Let $n \geqslant 3$ 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 minimized. 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) Common remarks. In all solutions, we say an index set $X$ is $\left(a_{i}\right)$-minimizing if it has the property in the problem for the given sequence $\left(a_{i}\right)$. Write $X^{c}$ for the complement of $X$, and $[a, b]$ for the interval of integers $k$ such that $a \leqslant k \leqslant b$. Note that
$$\left|1-\sum_{i \in X} a_{i}\right|=\left|1-\sum_{i \in X^{c}} a_{i}\right|,$$
so we may exchange $X$ and $X^{c}$ where convenient. Let
$$\Delta=\sum_{i \in X^{c}} a_{i}-\sum_{i \in X} a_{i}$$
and note that $X$ is $\left(a_{i}\right)$-minimizing if and only if it minimizes $|\Delta|$, and that $\sum_{i \in X} a_{i}=1$ if and only if $\Delta=0$.
In some solutions, a scaling process is used. If we have a strictly increasing sequence of positive real numbers $c_{i}$ (typically obtained by perturbing the $a_{i}$ in some way) such that
$$\sum_{i \in X} c_{i}=\sum_{i \in X^{c}} c_{i}$$
then we may put $b_{i}=2 c_{i} / \sum_{j=1}^{n} c_{j}$. So it suffices to construct such a sequence without needing its sum to be 2.
The solutions below show various possible approaches to the problem. Solutions 1 and 2 perturb a few of the $a_{i}$ to form the $b_{i}$ (with scaling in the case of Solution 1, without scaling in the case of Solution 2). Solutions 3 and 4 look at properties of the index set $X$. Solution 3 then perturbs many of the $a_{i}$ to form the $b_{i}$, together with scaling. Rather than using such perturbations, Solution 4 constructs a sequence $\left(b_{i}\right)$ directly from the set $X$ with the required properties. Solution 4 can be used to give a complete description of sets $X$ that are $\left(a_{i}\right)$-minimizing for some $\left(a_{i}\right)$.