maximum and minimum possible values of S

[orl]

Let a_1, a_2,...,a_1999 be non-negative real numbers satisfying the following two conditions:

(a) \sum^{1999}_{k=1} a_k = 2,

(b) a_1*a_2 + a_2*a_3 + ... + a_1998*a_1999 + a_1991*a_1 = 1

Let S = \sum^{1999}_{k=1} (a_k)^2. Find the maximum and minimum possible values of S.

Also consider the ILL problem 3 at: http://www.kalva.demon.co.uk/short/sh82.html

[harazi]

The following fact is very interesting:
We have already discussed the fact that (x_1+...+x_n))^2>=4(x_1*x_2+...+x_n*x_1) for all reals x_1,...,x_n. But the sequence in the problem verifies the above relation with equality. So, we should think about the cases of equality, which I believe it's not an easy job.

[A1lqdSchool]

I think the condition (b) is : a_1*a_2+...+a_1999*a_1=1

[harazi]

Yes, of course. I've been thinking about this problem, but I really don't know how to deduce the case of equality.