Given that $a,b,c,d,e$ are real numbers such that
$a+b+c+d+e=8$,
$a^2+b^2+c^2+d^2+e^2=16$.
Determine the maximum value of $e$.
By Cauchy Schwarz, we can see that $(1+1+1+1)(a^2+b^2+c^2+d^2)\geq (a+b+c+d)^2$
thus $4(16-e^2)\geq (8-e)^2$ 
Finally, $e(5e-16) \geq 0$ which means $\frac{16}{5} \geq e \geq 0$
so the maximum value of $e$ is $\frac{16}{5}$.
from: [Image from Gon Mathcenter.net](https://artofproblemsolving.comhttp://image.ohozaa.com/view2/vUGiXdRQdAPyw036)