Difference between revisions of "2023 IMO Problems/Problem 6"
(One intermediate revision by one other user 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> | ||
Line 9: | Line 10: | ||
==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] | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2023|num-b=5|after=Last Problem}} |
Latest revision as of 01:59, 19 November 2023
Problem
Let be an equilateral triangle. Let be interior points of such that , , , and Let and meet at let and meet at and let and meet at
Prove that if triangle is scalene, then the three circumcircles of triangles and 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]
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 |