2007 iTest Problems/Problem 9
Problem
Suppose that and are positive integers such that , the geometric mean of and is greater than , and the arithmetic mean of and is less than . How many pairs satisfy these conditions?
Solution
Since the arithmetic mean is less than 2007 and the geometric mean is greater than 2007, the arithmetic mean must be less than the geometric mean. But by the AM-GM inequality, this is impossible. Therefore no such pairs exist, and the answer is .
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 8 |
Followed by: Problem 10 | |
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