67th Romanian Mathematical Olympiad

The sum of the measures of the angles directly proportional with $5$, $5$ and $7$ is $170^{\circ }$. Their measures are $5x$, $5x$, $7x$, with $5x+5x+7x=170^{\circ }$, hence $x=10^{\circ }$.

There are three pairs of opposite angles around the orthocenter, so they must be equal. On one side of an altitude must be one angle of each pair. Since three of the angles have measures $50^{\circ }$, $50^{\circ }$, $70^{\circ }$, the other three must have measures $70^{\circ }$, $60^{\circ }$, $60^{\circ }$.



Using these angles we find the measures of the angles between the altitudes and the sides: $20^{\circ }$, $30^{\circ }$, $40^{\circ }$, hence the angles of the triangle have measures $50^{\circ }$, $60^{\circ }$, $70^{\circ }$.