RU-ALLRUSSIAN 1982 Problem 20

Let $\{\mathbf{a}\}$ denote the difference between a and the nearest integer. For example $\{3.8\} = 0.2, \{- 5.4\} = 0.4$ . Show that $\mathrm{la} \mathrm{la} - 11 \mathrm{la} - 2 \mathrm{l} \ldots \mathrm{la} - \mathrm{nl} \geq = \{\mathrm{a} \} \mathrm{n}!/2^{\mathrm{n}}$ .