Saturday, October 2, 2010

Australian Mathematics Competition Junior Q30

I have a list of twelve numbers where the first number is 1, the last number is 12 and each of the other numbers is one more than the average of its two neighbours. What is the largest number in the list?


1 , a , b , c , d , e , f , g , h , i , j , 12


a = (1 + b)/2 + 1
1 + b = 2 a - 2
b = 2 a - 3


c = 2 b - 2 - a
   = 4 a - 6 - 2 - a
  = 3 a - 8                     2 a + a - 3 - 5
 d = 2 c - 2 - b
   = 6 a - 16 - 2 - 2a + 3
   = 4 a - 15                 2 a + a + a - 3 - 5 - 7


f = 6 a - 15 - 9 - 11
  = 6 a - 35



i = 9 a - 35 - 13 - 15 - 17
   = 9 a - 80


j = 10 a - 80 - 19
   = 10 a - 99




j = (i + 12)/2 + 1
9 a - 80 + 12 = 2 x (10 a - 99 - 1)
11 a = 132
a = 132/11
   = 12

f = 6 x 12 - 35
   = 37    Central number between a and 12 is the largest

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