Phillips101

Question:

Given a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, (all real numbers)

find the maximal value of a1*a2+a2*a3+...+an*a1

Thoughts so far:

I've treated the expression as a combination of n variables and differentiated - when it came to putting the constraints in it got to be a hideous mess.

It is easily factorisable as 0.5*( (a1+a2)^2 + (a2+a3)^2 +...+ (an+a1)^2 ) -1, but then maximising the inside is just as hard.

Help would be appreciated! (Also, sorry about the lack of LATEX knowhow).