Difference between revisions of "2024 INMO"
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\text {In} triangle ABC with <math>CA=CB</math>, \text{point E lies on the circumcircle of} \text{triangle ABC such that} <math>\angle ECB=90^\circ</math>. \text{The line through E parallel to CB intersect CA in F} \text{and AB in G}.\text{Prove that}\\ \text{the centre of the circumcircle of} triangle EGB \text{lies on the circumcircle of triangle ECF.} | \text {In} triangle ABC with <math>CA=CB</math>, \text{point E lies on the circumcircle of} \text{triangle ABC such that} <math>\angle ECB=90^\circ</math>. \text{The line through E parallel to CB intersect CA in F} \text{and AB in G}.\text{Prove that}\\ \text{the centre of the circumcircle of} triangle EGB \text{lies on the circumcircle of triangle ECF.} | ||
==Solution== | ==Solution== | ||
| − | To Prove: Points E,F,P,C are concyclic | + | https://i.imgur.com/ivcAShL.png |
| − | + | To Prove: Points E, F, P, C are concyclic | |
Observe: <cmath>\angle CAB=\angle CBA=\angle EGA</cmath> <cmath>\angle ECB=\angle CEG=\angle EAB= 90^\circ</cmath> | Observe: <cmath>\angle CAB=\angle CBA=\angle EGA</cmath> <cmath>\angle ECB=\angle CEG=\angle EAB= 90^\circ</cmath> | ||
| − | Notice that <cmath>\angle CBA = \angle FGA</cmath> because <math>CB \parallel EG</math> <math>\Longrightarrow \angle FAG =\angle FGA | + | Notice that <cmath>\angle CBA = \angle FGA</cmath> because <math>CB \parallel EG</math> <math>\Longrightarrow \angle FAG =\angle FGA \Longrightarrow FA= FG</math>. |
| − | + | Here F is the circumcentre of <math>\triangle EAG</math> because <math>F</math> lies on the Perpendicular bisector of AG | |
| − | + | <math>\Longrightarrow F</math> is the midpoint of <math>EG \Longrightarrow FP</math> is the perpendicular bisector of <math>EG</math>. | |
| − | + | This gives <cmath>\angle EFP =90^\circ</cmath> | |
| − | This gives<cmath>\angle EFP =90^\circ</cmath> | + | And because <cmath>\angle EFP+\angle ECP=180^\circ</cmath> Points E, F, P, C are concyclic. |
| − | And because <cmath>\angle EFP+\angle ECP=180^\circ</cmath> | ||
Hence proven that the centre of the circumcircle of <math>\triangle EGB</math> lies on the circumcircle of <math>\triangle ECF</math>. | Hence proven that the centre of the circumcircle of <math>\triangle EGB</math> lies on the circumcircle of <math>\triangle ECF</math>. | ||
| + | |||
| + | ∼Lakshya Pamecha | ||
| + | ==Problem 3== | ||
| + | Let p be an odd prime number and <math>a,b,c</math> be integers so that the integers <cmath>a^{2023}+b^{2023}, b^{2024}+c^{2024}, c^{2025}+a^{2025}</cmath> are all divisible by p. Prove that p divides each of <math>a,b,c</math>. | ||
| + | |||
| + | ==Solution== | ||
Latest revision as of 13:28, 25 April 2024
==Problem 1
\text {In} triangle ABC with 
, \text{point E lies on the circumcircle of} \text{triangle ABC such that} 
. \text{The line through E parallel to CB intersect CA in F} \text{and AB in G}.\text{Prove that}\\ \text{the centre of the circumcircle of} triangle EGB \text{lies on the circumcircle of triangle ECF.}
Solution
https://i.imgur.com/ivcAShL.png To Prove: Points E, F, P, C are concyclic
Observe: ![\[\angle CAB=\angle CBA=\angle EGA\]](http://latex.artofproblemsolving.com/8/0/4/804eea72b35117cbeb648bd29eb74cd965fc3103.png)
![\[\angle ECB=\angle CEG=\angle EAB= 90^\circ\]](http://latex.artofproblemsolving.com/7/2/2/722712b65d8613d3797ea63c4f56a91c07bc74bb.png)
Notice that ![\[\angle CBA = \angle FGA\]](http://latex.artofproblemsolving.com/d/2/a/d2a5044184909fe92d8505f40511b9321c91a798.png)
because 
.
Here F is the circumcentre of 
because 
lies on the Perpendicular bisector of AG
is the midpoint of 
is the perpendicular bisector of 
.
This gives ![\[\angle EFP =90^\circ\]](http://latex.artofproblemsolving.com/0/8/f/08ffa237cee2e795c91719df6617a5e70836a3a0.png)
And because ![\[\angle EFP+\angle ECP=180^\circ\]](http://latex.artofproblemsolving.com/7/9/e/79ec6002f0be9c7e10f9e7c7b5fc805c0bc90903.png)
Points E, F, P, C are concyclic.
Hence proven that the centre of the circumcircle of 
lies on the circumcircle of 
.
∼Lakshya Pamecha
Problem 3
Let p be an odd prime number and 
be integers so that the integers ![\[a^{2023}+b^{2023}, b^{2024}+c^{2024}, c^{2025}+a^{2025}\]](http://latex.artofproblemsolving.com/2/3/3/23366379c5800a35a8a898097ea72ba86a02ebf1.png)
are all divisible by p. Prove that p divides each of .
