Question

Let $x,y,z$ be positive reals. Which of the following implies $x=y=z$?
$\left(I\right)$ $x^3+y^3+z^3=3xyz$
$\left(II\right)$ $x^3+y^{2}z+y{z}^2=3xyz$
$\left(III\right)$ $x^3+y^{2}z+z^{2}x=3xyz$
$\left(IV\right)$ ${(x+y+z)}^3=27xyz$

• A. $I.IV$ only
• B. $I,II,IV$ only
• C. $I,II$ and $III$ only
• D. All of them

Answer 1

The correct option is B $I,II,IV$ only

For option (3) if $x=z=1$ and $y=2$ then option (3) is right
So by option (3) we can't say $x=y=z$. Remaining options implies $x=y=z$