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

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Problem 1. Consider the convex quadrilateral ABCD. The point P is in the interior of ABCD.
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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:
The following ratio equalities hold:
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<cmath>\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC</cmath>
∠P AD : ∠P BA : ∠DP A = 1 : 2 : 3 = ∠CBP : ∠BAP : ∠BP C.
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Prove that the following three lines meet in a point: the internal bisectors of angles <math>\angle ADP</math> and
Prove that the following three lines meet in a point: the internal bisectors of angles ∠ADP and
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<math>\angle PCB</math> and the perpendicular bisector of segment <math>AB</math>.
∠P CB and the perpendicular bisector of segment AB.
 
 
== Video solution ==
 
== Video solution ==
  

Revision as of 16:33, 8 October 2020

$\textbf{Problem 1}$. 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 $AB$.

Video solution

https://youtu.be/rWoA3wnXyP8

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

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