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2021 OIM Problems/Problem 2

Revision as of 02:53, 14 December 2023 by Tomasdiaz (talk | contribs) (Created page with "== Problem == Consider an acute triangle <math>ABC</math>, with <math>AC > AB</math>, and let <math>\Gamma</math> be its circumcircle. Let <math>E</math> and <math>F</math>...")
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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 $X4 and$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

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See also

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

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