Problem:
How many positive integers can fill the blank in the sentence below? “One positive integer is more than twice another, and the sum of the two numbers is .”
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let the first number be . Then the second number is where both and are positive integers. Since their sum is , we know . This also means . With , the smallest possible value of is . With , we know
This means the maximum value of is . Since can be any number from to , we have solutions.
Thus, the answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions