Let $x=(x_0,x_1,\dots,x_{n-1})$ with discrete fourier transform $\hat{x}_k=r_ke^{i\phi_k}$. Show that $x$ is real data if and only if $r_{n-k}=r_k$, $\phi_{n-k}\equiv-\phi_k$ mod $2\pi$, and $\phi_0=0$.
The second direction doesn't seem too bad, use the definition of the inverse DFT and use the assumed equivalences to have the complex parts cancel each other out. I'm very stuck on the first direction: Assume that $x$ is real data and show the equivalences.