Given the set $\{1,2,3,5,8,13,21,34,55\}$, how many integers between 3 and 89 cannot be written as the sum of exactly two elements of the set?
(A) 51
(B) 57
(C) 55
(D) 34
(E) 43
Solution

First count the number of integers between 3 and 89 that can be written as the sum of exactly two elements. Since each element in the set is the sum of the two previous
elements, 55 can be added to each of the seven smallest elements to form seven unique integers smaller than 89 .

In the same way, 34 can be added to each of the seven smaller elements, 21 can be added to each of the six smaller elements, and so on.

The number of integers between 3 and 89 that can be written as the sum of two elements of the set is $7+7+6+5+4+3+2=34$. Since there are 85 integers between 3 and 89 , then $85-34=51$ integers cannot be written as the sum of exactly two elements in the set.

ANSWER: (A)

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