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