### International Olympic Revenge (IMOR) 2017-18 2p

geometry problems from International Mathematics Olympic Revenge (IMOR)
with aops links in the names

It is a competition for teachers,  written by students.

2017-2018

Let $ABC$ be a triangle, and let $P$ be a distinct point on the plane. Moreover, let $A'B'C'$ be a homothety of $ABC$ with ratio $2$ and center $P$, and let $O$ and $O'$ be the circumcenters of $ABC$ and $A'B'C'$, respectively. The circumcircles of $AB'C'$, $A'BC'$, and $A'B'C$ meet at points $X$, $Y$, and $Z$, different from $A'$, $B'$, and $C'$. In a similar way, the circumcircles of $A'BC$, $AB'C$, and $ABC'$ meet at $X'$, $Y'$, and $Z'$, different from $A$, $B$, $C$. Let $W$ and $W'$ be the circumcenters of $XYZ$ and $X'Y'Z'$, respectively. Prove that $OW$ is parallel to $O'W'$.

Proposed by Mateus Thimóteo, Brazil.

Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively.
Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles
$(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$.

Proposed by Ivan Chai, Malaysia.