I was reading a book on statistics and I found that the following must hold. I tested it in Mathematica for n up to 5 and it passes that test (btw it has nothing to do with stats anymore, only polynomials. For anybody into statistics this is the variance of the x's):
For $x_1,x_2,\dots,x_n$ reals
$$ \sum_{k=1}^n (x_k-a)^2=\frac{1}{2n}\sum_{i,j=1}^n(x_i-x_j)^2 $$
where $a$ is just the average of the x's,
$$ a=\frac{1}{n}\sum_{k=1}^nx_k $$
Any ideas fellas?
Write $(x_i-x_j)$ as $(x_i-a)+(a-x_j)$ and expand the lhs. The rest is straight forward.