Difference between revisions of "1984 USAMO Problems/Problem 2"
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| + | == Problem == | ||
| + | The geometric mean of any set of <math>m</math> non-negative numbers is the <math>m</math>-th root of their product. | ||
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| + | <math>\quad (\text{i})\quad</math> For which positive integers <math>n</math> is there a finite set <math>S_n</math> of <math>n</math> distinct positive integers such that the geometric mean of any subset of <math>S_n</math> is an integer? | ||
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| + | <math>\quad (\text{ii})\quad</math> Is there an infinite set <math>S</math> of distinct positive integers such that the geometric mean of any finite subset of <math>S</math> is an integer? | ||
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| + | ==Solution== | ||
| + | {{solution}} | ||
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| + | == See Also == | ||
| + | {{USAMO box|year=1984|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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| + | [[Category:Olympiad Number Theory Problems]] | ||
Revision as of 11:29, 18 July 2016
Problem
The geometric mean of any set of 
non-negative numbers is the -th root of their product.
For which positive integers 
is there a finite set 
of distinct positive integers such that the geometric mean of any subset of is an integer?
Is there an infinite set 
of distinct positive integers such that the geometric mean of any finite subset of is an integer?
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
| 1984 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
