Inverse $\mathcal Z$-transform problem
Just use the definition of the $\mathcal{Z}$-transform:

$$B(z)=\sum_{n=-\infty}^{\infty}b[n]z^{-n}\tag{1}$$

from which it follows that

$$B(-z)=\sum_{n=-\infty}^{\infty}b[n](-z)^{-n}=\sum_{n=-\infty}^{\infty}b[n](-1)^nz^{-n}\tag{2}$$

From $(2)$, the sequence that corresponds to $B(-z)$ is $b[n](-1)^n$. Now taking the inverse $\mathcal{Z}$-transform of

$$B(z)+B(-z)=2c\tag{3}$$

gives

$$b[n]+b[n](-1)^n=2c\delta[n]\tag{4}$$

For odd $n$, the left side of $(4)$ equals zero. For even $n$ it equals $2b[n]$. Consequently, equation $(4)$ can only be satisfied if $b[n]=0$ for even $n$ except for $n=0$, where we require $b[0]=c$:

$$b[n]=\begin{cases}c,&amp;n=0\\
0,&amp;n\text{ even}\end{cases}$$