Found problems: 2349
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]
[i]Proposed by Aritra Mondal[/i]
Let $f : Z^2 \to C$ be a function such that $f(x+11, y) = f(x, y+11) = f(x, y)$, and $f(x, y)f(z,w) = f(xz - yw,xw + yz)$. How many possible values can $f(1, 1)$ have?
Given real numbers $a, b$, find all functions $f: R \to R$ satisfying
$f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\]
for all $x,y\in\mathbb{R}$
Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$
Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that
\[ x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]
Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that
\[
f(ab) = f(a)b + af(b)
\]
for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$?
[i]
(Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.)[/i]
Find all functions $f:\mathbb R\rightarrow \mathbb R$ such that for all real numbers $x,y$:
$f(x)f(y)+f(xy)\leq x+y$.
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$,
\[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]
For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have
$\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
Let H be a Hilbert space over the field of real numbers $\Bbb R$. Find all $f: H \to \Bbb R$ continuous functions for which
$$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$
$$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$
is satisfied for any $x , y , z \in H$.
Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties:
(a) $f(1) = 0$,
(b) $f(x) > 0$ for all $x > 1$,
(c) For all $x, y\ge 0$ with $x + y > 0$ holds
$$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.