Difference between revisions of "2022 IMO Problems/Problem 4"
m |
|||
(9 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
− | Problem | + | ==Problem== |
− | point T inside ABCDE with | + | 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 | + | 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 | + | 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 | + | 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 |
+ | 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. | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=-AII0ldyDww [Video contains solutions to all day 2 problems] | ||
+ | |||
+ | https://youtu.be/WpM0mLyPyLg?si=yi9AZPVdYSPMCcHa | ||
+ | [Video Solution by little fermat] | ||
+ | |||
+ | ==Solution== | ||
+ | [[File:2022 IMO 4.png|400px|right]] | ||
+ | <cmath>TB = TD, TC = TE, BC = DE \implies</cmath> | ||
+ | <cmath>\triangle TBC = \triangle TDE \implies \angle BTC = \angle DTE.</cmath> | ||
+ | <cmath>\angle BTQ = 180^\circ - \angle BTC = 180^\circ - \angle DTE = \angle STE</cmath> | ||
+ | <cmath>\angle ABT = \angle AET \implies \triangle TQB \sim \triangle TSE \implies</cmath> | ||
+ | <cmath>\angle PQC = \angle EST, \hspace{18mm}\frac {QT}{ST}= \frac {TB}{TE} \implies</cmath> | ||
+ | <cmath>QT \cdot TE =QT \cdot TC = ST \cdot TB= ST \cdot TD \implies</cmath> | ||
+ | <math>\hspace{28mm}CDQS</math> is cyclic <math>\implies \angle QCD = \angle QSD.</math> | ||
+ | <cmath>\angle QPR =\angle QPC = \angle QCD - \angle PQC =</cmath> | ||
+ | <cmath>\angle QSD - \angle EST = \angle QSR \implies</cmath> | ||
+ | <math>\hspace{43mm}PRQS</math> is cyclic. | ||
+ | |||
+ | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{IMO box|year=2022|num-b=3|num-a=5}} |
Latest revision as of 01:55, 19 November 2023
Contents
Problem
Let be a convex pentagon such that
. Assume that there is a
point
inside
with
,
and
. Let line
intersect
lines
and
at points
and
, respectively. Assume that the points
occur on their
line in that order. Let line
intersect lines
and
at points
and
, respectively. Assume
that the points
occur on their line in that order. Prove that the points
lie on
a circle.
Video Solution
https://www.youtube.com/watch?v=-AII0ldyDww [Video contains solutions to all day 2 problems]
https://youtu.be/WpM0mLyPyLg?si=yi9AZPVdYSPMCcHa [Video Solution by little fermat]
Solution
is cyclic
is cyclic.
vladimir.shelomovskii@gmail.com, vvsss
See Also
2022 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |