MT【149】和式变形

(2018浙江省赛14题)
将$2n(n\ge2)$个不同的整数分成两组$a_1,a_2,\cdots,a_n;b_1,b_2,\cdots,b_n$.
证明:$\sum\limits_{1\le i\le n;1\le j\le n}|a_i-b_j|-\sum\limits_{1\le i<j\le n}{\left(|a_j-a_i|+|b_j-b_i|\right)}\ge n$

$\textbf{证明:}$不妨设$a_1<a_2<\cdots<a_n;b_1<b_2<\cdots<b_n$
$$\begin{align*} \sum\limits_{1\le i\le n;1\le j\le n}|a_i-b_j| &=\sum\limits_{1\le i<j\le n}{\left(|a_i-b_j|+|a_j-b_i|\right)}+\sum\limits_{i=j}|a_i-b_j| \\ &\ge\sum\limits_{1\le i<j\le n}{\left(|a_i-b_j|+|a_j-b_i|\right)}+n\\ &\ge\sum\limits_{1\le i<j\le n}{\left(b_j-a_i+a_j-b_i\right)}+n\\ &=\sum\limits_{1\le i<j\le n}{\left(|a_j-a_i|+|b_j-b_i|\right)}+n\\ \end{align*}$$