# Floors and Ceilings 1- Floor and Ceiling Together

Algebra Level 5

$\large \left\lfloor x \right\rfloor \left\lceil x \right\rceil =2015\left\{ x \right\}$

Suppose there are $$n$$ real values of $$x$$ that satisfy the equation above, and we define $$S$$ as the sum of all these solutions. Evaluate $$n + \lfloor 1000S \rfloor$$.

Details and Assumptions:

• $$\lfloor x \rfloor$$ and $$\lceil x \rceil$$ denote the floor function and ceiling function respectively.

• $$\{ x \}$$ denotes the fractional part function: $$\{ x\} = x - \lfloor x \rfloor$$ for all real values of $$x$$. Thus it is always positive. For example, $$\{-1.7\}$$ $$=$$ $$-1.7-(-2)$$ $$=$$ $$0.3$$.

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