Maximisation problem I am trying the following question:

If$$a+b+c+d=0,\;a^2+b^2+c^2+d^2=1$$ Then what is the maximum value of
  $ab+bc+cd+da?$

By the rearrangement inequality I can get $ab+bc+cd+da\leq 1$ but I am having trouble bringing the first condition into this. I have tried squaring it but that does not look like it leads anywhere. By playing with some numbers it looks like the answer might be $-½$. Help?

$ab+bc+cd+da = (a+c)(b+d) = -(a+c)^2 \leq 0$, and $\text{max} = 0$ when $a = -c$,
and so $b = -d$. Thus $a^2+b^2+c^2+d^2 = 2(a^2+b^2) = 1 \to a^2+b^2 = \dfrac{1}{2}$
