What is the maximum value of
$x_1x_2 +x_3x_4 + ... x_7x_8$
if $x_1+x_2... +x_8 = 8$
I tried using Cauchy-Schwarz
$(x_1 + x_2 .. +x_7)(x_2 +x_3 ... +x_8)= (\sqrt{x_1x_2} +... \sqrt{x_7x_8})$
$=(8-x_8)(8-x_1)$
Where I am stuck. The answer is really easy to guess but I do not know a good solution.