RIOPLATENSE 2007 Problem 6

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$ -smallif there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $n - 1$ $n$ -small sets .

Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .