PL-OM 2009 Problem 4

Let $x_1,x_2,..,x_n$ be non-negative numbers whose sum is $1$ . Show that there exist numbers $a_1,a_2,\ldots ,a_n$ chosen from amongst $0,1,2,3,4$ such that $a_1,a_2,\ldots ,a_n$ are different from $2,2,\ldots ,2$ and $2\leq a_1x_1+a_2x_2+\ldots+a_nx_n\leq 2+\frac{2}{3^n-1}$ .