IMC_2012_web_solutions

Euler lived from 1707 to 1783 but his theorem about

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Euler lived from 1707 to 1783, but his theorem about perfect numbers was not published until 1849. It is still not known whether there are infinitely many even perfect numbers, as we don’t know whether there are infinitely many primes of the form 1 2 p . It is also not known whether there are any odd perfect numbers.
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7 11. In the diagram, PQRS is a parallelogram; 0 50 QRS ; 0 62 SPT and PT PQ . What is the size of TQR ? A 0 84 B 0 90 C 0 96 D 0 112 E 0 124 Solution: C Because PQRS is a parallelogram, 0 50 QRS SPQ . Therefore 0 0 112 ) 50 62 ( TPQ . Therefore, as the angles in a triangle add up to 0 180 , 0 0 0 68 112 180 PTQ PQT . Because PQ=PT , the triangle QPT is isosceles, and so PTQ PQT . Therefore 0 34 PTQ PQT . Because PQRS is a parallelogram, 0 180 QRS PQR , and therefore 0 0 0 130 50 180 PQR . Therefore, . 96 34 130 0 0 0 PQT PQR TQR 12. Which 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 Solution: B We have that 40 . 5 £ ) 30 £( 30 £ of % 18 100 18 . Similarly, 12% of £50 is £6.00, and 6% of £90 is £5.40. We already see that option B must be the odd one out. It is easy to check that 4% of £135 and 2% of £270 are also both £5.40. 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 Solution: D Since they each hit about 45 shots in one minute, between them they hit about 90 shots per minute. Now 2 hours and 12 minutes is 132 minutes. So the total number of shots in the match is 132 90 , and 132 90 is approximately 000 12 120 100 . Extension Problem 13.1 Note that 90 is 90% of 100 and 132 is 110% of 120. What is the percentage error in approximating 132 90 by 120 100 ? 62 0 50 0 P Q R S T
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8 14. What value of x makes the mean of the first three numbers in this list equal to the mean of the last four? 15 5 x 7 9 17 A 19 B 21 C 24 D 25 E 27 Solution: A The mean of the first three numbers in the list is ) 5 15 ( 3 1 x and the mean of the last four is ) 17 9 7 ( 4 1 x . Now, ) 17 9 7 ( 3 ) 5 15 ( 4 ) 17 9 7 ( ) 5 15 ( 4 1 3 1 x x x x 99 3 4 80 x x 19 x . An alternative method in the context of the IMC would be just to try the given options in turn. This runs the risk of involving a lot of arithmetic, but here, as the first option is the correct answer, the gamble would pay off. 15. Which of the following has a value that is closest to 0? A 4 1 3 1 2 1 B 4 1 3 1 2 1 C 4 1 3 1 2 1 D 4 1 3 1 2 1 E 4 1 3 1 2 1 Solution: E When working out the values of these expressions it is important to remember the convention (sometimes known as BODMAS or BIDMAS) that tells us that Divisions and Multiplications are carried out before Additions and Subtractions.
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