Suppose that $x_{1}, x_{2}, \ldots, x_{n}(n>2)$ are real numbers such that $x_{i}=x_{n-i+1}$ for $1 \leq i \leq n .$ Consider the sum $S=\Sigma \Sigma \Sigma x_{i} x_{j} x_{k},$ where summations are taken over all i, $j, k: 1 \leq i, j, k \leq n$ and $i, j, k$ are all distinct. Then S equals
$S=\sum \Sigma x_{i} x_{j}\left(L-x_{i}-x_{j}\right) i \neq j$
where $L=x_{1}+x_{2}+\ldots+x_{n}$
$=L \sum \Sigma x_{i} x_{j}-\sum \Sigma x_{i}^{2} x_{j}-\sum \Sigma x_{i} x_{j}^{2}$
$=\mathrm{L} \sum \mathrm{x}_{\mathrm{i}}\left(\mathrm{L}-\mathrm{x}_{\mathrm{i}}\right)-\Sigma \mathrm{x}_{\mathrm{i}}^{2}\left(\mathrm{L}-\mathrm{x}_{\mathrm{i}}\right)-\Sigma \mathrm{x}_{\mathrm{i}}\left(\mathrm{M}-\mathrm{x}_{\mathrm{i}}^{2}\right)$
where $\mathrm{M}=\mathrm{x}_{1}^{2}+\mathrm{x}_{2}^{2}+\ldots .+\mathrm{x}_{\mathrm{n}}^{2}$
$=\mathrm{L}^{3}-3 \mathrm{LM}+2 \Sigma \mathrm{x}_{\mathrm{i}}^{3}$
What to do next?