Let $x_1\ge x_2\ge ...x_r\ge 0>x_{r+1}\ge...\ge x_n$. Let $s=x_1+...+x_r$. Then \[1=x_1^2+....+x_r^2+(x_{r+1}^2+...+x_n^2))\ge r(\frac{x_1+...+x_r}{r})^2+(n-r)(\frac{x_{r+1}+...+x_n}{n-r})^2=\frac{ns^2}{r(n-r)}\] and \[0=(x_1+...+x_r)^2+(x_{r+1}+...+x_n)^2+2\sum_{i\le r<j}x_ix_j \ge 2s^2-2\frac{r(n-r)}{n}.\] Equality only if $x_1=...=x_r=\frac{s}{r},x_{r+1}=...=x_n=\frac{-s}{n-r}.$ Therefore $n^2s^2=nr(n-r)$ must be perfect square. In these case $n=c^2,r=a^2,n-r=b^2$. Minimal value for n is $n=25$.