IMC_2012_web_solutions

Some work can be saved by noting that the expressions

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Some work can be saved by noting that the expressions A and B have values greater than 2 1 , whereas the value of expression E lies between 0 and 2 1 . So it must be C, D or E that has the value closest to 0. Now, noting that 3 4 1 4 3 1 4 1 3 1 , we obtain that the value of C is 3 2 3 4 2 1 4 1 3 1 2 1 ; that of D is 6 5 3 4 2 1 4 1 3 1 2 1 ; and that of E is 12 5 12 1 2 1 4 1 3 1 2 1 . From these calculations we see that E gives the value closest to 0. D 0 E 2 1 A C B 6 5 12 5 12 7 3 2 6 11 [The value of A is 12 7 12 1 2 1 4 1 3 1 2 1 ; and that of B is B 6 11 3 4 2 1 4 1 3 1 2 1 .]
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9 16. The diagram shows a large equilateral triangle divided by three straight lines into seven regions. The three grey regions are equilateral triangles with sides of length 5 cm and the central black region is an equilateral triangle with sides of length 2 cm. What is the side length of the original large triangle? A 18 cm B 19 cm C 20 cm D 21 cm E 22cm Solution : B Let P, Q, R, S, T, U and V be the points shown. All the angles in all the triangles are 0 60 . So PSU QRT and hence RT is parallel to SU. Similarly, as TUV RSV , RS is parallel to TU. Therefore RSUT is a parallelogram. Therefore RS has the same length as TU , namely, 7 5 2 cm. Similarly PQ has length 7 cm. So the length of PS which is the sum of the lengths of PQ, QS and RS is 19 7 5 7 cm. 17. The first term in a sequence of positive integers is 6. The other terms in the sequence follow these rules: if a term is even then divide it by 2 to obtain the next term; if a term is odd then multiply it by 5 and subtract 1 to obtain the next term. For which values of n is the n th term equal to n ? A 10 only B 13 only C 16 only D 10 and 13 only E 13 and 16 only Solution: E Since the options refer only to the 10th, 13th and 16th terms of the sequence, as far as this IMC question is concerned it is only necessary to check the first 16 terms in the sequence. These are as shown in the table below: From this we see that the 13th term is 13, and the 16th term is 16, and that these are the only cases where the n th term is equal to n . However, a complete answer requires a proof that for all 16 n , the n th term is not equal to n . It can be seen that after the 16th term the sequence continues 8, 4, 2, 1, 4, 2, 1… with the cycle 4, 2, 1 now repeating for ever. It follows that, for 17 n , the only values taken by the n th term are 8, 4, 2 and 1. We deduce that for 16 n , the n th term is not equal to n. P Q R S T U V 16 16 13 13 2 32 42 2 84 8 32 2 64 15 84 1 5 17 7 64 1 5 13 14 17 2 34 6 2 26 34 1 5 7 5 26 2 52 12 7 2 14 4 52 2 104 11 14 1 5 3 3 104 1 5 21 10 3 2 6 2 21 2 42 9 6 1 th term th term n n n n
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10 18. Peri the winkle starts at the origin and slithers anticlockwise around a semicircle with centre ) 0 , 4 ( . Peri then slides anticlockwise around a second semicircle with centre ) 0 , 6 ( , and finally
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  • Spring '13
  • MRR
  • Math, Prime number, triangle, Divisor, perfect number

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