Found problems: 2349
Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$
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 $x,y,z\in\mathbb R_{\ge0}$.
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\
• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\
• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied
$f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$
If $g(x)=f(x)-x+1$ then find $g(2016)$ .
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ :
$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$.
[i]Proposed by Amirhossein Zolfaghari [/i]
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that
$$f(x + f(y)) = f(x) + f(y)$$
for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
A function $f:\mathbb{Z}\implies\mathbb{Z}$ satisfies
$f(x+4)-f(x)=8x+20$
$f(x^2-1)=(f(x)-x)^2+x^2-2$
Find $f(0)$ and $f(1)$.
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that
\[f(x + y) + f(x + z) - f(x)f(y + z) \ge 1\]
for all $x,y,z \in \mathbb{R}$
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property:
for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$.
Prove that $f (x) = x$ for all $x \in S$.
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
The function f is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1)=2$ and $f(xy)=f(x)f(y)-f(x+y)+1$ for all $x,y \in \mathbb{Q}$. Determine f (with proof)
Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions:
1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$;
2. $f$ takes all integer values.
[i](I. Voronovich)[/i]
Find all functions $f : \mathbb Z \to \mathbb Z$ such that $$(f(x + y))^2 = f(x^2) + f(y^2)$$ for all $x, y \in \mathbb Z$.
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$
holds for all $x,y\in\mathbb{R^{+}}.$
($f^m(x) =$ $f$ applies $m$ times.)
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.