VMO 2018 Problem 7

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a, H_b, H_c$ be the projections of $A, B, C$ on respective opposite sides and $D, E, F$ be the midpoints of $BC, CA, AB$ in that order. $\overrightarrow{GH_a}, \overrightarrow{GH_b}, \overrightarrow{GH_c}$ intersect $(O)$ at $X, Y, Z$ respectively. a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$ b. Let $M, N, P$ be the midpoints of $AX, BY, CZ$ respectively. Prove that $\overleftrightarrow{DM}, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.