2024 INMO
==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 a,b,c be integers so that the integers are all divisible by p. Prove that p divides each of .
Solution
If \Rightarrow and \Rightarrow p\vert b \Rightarrow and \Rightarrow .\\ Therefore, if divides one of it will divide all of them.\\ Assume that does not divide or Set $$ (Error compiling LaTeX. Unknown error_msg) a^{2023} &\equiv k \pmod{p} \Rightarrow b^{2023} \equiv -k \pmod{p} \\ b^{2024} &\equiv -bk \pmod{p} \Rightarrow c^{2024} \equiv kb \pmod{p}\\ c^{2025} &\equiv kbc \pmod{p}\Rightarrow a^{2025} \equiv -kbc \pmod{p}\$$<cmath>\Rightarrow \boxed{a^2 &\equiv -bc \pmod{p}}</cmath> Now we see that$ (Error compiling LaTeX. Unknown error_msg)$(a^{2023})^2 &\equiv (b^{2023})^2 \pmod{p}\\ (-bc)^{2023} &\equiv (b^2)^{2023} \pmod{p}\\ \Rightarrow -c^{2023} &\equiv b^{2023} \pmod{p}\; \text{and} \; b^{4048} \equiv c^{4048}\pmod{p}$ (Error compiling LaTeX. Unknown error_msg) \text{So},
\[\boxed{b^2 &\equiv c^2 \pmod{p}}\] (Error compiling LaTeX. Unknown error_msg)
This gives us to 2 cases:\\ Case I:
\[b-c \equiv 0 \pmod{p} \Rightarrow b^{2024} \equiv c^{2024} \pmod{p} \Rightarrow 2b^{2024} &\equiv 0 \pmod{p} \Rightarrow p\vert b\] (Error compiling LaTeX. Unknown error_msg)
Case II: On checking for both cases we get which implies and .