Five five five and five!

We write : $a+b+c+d+e=8$ and

${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 }+{ e }^{ 2 }=16$ rewriting a little bit we have:

$a+b+c+d=8-e$ (i)

${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 }=16-{ e }^{ 2 }$ (ii)

Applying cauchy schwarz inequality we have :

$4({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 })\ge {(a+b+c+d)}^{2}$

Substituting the values form (i) and (ii) we get :

$4(16-{e}^{2})\ge{(8-e)}^{2}$

Solving we get $e\epsilon (0,16/5]$

$\boxed { \Rightarrow { e }_{ max }=\frac { 16 }{ 5 } }$

This problem also appears here https://brilliant.org/problems/2-equations-5-variables/#!/solution-comments/182807/