Found problems: 30
Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and
$$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$
for any nonnegative real number $ x. $
Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify
$$ f(x+a+f(y))=f(x+b)+y, $$
for any rational $ x,y. $
[i]Vasile Pop[/i]
Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that
$$ (f(\sin x))^2-3f(x)=-2, $$
for any real numbers $ x. $
[i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]
Find all functions $f: R \to R$ such that, for any $x, y \in R$:
$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and
$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the equality
$$ \int_a^b f(x)dx=f(b)-f(a), $$
for any real numbers $ a,b. $
[i]Cosmin Nitu[/i]
[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective.
[b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation
$$ xf(x/a)-f(a/x)=b, $$
where $ a\neq 0,b $ are two real numbers.
[i]Dan Popescu[/i]
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation
$$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$
for any rational numbers $ x,y. $
[i]Mihai Onucu Drîmbe[/i]
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $
[b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $
[b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be.
[i]Bogdan Blaga[/i]
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$
for all real numbers $ x,y. $
Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that
$$ f'(x)+F(x)=2f(x)+x^2/2, $$
for any real numbers $ x; $ where $ F $ is a primitive of $ f. $
[i]Carmen Botea[/i]
Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$
for any real numbers $ x,y. $
Prove that there exists a nonconstant function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ verifying the following system of relations:
$$ \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,& \quad \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,& \quad \forall x,y\in\mathbb{R} \end{matrix} \right. $$
Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties:
$ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $
$ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $
Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation
$$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$
[b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational.
[b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?
Find the functions $ f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0} $ that satisfy the following two conditions:
$ \text{(a)} f(m+n)=f(n)+f(m)+2mn,\quad\forall m,n\in\mathbb{Z} $
$ \text{(b)} f(f(1))-f(1) $ is a perfect square
[i]Marin Ionescu[/i]
Let $ 0 $ be a root for a polynom $ f\in\mathbb{R}[X] $ that has the property that $ f(X^2-X+1) =f^2(X)-f(X)+1. $
Determine this polynom.
[i]Nelu Chichirim[/i]
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $
Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds.
$$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$