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Difference between revisions of "2025 IMO Problems/Problem 2"

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Let <math>\Omega</math> and <math>\Gamma</math> be circles with centres <math>M</math> and <math>N</math>, respectively, such that the radius of <math>\Omega</math> is less than the radius of <math>\Gamma</math>. Suppose <math>\Omega</math> and <math>\Gamma</math> intersect at two distinct points <math>A</math> and <math>B</math>. Line <math>MN</math> intersects <math>\Omega</math> at <math>C</math> and <math>\Gamma</math> at <math>D</math>, so that <math>C, M, N, D</math> lie on <math>MN</math> in that order. Let <math>P</math> be the circumcentre of triangle <math>ACD</math>. Line <math>AP</math> meets <math>\Omega</math> again at <math>E\neq A</math> and meets <math>\Gamma</math> again at <math>F\neq A</math>. Let <math>H</math> be the orthocentre of triangle <math>PMN</math>.
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Let <math>\Omega</math> and <math>\Gamma</math> be circles with centers <math>M</math> and <math>N</math>, respectively, such that the radius of <math>\Omega</math> is less than the radius of <math>\Gamma</math>. Suppose circles <math>\Omega</math> and <math>\Gamma</math> intersect at two distinct points <math>A</math> and <math>B</math>. Line <math>MN</math> intersects <math>\Omega</math> at <math>C</math> and <math>\Gamma</math> at <math>D</math>, such that points <math>C</math>, <math>M</math>, <math>N</math>, and <math>D</math> lie on the line in that order. Let <math>P</math> be the circumcenter of triangle <math>ACD</math>. Line <math>AP</math> intersects <math>\Omega</math> again at <math>E\neq A</math>. Line <math>AP</math> intersects <math>\Gamma</math> again at <math>F\neq A</math>. Let <math>H</math> be the orthocenter of triangle <math>PMN</math>.
  
 
Prove that the line through <math>H</math> parallel to <math>AP</math> is tangent to the circumcircle of triangle <math>BEF</math>.
 
Prove that the line through <math>H</math> parallel to <math>AP</math> is tangent to the circumcircle of triangle <math>BEF</math>.
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(The <i>orthocenter</i> of a triangle is the point of intersection of its altitudes.)
  
 
==Video Solution==
 
==Video Solution==
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Solution from Edutube:
 
Solution from Edutube:
 
https://www.youtube.com/watch?v=-Fj7CMJ6iMo
 
https://www.youtube.com/watch?v=-Fj7CMJ6iMo
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==See Also==
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* [[2025 IMO]]
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competitions resources]]
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{{IMO box|year=2025|num-b=1|num-a=3}}

Latest revision as of 19:40, 19 July 2025

Let $\Omega$ and $\Gamma$ be circles with centers $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C$, $M$, $N$, and $D$ lie on the line in that order. Let $P$ be the circumcenter of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E\neq A$. Line $AP$ intersects $\Gamma$ again at $F\neq A$. Let $H$ be the orthocenter of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

(The orthocenter of a triangle is the point of intersection of its altitudes.)

Video Solution

Solution from channel Dedekind cuts https://www.youtube.com/watch?v=A4_bYF97IQI

Solution and expansion from channel Olympiad Geometry Club: https://www.youtube.com/watch?v=0TcMSrOYZ7c&t=772s

Solution from Edutube: https://www.youtube.com/watch?v=-Fj7CMJ6iMo

See Also

2025 IMO (Problems) • Resources
Preceded by
Problem 1 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 3
All IMO Problems and Solutions
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