Search found 46 matches
- Sun Jan 17, 2021 7:43 pm
- Forum: Secondary Level
- Topic: Cute little Diophantine!!
- Replies: 3
- Views: 5639
Re: Cute little Diophantine!!
Find all integers $x,y$ such that $x^3+y^3=(x+y)^2$ :P $\textbf{Solution :}$ $\textbf{Claim:}$ The equation is only true for non-negative integers. Proof : Assume that both of the integers are negative. Hence the right side is positive, where the left side is negative, which is not possible. Now, A...
- Fri Jan 15, 2021 6:25 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 646573
Re: FE Marathon!
$\textbf{Problem 10 :}$
Determine all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
\[f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))\ ,\ \forall x,y\in \mathbb{R}\]
Determine all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
\[f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))\ ,\ \forall x,y\in \mathbb{R}\]
- Fri Jan 15, 2021 3:18 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 646573
Re: FE Marathon!
Problem 9: $f:R\rightarrow R $ such that $f(xf(x-y))+yf(x)=x+y+f(x^2)$ for all real x,y. $\textbf{Solution :}$ Let $P(x,y)$ be the assertion of the functional equation. $P(0,y) \Rightarrow f(0)=1 $. $P(x,x) \Rightarrow f(x)+xf(x)=2x+f(x^2) \dots \dots (1)$ Plugging in $x=1$ in $(1)$, we have $f(1)=...
- Thu Jan 14, 2021 11:48 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 646573
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:
$$f(f(x)+2f(y))=f(x)+y+f(y)$$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ satisfy:
$$f(f(x)+2f(y))=f(x)+y+f(y)$$
- Thu Jan 14, 2021 11:35 pm
- Forum: Secondary Level
- Topic: Cute little Diophantine!!
- Replies: 3
- Views: 5639
Re: Cute little Diophantine!!
Find all integers $x,y$ such that $x^3+y^3=(x+y)^2$ :P $\textbf{Solution :}$ $\textbf{Claim:}$ The equation is only true for non-negative integers. Proof : Assume that both of the integers are negative. Hence the right side is positive, where the left side is negative, which is not possible. Now, A...
- Wed Jan 13, 2021 11:14 pm
- Forum: Algebra
- Topic: FE Marathon!
- Replies: 98
- Views: 646573
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. $\textbf{Solution :}$ Let $P(x,y)$ be the assertion of the functional equation $P(x,0) \Rightarrow f(x^2)=xf(x) \dots \dots (1)$ So $P(0,0)\Rightarrow f(0)=0$. $\textbf{Case 1:}$ Assume th...