2007 Alabama ARML TST Problems/Problem 7
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.
