<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] $$0=3(a+b+c+d)^2-8(ab+ac+ad+bc+bd+cd)=$$
$$=(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2\geq0,$$
where the equality occurs for $a=b=c=d.$
Thus, $a=b=c=d=5.$
</s>