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\sum_{i \notin X} b_i= \sum_{i \in X} b_i with 0<b_1 <b_2 <...<b_n wanted

Source: 2020 Vietnam Team Selection Test p1 VNTST

November 13, 2020
algebraSummin

Problem Statement

Given that n>2n> 2 is a positive integer and a sequence of positive integers a1<a2<...<ana_1 <a_2 <...<a_n. In the subsets of the set {1,2,...,n}\{1,2,..., n\} , there a subset XX such that ∣∑i∉Xai−∑i∈Xai∣| \sum_{i \notin X} a_i -\sum_{i \in X} a_i | is the smallest . Prove that there exists a sequence of positive integers 0<b1<b2<...<bn0<b_1 <b_2 <...<b_n such that ∑i∉Xbi=∑i∈Xbi\sum_{i \notin X} b_i= \sum_{i \in X} b_i.
In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese.
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