Difference between revisions of "2008 AIME I Problems/Problem 7"
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== Solution == | == Solution == | ||
| − | The difference between consecutive squares is | + | The difference between consecutive squares is <math>(x + 1)^2 - x^2 = 2x + 1</math>, which means that all squares above <math>50^2 = 2500</math> are more than <math>100</math> apart. |
| − | < | + | |
| − | which means that all squares above <math>50^2 = 2500</math> are more than 100 apart. | + | Then the first <math>26</math> sets (<math>S_0,\cdots S_{25}</math>) each have at least one perfect square. Also, since <math>316^2 < 10000 < 317^2</math>, there are <math>316 - 50 = 266</math> other sets after <math>S_{25}</math> that have a perfect square. |
| − | Then the first 26 sets (<math>S_0,\cdots S_{25}</math>) each have at least one perfect square. Also, since <math>316^2 < 10000 < 317^2</math>, there are <math>316 - 50 = 266</math> other sets after <math>S_{25}</math> that have a perfect square. | + | |
| + | There are <math>1000 - 266 - 26 = \boxed{708}</math> sets without a perfect square. | ||
== See also == | == See also == | ||
Revision as of 14:25, 19 April 2008
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 
, 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 | ||
