Found problems: 1513
Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that
\[ n^2+4f(n)=f(f(n))^2 \]
for all $n\in \mathbb{Z}$.
[i]Proposed by Sahl Khan, UK[/i]
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions:
$i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$
$ii) f$ takes all integer values
A function $f: Z \to Z$ has the following properties:
$f (92 + x) = f (92 - x)$
$f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$)
$f (1992 + x) = f (1992 - x)$
for all integers $x$. Can all positive divisors of $92$ occur as values of f?
Let $f: N \to N$ be a function satisfying the following:
$\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$.
$\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes.
Determine all possible values of $f(2002)$. Justify your answers.
Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying
$\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.
Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$
$$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$
[i]Created by Liang Xiao, Yunhao Fu[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and
\[f(x+f(y)^2 ) = {f(x+y)}^2.\]
(25 points)
Find all functions $f: (0, +\infty)\cap\mathbb{Q}\to (0, +\infty)\cap\mathbb{Q}$ satisfying thefollowing conditions:
[list=1]
[*] $f(ax) \leq (f(x))^a$, for every $x\in (0, +\infty)\cap\mathbb{Q}$ and $a \in (0, 1)\cap\mathbb{Q}$
[*] $f(x+y) \leq f(x)f(y)$, for every $x,y\in (0, +\infty)\cap\mathbb{Q}$
[/list]
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i]Proposed by Paisan Nakmahachalasint, Thailand[/i]
Find all functions $f:[0, \infty) \to [0,\infty)$, such that for any positive integer $n$ and and for any non-negative real numbers $x_1,x_2,\dotsc,x_n$
\[f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.\]
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
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 all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying
$$f(f(x+y)-f(x-y))=y^2f(x)$$
for all $x, y \in \mathbb{R}$.
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \]
[i]Mihai Piticari[/i]
Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions:
[list]
[*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$.
[*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$.
[/list]
Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that
$$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$
for all $x, y \in\mathbb{R^+}.$
[i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.
Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying
$f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.