X,y,z,m real numbers are so positive in which. Xyz=1 & 2xm/xy+x+1=2ym/yz+y+1=2zm/xz+z+1=1. Prove that m=1/2 ?

1 Answer
Jul 29, 2018

Please see Explanation.

Explanation:

xyz=1 :. xy=1/z.

Utilising this in (2xm)/(xy+x+1)=1, we get,

(2xm)/(1/z+x+1)=1, or, (2xzm)/(1+xz+z)=1.

Since, (2zm)/(xz+z+1)=1...........[Given], we have,

(2xzm)/(1+xz+z)=1=(2zm)/(xz+z+1), i.e.,

x=1.

Similarly, we can prove y=1 and z=1.

Thus, altogether, x=y=z=1.

Subst.ing these in (2xm)/(xy+x+1)=1, we find,

m=(xy+x+1)/(2x)=(1+1+1)/2=3/2.