International Mathematical Olympiad 1987 Problem 3

Let $x_1, x_2, \ldots, x_n$ be real numbers satisfying $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$. Prove that for every integer $k \geq 2$ there are integers $a_1, a_2, \ldots, a_n$, not all 0, such that $|a_i| \leq k - 1$ for all $i$ and

$$\left| a_1 x_1 + a_1 x_2 + \cdots + a_n x_n \right| \leq \frac{(k - 1) \sqrt{n}}{k^n - 1}.$$