IR-MO 1993 Problem 6

Let $x_1, x_2, \ldots, x_{12}$ be twelve real numbers such that for each $1 \leq i \leq 12$ , we have $|x_i| \geq 1$ . Let $I=[a,b]$ be an interval such that $b-a \leq 2$ . Prove that number of the numbers of the form $t= \sum_{i=1}^{12} r_ix_i$ , where $r_i=\pm 1$ , which lie inside the interval $I$ , is less than $1000$ .