Found problems: 2349
Known $f:\mathbb{N}_0 \to \mathbb{N}_0$ function for $\forall x,y\in \mathbb{N}_0$ the following terms are paid
$(a). f(0,y)=y+1$
$(b). f(x+1,0)=f(x,1)$
$(c). f(x+1,y+1)=f(x,f(x+1,y)).$
Find the value if $f(4,1981)$
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy
\[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.
How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and
$$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$
for all non-negative integers $x$, $y$?
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$
holds for all $p,q\in\mathbb{P}$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that
\[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\]
for every $x$, $y\in \mathbb{R}_+$.
[i]Proposed by Li4 and Untro368.[/i]
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that
\[ n^2+4f(n)=f(f(n))^2 \]
for all $n\in \mathbb{Z}$.
[i]Proposed by Sahl Khan, UK[/i]
Does there exist a surjective function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which
$f(x+y)-f(x)-f(y)$
takes only 0 and 1 for values for random $x$ and $y$?
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions:
$i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$
$ii) f$ takes all integer values
A function $f: Z \to Z$ has the following properties:
$f (92 + x) = f (92 - x)$
$f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$)
$f (1992 + x) = f (1992 - x)$
for all integers $x$. Can all positive divisors of $92$ occur as values of f?
Let $f: N \to N$ be a function satisfying the following:
$\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$.
$\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes.
Determine all possible values of $f(2002)$. Justify your answers.
Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$,
$f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$
Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ a function having the property that
$$ \left| f(x+y)+\sin x+\sin y \right|\le 2, $$
for all real numbers $ x,y. $
[b]a)[/b] Prove that $ \left| f(x) \right|\le 1+\cos x, $ for all real numbers $ x. $
[b]b)[/b] Give an example of what $ f $ may be, if the interval $ \left( -\pi ,\pi \right) $ is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.[/url]
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying
$\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.
Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$
$$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$
[i]Created by Liang Xiao, Yunhao Fu[/i]
Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions:
(i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$;
(ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and
\[f(x+f(y)^2 ) = {f(x+y)}^2.\]
(25 points)
Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions:
[list=1]
[*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$
[*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$
[/list]
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.