## FE Marathon!

For discussing Olympiad Level Algebra (and Inequality) problems
Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

I am giving up
I tried a lot for the last 2 days but couldn't solve it

My progress:
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: 65
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Fri Dec 18, 2020 9:37 pm
I am giving up
I tried a lot for the last 2 days but couldn't solve it

My progress:
But i won't give you the solution, you can find the solution of this problem. Its IMO SL A7 2009.
You can give a new problem to continue this marathon.

Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

Find all functions $f$ $:$ $\Bbb Q \to \Bbb Q$ that satisfies,

$f(x+y)=f(x)+f(y)+xy$
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: 65
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Dustan
Posts: 65
Joined: Mon Aug 17, 2020 10:02 pm

### Re: FE Marathon!

Problem 5: $f:R\rightarrow R$
Find all the function such that for all real valued x,y
$f(x^4+ y)$= $x^3\cdot f(x) + f(f(y))$

Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

I am not sure this solution is correct. Bhaya plz check it if right then I will post a new problem if not then I will probably try more to solve 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

tanmoy
Posts: 305
Joined: Fri Oct 18, 2013 11:56 pm

### Re: FE Marathon!

Mehrab4226 wrote:
Tue Dec 22, 2020 11:33 pm
I am not sure this solution is correct. Bhaya plz check it if right then I will post a new problem if not then I will probably try more to solve it.
The solution is not correct . You haven't proved that the function is injective, so, you can't say that the values of $q$ are different for different values of $t$. For example, suppose, $f(10) = f(15) = 15$, then $15 = f(10) = f(15) = f(f(10))$. It still holds the problem's conditions but $f(10) \neq 10$.
"Questions we can't answer are far better than answers we can't question"

Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

tanmoy wrote:
Tue Dec 22, 2020 11:46 pm
Mehrab4226 wrote:
Tue Dec 22, 2020 11:33 pm
I am not sure this solution is correct. Bhaya plz check it if right then I will post a new problem if not then I will probably try more to solve it.
The solution is not correct . You haven't proved that the function is injective, so, you can't say that the values of $q$ are different for different values of $t$. For example, suppose, $f(10) = f(15) = 15$, then $15 = f(10) = f(15) = f(f(10))$. It still holds the problem's conditions but $f(10) \neq 10$.
Ok. It looked too good to be true anyway.
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

Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm

### Re: FE Marathon!

This one kinda looks like an okay solution.[This looks big because of the spacings]
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

Mehrab4226
Posts: 212
Joined: Sat Jan 11, 2020 1:38 pm
Find all functions $f : \Bbb R \to \Bbb R$ that satisfies,
$f((x-y)^2)=f(x)^2-2xf(y)+y^2$