<s>[INST] maximum value of $\sum ab-2abc$ If $a+b+c=1$ and $a,b,c\in(0,1)$, then what is the maximum value of $(ab+bc+ca-2abc)$?
What I've tried:
$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\geq 4(ab+bc+ca)$
$(a-b)^2=a^2+b^2-2ab\geq 0$
$a^2+b^2\geq 2ab,b^2+c^2\geq 2bc,c^2+a^2\geq 2ca$
$ab+bc+ca\leq\frac14$
How do I solve it help me please.
 [/INST] Σ(ab) =  Σ ( a²b+b²a )   +  abc  and by using AM-GM to the terms under the modified sigma we get that it is always greater than 6abc and using AM -  GM on a,b,c  we get abc<1/27  hence the maximal value must be 7/27
</s>