Found problems: 2349
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying
\[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \]
for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)
Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$
$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$
for all real numbers $x, y$.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that \[f(n+1)>\frac{f(n)+f(f(n))}{2}\] for all $n\in\mathbb{N}$, where $\mathbb{N}$ is the set of strictly positive integers.
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$ for all $x>0, y>x+1$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]
for all $x,y\in\mathbb{R}$.
[i]Proposed by Carl Schildkraut[/i]
Let $g : \mathbb{N} \to \mathbb{N}$ be a function satisfying:
[list]
[*] $g(xy) = g(x)g(y)$ for all $x, y \in \mathbb{N}$,
[*] $g(g(x)) = x$ for all $x \in \mathbb{N}$, and
[*] $g(x) \neq x$ for $2 \leq x \leq 2018$.
[/list]
Find the minimum possible value of $g(2)$.
The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$.
a) If $A = \mathbb{R}$, find all functions satisfying the conditions.
b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions.
[i](With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)[/i]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any reals $x \neq y$ the following equality is true:
$$f(x+y)^2=f(x+y)+f(x)+f(y)$$
[i]D. Zmiaikou[/i]
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that
$f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$.
by Athanasios Kontogeorgis (aka socrates)
Consider functions $f$ satisfying the following four conditions:
(1) $f$ is real-valued and defined for all real numbers.
(2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$.
(3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$.
(4) We have $f(2)=4$.
Prove that:
a) There is a function $f$ with $f(3)=9$ satisfying the four conditions.
b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that
a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$;
b) $f(x+1,x)=2$ for all integers $x$.
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which
\[ x(f(x+1)-f(x)) = f(x), \]
for all $x\in\mathbb{R}$ and
\[ | f(x) - f(y) | \leq |x-y| , \]
for all $x,y\in\mathbb{R}$.
[i]Mihai Piticari[/i]
Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$
for all $x,y\in{\mathbb{R}}$
Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.
Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
[i]Proposed by Athanasios Kontogeorgis, Greece[/i]
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all monotone functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying the equation
$$
f(4x)-f(3x) = 2x \ \ \text{ for } \ \ x \in \mathbb{R}.$$