2019 OIM Problems/Problem 3

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Problem

Let $\Gamma$ be the circumcircle of triangle $ABC$. The parallel to $AC$ that passes through $B$ cuts $\Gamma$ at $D$ ($D \ne B$) and the parallel to $AB$ that passes through $C$ cuts $\Gamma$ at $E$ ($E \ne C$). The straight lines $AB$ and $CD$ intersect at $P$, and lines $AC$ and $BE$ intersect at $Q$. Let $M$ be the midpoint of $DE$. The line $AM$ cuts $\Gamma$ at $Y$ ($Y \ne A$) and the line $PQ$ at $J$. The line $PQ$ cuts the circumcircle of triangle $BCJ$ at $Z$ ($Z \ne J$). If the lines $BQ$ and $CP$ intersect at $X$, show that $X$ belongs to the line $YZ$.

Note: The circumcircle of a triangle is the circle that passes through the vertices of the triangle.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions