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

Difference between revisions of "2023 IMO Problems/Problem 2"

(Created page with "==Problem== Let <math>ABC</math> be an acute-angled triangle with <math>AB < AC</math>. Let <math>\Omega</math> be the circumcircle of <math>ABC</math>. Let <math>S</math> be...")
 
(→‎Problem)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 +
[[File:2023 IMO 2o.png|250px|right]]
 
Let <math>ABC</math> be an acute-angled triangle with <math>AB < AC</math>. Let <math>\Omega</math> be the circumcircle of <math>ABC</math>. Let <math>S</math> be the midpoint of the arc <math>CB</math> of <math>\Omega</math> containing <math>A</math>. The perpendicular from <math>A</math> to <math>BC</math> meets <math>BS</math> at <math>D</math> and meets <math>\Omega</math> again at <math>E \neq A</math>. The line through <math>D</math> parallel to <math>BC</math> meets line <math>BE</math> at <math>L</math>. Denote the circumcircle of triangle <math>BDL</math> by <math>\omega</math>. Let <math>\omega</math> meet <math>\Omega</math> again at <math>P \neq B</math>. Prove that the line tangent to <math>\omega</math> at <math>P</math> meets line <math>BS</math> on the internal angle bisector of <math>\angle BAC</math>.
 
Let <math>ABC</math> be an acute-angled triangle with <math>AB < AC</math>. Let <math>\Omega</math> be the circumcircle of <math>ABC</math>. Let <math>S</math> be the midpoint of the arc <math>CB</math> of <math>\Omega</math> containing <math>A</math>. The perpendicular from <math>A</math> to <math>BC</math> meets <math>BS</math> at <math>D</math> and meets <math>\Omega</math> again at <math>E \neq A</math>. The line through <math>D</math> parallel to <math>BC</math> meets line <math>BE</math> at <math>L</math>. Denote the circumcircle of triangle <math>BDL</math> by <math>\omega</math>. Let <math>\omega</math> meet <math>\Omega</math> again at <math>P \neq B</math>. Prove that the line tangent to <math>\omega</math> at <math>P</math> meets line <math>BS</math> on the internal angle bisector of <math>\angle BAC</math>.
  
 
==Solution==
 
==Solution==
 
https://www.youtube.com/watch?v=JhThDz0H7cI [Video contains solutions to all day 1 problems]
 
https://www.youtube.com/watch?v=JhThDz0H7cI [Video contains solutions to all day 1 problems]

Revision as of 08:39, 23 July 2023

Problem

2023 IMO 2o.png

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Solution

https://www.youtube.com/watch?v=JhThDz0H7cI [Video contains solutions to all day 1 problems]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_IMO_Problems/Problem_2&oldid=195986"