I'm preparing for some comprehensive exams and this is a question from a past year.
The $n$-torus $T^n = \mathbb{R}^n/\mathbb{Z}^n$ is both a smooth n-manifold and an Abelian group, by virtue of structures inherited from the real vector space $\mathbb{R}^n$. In particular, $T^n$ acts on itself by smooth maps corresponding to translations of $\mathbb{R}^n$. By averaging, prove that each de Rham cohomology class on $T^n$ contains a unique differential form which is translation-invariant. Then use this observation to calculate the dimensions of the de Rham cohomology vector spaces $H^p(T^n)$ for all $p$.
I can compute the de Rham cohomology using Kunneth's formula and some induction (I think the dimensions are something like $n \choose p$) but that isn't the point. Plus, I'm quite curious about this idea of averaging so that we get something invariant under group actions. I've seen a similar question on showing that a Lens space is orientable by looking at some averaged form and then showing it is invariant under some orientation preserving transformations, or something like that.