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. $Y(e^{jomega})|_{omega = 2pi frac{k}{5}}$, matches with $X(e^{jomega})$ evaluated at those $omega$ exactly.

This should draw your attention towards upsampling of 5-point DFT $X(e^{jomega})|_{omega = 2pi frac{k}{5}}$ 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.

Hint: Solve for the Z-transform of $x(n)$ which is $X(z)$. Then from this solve from $X(e^{jomega})$ with $omega = 2pi k/5$ (the missing j is certainly a typo) and the k's given.

If you are unable to solve the Z-transform, try to use the geometric series directly with the equation for the z-transform and this should help:

$$sum_{n=0}^infty r^n = frac{1}{1-r} space space r<1$$

This should get you past the clueless point.