jmc

Let $f$ be the flat fee for the first night and $n$ be the fixed fee for each night thereafter. Notice that the first night is incorporated into the flat fee. We can create a system of two equations to represent the given information as follows:

$$\begin{array}{rl}f+2n& =155\\ f+5n& =290\end{array}$$It's easiest to eliminate $f,$ solve for $n$ and then solve for $f$ using that value. To solve for $n$, subtract the first equation from the second, obtaining $3n=135$, or $n=45$. Substitute for $n$ in the first equation to obtain $f=155-90$, or $f=65$. Thus, the flat fee for the first night is \boxed{\65}$.