2007 Alabama ARML TST Problems/Problem 7

Problem

Find the number of distinct integers in the list

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

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.

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

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2007 Alabama ARML TST Problems/Problem 7

Problem

Solution

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