2007 Alabama ARML TST Problems/Problem 7

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Problem

Find the number of distinct integers in the list

\[\left\lfloor \dfrac{1^2}{2007}\right\rfloor , \left\lfloor \dfrac{2^2}{2007}\right\rfloor , \left\lfloor \dfrac{3^2}{2007}\right\rfloor , \left\lfloor \dfrac{4^2}{2007}\right\rfloor , \cdots , \left\lfloor \dfrac{2007^2}{2007}\right\rfloor ,\]

where $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$.

Solution

The first time that the difference of two consecutive squares is greater than or equal to 2007 is $1004^2-1003^2=2007$. Below $\left\lfloor \frac{1003^2}{2007}\right\rfloor =501$, every non-negative integer can be reached. Then above that, each number is distinct. So there are $502+(2007-1004+1)=\boxed{1506}$ distinct integers in the given list.

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 6
Followed by:
Problem 8
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