AT-OEMO-FED1 2006 Problem 4

Given is the function $f= \lfloor x^2 \rfloor + \{ x \}$ for all positive reals $x$ . ( $\lfloor x \rfloor$ denotes the largest integer less than or equal $x$ and $\{ x \} = x - \lfloor x \rfloor$ .) Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $3$ , if they are a reduced fraction, and don't lie in the range of the function $f$ .