Question

if (x+1/y)=(y+1/z)=(z+1/x) then show that x²+y²+z²=1​

Answer 1

Answer:

1.I equaled the equation to 'k'. Using the AM-GM inequality, I found that k>2 (the equality does not hold because all are distinct). However, this, I couldn't put to much use.

2. For the next attempt, I substituted the values of x and y in terms of z and k in xyz. What I am getting is xyz=zk2−k−z. That's the farthest I could do..Please, help.

Answer 2

Note that x−y=y−zyz, y−z=z−xzx and z−x=x−yxy. Multiply to give (x−y)(y−z)(z−x)=(x−y)(y−z)(z−x)(xyz)2. As x, y and z distinct this means (xyz)2=1. As x, y and z