RU-ORAL-GEOMETRY 2015

In triangle $ABC$ , the altitude $AH$ passes through midpoint of the median $BM$ . Prove that in the triangle $BMC$ also one of the altitudes passes through the midpoint of one of the medians.

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$ .

In triangle $ABC$ , points $D, E$ , and $F$ are marked on sides $AC, BC$ , and $AB$ respectively, so that $AD = AB$ , $EC = DC$ , $BF = BE$ . After that, they erased everything except points $E, F$ and $D$ . Reconstruct the triangle $ABC$ (no study required).