(2018浙江省赛14题)
将$2n(nge2)$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:$sumlimits_{1le ile n;1le jle n}|a_i-b_j|-sumlimits_{1le i<jle n}{left(|a_j-a_i|+|b_j-b_i|
ight)}ge n$
$ extbf{证明:}$不妨设$a_1<a_2<cdots<a_n;b_1<b_2<cdots<b_n$
$$egin{align*}
sumlimits_{1le ile n;1le jle n}|a_i-b_j|
&=sumlimits_{1le i<jle n}{left(|a_i-b_j|+|a_j-b_i|
ight)}+sumlimits_{i=j}|a_i-b_j| \
&gesumlimits_{1le i<jle n}{left(|a_i-b_j|+|a_j-b_i|
ight)}+n\
&gesumlimits_{1le i<jle n}{left(b_j-a_i+a_j-b_i
ight)}+n\
&=sumlimits_{1le i<jle n}{left(|a_j-a_i|+|b_j-b_i|
ight)}+n\
end{align*}$$