55rd Ukrainian National Mathematical Olympiad - Third Round

Three cyclists start off at the same time and ride along the sides of a triangle $ABC$ along the route $AB\to BC\to CA$. Their speeds on each of the segments $AB$, $BC$, $CA$ are known: the first cyclist has speeds $12$, $10$ and $20$ mph respectively on the three sides, the second one rides $15$, $15$ and $10$ mph, the third one rides $10$, $20$ and $12$ mph respectively. What can be the angle measure of $\angle ABC$, if all three cyclists arrived back at the point $A$ simultaneously?