Baltic Way 2022 · Problem 14

Let $\Gamma$ denote the circumcircle and $O$ the circumcentre of the acute-angled triangle $ABC$, and let $M$ be the midpoint of the segment $BC$.

Let $T$ be the second intersection point of $\Gamma$ and the line $AM$, and $D$ the second intersection point of $\Gamma$ and the altitude from $A$. Let further $X$ be the intersection point of the lines $DT$ and $BC$. Let $P$ be the circumcentre of the triangle $XDM$. Prove that the circumcircle of the triangle $OPD$ passes through the midpoint of $XD$.