MathNet 2023-part-06 — 191 problems (EN–VI) Question 90

Let $n \ge 3$ be a positive integer. Show that for any real numbers $0 \le x_1, x_2, \dots, x_n \le 1$ satisfying $x_1 + x_2 + \dots + x_n = 3$, there always exist $x_i$, $x_j$ such that $x_ix_j \gt 2^{-|i-j|}$.