Question 1
Let $a_1,a_2,\ldots,a_n$ be $n$ real numbers whose squares sum to $1$. Prove that for any integer $k\ge2$, there exist $n$ integers $x_1,x_2,\ldots,x_n$, each with absolute value at most $k-1$ and not all zero, such that
$$\left|\sum_{i=1}^n a_ix_i\right|\le\frac{(k-1)\sqrt n}{k^n-1}.$$
$$\left|\sum_{i=1}^n a_ix_i\right|\le\frac{(k-1)\sqrt n}{k^n-1}.$$