International Mathematical Olympiad 1961 Problem 5

Construct triangle $ABC$ if $AC = b$, $AB = c$ and $\measuredangle AMB = \omega$, where $M$ is the midpoint of segment $BC$ and $\omega < 90°$. Prove that a solution exists if and only if $$b \tan \frac{\omega}{2} \leq c < b.$$ In what case does the equality hold?