Impulse Response of real coefficient, LTI System
It must be added to the problem that $R(\omega)$ is a real-valued, possibly bipolar function. In that case, its inverse discrete-time Fourier transform must be even:
$$r[n]=r[-n]\tag{1}$$
From the given relation between $H(e^{j\omega})$ and $R(\omega)$ it is clear that
$$h[n]=r[n-25]\tag{2}$$
must hold. I'm sure that you'll manage to combine $(1)$ and $(2)$ to come up with the desired result.