Difference between revisions of "2024 IMO Problems/Problem 4"
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Prove that <math>\angle KIL + \angle YPX = 180^{\circ}</math> | Prove that <math>\angle KIL + \angle YPX = 180^{\circ}</math> | ||
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+ | ==Video Solution== | ||
+ | https://youtu.be/WnZv3fdpFXo | ||
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+ | ==Video Solution (AI solution)== | ||
+ | https://youtu.be/cjnJ6EXKWW4 | ||
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==Video Solution(In Chinese)== | ==Video Solution(In Chinese)== | ||
https://youtu.be/QphkkutmY5M | https://youtu.be/QphkkutmY5M | ||
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==Video Solution== | ==Video Solution== |
Revision as of 20:22, 29 August 2024
Let be a triangle with . Let the incentre and incircle of triangle be and , respectively. Let be the point on line different from such that the line through parallel to is tangent to . Similarly, let be the point on line different from such that the line through parallel to is tangent to . Let intersect the circumcircle of triangle again at . Let and be the midpoints of and , respectively. Prove that .
Contents
[hide]Video Solution
Video Solution (AI solution)
Video Solution(In Chinese)
Video Solution
Part 1: Derive tangent values and with trig values of angles , ,
Part 2: Derive tangent values and with side lengths , , , where is the midpoint of
Part 3: Prove that and .
Comments: Although this is an IMO problem, the skills needed to solve this problem have all previously tested in AMC and its system math contests, such as HMMT.~ also proved by Kislay Kai
Evidence 1: 2020 Spring HMMT Geometry Round Problem 8
I used the property that because point is on the angle bisector , is isosceles. This is a crucial step to analyze . This technique was previously tested in this HMMT problem.
Evidence 2: 2022 AMC 12A Problem 25
The technique in this AMC problem can be easily and directly applied to this IMO problem to quickly determine the locations of points and . If you read my solutions to both this AMC problem and this IMO problem, you will find that I simply took exactly the same approach to solve both.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution with discussion of a generalized case
https://youtu.be/NJc79Ccg82E?si=J0YdHAz-46miJIO2
See Also
2024 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |