<s>[INST] max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ Given that a,b,c,d,e are real number such that:
$\begin{cases}
a+b+c+d+e=8\\
a^2+b^2+c^2+d^2+e^2=16
\end{cases}$
determine the maximun value of $e$.
I started like that :
$\max(e)=\max(8-a-b-c-d)=\min(a+b+c+d)$
but i don't kow how to carry on 
can you help please ?
 [/INST] The geometry is the same as in 3 dimensions, when taking the plane section $x+y+z = c$ of a sphere $x^2+y^2+z^2=C$.  Extrema of $z$ are when $x=y$.  The only remaining thing to understand is why the answer for the parameters given in the problem does not involve square roots.
Let $n$, which equals $4$ in the posted problem, be the number of variables other than the last one, which I will continue to call "$e$".
Max/min $e$ in $(n+1)$ dimensions occur at the two solutions of 
$na+e=c$ and $na^2+e^2=C$, so $(c-e)^2 = n(C - e^2)$ or $$(n+1)e^2 - 2ce + (c^2 - nC) = 0 .$$
To get a problem with extremal $e$ rational in the parameters, 
$c^2 - (n+1)(c^2 - nC^2) = n((n+1)C - c^2)$ must be a perfect square. 
There is no special perfect square expression that appears in the algebra, and the problem was just based on a choice of small integer values that lead to a square.
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