Algebra · Floor function · Monotone sequences · Counting distinct values
Consider the finite sequence of \(2003\) numbers given by
\[ a_n = \left\lfloor \frac{n^2}{2004} \right\rfloor, \qquad n = 1, 2, \ldots, 2003, \]
where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding \(x\). How many distinct terms does this sequence contain?
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Serbian Regional Competition (Okruzno takmicenje) 2004, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source