Difference between revisions of "2008 AIME I Problems/Problem 7"
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{{AIME box|year=2008|n=I|num-b=6|num-a=8}} | {{AIME box|year=2008|n=I|num-b=6|num-a=8}} | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
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Revision as of 04:11, 4 May 2020
Problem
Let 
be the set of all integers 
such that 
. For example, 
is the set 
. How many of the sets 
do not contain a perfect square?
Solution
The difference between consecutive squares is 
, which means that all squares above 
are more than 
apart.
Then the first 
sets (
) each have at least one perfect square. Also, since 
(which is when 
), there are 
other sets after 
that have a perfect square.
There are 
sets without a perfect square.
See also
| 2008 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
Video Solution:
https://www.youtube.com/watch?v=6eBLXnzK0n4
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
