Problem 4

Assume $\theta$ is not of the form $180^\circ/n$, so $180^\circ\notin\mathbb{Z}\theta$. Call an angle safe when it is not an integer multiple of $\theta$. A safe initial triangle exists: only finitely many multiples of $\theta$ lie in $(0^\circ,180^\circ)$, so after choosing a generic positive angle $A$ one can choose $B$ outside the finitely many forbidden values for $B$ and $180^\circ-A-B$, making all three angles safe. Now let $ABC$ be safe and let a legal cut $AD$ with $D$ in the interior of $BC$ produce $ABD$ and $ACD$. Write $x=\angle BAD$ and $y=\angle ADB$. If $ABD$ is safe, Shan-Yu keeps it. Otherwise $\angle B$ is safe, so at least one of $x,y$ is an integer multiple of $\theta$. If $x$ is such a multiple, then $\angle DAC=\angle A-x$ is safe (safe minus a multiple), and $\angle CDA=\angle B+x$ is safe (safe plus a multiple); together with the original safe angle $C$, this makes $ACD$ safe. If $y$ is such a multiple, then $\angle CDA=180^\circ-y$ is safe because $180^\circ$ is not a multiple, while $\angle DAC=y-\angle C$ is safe because $\angle C$ is safe. Thus after every cut Shan-Yu can retain a safe child, and induction gives a safe triangle after every finite number of cuts; no triangle ever contains an angle equal to $\theta$, so Mulan cannot force a win.