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2021 OIM Problems/Problem 2

Problem

Consider an acute triangle $ABC$, with $AC > AB$, and let $\Gamma$ be its circumcircle. Let $E$ and $F$ be the midpoints of the sides $AC$ and $AB$, respectively. The circumcircle of triangle $CEF$ intersects $\Gamma$ at $X$ and $C$, with $X \ne C$. The line $BX$ and the line tangent to $\Gamma$ at $A$ intersect at $Y$. Let $P$ be the point on segment $AB$ such that $YP = YA$, with $P \ne A$, and let $Q$ be the point where $AB$ intersects the line parallel to $BC$ passing through $Y$. Show that $F$ is the midpoint of $PQ$.

Note: The circumcircle of a triangle is the circle passing through its three vertices.

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also

https://olcoma.ac.cr/internacional/oim-2021/examenes

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