Found problems: 2349
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have
$$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$.
Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(f\left(x\right)+y\right) = f\left(x^2-y\right)+4\left(y-2\right)\left(f\left(x\right)+2\right) $$ holds for all $ x, y \in \mathbb{R} $
The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$
(i) $ F(4n) \equal{} F(2n) \plus{} F(n),$
(ii) $ F(4n \plus{} 2) \equal{} F(4n) \plus{} 1,$
(iii) $ F(2n \plus{} 1) \equal{} F(2n) \plus{} 1.$
Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) \equal{} F(3n)$ is $ F(2^{m \plus{} 1}).$
Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies:
$\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$.
$\bullet$ $f (1998) = 2$
Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,
Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$
Find the differentiable functions $ f:\mathbb{R}\longrightarrow (-\infty ,1) $ with the property $ f(1)=-1 $ and
$$ f(x+y)=f(x)+f(y)-f(x)f(y) , $$
for any reals $ x,y. $
[i]Vasile Pop[/i]
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$:
$$f(mf(n)) + f(n) | mn + f(f(n)).$$
Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$.
[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
[b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$
[i]Proposed by Carl Schildkraut[/i]
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.
Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$
i) $f(1)=2016$
ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$
Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that:
$f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$
Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that:
$ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.
Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and \[f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.\]
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$
.
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]
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. $
Let \( \mathbb{R} \) be the set of real numbers. Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for any real numbers \( x \) and \( y \),
\[
f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1.
\]
If $f:\mathbb N\to\mathbb R$ is a function such that
$$\prod_{d\mid n}f(d)=2^n$$holds for all $n\in\mathbb N$, show that $f$ sends $\mathbb N$ to $\mathbb N$.
Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions
$\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$
$\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function.
holds
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.