Found problems: 66
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$,
$$f(xf(y)+y)=yf(x)+f(y).$$
Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.
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]
If $f:\mathbb N\to\mathbb R$ is a function such that
$$\prod_{d\mid n}f(d)=2^n$$holds for all $n\in\mathbb N$, show that $f$ sends $\mathbb N$ to $\mathbb N$.
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions?
1. $f(0,0)=1$;
2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$;
3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$.
Justify your answer.
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f\left(x^2+f(y)\right)-y=(f(x+y)-y)^2$$holds for all $x,y\in\mathbb R$.
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$,
$$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$
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 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 polynomials $P(x)$ with real coefficients that satisfy the relation
$$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
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 functions $f:(0,\infty)\to(0,\infty)$ such that
$$f(f(f(x)))+4f(f(x))+f(x)=6x.$$
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$.
Find all polynomials $p$ with real coefficients such that for all real $x$
$$(x-8)p(2x)=8(x-1)p(x).$$
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,
$$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$.
(b) Calculate $f(n)$ for any natural number $n$.
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy
$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$
for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.
[i]L. Davidov[/i]
We consider a function $f:\mathbb R\to\mathbb R$ such that
$$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$
for each $x,y\in\mathbb R$.
i) Calculate $f(0)$ and $f(-1)$.
ii) Prove that $f$ is an even function.
iii) Give an example of such a function.
iv) Find all monotone functions with the above property.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.