Difference between revisions of "2024 INMO"
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∼Lakshya Pamecha | ∼Lakshya Pamecha | ||
==Problem 3== | ==Problem 3== | ||
− | Let p be an odd prime number and a,b,c 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>. | + | 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>. |
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==Solution== | ==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: Notice that 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 And because 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 are all divisible by p. Prove that p divides each of .