Let $x, y, z$ be positive reals. Which of the following implies $x=y=z$ ?

$I.$ $x^3+y^3+z^3=3 x y z$

$II.$ $x^3+y^2 z+y z^2=3 x y z$

$III.$ $x^3+y^2 z+z^2 x=3 x y z$

$IV.$ $(x+y+z)^3=27 x y z$

  • [KVPY 2015]
  • A

    $I, IV$ only

  • B

    $I, II$ and $IV$ only

  • C

    $I, II$ and $III$ only

  • D

    All of them

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