This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.

Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, define a real $f_k\left(x_1,x_2,...,x_n\right)$ as the sum

$\sum\limits_{T\subseteq\left\lbrace 1,2,...,n\right\rbrace ;\\ \ \left|T\right|=k} \left|\sum\limits_{t\in T}x_t - \sum\limits_{t\in\left\lbrace 1,2,...,n\right\rbrace \setminus T} x_t\right|$.

We mostly care about the case of $n$ even and $k=\frac n 2$; in this case, $f_k\left(x_1,x_2,...,x_n\right)$ is a kind of measure for the dispersion of the reals $x_1$, $x_2$, ..., $x_n$ (more precisely, of their $\frac n 2$-element sums).

Now my conjecture is that if $n$ is even and $k=\frac n 2$, then

$f_k\left(x_1,x_2,...,x_n\right)\geq f_k\left(\left|x_1\right|,\left|x_2\right|,...,\left|x_n\right|\right)$

for any reals $x_1$, $x_2$, ..., $x_n$.

I think I have casebashed this for $n=4$ and maybe $n=6$; I don't remember anymore - it's too long ago. Sorry. I still have no idea what to do in the general case, although my attempts at big-$n$ counterexamples weren't of much success either.