X>Y>Z,试确定1/(X-Y)+1/(Y-Z)+1/(Z-X)的符号

X>Y>Z => X-Z>Y-Z>0 =>0<1/(X-Z)<1/(Y-Z) =>1/(Y-Z)-1/(X-Z)>0
1/(X-Y)+1/(Y-Z)+1/(Z-X)=1/(X-Y)+1/(Y-Z)-1/(X-Z)
=1/(X-Y)+(1/(Y-Z)-1/(X-Z))
>1/(X-Y)+0 >0
1/(X-Y)+1/(Y-Z)+1/(Z-X)的符号为正

特殊值法就可以了；）

特殊值法
设x=3,y=2,z=1，代入结果大于0，为正

X>Y>Z
X-Y>0;Y-Z>0;Z-X>0

1/(X-Y)+1/(Y-Z)+1/(Z-X)三项都大于零

应此为正。