We have:

\(\left(3x-5\right)^{2016}\ge0\) with \(\forall x\in R\)

\(\left(y^2-1\right)^{2008}\ge0\) with \(\forall y\in R\)

\(\left(x-z\right)^{2100}\ge0\) with \(\forall z\in R\)

\(\Rightarrow\left(3x-5\right)^{2006}+\left(y^2-1\right)^{2008}+\left(x-z\right)^{2100}\ge0\) with \(\forall x,y,z\in R\)

Sign "=" occur 

\(\Leftrightarrow\left\{{}\begin{matrix}3x-5=0\\y^2-1=0\\x-z=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{3}\\\left[{}\begin{matrix}y=1\\y=-1\end{matrix}\right.\\z=\dfrac{5}{3}\end{matrix}\right.\)

\(\Rightarrow\left(x;y;z\right)\in\left\{\left(\dfrac{5}{3};1;\dfrac{5}{3}\right);\left(\dfrac{5}{3};-1;\dfrac{5}{3}\right)\right\}\)