IMC_2012_web_solutions

# We have now used the digits 2 3 4 5 8 and 9 leaving 1

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We have now used the digits 2, 3, 4, 5, 8 and 9, leaving 1, 6 and 7 . From the middle ring we have that 11 4 x w , and so 7 x w . From the second ring from the right 11 3 y x , and so 8 y x . So we need to solve the equations 7 x w and 8 y x , using 1, 6 and 7. It is easy to see that the only solution is 1 x , 7 y and 6 w . So 6 goes in the region marked *. 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 Solution: B Since no more than 1% of the fleas will remain, at least 99% of them will be eradicated. Now 99% of a million is 000 990 000 10 99 000 000 1 100 99 . 9 5 * 8 9 5 w * 8 v u v x y z

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6 10. An ‘abundant’ number is a positive integer N , such that the sum of the factors of N (excluding N itself) is greater than N. . What is the smallest abundant number? A 5 B 6 C 10 D 12 E 15 Solution: D In the IMC, it is only necessary to check the factors of the numbers given as the options. However, to be sure that the smallest of these which is abundant, is the overall smallest abundant number, we would need to check the factors of all the positive integers in turn, until we find an abundant number. The following table gives the sum of the factors of N (excluding N itself), for 12 1 N . N 1 2 3 4 5 6 7 8 9 10 11 12 factors of N, excluding N - 1 1 1,2 1 1,2,3 1 1,2,4 1,3 1,2,5 1 1,2,3,4,6 sum of these factors 0 1 1 3 1 6 1 7 4 8 1 16 From this table we see that 12 is the smallest abundant number. Extension Problems 10.1. Which is the next smallest abundant number after 12? 10.2. Show that if n is a power of 2, and 2 n (that is, n 4, 8, 16, .. etc) then 3 n is an abundant number. 10.3 Prove that if n is an abundant number, then so too is each multiple of n . 10.4 A number, N , is said to be deficient if the sum of the divisors of N , excluding N itself, is less than N. Prove that if N is a power of 2, then N is a deficient number. 10.5 A number, N , is said to be perfect if the sum of the divisors of N , excluding N itself, is equal to N. We see from the above table that 6 is the smallest perfect number. Find the next smallest perfect number. Note: It follows from Problems 10.2 and 10.4 that there are infinitely many abundant numbers and infinitely many deficient numbers. It remains an open question as to whether there are infinitely many perfect numbers. In Euclid’s Elements (Book IX, Proposition 36) it is proved that even integers of the form ) 1 2 ( 2 1 p p , where 1 2 p is a prime number are perfect (for example, the perfect number 6 corresponds to the case where 2 p ). Euclid lived around 2300 years ago. It took almost 2000 years before the great Swiss mathematician Leonard Euler showed that, conversely, all even perfect numbers are of the form ) 1 2 ( 2 1 p p , where 1 2 p is prime.
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