More fun with quadratic forms

$\color{forestgreen}{\text{Applying Cauchy Shwarz inequality}}$ $S=\displaystyle \sum_{k=1}^{n}x_0x_k\leq \sqrt{(n x_0^2)\displaystyle \sum_{k=1}^{n}x_k^2}=\sqrt{(nx_0^2)(1-x_0^2)}$ $\color{forestgreen}{\text{Applying}~GM\leq AM}$ $\leq\sqrt{n\times \frac{(x_0^2+1-x_0^2)}{4}}$ $=\sqrt{\dfrac{n}{4}}>1~ or ~ \Large n>4$ Hence smallest integral n= $\boxed 5$ $\color{#D61F06}{\text{For equality}~x_1=x_2=....=x_n=\dfrac{1}{\sqrt{2n}} ~and~x_0=\dfrac{1}{\sqrt{2}}}$ For n= $5$ , we get $x_1=x_2=....=x_n=\dfrac{1}{\sqrt{10}} ~and~x_0=\dfrac{1}{\sqrt{2}}$