Difference between revisions of "2020 IMO Problems/Problem 1"

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<math>\textbf{Problem 1}</math>. Consider the convex quadrilateral <math>ABCD</math>. The point <math>P</math> is in the interior of <math>ABCD</math>. The following ratio equalities hold:
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== Problem ==
<cmath>\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC</cmath>
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Consider the convex quadrilateral <math>ABCD</math>. The point <math>P</math> is in the interior of <math>ABCD</math>. The following ratio equalities hold:
Prove that the following three lines meet in a point: the internal bisectors of angles <math>\angle ADP</math> and
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<cmath>\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC.</cmath> Prove that the following three lines meet in a point: the internal bisectors of angles <math>\angle ADP</math> and <math>\angle PCB</math> and the perpendicular bisector of segment <math>\overline{AB}</math>.
<math>\angle PCB</math> and the perpendicular bisector of segment <math>AB</math>.
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== Video solution ==
 
== Video solution ==
  
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https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
 
https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]
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==See Also==
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{{IMO box|year=2020|before=First Problem|num-a=2}}
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[[Category:Olympiad Geometry Problems]]

Latest revision as of 11:32, 14 May 2021

Problem

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: \[\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC.\] Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $\overline{AB}$.

Video solution

https://youtu.be/rWoA3wnXyP8

https://youtu.be/bDHtM1wijbY [Shorter solution, video covers all day 1 problems]

See Also

2020 IMO (Problems) • Resources
Preceded by
First Problem
1 2 3 4 5 6 Followed by
Problem 2
All IMO Problems and Solutions
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