Found problems: 1513
Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following:
$$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$
[i]Proposed by Vadym Koval[/i]
Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that:
$ f(x)+f(\frac1x)=1$
$ 2f(f(x))=f(2x)$
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true:
$$f(x+y+f(y))=f(x)+2y$$
[I]proposed by F. Petrov[/i]
Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$
Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that
\[
f(xf(x)+y^2) = x^2+yf(y)
\]
for any positive reals $x,y$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.
Solve for $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that
$$g(m + n) = g(m) + mn(m + n) + g(n)$$
Show that $g(n)$ is of the form $\sum_{i=0}^{d} {c_i n^i}$ \\
and find necessary and sufficient conditions on $d$ and $c_0, c_1, \cdots , c_d$
Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that
$$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$
for all $f,g:\mathbb R\to\mathbb R$.
Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that
$$F(f)=\sigma\circ f\circ\sigma$$
for all $f:\mathbb R\to\mathbb R$.
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\][i]Proposed by Dániel Dobák, Budapest[/i]
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions:
(i) $f(x)$ is an integer if and only if $x$ is an integer;
(ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.
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)
Let $\mathbb{R}$ denote the set of all real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
\[f(xy+xf(x))=f(x)\left(f(x)+f(y)\right).\]
Prove that a function $f:\mathbb{R}_+\to\mathbb{R}$ satisfies \[f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)\]if and only if it satisfies $f(xy)=f(x)+f(y)$.
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$:
$f(f(f(n)))=n+2f(n)$?
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with
the properties:
[list=a]
[*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $
[*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i]Proposed by Thomas Huber, Switzerland[/i]
Find all functions $h: \mathbb{Z}\to \mathbb{Z}$ such that for all $x,y\in \mathbb{Z}$: \[h(x+y)+h(xy)=h(x)h(y)+1.\]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[ f(xy) = yf(x) + x + f(f(y) - f(x)) \]
for all $x,y \in \mathbb{R}$.
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation
$$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]