50th Mathematical Olympiad in Ukraine, Fourth Round (March 24, 2010)

Sum of all angles of convex $n$-gon is $180^{\circ }(n-2)$, maximal angle, which has integer degree measure is $179^{\circ }$. So we can write the following inequality $180^{\circ }(n-2)\le 179^{\circ }n\Rightarrow n\le 360$. Hence, $n=360$ maximal possible value. Regular 360-gon provides an example.