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I have been trying to use the logic that both X and Y should have same Z transform, but according to the definition, Y is not anti causal. So, I am clueless! 

#Fourier #transform #question #discrete

Have you tried just finding $$ Y(e^\omega) $$ symbolically, and then solving?  For $$ \omega \in \left \lbrace 2 \pi k / 5 \right \rbrace $$ there should only be ten equations in ten unknowns.