Difference between revisions of "2022 IMO Problems/Problem 4"

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Problem 4. Let ABCDE be a convex pentagon such that BC = DE. Assume that there is a
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==Problem==
point T inside ABCDE with TB = TD, TC = TE and ∠ABT = ∠TEA. Let line AB intersect
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Let <math>ABCDE</math> be a convex pentagon such that <math>BC = DE</math>. Assume that there is a
lines CD and CT at points P and Q, respectively. Assume that the points P, B, A, Q occur on their
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point <math>T</math> inside <math>ABCDE</math> with <math>TB = TD</math>, <math>TC = TE</math> and <math>\angle ABT = \angle TEA</math>. Let line <math>AB</math> intersect
line in that order. Let line AE intersect lines CD and DT at points R and S, respectively. Assume
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lines <math>CD</math> and <math>CT</math> at points <math>P</math> and <math>Q</math>, respectively. Assume that the points <math>P, B, A, Q</math> occur on their
that the points R, E, A, S occur on their line in that order. Prove that the points P, S, Q, R lie on
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line in that order. Let line <math>AE</math> intersect lines <math>CD</math> and <math>DT</math> at points <math>R</math> and <math>S</math>, respectively. Assume
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that the points <math>R, E, A, S</math> occur on their line in that order. Prove that the points <math>P, S, Q, R</math> lie on
 
a circle.
 
a circle.
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==Solution==
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https://www.youtube.com/watch?v=-AII0ldyDww [Video contains solutions to all day 1 problems]

Revision as of 05:47, 23 July 2022

Problem

Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.

Solution

https://www.youtube.com/watch?v=-AII0ldyDww [Video contains solutions to all day 1 problems]