Problem:
Let , and be digits with . The three-digit integer lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer lies two thirds of the way between the same two squares. What is ?
Answer Choices:
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E.
Solution:
Let be the smaller of the two squares. Then the difference between the two squares is . The given conditions state that
Subtraction yields , from which . If or 2 , then is not an integer. If , then , so is not a three-digit integer. If , then . The numbers that are, respectively, one third of the way and two thirds of the way from to are 178 and 187 , so .
The problems on this page are the property of the MAA's American Mathematics Competitions