FE Marathon!
Re: FE Marathon!
can you suggest me any book for solving this kind of function problem.
 Anindya Biswas
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Re: Solution to P24
Warning : Don't click, graphical violence ahead...
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
 Anindya Biswas
 Posts: 196
 Joined: Fri Oct 02, 2020 8:51 pm
 Location: Magura, Bangladesh
 Contact:
Problem 25
Find functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(n)+f(f(n))+f(f(f(n)))=3n\ \ \ \forall n\in\mathbb{N}\]
Where $\mathbb{N}$ is the set of all positive integers
Where $\mathbb{N}$ is the set of all positive integers
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann

 Posts: 170
 Joined: Sat Jan 02, 2021 9:28 pm
Re: Problem 25
Again 2 days passed no solutionAnindya Biswas wrote: ↑Sat Apr 03, 2021 9:54 pmFind functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(n)+f(f(n))+f(f(f(n)))=3n\ \ \ \forall n\in\mathbb{N}\]
Where $\mathbb{N}$ is the set of all positive integers
Hmm..Hammer...Treat everything as nail
Re: Problem 25
$\textbf{Solution 25}$Anindya Biswas wrote: ↑Sat Apr 03, 2021 9:54 pmFind functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(n)+f(f(n))+f(f(f(n)))=3n\ \ \ \forall n\in\mathbb{N}\]
Where $\mathbb{N}$ is the set of all positive integers
 Anindya Biswas
 Posts: 196
 Joined: Fri Oct 02, 2020 8:51 pm
 Location: Magura, Bangladesh
 Contact:
Re: Problem 25
Yeah, but this last statement can be nicely proven by induction.~Aurn0b~ wrote: ↑Sat Apr 10, 2021 10:31 pm$\textbf{Solution 25}$Anindya Biswas wrote: ↑Sat Apr 03, 2021 9:54 pmFind functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(n)+f(f(n))+f(f(f(n)))=3n\ \ \ \forall n\in\mathbb{N}\]
Where $\mathbb{N}$ is the set of all positive integers
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
Re: FE Marathon!
$\textbf{Problem 25}$
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\](Here $\mathbb{Z}$ denotes the set of integers.)
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\](Here $\mathbb{Z}$ denotes the set of integers.)

 Posts: 170
 Joined: Sat Jan 02, 2021 9:28 pm
Problem 26
For the sake to continue the marathon, This is 2012 IMO P4. I am posting the next problem.~Aurn0b~ wrote: ↑Sun Apr 11, 2021 10:10 am$\textbf{Problem 25}$
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\](Here $\mathbb{Z}$ denotes the set of integers.)
Problem 26
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x^2y^2)=xf(x)yf(y)$
Hmm..Hammer...Treat everything as nail