Evangelos Katsoulis and Titu Andreescu have published verified collections (e.g., Russian Mathematical Olympiad 1993-2002 ). While commercial, verified PDFs are available through institutional access (e.g., via Springer or the Isaacs Archive). These are gold-standard because they include official solution keys.

Let $\angle AMB = \alpha$ and $\angle AMC = \beta$. Since $M$ is the midpoint of $BC$, we have $\angle BAM = \angle CAM$. Let $\angle BAM = \angle CAM = \gamma$. Then $\alpha + \gamma = \pi - \angle ABM$ and $\beta + \gamma = \pi - \angle ACM$. Adding these two equations, we get $\alpha + \beta + 2\gamma = 2\pi - (\angle ABM + \angle ACM)$. Since $\angle ABM + \angle ACM \leq \pi$, we have $\alpha + \beta \geq \pi$.

: AoPS maintains a vast community-verified database of All-Russian Olympiad problems (grades 9-11) with printable PDF collections, such as the 2017 All-Russian Olympiad PDF Mathematics Via Problems (AMS Library)