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 6"

(Created page with "==Problem== Let <math>ABC</math> be an equilateral triangle. Let <math>A_1,B_1,C_1</math> be interior points of <math>ABC</math> such that <math>BA_1=A_1C</math>, <math>CB_1=B...")
 
(→‎Solution)
 
(3 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
 +
[[File:2023 IMO 6.png|300px|right]]
 
Let <math>ABC</math> be an equilateral triangle. Let <math>A_1,B_1,C_1</math> be interior points of <math>ABC</math> such that <math>BA_1=A_1C</math>, <math>CB_1=B_1A</math>, <math>AC_1=C_1B</math>, and
 
Let <math>ABC</math> be an equilateral triangle. Let <math>A_1,B_1,C_1</math> be interior points of <math>ABC</math> such that <math>BA_1=A_1C</math>, <math>CB_1=B_1A</math>, <math>AC_1=C_1B</math>, and
 
<cmath>\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ</cmath>Let <math>BC_1</math> and <math>CB_1</math> meet at <math>A_2,</math> let <math>CA_1</math> and <math>AC_1</math> meet at <math>B_2,</math> and let <math>AB_1</math> and <math>BA_1</math> meet at <math>C_2.</math>
 
<cmath>\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ</cmath>Let <math>BC_1</math> and <math>CB_1</math> meet at <math>A_2,</math> let <math>CA_1</math> and <math>AC_1</math> meet at <math>B_2,</math> and let <math>AB_1</math> and <math>BA_1</math> meet at <math>C_2.</math>
 +
 
Prove that if triangle <math>A_1B_1C_1</math> is scalene, then the three circumcircles of triangles <math>AA_1A_2, BB_1B_2</math> and <math>CC_1C_2</math> all pass through two common points.
 
Prove that if triangle <math>A_1B_1C_1</math> is scalene, then the three circumcircles of triangles <math>AA_1A_2, BB_1B_2</math> and <math>CC_1C_2</math> all pass through two common points.
  
Line 7: Line 9:
  
 
==Solution==
 
==Solution==
−
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems]
+
https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems, it was created by a man known as Najeeb Abdullah, who qualified in the IMO from the time he was 6 years old]
 +
 
 +
==See Also==
 +
 
 +
{{IMO box|year=2023|num-b=5|after=Last Problem}}

Latest revision as of 21:10, 23 November 2024

Problem

2023 IMO 6.png

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and \[\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ\]Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$

Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

Solution

https://www.youtube.com/watch?v=jZNIpapyGJQ [Video contains solutions to all day 2 problems, it was created by a man known as Najeeb Abdullah, who qualified in the IMO from the time he was 6 years old]

See Also

2023 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_IMO_Problems/Problem_6&oldid=235381"