Found problems: 2349
A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have
$$
|f(x)+g(y)| = | f(y) + g(x)|.
$$
Proposed by [i]Mojtaba Zare, Amirabbas Mohammadi[/i]
Let the function $f:R \to R$ satisfies the following conditions:
1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$
2)$ f(1)=1$
3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$
Prove that for all $x \in R$, $f(x) = x$.
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.
Let $A_n = \{1,2,...,n\}$. Prove or disprove:
For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)f(a+b)-ab$ is a perfect square for all $a, b \in \mathbb{N}$.
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\]
for all $x,y\in\mathbb{R}$.
Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds:
\[f(f(x)-y)=f(y)-f(f(x)).\]
Show that $f$ is bounded.
Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R}\longrightarrow \mathbb{R}$ so that the equality $$f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2$$
is true for any real numbers $x,y$.
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have
\[ f(f(f(x)) + f(y)) = f(y) - f(x) \]
Function $f(x)$. which is defined on the set of non-negative real numbers, acquires real values. It is known that $f(0)\le 0$ and the function $f(x)/x$ is increasing for $x>0$. Prove that for arbitrary $x\ge 0$ and $y\ge 0$, holds the inequality $f(x+y)\ge f(x)+ f(y)$ .
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true:
$f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$
[url=http://matol.kz/comments/3373/show]source[/url]
Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that
$$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$
for all positive real numbers $x$ and $y$.
Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
Show that for all non-constant functions $f:\mathbb R\to\mathbb R$, there are two real numbers $x,y$ such that
$$f(x+f(y))>xf(y)+x.$$
Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$.
The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$.
$\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$
$\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$.
Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,
\[f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).\]
Let $f$ be a function from the set $Q$ of the rational numbers onto itself such that $f(x+y)=f(x)+f(y)+2547$ for all rational numbers $x,y$.
Moreover $f(2004) = 2547$.
Determine $f(2547).$
Pierre.