RU-ORAL-GEOMETRY 2019

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$ , with $\angle BIM = 90^o$ . Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.

In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and $H$ is the orthocenter. The angle bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.