Difference between revisions of "2007 Alabama ARML TST Problems/Problem 7"
(New page: =Problem== Find the number of distinct integers in the list <cmath>\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{200...) |
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| − | =Problem== | + | ==Problem== |
Find the number of distinct integers in the list | Find the number of distinct integers in the list | ||
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==Solution== | ==Solution== | ||
| − | The first time that the difference of two consecutive squares is greater than or equal to 2007 is <math>1004^2-1003^2=2007</math>. Below <math>\lfloor \frac{1003^2}{2007}\rfloor =501</math>, every non-negative integer can be reached. Then above that, each number is distinct. So there are <math>502+(2007-1004+1)=\boxed{1506}</math> distinct integers in the given list. | + | The first time that the difference of two consecutive squares is greater than or equal to 2007 is <math>1004^2-1003^2=2007</math>. Below <math>\left\lfloor \frac{1003^2}{2007}\right\rfloor =501</math>, every non-negative integer can be reached. Then above that, each number is distinct. So there are <math>502+(2007-1004+1)=\boxed{1506}</math> distinct integers in the given list. |
==See also== | ==See also== | ||
| + | {{ARML box|year=2007|state=Alabama|num-b=6|num-a=8}} | ||
| + | |||
[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
Latest revision as of 17:35, 28 January 2009
Problem
Find the number of distinct integers in the list
![\[\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{2007}\right\rfloor , \left\lfloor \dfrac{4^2}{2007}\right\rfloor , \cdots , \left\lfloor \dfrac{2007^2}{2007}\right\rfloor ,\]](http://latex.artofproblemsolving.com/1/8/7/18743e9edb1ccf2a25c6f2bc2cd41311b571def0.png)
where 
represents the greatest integer less than or equal to 
.
Solution
The first time that the difference of two consecutive squares is greater than or equal to 2007 is 
. Below 
, every non-negative integer can be reached. Then above that, each number is distinct. So there are 
distinct integers in the given list.
See also
| 2007 Alabama ARML TST (Problems) | ||
| Preceded by: Problem 6 |
Followed by: Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
