Difference between revisions of "2024 IMO Problems/Problem 4"
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− | Let ABC be a triangle with AB < AC < BC. Let the incentre and incircle of triangle | + | Let <math>ABC</math> be a triangle with <math>AB < AC < BC</math>. Let the incentre and incircle of triangle |
− | ABC be I and | + | <math>ABC</math> be <math>I</math> and <math>\omega</math>, respectively. Let <math>X</math> be the point on line <math>BC</math> different from <math>C</math> such that the line |
− | through X parallel to AC is tangent to | + | through <math>X</math> parallel to <math>AC</math> is tangent to <math>\omega</math>. Similarly, let <math>Y</math> be the point on line <math>BC</math> different from |
− | B such that the line through Y parallel to AB is tangent to | + | <math>B</math> such that the line through <math>Y</math> parallel to <math>AB</math> is tangent to <math>\omega</math>. Let <math>AI</math> intersect the circumcircle of |
− | triangle ABC again at P | + | triangle <math>ABC</math> again at <math>P \neq A</math>. Let <math>K</math> and <math>L</math> be the midpoints of <math>AC</math> and <math>AB</math>, respectively. |
− | Prove that | + | Prove that <math>\angle KIL + \angle YPX = 180^{\circ}</math> |
. | . | ||
+ | ==Video Solution(In Chinese)== | ||
+ | https://youtu.be/QphkkutmY5M | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/WnZv3fdpFXo | ||
+ | |||
+ | ==Video Solution== | ||
+ | Part 1: Derive tangent values <math>\angle AIL</math> and <math>\angle AIK</math> with trig values of angles <math>\frac{A}{2}</math>, <math>\frac{B}{2}</math>, <math>\frac{C}{2}</math> | ||
+ | |||
+ | https://youtu.be/p_AmooMMln4 | ||
+ | |||
+ | Part 2: Derive tangent values <math>\angle XPM</math> and <math>\angle YPM</math> with side lengths <math>AB</math>, <math>BC</math>, <math>CA</math>, where <math>M</math> is the midpoint of <math>BC</math> | ||
+ | |||
+ | https://youtu.be/MgrghZ2ESAg | ||
+ | |||
+ | Part 3: Prove that <math>\angle AIL + \angle XPM = 90^\circ</math> and <math>\angle AIK + \angle YPM = 90^\circ</math>. | ||
+ | |||
+ | https://youtu.be/iOp9mnmZyzU | ||
+ | |||
+ | 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 <math>P</math> is on the angle bisector <math>AI</math>, <math>\triangle BPC</math> is isosceles. This is a crucial step to analyze <math>\angle XPY</math>. 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 <math>X</math> and <math>Y</math>. 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) | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2024|num-b=3|num-a=5}} |
Revision as of 10:59, 4 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 .
Video Solution(In Chinese)
Video Solution
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)
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 |