Found problems: 2349
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :
$$f(n)+1400m^2|n^2+f(f(m))$$
Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function.
(a) Prove that every cool function is periodic.
(b) Give an example of a periodic function that is not cool.
[b]4.[/b] Find all functions of two variables defined over the entire plane that satisfy the relations $f(x+u,y+u)=f(x,y)+u$ and $f(xv,yv)= f(x,y) v$ for any real numbers $x,y,u,v$. [b](R.12)[/b]
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that
\[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\]
a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$.
b) Prove that $\mathcal{F}$ has exactly two elements.
[i]Nelu Chichirim[/i]
Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$.
[i]Proposed by Sardor Gafforov, Uzbekistan[/i]
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
Find all function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that
$f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)$
for all integer $x,y$
Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$, where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$, such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$.
[i]Proposed by Grant Yu[/i]
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$
for all real numbers $ x,y,t $ with $ t\in [0,1] . $
Prove that:
[b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
[b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ the following equation holds:$$1+f(xy)=f(x+f(y))+(y-1)f(x-1)$$
[i]M. Zorka[/i]
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$
Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.
Find all functions $f: \mathbb R^{+} \to \mathbb R^{+}$ satisfying
\begin{align*}
f(f(x)+y) = f(y) + x, \qquad \text{for all } x,y \in \mathbb R^{+}.
\end{align*}
Note that $\mathbb R^{+}$ is the set of all positive real numbers.
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation
$$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$
for all $x,y \in \mathbb{R}. $
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$.
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds:
$a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$
$b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$