Found problems: 2349
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]
Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that
\[
\frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1
\]
for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon:
1- $f(0,x)=x+1$
2- $f(x+1,0)=f(x,1)$
3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997)
find $f(3,1997)$
Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying
\[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95)
\]
for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x)-y^{2})=f(x)^{2}-2f(x)y^{2}+f(f(y)).\]
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality
$$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$
holds for all real numbers $x$, $y$.
Proposed by Athanasios Kontogeorgis
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$
Where $f^n(m)$ is defined in the standard iterative manner.
Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that
$$f(x-2f(y))= xf(y)-yf(x)+g(x)$$ for all real $x,y$
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied
\[f(x+y) + f(x)f(y) = f(xy) + 1 \]
$\forall x, y \in \mathbb{R}$
Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions:
1) $f(f(x)) =xf(x)-ax$ for all real $x$
2) $f$ is not constant
3) $f$ takes the value $a$
Find all functions $f : R \to R$ that satisfy $f((x-y)^2)= f(x)^2 -2x f(y)+y^2$ for all $x,y \in R$
Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\]
[i]Proposed by Sutanay Bhattacharya[/i]
Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$.
(i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$.
(ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$.
(iii) $f(1)=1$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality
$$
f(f(x)+f(y))=(x+y)f(x+y)
$$
for all real $x$ and $y$.
[i](B. Serankou)[/i]
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Let $\mathbb{Q}^+$ be the set of all positive rational numbers. Find all functions $f:\mathbb{Q}^+\rightarrow \mathbb{Q}^+$ satisfying $f(1)=1$ and
\[ f(x+n)=f(x)+nf(\frac{1}{x}) \forall n\in\mathbb{N},x\in\mathbb{Q}^+\]