Difference between revisions of "2022 IMO Problems/Problem 4"
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'''vladimir.shelomovskii@gmail.com, vvsss''' | '''vladimir.shelomovskii@gmail.com, vvsss''' | ||
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| + | {{IMO box|year=2022|num-b=3|num-a=5}} | ||
Latest revision as of 00:55, 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
![\[TB = TD, TC = TE, BC = DE \implies\]](http://latex.artofproblemsolving.com/6/8/e/68ea05a51c9d42db04e1b0c91ba34921b99c0e64.png)
![\[\triangle TBC = \triangle TDE \implies \angle BTC = \angle DTE.\]](http://latex.artofproblemsolving.com/b/f/2/bf211f5bde0286c839829cbfb266b0b7a47fa30a.png)
![\[\angle BTQ = 180^\circ - \angle BTC = 180^\circ - \angle DTE = \angle STE\]](http://latex.artofproblemsolving.com/0/9/6/096cb624ceca29c46fe7e23a28a6bb00ce4eac2f.png)
![\[\angle ABT = \angle AET \implies \triangle TQB \sim \triangle TSE \implies\]](http://latex.artofproblemsolving.com/7/8/0/780a0ed94981f46136ac44ee6df9e53324cdbb60.png)
![\[\angle PQC = \angle EST, \hspace{18mm}\frac {QT}{ST}= \frac {TB}{TE} \implies\]](http://latex.artofproblemsolving.com/5/5/f/55f09ddeed3125faac91f8e3d560ec4413ba2110.png)
![\[QT \cdot TE =QT \cdot TC = ST \cdot TB= ST \cdot TD \implies\]](http://latex.artofproblemsolving.com/2/6/f/26f2357e000386f6a255dc86fb6599f9cb4e1f64.png)
is cyclic 
![\[\angle QPR =\angle QPC = \angle QCD - \angle PQC =\]](http://latex.artofproblemsolving.com/a/4/f/a4f432c1d2d16eabfca2a95129fd6cb99d3048bf.png)
![\[\angle QSD - \angle EST = \angle QSR \implies\]](http://latex.artofproblemsolving.com/b/2/8/b2849a1f2ea4cb7b3b5de500ebc5877020932735.png)
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 | ||
