Found problems: 2349
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\]
for all $x,y \in \mathbb{R}$.
Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
(1) $f(1)=1$;
(2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;
(3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$.
Find all solutions of equation $f(k) +f(l)=293$, where $k<l$.
($\mathbb{N}$ denotes the set of all natural numbers).
Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$.
I. Gorodnin
Let $S$ be the set of all 9-digit natural numbers, which are written only with the digits 1, 2, and 3. Find all functions $f:S\rightarrow \{1,2,3\}$ which satisfy the following conditions:
(1) $f(111111111)=1$, $f(222222222)=2$, $f(333333333)=3$, $f(122222222)=1$;
(2) If $x,y\in S$ differ in each digit position, then $f(x)\neq f(y)$.
Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying
\[D(f\circ g)=(D(f)\circ g)\cdot D(g)\]
for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$
Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.
Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$
.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
\[f(x + y)=f(f(x)) + y + 2022\]
Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$.
(Walther Janous, Austria)
Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied:
$\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$
where $a_{n+1} = a_1$.
Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers
such that
\[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \]
for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$.
[i]David Yang[/i]
Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\]
[i]Proposed by Holden Mui[/i]
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that
\[f(xf(y)+f(x)) = 2f(x)+xy\]
for every reals $x,y$.
Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.
Let $f(x)$ be a continuous function, where $f(0)\neq0$. Find $\lim_{x \to 0} \frac{2\int_{0}^{x}(x-t)f(t)dt}{x\int_{0}^{x}f(x-t)dt}$.
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function.
a) Prove that $f$ is continous;
b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$
\[f(m)+f(n)+f(f(m^2+n^2))=1.\]
We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$.
[i]Proposed by Amirhossein Gorzi[/i]
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that $$f(xy)+f(x+y)=f(x)f(y)+f(x)+f(y)$$ for all $x,y\in\mathbb{Q}$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.