## FE Marathon!

For discussing Olympiad Level Algebra (and Inequality) problems
Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm
Re: FE Marathon!
~Aurn0b~ wrote:
Sat Feb 27, 2021 7:02 pm
$\textbf{Problem 17}$
Find all functions $f: \mathbb R \to \mathbb R$ such that$f( xf(x) + f(y) ) = f(x)^2 + y$for all $x,y\in \mathbb R$.
DO MENTION THE SOURCE
Proof(Not sure this the final version ):
Last edited by Asif Hossain on Tue Mar 02, 2021 9:18 am, edited 8 times in total.
Hmm..Hammer...Treat everything as nail

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

Find all functions $f:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x+1))=f(x)+1$
for all real $x$.
Update: this has no specific or general solution thanks to Dustan.
Hmm..Hammer...Treat everything as nail

~Aurn0b~
Posts:46
Joined:Thu Dec 03, 2020 8:30 pm

### Re: FE Marathon!

Asif Hossain wrote:
Mon Mar 01, 2021 3:51 pm
~Aurn0b~ wrote:
Sat Feb 27, 2021 7:02 pm
$\textbf{Problem 17}$
Find all functions $f: \mathbb R \to \mathbb R$ such that$f( xf(x) + f(y) ) = f(x)^2 + y$for all $x,y\in \mathbb R$.
DO MENTION THE SOURCE
Proof(Not sure this the final version ):
Source:BMO 2000

$f(x)^2=x^2$ doesnt imply $f(x)=\pm x, \forall x$

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

~Aurn0b~ wrote:
Tue Mar 02, 2021 10:06 am
Asif Hossain wrote:
Mon Mar 01, 2021 3:51 pm
~Aurn0b~ wrote:
Sat Feb 27, 2021 7:02 pm
$\textbf{Problem 17}$
Find all functions $f: \mathbb R \to \mathbb R$ such that$f( xf(x) + f(y) ) = f(x)^2 + y$for all $x,y\in \mathbb R$.
DO MENTION THE SOURCE
Proof(Not sure this the final version ):
Source:BMO 2000

$f(x)^2=x^2$ doesnt imply $f(x)=\pm x, \forall x$
I meant either $f(x)=x$ or $f(x)=-x$ $\forall x \in \mathbb{R}$???
Hmm..Hammer...Treat everything as nail

Dustan
Posts:71
Joined:Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Pointwise trap .

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

Dustan wrote:
Tue Mar 02, 2021 10:33 am
Pointwise trap .
ooooo now i understand thanks to aops (this was subtle )
Here is the rest credit goes le qk's math
Assume there exists $a,b$ such that $f(a)=a ,f(b)=-b$ now we provide a contradiction
$P(a,b) \Rightarrow f(a^2 -b )=a^2 +b$ so, happy contradiction....
Hmm..Hammer...Treat everything as nail

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

Nobody posting any problem
Find all functions $f:\mathbb{Q} \rightarrow \mathbb{R}$
$1$. $f(x+y)-yf(x)-xf(y)=f(x)f(y)-x-y+xy; \forall x,y \in \mathbb{Q}$
$2$. $f(x)=2f(x+1)+x+2; \forall x \in \mathbb{Q}$
$3$. $f(1)+1>0$
Source:
Hmm..Hammer...Treat everything as nail

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

Partial Solution:
Hmm..Hammer...Treat everything as nail

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

### Re: FE Marathon!

Since Nobody posting any problem here is one...
Problem 19
Determine all functions $\mathbb{Q} \to \mathbb{C}$ ST
1) for any rational $x_1,X_2,...,x_2010,$ $f(x_1+x_2+...+x_2010)=f(x_1)f(x_2)...f(x_2010)$
2) $\forall x \in \mathbb{Q}$ , conjugate of $f(2010)$ $\times$ $f(x)=f(2010)$ $\times$ conjugate of $f(x)$ where $\times$ represent normal multiplication.
Hmm..Hammer...Treat everything as nail

Anindya Biswas
Posts:264
Joined:Fri Oct 02, 2020 8:51 pm
2) $\forall x \in \mathbb{Q}$ , conjugate of $f(2010)$ $\times$ $f(x)=f(2010)$ $\times$ conjugate of $f(x)$ where $\times$ represent normal multiplication.
Did you mean $\overline{f(2010)}\cdot f(x)$ or $\overline{f(2010)\cdot f(x)}$? I guess the first one.