Evaluating different inputs into a given system
The output will be equal to the input $(y[n]=x[n])$ if no aliasing occurs as a result of downsampling. Which means $X(e^{j\omega})=0$ for $\frac{\pi}{4} \leq |\omega| \leq \pi$.   
a) $X(e^{j\omega})$ has impulses at $\omega = \pm \frac{2\pi}{7}$ $=&gt;$ aliasing exists $=&gt;$ $y[n]\neq x[n]$ 
b) $X(e^{j\omega})$ has impulses at $\omega = \pm \frac{2\pi}{9}$ $=&gt;$ no aliasing $=&gt;$ $y[n]= x[n]$ 
c) $X(e^{j\omega})$ has impulses at $\omega = \pm \frac{\pi}{8}$ $=&gt;$ no aliasing $=&gt;$ $y[n]= x[n]$ 
d) You need to sketch $X(e^{j\omega})$ to see if aliasing exists  hint: the impulse response of your filter is $h[n]=\frac{sin(\pi n/4)}{\pi n}$