Found problems: 2349
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation
\[ f(f(f(n))) = f(n+1 ) +1 \]
for all $ n\in \mathbb{Z}_{\ge 0}$.
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
[b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$.
[b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$.
[i]Proposed by Nikola Velov[/i]
Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
For which integers $k$ does there exist a function $f : N \to Z$ such that
$f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$.
For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:
$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.
$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.
Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/i]
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation:
$ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$
for any two real numbers $ x$ and $ y$.
Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity
$$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$
holds for all $x,y \in \mathbb N$
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
$f$ is a continuous real-valued function such that $f(x+y)=f(x)f(y)$ for all real $x$, $y$. If $f(2)=5$, find $f(5)$.
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$
Find the sum of all possible values of $f(2023)$.
Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation
\[(y+1)f(x+y) = f\left(xf(y)\right)\]
For all non-negative real numbers $x$ and $y.$
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that
i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$
ii) $f (f(x)) = f(x)$
where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$