Found problems: 1513
Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.
(Walther Janous)
Let $\Bbb{Z}^+$ denote the set of positive integers. Find all functions $f:\Bbb{Z}^+\to \Bbb{Z}^+$ that satisfy the following condition: for each positive integer $n,$ there exists a positive integer $k$ such that $$\sum_{i=1}^k f_i(n)=kn,$$ where $f_1(n)=f(n)$ and $f_{i+1}(n)=f(f_i(n)),$ for $i\geq 1. $
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds:
$$f(2^x+2y) =2^yf(f(x))f(y).$$
Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have
\[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \]
Find the number of [i]good[/i] functions.
Find all functions $f:\mathbb R \rightarrow \mathbb R$, such that
$2xyf(x^2-y^2)=(x^2-y^2)f(x)f(2y)$
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
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 $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions:
1) $|f(x)|\ge 1$,
2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.
Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions:
[list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$
[*] For all real number $x$, $f(g(x))=g(f(x))=0$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$
[/list]
Let $a \ne 0$ be a real number.
Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$
for all $x, y \in R_{>0}$.
[i](Proposed by Walther Janous)[/i]
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.
Is there a function $f(x)$ defined and continuous on $R$ such that:
a) $f(f(x)) = 1 + 2x$ ?
b) $f(f(x)) = 1 - 2x $?
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
[i]Proposed by Jason Prodromidis, Greece[/i]
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$
[b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $f(0) = 0$ and for any real numbers $x, y$ the following equality holds \[f(x^2+yf(x))+f(y^2+xf(y))=f(x+y)^2.\]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
$f$ 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 $x$.
Let's call a function $f:\mathbb R\to\mathbb R$[i] weakly periodic[/i] if it is continuous and $f(x+1)=f(f(x))+1$ for all $x\in\mathbb R$.
a) Does there exist a weakly periodic function such that $f(x)>x$ for all $x\in\mathbb R$?
b) Does there exist a weakly periodic function such that $f(x)<x$ for all $x\in\mathbb R$?
[i]Proposed by: András Imolay, Budapest[/i]
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that
\[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}
\]