<s>[INST] Equation System with 4 real variables I need to solve the next equation system:
Find all real numbers $a,b,c,d$ such that:
$$
\left\{ 
\begin{array}{c}
a+b+c+d=20 \\ 
ab+ac+ad+bc+bd+cd=150 \\ 
\end{array}
\right. 
$$
I tried something like this:
$b+c+d=20-a$
And i put the second equation like this
$a(b+c+d) + bc+bd+cd=150$
Getting 
$20a-a^2 +bc+bd+cd=150$
But i see that this is useless, so i don't know how to start this problem.
 [/INST] By $(a+b+c+d)^2=a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd)$, we find $$ a^2+b^2+c^2+d^2 = 100 $$
Also by Cauchy-Schwarz inequality, $(a+b+c+d)^2 \leq (1^2 + 1^2 + 1^2 +1^2)(a^2+b^2+c^2+d^2)$ and therefore we yields: $$ 400 \leq 400 $$
Then, the equality condition occurs if and only if $a=b=c=d=5$.
</s>