Found problems: 2349
Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.
Determine all functions $f : R_+ \to R_+$ such that:
a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$
b) $f(2) = 0$
c) $f(x) \ne 0$ for $0 \le x < 2$.
Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that:
a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$.
b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$.
[i]Mihai Piticari[/i]
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$
a) Prove that there exist nonconstant functions in $\mathcal{F}.$
b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]
Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$.
$$f(f(x)f(y)-1) = xy - 1$$
A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.
For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation
$$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$
for any two real numbers $x$ and $y$.
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$
Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$
Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$
Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition
\[f(f(x) + y) = 2x + f(f(y) - x)\quad \text{ for all } x, y \in\mathbb{R}.\]
Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that
$f(x + y) = f(x)g(y) + f(y)$.
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list]
a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$
b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$.
$$(x+y)f(x-y) = f(x^2-y^2).$$
Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies:
$$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$.
a) Determine the value of $f(0)$.
b) Prove that $f(x) = 2-x$ for every real number $x$.