Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2021 OIM Problems/Problem 2

Revision as of 03:53, 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>...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_OIM_Problems/Problem_2&oldid=207210"