Problem 4. Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^\circ < \theta < 180^\circ$ known to both players. Initially, Shan-Yu makes a paper triangle $\mathcal{T}$ with measurements of his choice. Then, they repeatedly perform the following steps:
- If $\mathcal{T}$ has at least one angle measuring exactly $\theta$, then the game stops and Mulan wins.
- Otherwise, Mulan chooses a point $P$ on the perimeter of $\mathcal{T}$, different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $\mathcal{T}$, splitting it into two triangles.
- Shan-Yu discards one of the two triangles. The remaining triangle becomes the new $\mathcal{T}$.
For which real values of $\theta$ can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?
