Question

Let $A,B,C$ and $D$ be four points in space whose coordinates $(x,y,z)$ satisfies $x^4+y^4+z^4+1=4xyz,$
and volume of tetrahendron $ABCD$ is $V$ then the value of $3V$ is

Answer 1

Since $x^4+y^4+z^4+1=4xyz$
$\Rightarrow xyz\ge \mathrm{0......}\left(1\right)$
For $|x|,|y|,|z|$
AM$\ge G.M.\Rightarrow \frac{x^4+y^4+z^4+1}{4}\ge |x||y||z|$
equality holds when all numbers are equal. $\Rightarrow |x|=|y|=|z|=\mathrm{1......}\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
Vertices are $(1,1,1),(1,-1,-1),(-1,1,-1)$ & $(-1,-1,1)$
$\therefore V=\frac{1}{6}\left|\begin{array}{ccc}0& 2& 2\\ 2& 0& 2\\ 2& 2& 0\end{array}\right|=\frac{8}{3}$