Found problems: 1513
Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$
Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.
Determine all functions $f: R\to R$, such that $f (2x + f (y)) = x + y + f (x)$ for all $x, y \in R$.
(Gerhard Kirchner)
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Find all non-constant functions $f : Q^+ \to Q^+$ satisfying the equation $$f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)$$
for all $a, b,c \in Q^+$ .
Find all functions $f:\mathbb R \to \mathbb R$ such that
\[
xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2,
\]
for all real numbers $x,y$.
[i]Proposed by B.J. Venkatachala[/i]
Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$.
(I. Voronovich)
Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$.
[i]Pakawut Jiradilok[/i]
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
Given a (fixed) positive integer $N$, solve the functional equation
\[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\]
[i](Dan Schwarz)[/i]
Find all functions $f : \mathbb{N} \to \mathbb{N}$ with
$$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$
holds for all positive integers $x,y.$
Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy
$$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$
for all integers $n\ge 1$ and rational numbers $q>0$.
Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties:
(a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$,
(b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite
Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$
(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.$$
(Albania)
Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that
$f(x,y)+f(y,z)+f(z,x)=0$.
Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.
find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
$f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
Find all functions $ f: \mathbb{N} \to \mathbb{N} $ such that $$ f\left(x+yf\left(x\right)\right) = x+f\left(y\right)f\left(x\right) $$ holds for all $ x,y \in \mathbb{N} $
Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which
\[f(f(x)+y)=f(f(y)+x)\]
holds for all $x, y\in \mathbb{R}$.
Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$
Find $f (1)$.
Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfi ed $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$