IDTFT of convolution in the frequency domain
It seems to me that this exercise is meant to combines  the basic properties of the DTFT (gathered here: Table of DTFT Properties). Matt described the product/convolution property. You also get the time-shift/complex modulation. I suspect (thought I did not do the calculations) that the issue with the factor $10$ could be tackled with a change in variable: $10\Omega = 5\times (2\Omega)$, and the use of the time-scaling (expansion) property (see Time scaling of discrete-time sequences and the DTFT):
$$ S(c\Omega) \Longleftrightarrow \left\{
                \begin{array}{ll}
                  x[n/c] \textrm{ if } n/c \textrm{ is an integer } \\
                  0 \textrm{ otherwise.}
                \end{array}
              \right.$$
Other hints hinge around Dirichlet kernels:
$$D_N(x) =\frac{\sin\left(\left(N +1/2\right) x \right)}{\sin(x/2)}$$
They are also called asinc or psinc (aliased or periodic cardinal sine or sinc), and related to finite-support discrete windows. If $*$ is the Convolution sign, the resolution may use the product/convolution properties of Fourier.
And more generally (at the bottom of the page on Dirichlet kernels), you have the identity:
$$\sum_{n=0}^{N-1} e^{jn\Omega} = e^{j(N-1)\Omega/2}\frac{\sin(N \, \Omega/2)}{\sin(\Omega/2)}\,,$$
which relates the DTFT of a discrete time window $w_{[0,N-1]}$ (from index $n=0$ to $n=N-1$) to ratios of sines with a phase correcting term. You can check details at Discrete-time Fourier transform of a window function.