Find the $\sum\limits_{sym}ab$ maximum of the value Let $a,b,c,d,e\in (0,1)$ and such $$a+b+c+d+e=1$$
find the maximun of the value
$$S=ab+ac+ad+ae+bc+bd+be+cd+ce+de$$
I Conjecture the maximun is $\dfrac{2}{5}?$,such  $a=b=c=d=e=\dfrac{1}{5}$,so $$S\le\dfrac{10}{25}=\dfrac{2}{5}?$$
right?Thanks,

Since 
$$4(a+b+c+d+e)^2-10(ab+ac+ad+ae+bc+bd+ce+cd+ce+de)=$$
$$=(a-b)^2+(a-c)^2+(a-d)^2+(a-e)^2+(b-c)^2+(b-d)^2+(b-e)^2+(c-d)^2+(c-e)^2+(d-e)^2\geq0$$
so the answer is $\frac{2}{5}$.
