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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