Found problems: 1513
Denote by $R_{>0}$ the set of all positive real numbers. Find all functions $f: R_{>0} \to R_{>0}$ such that for all $x,y \in R_{>0}$ the following equation holds $$f(y)f(x+f(y))=f(1+xy)$$
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
Does there exist a function $f:\mathbb N\to\mathbb N$ such that
$$
f(f(n+1))=f(f(n))+2^{n-1}
$$
for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.)
[i](I. Gorodnin)[/i]
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:
(i) $f(1) = 2$;
(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.
Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$): $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$
$$f(y)f(x+f(y))=f(x)f(xy)$$
Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$.
Then the value of $f(2012)$ is
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above.
Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
[i]Fajar Yuliawan, Indonesia[/i]
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f\left( {yf(x + y) + f(x)} \right) = 4x + 2yf(x + y)\] for all $x,y\in\mathbb{R}$.
[i]Netherlands (Birgit van Dalen)[/i]
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that
$f(x+y) = f(y) f(x f(y))$
for every two real numbers $x$ and $y$.
Find all functions $f$ mapping non-empty finite sets of integers, to integers, such that
$$f(A+B)=f(A)+f(B)$$
for all non-empty sets of integers $A$ and $B$.
$A+B$ is defined as $\{a+b: a \in A, b \in B\}$.
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds:
$\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and
$\bullet$ $f(f(x)) = x$, for all $x \in R$.
Find all real functions of a real numbers, such that for any $x$, $y$ and $z$ holds the equality $$ f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.$$
Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$
\[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \]
[i]Proposed by F. Petrov[/i]
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$.
(Here $\sqrt[3]{x}$ is defined all over $R$.)
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$:
$$0\le y+f(x)-f^{f(y)}(x)\le1$$
that here
$$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$
For an integer $ m$, denote by $ t(m)$ the unique number in $ \{1, 2, 3\}$ such that $ m \plus{} t(m)$ is a multiple of $ 3$. A function $ f: \mathbb{Z}\to\mathbb{Z}$ satisfies $ f( \minus{} 1) \equal{} 0$, $ f(0) \equal{} 1$, $ f(1) \equal{} \minus{} 1$ and $ f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m)$ for all integers $ m$, $ n\ge 0$ with $ 2^n > m$. Prove that $ f(3p)\ge 0$ holds for all integers $ p\ge 0$.
[i]Proposed by Gerhard Woeginger, Austria[/i]
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$