已知1/m+1/n=1/m+n,则m*m/n*n+n*n/m*m

题目

已知1/m+1/n=1/m+n,则m*m/n*n+n*n/m*m

答案

1/m+1/n=1/m+n
(m+n)/mn=1/(m+n)
mn=(m+n)^2
m^2n^2=(m+n)^4
m*m/n*n+n*n/m*m
=m^2/n^2+n^2/m^2
=(m^4+n^4)/(m^2n^2)
=[(m^2+n^2)^2-2m^2n^2]/(m^2n^2)
=(m^2+n^2)^2/(m^2n^2)-2
=[(m+n)^4+4m^2n^2-4mn(m+n)^2]/(m^2n^2)-2
=1+4-4-2
=-1