Synthesis discrete time signal from fourier coefficients
As you indicated, Fourier coefficients are even from first point. This tells you that $a_9 = a_1$, so $a_9 = 5$.

Then from Parseval's theorem, you get

$$
   \begin{align}
    \frac{1}{10} \sum_{n=0}^9 \left|x[n]\right|^2 &amp;= \sum_{k=0}^9 \left|a_k\right|^2 \\
     &amp;= 50
   \end{align}
   $$

Since $a_1^2 + a_9^2 = 50$, the other coefficients must be 0.

Then $x[n]$ can be computed as:

$$
   \begin{align}
    x[n] &amp;= \sum_{k=0}^9 a_k \exp(2\pi j n k / 10) \\
         &amp;= a_1 \exp(2\pi j n / 10) + a_9 \exp(2\pi j n \cdot 9 /10) \\
         &amp;= a_1 \exp(2\pi j n / 10) + a_9 \exp(-2\pi j n /10) \\
         &amp;= a_1\left[\exp(2\pi j n / 10) + \exp(-2\pi j n /10)\right] \\
         &amp;= 2 a_1 \cos(2\pi n/10) \\
         &amp;= 10 \cos(2\pi n/10)
   \end{align}
   $$