Question 5
Let $n \ge 3$ be an integer and let $x_1, x_2, \dots, x_n$ be real numbers in the interval $[0, 1]$. Let $s = x_1 + x_2 + \dots + x_n$, with $s \ge 3$. Prove that there exist integers $i$ and $j$ with $1 \le i \lt j \le n$ such that
$$
2^{j-i} x_i x_j \gt 2^{s-3}.
$$
$$
2^{j-i} x_i x_j \gt 2^{s-3}.
$$