Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"

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Anindya Biswas
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Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"

Unread post by Anindya Biswas » Thu May 13, 2021 12:12 am

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
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Dustan
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Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"

Unread post by Dustan » Thu May 13, 2021 2:36 pm

Easy!😐 Ans
No such function [\hide]

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Anindya Biswas
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Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"

Unread post by Anindya Biswas » Thu May 13, 2021 3:38 pm

Dustan wrote:
Thu May 13, 2021 2:36 pm
Easy!😐 Ans
No such function [\hide]
I also thought this, but found no contradiction.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

~Aurn0b~
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Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"

Unread post by ~Aurn0b~ » Fri May 14, 2021 11:21 am

Anindya Biswas wrote:
Thu May 13, 2021 12:12 am
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
$P(x,f(y))\Rightarrow f(f(f(x)+f(y)))=f(x+f(y))+f(x)+f(y)$
$\Rightarrow f(x+f(y))=f(y+f(x))$
$\Rightarrow f(x+y)+f(y)+x=f(x+y)+f(x)+y$
$\Rightarrow f(x)=x+f(0)$
Plugging this into the main equation wee see that no such function exists.$\blacksquare$

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