Given that $n>i$ satisfies $\sum_{i=1}^n x_i^2 + \sum_{i=0}^{n-1} x_ix_{i+1} = 1 $, what is the largest value of $|x_k|$?
I multiplied the equation by 2 then rearranged the terms,
$x^2_1+(x_1+x_2)^2 + (x_2+x_3)^2 ... (x_{n-1} + x_n)^2 + x_n = 2$
After this I think we will use the Quadratic Mean > Arithmetic Mean?
$\sqrt{(x_1+x_2)^2 ... (x_{n-1}+x_n)}/\sqrt{n} > \frac{x_1+x_2...x_n}{n}$
I don't know how to move forward from this,
I can't prove it, but we can let $x_1$ be as large as possible and step down evenly so that all the other squares are the same size. Specifically, let $$x_i=\pm \frac {n-i+1}nx_1$$ where the even terms are negative and the odd terms are positive. All the squares after the first are $(\frac {x_1}{n})^2$ so our equation becomes $$x_1^2+(n-1)\frac {x_1^2}{n^2}=2\\x_1^2=\frac {2n^2}{n^2+n-1}\\ x_1=\sqrt{\frac {2n^2}{n^2+n-1}}$$ Which starts about $1.265$ for $n=2$ and rises to $\sqrt 2$ as $n\to \infty$