is the size of ?∠FAB ABCDE48°63°66°69°72° 11. For which of the following numbers is the sum of all its factors not equal to a square number? A 3 B 22 C 40 D 66 E 70

12. The sum one + four = seventy becomes correct if we replace each word by the number of letters in it to give .3+4=7 Using the same convention, which of these words could be substituted for to make the sum three + five true?x =x A eight B nine C twelve D seventeen E eighteen 13. Four congruent isosceles trapeziums are placed so that their longer parallel sides form the diagonals of a square , as shown. The point divides in the ratio 3:1. What fraction of the square is shaded? PQRS XPQ ABCDE 5 16 3 8 7 16 5 12 1 2 PQ R SX 14. Which of the following has the greatest value? ABCDE ( 11 7 ) 3 ( 5 3 ) 3 ( 7 4 ) 3 ( 9 5 ) 3 ( 3 2 ) 3 15. I have a bag of coins. In it, one third of the coins are gold, one fifth of them are silver, two sevenths are bronze and the rest are copper. My bag can hold a maximum of 200 coins. How many coins are in my bag? A 101 B 105 C 153 D 195 E more information is needed  :KLFKGLDJUDPVKRZVWKHJUDSKRI DIWHULWKDVEHHQURWDWHG FORFNZLVHDERXWWKH SRLQW  "y=x90° ABCDE      x y 2 2 1 1 -1 -1x y 2 2 1 1 -1 -1x y 2 2 1 1 -1 -1x y 2 2 1 1 -1 -1x y 17. The diagram shows four equal discs and a square. Each disc touches its two neighbouring discs. Each corner of the square is positioned at the centre of a disc. The side length of the square is . What is the length of the perimeter of the figure? 2/π A3 B 4 C D6 E 3π 22π 18. The triangle has sides of length 6, 5, 5. The triangle has sides of length 8, 5, 5. TU What is the ratio area : area ?TU A 9 : 16 B 3 : 4 C 1 : 1 D 4 : 3 E 16 : 9 19. Which of the expressions below is equivalent to ? (x ÷ (y ÷ z))÷ ((x ÷ y) ÷ z) A1BCDE 1 xyzx 2 y2 z2

20. Jack's teacher asked him to draw a triangle of area . Two sides are to be of length 6cm and 8cm. How many possibilities are there for the length of the third side of the triangle? 7 2 cm A 1 B 2 C 3 D 4 E more than 4  7KHVTXDUH KDVDQDUHDRI,WFRQWDLQVWZRRYHUODSSLQJ VTXDUHVWKHODUJHURIWKHVHVTXDUHVKDVDQDUHDWLPHVWKDWRIWKH VPDOOHUDQGWKHDUHDRIWKHLURYHUODSLV:KDWLVWKHWRWDODUHDRI WKHVKDGHGUHJLRQV"ABC D A 44 B 72 C 80 D 152 E more information is needed AB C D 22. The diagrams show squares placed inside two identical semicircles. In the lower diagram the two squares are identical. What is the ratio of the areas of the two shaded regions? A 1 : 2 B 2 : 3 C 3 : 4 D 4 : 5 E 5 : 6 23. Four brothers are discussing the order in which they were born. Two are lying and two are telling the truth. Which two are telling the truth? Alfred: “Bernard is the youngest.” Horatio: “Bernard is the oldest and I am the youngest.” Inigo: “I was born last.” Bernard: “I'm neither the youngest nor the oldest.” A Bernard and Inigo B Horatio and Bernard C Alfred and Horatio D Alfred and Bernard E Inigo and Horatio 24. The diagram shows a shaded shape bounded by circular arcs with the same radius. The centres of three arcs are the vertices of an equilateral triangle; the other three centres are the midpoints of the sides of the triangle. The sides of the triangle have length 2. What is the difference between the area of the shaded shape and the area of the triangle? ABCDEπ π 6 π 4 π 3 π 2 25. In 1984 the engineer and prolific prime-finder Harvey Dubner found the biggest known prime each of whose digits is either a one or a zero. The prime can be expressed as . How many digits does this prime have? 10 641 × (10 640 −1 ) 9+1 A 640 B 641 C 1280 D 1281 E 640×641