PAMO 2006 Problem 3

For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$ . If $a, b, c$ are distinct real numbers, prove that $$\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$$ is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.