I investigated a family of polynomials given by for $n>1$:
$x^n+(x+1)^n+...+(x+n-1)^n=(x+n)^n$
And I found, with the help of Magma, that the Galois group is $S_n$ for $n$ up to a hundred. The small examples up to $n=10$ also have a unique positive solution and I think proving this fact will not be hard.
But my main question is, whether the Galois group is $S_n$ for all $n$. I have almost no idea how to compute the Galois group, but maybe the special form of the equation will allow a moderately hard solution.
I am also interested in a softer problem: Is it true, that for $n>5$ the Galois group of the equation above is non-solvable?