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

Revision as of 13:56, 19 July 2025

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

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

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

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

Revision as of 13:56, 19 July 2025

Video Solution

@@ Line 6: / Line 6: @@
 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
+Solution from Edutube:
+https://www.youtube.com/watch?v=-Fj7CMJ6iMo