Difference between revisions of "2007 iTest Problems/Problem 9"
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==Solution== | ==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 <math>(m, n)</math> exist, and the answer is <math>0\Rightarrow\boxed{A}</math>. | 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 <math>(m, n)</math> exist, and the answer is <math>0\Rightarrow\boxed{A}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{iTest box|year=2007|num-b=8|num-a=10}} | ||
+ | |||
+ | [[Category:Introductory Algebra Problems]] |
Latest revision as of 20:24, 10 January 2019
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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