Question on N point DTFT - Fourier transform
Hint: Question says $y[n]$ has length $10$, but the alternate DFT coefficients of $y[n]$ i.e. $Y(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$, matches with $X(e^{j\omega})$ evaluated at those $\omega$ exactly.
This should draw your attention towards upsampling of 5-point DFT $X(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$ or equivalently periodization of a length $5$ segment of $x[n]$.
Like upsampling of time-domain sequence by $N$, by inserting $(N-1)$ zeros between samples, shrinks the spectrum in frequency domain and brings $N-1$ more copies of spectrum inside $[-\pi, \pi]$, similarly, upsampling in frequency domain by inserting $(N-1)$ zeros between DFT samples will create more copies of time-domain sequence.