## FE Marathon!

For discussing Olympiad Level Algebra (and Inequality) problems
tanmoy
Posts: 305
Joined: Fri Oct 18, 2013 11:56 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Wed Dec 23, 2020 7:15 pm
Mehrab4226 wrote:
Wed Dec 23, 2020 7:15 pm
"Questions we can't answer are far better than answers we can't question"

Dustan
Posts: 64
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Wed Dec 23, 2020 8:24 pm
If my solution is correct then the next problem in the marathon, if my solution is not correct then ignore this message and plz tell me where I am wrong.

Problem 6:
Find all functions $f : \Bbb R \to \Bbb R$ that satisfies,

$f((x-y)^2)=f(x)^2-2xf(y)+y^2$

Mehrab4226
Posts: 208
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Wed Dec 23, 2020 7:15 pm
This one kinda looks like an okay solution.[This looks big because of the spacings]
Bug 1:$f(0)=?$
Bug 2: (t can be negative as well)
Bug 3: Cauchy function application criteria
But how do I do that?
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Dustan
Posts: 64
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Thu Dec 24, 2020 11:42 pm
Mehrab4226 wrote:
Wed Dec 23, 2020 7:15 pm
This one kinda looks like an okay solution.[This looks big because of the spacings]
Bug 1:$f(0)=?$
Bug 2: (t can be negative as well)
Bug 3: Cauchy function application criteria
But how do I do that?

I think if we will show that $f(x+y)=f(x)+f(y)$ for all x greater than 0
And $f(x+y)=f(x)+f(y)$ for x less then 0 then it will be ok maybe.

Mehrab4226
Posts: 208
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

Dustan wrote:
Fri Dec 25, 2020 11:54 am
Mehrab4226 wrote:
Thu Dec 24, 2020 11:42 pm
Mehrab4226 wrote:
Wed Dec 23, 2020 7:15 pm
This one kinda looks like an okay solution.[This looks big because of the spacings]
Bug 1:$f(0)=?$
Bug 2: (t can be negative as well)
Bug 3: Cauchy function application criteria
But how do I do that?

I think if we will show that $f(x+y)=f(x)+f(y)$ for all x greater than 0
And $f(x+y)=f(x)+f(y)$ for x less then 0 then it will be ok maybe.
But we showed that $f(x+y)=f(x)+f(y)$ for all real $x,y$. That would include your 2 statements together, wouldn't it?
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Dustan
Posts: 64
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Problem 7: Find all function such that
$f:R\rightarrow R$ and
$f(x^2+yf(x))=xf(x+y)$ for all real valued x,y.

~Aurn0b~
Posts: 43
Joined: Thu Dec 03, 2020 8:30 pm

### Re: FE Marathon!

Dustan wrote:
Fri Jan 08, 2021 5:57 pm
Problem 7: Find all function such that
$f:R\rightarrow R$ and
$f(x^2+yf(x))=xf(x+y)$ for all real valued x,y.
$\textbf{Solution :}$

Dustan
Posts: 64
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

~Aurn0b~ wrote:
Wed Jan 13, 2021 11:14 pm
Dustan wrote:
Fri Jan 08, 2021 5:57 pm
Problem 7: Find all function such that
$f:R\rightarrow R$ and
$f(x^2+yf(x))=xf(x+y)$ for all real valued x,y.
$\textbf{Solution :}$
Post Problem 8

~Aurn0b~
Posts: 43
Joined: Thu Dec 03, 2020 8:30 pm

### Re: FE Marathon!

$\textbf{Problem 8 :}$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ satisfy:
$$f(f(x)+2f(y))=f(x)+y+f(y)$$

Dustan
Posts: 64
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

~Aurn0b~ wrote:
Thu Jan 14, 2021 11:48 pm
$\textbf{Problem 8 :}$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ satisfy:
$$f(f(x)+2f(y))=f(x)+y+f(y)$$