已知x/(y+z+u)=y/(z+u+x)=z/(u+x+y)=u/(x+y+z),求(x+y)/(z+u)+(y+z)/(x+u)+(z+u)/(x+y)+(u+x)/(y+z)要求用设参数法

Question

已知x/(y+z+u)=y/(z+u+x)=z/(u+x+y)=u/(x+y+z),求(x+y)/
(z+u)+(y+z)/(x+u)+(z+u)/(x+y)+(u+x)/(y+z)
要求用设参数法

Answer 1

设x/（y+z+u)=y/（z+u+x)=z/(u+x+y)=u/(x+y+z)=a
x=a（y+z+u）
y=a（z+u+x）
z=a（x+y+u）
u=a（x+y+z）
四个式子相加
x+y+u+z=3a（x+y+z+u）
3a=1
a=1/3
x/(y+z+u)=y/(z+u+x)=z/(u+x+y)=u/(x+y+z)=(x+y+z+u)/(3x+3y+3z+3u)=1/3
∴3x=y+z+u.(1)
3y=z+u+x.(2)
3z=u+x+y.(3)
3u=x+y+z.(4)
(1)-(2),得
3(x-y)=y-x
∴x=y
同理y=z,z=u
∴x=y=z=u
代入
(x+y)/(z+u)+(y+z)/(x+u)+(z+u)/(x+y)+(u+x)/(y+z)
=1+1+1+1
=4