International Mathematical Olympiad 1961 — Contest Problems Question 5

(CZS) Construct a triangle $ABC$ if the following elements are given: $AC=b$, $AB=c$, and $\angle AMB=\omega$ ($\omega<90^\circ$), where $M$ is the midpoint of $BC$. Prove that the construction has a solution if and only if
$$b\tan\frac\omega2\le c<b.$$
In what case does equality hold?