2022 IMO Shortlist A4

Let $n \geq 3$ be an integer, and let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers in the interval $[0,1]$. Let $s=x_{1}+x_{2}+\ldots+x_{n}$, and assume that $s \geq 3$. Prove that there exist integers $i$ and $j$ with $1 \leq i<j \leq n$ such that $$2^{j-i} x_{i} x_{j}>2^{s-3}$$ (Trinidad and Tobago)

令 $n \geq 3$ 为整数，并令 $x_{1}, x_{2}, \ldots, x_{n}$ 为区间 $[0,1]$ 内的实数。设$s=x_{1}+x_{2}+\ldots+x_{n}$，并假设$s \geq 3$。证明存在整数 $i$ 和 $j$ 且 $1 \leq i<j \leq n$ 使得 $$2^{j-i} x_{i} x_{j}>2^{s-3}$$ (特立尼达和多巴哥)