Found problems: 2349
Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that:
$a)$ $f(1)+2>0$
$b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$
$c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.
Find all strictly monotone functions $f : \mathbb{R} \to \mathbb{R}$ such that some polynomial $P(x, y)$
satisfies the equality
$$f(x + y) = P(f(x), f(y))$$
for all real numbers $x$ and $y$
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 $x \neq 0$.
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying
\[xf(f(x)+y)=f(xy)+x^2\]
for all $x,y \in \mathbb{R}$.
Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil
\[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that
$$m+f(n) \mid f(m)^2 - nf(n)$$
for all positive integers $m$ and $n$.
(Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)
Let $ f (n) $ be a function that fulfills the following properties:
$\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $.
$\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $.
$\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $.
Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.
The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)
[b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$.
[b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.
Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.
Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$,
$$f(x-f(y))=1-x-y.$$
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$
and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(a-b)f(c-d) + f(a-d)f(b-c) \leq (a-c)f(b-d) \] for all real numbers $a, b, c,$ and $d$.
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying
$$p(x)p(4-x)=p(x(4-x))$$
Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$
\[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:
\[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\]
[i]Proposed by Nikolay Nikolov[/i]
Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(f(x-y)-yf(x))=xf(y).$$
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.