Found problems: 2349
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:
$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$
$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $ \forall x\notin\{-1,1\}$ holds:
\[\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}\]
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.
Let $f$ be a function of the positive reals in the positive reals, such that
$$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$
(a) Find $f(1)$.
(b) Find $f(x)$.
Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$.
Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.
find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$.
Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
Determine all functions $f : R \times R \to R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$
Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.
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
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
Let $f:Z \to N -\{0\}$ such that:
$f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$.
Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$.
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.
Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$
Find all such functions.
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation
$$
(f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)
$$
for all $x ,y \in \mathbb{R}$.
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
$f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$
Given a rational number $a \neq 0$, find all functions $f:\mathbb{Q} \rightarrow \mathbb{Q}$ such that $$f(f(x)+ay)=af(y)+x$$ for all rational $x, y$.
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]