12.289. All lateral faces of the pyramid form the same angle with the base plane. Find this angle if the ratio of the total surface area of the pyramid to the area of the base is $k$. For what values of $k$ does the problem have a solution
Solution.

Let $S$ be the total surface area, $S_{1}$ be the lateral surface area of the pyramid, and $S_{2}$ be the area of its base.

Since all lateral faces form the same angle $\alpha$ with the plane of the base, then $S_{1}=\frac{S_{2}}{\cos \alpha}$.

Further, $S=S_{1}+S_{2}=\frac{S_{2}}{\cos \alpha}+S_{2}=\frac{S_{2}(1+\cos \alpha)}{\cos \alpha}$.

By the condition $\frac{S_{1}}{S_{2}}=k \Rightarrow \frac{1+\cos \alpha}{\cos \alpha}=k \Leftrightarrow(k-1) \cos \alpha=1 \Rightarrow \alpha=\arccos \frac{1}{k-1}$.

Since $k>0$ and $0<\alpha<\frac{\pi}{2}$, it follows that $k>2$.

Answer: $\arccos \frac{1}{k-1}, k>2$.