Question 1
Let $n\gt 2$ be an integer and $a_1\lt a_2\lt \cdots\lt a_n$ positive integers. Let $X\subseteq\{1,\ldots,n\}$ minimize $\left|\sum_{i\notin X}a_i-\sum_{i\in X}a_i\right|$. Prove that there exist positive integers $b_1\lt b_2\lt \cdots\lt b_n$ such that $\sum_{i\notin X}b_i=\sum_{i\in X}b_i$.