2019 IMO Shortlist A3

Let $n \geq 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2 . Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of $$\left|1-\sum_{i \in X} a_{i}\right|$$ is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that $$\sum_{i \in X} b_{i}=1$$ (New Zealand)

令 $n \geq 3$ 为正整数，并令 $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ 为 $n$ 正实数的严格递增序列，总和等于 2 。令 $X$ 为 $\{1,2, \ldots, n\}$ 的子集，使得 $$\left|1-\sum_{i \in X} a_{i}\right| 的值$$ 被最小化。证明存在 $n$ 正实数 $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ 的严格递增序列，总和等于 2，使得 $$\sum_{i \in X} b_{i}=1$$ (新西兰)