Found problems: 2349
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.
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
For each pair $(\alpha, \beta)$ of non-negative reals with $\alpha+\beta \geq 2$, determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(x)f(y) \leq f(xy)+\alpha x+\beta y$$ for all reals $x, y$.
Find all functions $f(x) = ax^2 + bx + c$, with $a \ne 0$, such that $f(f(1)) = f(f(0)) = f(f(-1))$ .
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$:
$(i)$: $x$ and $f(x)$ have the same number of positive divisors.
$(ii)$: If $x \nmid y$ and $y \nmid x$, then:
$$\gcd(f(x), f(y)) > f(\gcd(x, y))$$
Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$:
$$f(xy) \le \frac{xf(y) + yf(x)}{2}$$
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 $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.
Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$
$f(x+f(y))=2x+2f(y+1)$
Determine all functions $f$ from the reals to the reals for which
(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$,
where $g(x)$ is the composition inverse function to $f(x)$. (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)
Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.
Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ :
$$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$
are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$.
[i]Proposed by Amirhossein Zolfaghari [/i]
Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.
Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$.
(E. Barabanov)
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying relation :
$$f(xf(y)-f(x))=2f(x)+xy$$
$\forall x,y \in \mathbb{R}$
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions
$Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$ holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.
[i]Proposed by Ivan Chan Kai Chin[/i]
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for every positive integer $n$ the following is valid: If $d_1,d_2,\ldots,d_s$ are all the positive divisors of $n$, then $$f(d_1)f(d_2)\ldots f(d_s)=n.$$
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$
Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]