2025 IMO Problems/Problem 2

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

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

Video Solution

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