IMO 1987 Problem 3

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$ . Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$ , not all zero, such that $|a_i|\le k-1$ for all $i$ , and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$ .